Global attractors for 3-dimensional stochastic Navier-Stokes equations

Nigel J. Cutland and H. Jerome Keisler

April 17, 2003

Abstract

Sell’s approach [32] to the construction of attractors for the Navier– Stokes equations in 3-dimensions is extended to the 3-D stochastic equations with a general multiplicative noise.

1 Introduction

The existence of global attractors for deterministic time-homogeneous incompressible Navier–Stokes equations

\{ \begin{matrix}du = [νΔu − ⟨u, ∇⟩u + f(u) − ∇p]dt \\ div u = 0 \end{matrix}  (1)

on a bounded domain D ⊆ ℝ^{2} is well known (see Temam [33] for an exposition): there is a compact global attractor in H, the subspace of divergence free vector fields in L^{2}(D), which is the phase space for the equations (1). (The solutions to (1) have u(t) ∈ H for all t.)

Equally well known is the fundamental problem with the very idea of attractors for these equations in 3-dimensions; the question of uniqueness of weak solutions to the equations is still open, with the consequence that the conventional formulation of the notion of an attractor may not even make sense. It would require the existence of a semiflow of solutions S_{t} say, so that S_{t}v is the (unique) solution for the given initial condition v. If attention is restricted to strong solutions, where uniqueness is known, the fundamental open problem is that of existence for all time – which again means that the usual notion of attractor cannot be used. In order to overcome the difficulties mentioned above when considering attractors for the 3-dimensional deterministic Navier–Stokes equations, a number of approaches have been suggested, beginning with Foias & Temam [23], and more recently Sell [32], Cutland & Capiński [8] and Ball [2].

Sell’s radical approach [32] to the 3-D problem of attractors was to replace the phase space H by a space W of entire solutions to the Navier–Stokes equations. That is, each point in W is the complete trajectory in H of a solution. There is a simple semigroup action S_{t} on W – namely time translation. Thus, if u = u(·) ∈ W then S_{t}u = v ∈ W is given by

v(s) = u(t + s).

Clearly this is well defined, and has the crucial semi-flow property

S_{t_{1}} ∘ S_{t_{2}} = S_{t_{1}+t_{2}}

along with S_{0}u = u.

Using this idea, Sell was able to establish the existence of a global attractor for the 3-D (deterministic) Navier–Stokes equations.

The goal of this paper is to extend Sell’s approach to the general timehomogeneous stochastic Navier–Stokes equations (2) with multiplicative noise in dimensions 2 and 3, taking the form

\{ \begin{matrix}du = [νΔu − ⟨u, ∇⟩u + f(u) − ∇p]dt + g(u)dw_{t} \\ div u = 0 \end{matrix}  (2)

where u = u(t, ω) is now a random velocity field at time t ≥ 0 of a fluid in a bounded domain D ⊆ ℝ^{3} (or D ⊆ ℝ^{2}). For simplicity the driving noise w_{t} is taken to be a 1-dimensional Wiener process.

Even in 2-dimensions there are considerable difficulties when seeking stochastic attractors. These can be overcome in a number of ways – for example by considering measure attractors (see [31, 6]), or by working with the notion of stochastic attractor developed by Crauel & Flandoli [10]. For this it is necessary first to show the existence of a flow of solutions with a stochastic equivalent of the semi-group property – known as a perfect cocycle. This was achieved in [9] for a very special form of the noise g(u) when the system has periodic boundary conditions.

For the 3-dimensional stochastic case, Sell’s idea was used by Flandoli & Schmalfuss in the paper [12] for the Navier–Stokes equations with a special form of multiplicative noise, using a mild solution concept. The equation considered allowed essentially a pathwise solution, and then a random attractor was obtained by combining Sell’s approach with the idea of pulling back in time to −∞, as developed by Crauel & Flandoli [12]. In a later paper [13] Flandoli & Schmalfuss consider in the same framework 3-D-Navier–Stokes equations with an irregular forcing term, but no feedback.

In the current paper we consider 3-D stochastic Navier–Stokes equations with a general multiplicative noise g(u) as above. The idea is to use Sell’s approach at the level of processes rather than paths. In this way the idea of an attractor is formulated in the conventional sense, examining the long term behavior of solutions as t → ∞. To do this, it is necessary to have a single underlying probability space, rich enough to carry a supply of solutions to the 3-D stochastic Navier–Stokes equations that is sufficient for the concepts to make sense. For this we need a filtered Loeb space.

2 Preliminaries and assumptions

We will always state our definitions for the three dimensional case, but it should be understood that these definitions can be modified in the natural way to get the analogous notions for two dimensions. We use ℕ for the set of positive integers.

2.1 Stochastic Navier–Stokes equations

We consider the stochastic Navier–Stokes equations (2) in a bounded domain D ⊆ ℝ^{3} with boundary of class C^{2}, and adopt the conventional Hilbert space approach as follows.

Let H be the closure of the set

{u ∈ C_{0}^{∞}(D, ℝ^{3}) : div u = 0}

in the L^{2} norm |u| = (u, u)^{1/2}, where u = ⟨u_{1}, u_{2}, u_{3}⟩ and

(u, v) = ∑_{j=1}^{3} ∫_{D} u_{j}(x)v_{j}(x)dx.

The letters u, v, w will be used for elements of H. The subspace V is the closure of the set {u ∈ C_{0}^{∞}(D, ℝ^{3}) : div u = 0} in the stronger norm |u|+∥u∥ where ∥u∥ = ((u, u))^{1/2} and

((u, v)) = ∑_{j=1}^{3} (\frac{∂u}{∂x_{j}} , \frac{∂v}{∂x_{j}} ) .

H and V are Hilbert spaces with scalar products (·, ·) and ((·, ·)) respectively, and | · | ≤ c∥ · ∥ for some constant c.

By A we denote the self adjoint extension of the projection of −Δ in H. Classical theory shows that there is an orthonormal basis {e_{k} : k ∈ ℕ} of eigenfunctions of A with corresponding eigenvalues λ_{k} > 0 such that λ_{k} ↗ ∞. For u ∈ H we write u_{(k)} = (u, e_{k}), and write Pr_{m} for the projection of H on the subspace H_{m} spanned by {e_{1}, …, e_{m}}. Since each e_{k} ∈ V, then H_{m} ⊆ V. If u = ∑ u_{(k)}e_{k} ∈ V then ∥u∥^{2} = ∑ λ_{k}u_{(k)}^{2}, so that the constant c above is

λ1^{\frac{1}{2}} .

A trilinear form b is defined by

b(u, v, w) = ∑_{i,j=1}^{3} ∫_{D} u_{j}(x)\frac{∂v_{i}}{∂x_{j}} (x)w_{i}(x)dx = (⟨u, ∇⟩v, w)

whenever the integrals make sense. Note the following well-known properties of the trilinear form b, where c is a real constant.

b(u, v, w) = −b(u, w, v),
b(u, v, v) = 0,
|b(u, v, w)| ≤ c∥u∥ ∥v∥ ∥w∥.

The last is a continuity property of b with respect to the norm ∥ · ∥. There are a number of other important continuity properties for other norms which we will not need.

2.2 Functional formulation of the Navier–Stokes equations

In the above framework, the stochastic Navier–Stokes equations may be formulated as the following stochastic differential equation in V′ (the dual of V):

du = [−νAu − B(u) + f(u)]dt + g(u)dw_{t},  (3)

where B(u) = b(u, u, ·). Note that the pressure has disappeared, because ∇p = 0 in V′ (using div v = 0 in V and an integration by parts). Although equation (3) is regarded as an equation in V′, it turns out that solutions can be found that live in H (and in fact in V for almost all times).

We take w_{t} to be a 1-dimensional adapted Wiener process on a filtered probability space

Ω Ω = (Ω, F, (F_{t})_{t≥0}, P).

The term adapted is always taken to mean adapted to the filtration (F_{t})_{t≥0}. Further assumptions on Ω Ω that are needed to formulate the idea of an attractor for a class of stochastic processes are given in the next section.

The equation (3) is really an integral equation, with the first integral being the Bochner integral and the second an extension of the Itô integral to Hilbert spaces in the weak sense. Thus, when we write

u(t_{1}) = u(t_{0}) + ∫_{t_{0}}^{t_{1}}[−νAu(t) − B(u(t)) + f(u(t))]dt + ∫_{t_{0}}^{t_{1}} g(u(t))dw_{t}

we mean that for all v ∈ V we have

(u(t_{1}), v) =

(u(t_{0}), v)+∫_{t_{0}}^{t_{1}}[−ν(Au(t), v)−(B(u(t)), v)+(f(u(t)), v)]dt+∫_{t_{0}}^{t_{1}}(g(u(t)), v)dw_{t}

as a stochastic equation in ℝ.

We make the following assumptions on the coefficients f and g, where c_{0}, d_{1}, d_{2} are positive real constants.

(H1) f : H → H and |f(u)| ≤ c_{0} + d_{1}|u|.

(H2) g : H → H and |g(u)| ≤ c_{0} + d_{2}|u|.

(H3) f and g are continuous.

(H4) 2d_{1} + 3d_{2}^{2} < 2νλ_{1}.

The general theory of stochastic Navier–Stokes equations expounded in [7] shows that the equation (3) can be solved with only the assumptions (H1)–(H3). The additional growth restriction (H4) on f, g is needed here to obtain the attractor.

2.3 Truncation functions

For technical reasons (in connection with testing for S-integrability) we will need an explicit family of “truncation functions”. We give the details here but suggest that the reader refer back to this section only when needed later.

The following real C^{2} function ψ : [0, ∞) → [0, 1] is designed to be concave on [0, 1] and constant (with value 1) on [1, ∞).

Definition 2.1 (a)

ψ(x) = \{ \begin{matrix}(x − 1)^{3} + 1 & if 0 ≤ x ≤ 1 \\ 1 & if x ≥ 1 \end{matrix}

(b) For each n ∈ ℕ,

ψ_{n}(x) = ψ(x/n^{2}).

(c) For each n ∈ ℕ a function φ_{n}(u) is defined for u in any Hilbert space (finite or infinite dimensional) by

φ_{n}(u) = u^{2}ψ_{n}(u^{2}),

where we write u^{2} to mean |u|^{2} to ease the notation.

Direct calculation gives:

(d)

ψ′_{n}(x) = \{ \begin{matrix}3n^{−2}(1 − x/n^{2})^{2} & if 0 ≤ x ≤ n^{2} \\ 0 & if n^{2} ≤ x \end{matrix}

(e)

ψ′′_{n}(x) = \{ \begin{matrix}−6n^{−4}(1 − x/n^{2}) & if 0 ≤ x ≤ n^{2} \\ 0 & if n^{2} ≤ x \end{matrix}

(f)

φ′_{n}(u) = 2u(ψ_{n}(u^{2}) + u^{2}ψ′_{n}(u^{2})).

(g)

φ′′_{n}(u) = 2I[ψ_{n}(u^{2}) + u^{2}ψ′_{n}(u^{2})] + 4uu^{T} [2ψ′_{n}(u^{2}) + u^{2}ψ′′_{n}(u^{2})]

(again writing u^{2} for |u|^{2}).

The significance of the functions φ_{n} involves the concept of S-integrability from Loeb measure theory, to be defined later. We will show that given an internal random vector U(ω), |U|^{2} is S-integrable if and only if ^{∘}𝔼(φ_{n}(U)) → 0 as n → ∞. The application of this fact will involve the following particular properties of ψ_{n} and φ_{n}.

Lemma 2.2 (Properties of ψ_{n} and φ_{n}) The functions ψ_{n} and φ_{n} have the following properties:

(a) xψ′_{n}(x) ≤ ψ_{n}(x) all n and x ≥ 0.

(b) 2|u| ψ_{n}(u^{2}) ≤ |φ′_{n}(u)| ≤ 4|u|ψ_{n}(u^{2}).

(c) 2φ_{n}(u) ≤ |u| |φ′_{n}(u)| ≤ 4φ_{n}(u).

Proof Elementary calculation. □

Lemma 2.3 For any random vector u(ω) in a Hilbert space:

(a) 𝔼(|u|ψ_{n}(u^{2})) ≤ n^{−1/2}(3 + 𝔼(u^{2})).

(b) 𝔼(|φ′_{n}(u)|) ≤ 4n^{−1/2}(3 + 𝔼(u^{2})).

(c) 𝔼(ψ_{n}(u^{2})) ≤ n^{−1}(3 + 𝔼(u^{2})).

Proof (a) We have

𝔼(|u|ψ_{n}(u^{2})) = ∫_{|u|^{2}≤n} |u|ψ_{n}(u^{2}) + ∫_{|u|^{2}>n} |u|ψ_{n}(u^{2})
≤ n^{1/2}ψ_{n}(n) + ∫_{|u|>\sqrt{n}} |u| (since ψ_{n} ≤ 1)
≤ n^{1/2}ψ_{n}(n) + ∫_{|u|>\sqrt{n}} |u||u|/\sqrt{n}
≤ n^{1/2}ψ_{n}(n) + n^{−1/2}𝔼(u^{2})
≤ n^{−1/2}(3 + 𝔼(u^{2}))

using the fact that nψ_{n}(n) = n^{−2} − 3n^{−1} + 3 ≤ 3.

(b) This follows from (a) since |φ′_{n}(u)| ≤ 4|u|ψ_{n}(u^{2}).

(c) We have

𝔼(ψ_{n}(u^{2})) = ∫_{|u|^{2}≤n} ψ_{n}(u^{2}) + ∫_{|u|^{2}>n} ψ_{n}(u^{2})
≤ ψ_{n}(n) + ℙ(u^{2} > n) (since ψ_{n} ≤ 1)
≤ ψ_{n}(n) + n^{−1}𝔼(u^{2})
≤ n^{−1}(3 + 𝔼(u^{2})).
□

3 Semiflows and attractors

3.1 Semiflows

We now assume that the space Ω Ω is equipped with a family of measure preserving maps θ_{t} : Ω → Ω for t ≥ 0 with the following properties:

(θ1) θ_{0} = identity and θ_{t} ∘ θ_{s} = θ_{t+s};

(θ2) θ_{t}F_{s} = F_{t+s} for all s, t ≥ 0;

(θ3) w(t + s, θ_{t}ω) − w(t, θ_{t}ω) = w(s, ω) for all s ≥ 0.

Note that the property (θ3) tells us that for a fixed t the increments of the process w(t + s, θ_{t}ω) are the same as those of the process w(s, ω). Thus θ_{t} can be thought of as a shift of the noise to the right by t.

The family (θ_{t}) allows the following definition of a semiflow S_{r} of stochastic processes.

Definition 3.1 (Semiflow of Processes) (a) Suppose that u = u(t, ω) is a stochastic process defined for t > 0. Then for any r ≥ 0 the process v = S_{r}u is defined by

v(t, ω) = u(r + t, θ_{r}ω).

(b) By a semiflow (S_{t})_{t≥0} on a filtered space Ω Ω we mean that there is a measure preserving family (θ_{t})_{t≥0} obeying (θ1)−(θ3) from which S_{t} is defined as above.

Proposition 3.2

(a) S_{r} is a semigroup on the class of all stochastic processes u : [0, ∞)×Ω → H.

(b) If u is adapted (to the filtration (F_{t})_{t≥0} ) then so is v = S_{r}u.

(c) If u is adapted and v = S_{r}u then for t ≥ 0:

(i) For appropriate continuous g,

∫_{0}^{t} g(v(s, ω))dw_{s}(ω) = ∫_{r}^{r+t} g(u(s, θ_{r}ω))dw_{s}(θ_{r}ω)

(meaning that I(ω) = J(θ_{r}ω) as random variables, where I(ω) is the left-hand integral and J(ω) = ∫_{r}^{r+t} g(u(s, ω))dw_{s}(ω)).

(ii) For appropriate continuous f,

∫_{0}^{t} f(v(s, ω))ds = ∫_{r}^{r+t} f(u(s, θ_{r}ω))ds.

(By appropriate we mean that the integrals are defined.)

(d) If u is adapted and v = S_{r}u and t_{1} ≥ t_{0} ≥ 0 with

u(t_{1} + r) = u(t_{0} + r) + ∫_{t_{0}+r}^{t_{1}+r} f(u(s))ds + ∫_{t_{0}+r}^{t_{1}+r} g(u(s))dw_{s},

then

v(t_{1}) = u(t_{0}) + ∫_{t_{0}}^{t_{1}} f(v(s))ds + ∫_{t_{0}}^{t_{1}} g(v(s))dw_{s}.

Proof Elementary. □

3.2 Attractors for the 3-D stochastic Navier-Stokes equations

Suppose now that a filtered probability space Ω Ω and a Wiener process w is given, together with a family of measure preserving maps as above.

A natural space for paths of solutions to the stochastic Navier–Stokes equations is the space M defined as follows.

Definition 3.3 (a) For a measurable (deterministic) function ξ : [0, ∞) → H define a norm

|ξ| = (∫_{0}^{∞} ξ(t)^{2} exp(−t)dt)^{\frac{1}{2}} = (∫_{0}^{∞} ξ(t)^{2}μ(dt))^{\frac{1}{2}}

where μ(dt) = exp(−t)dt, and write

M = {ξ : |ξ| < ∞}

for this space of paths, which is a separable Hilbert space.

(b) For a process u(t, ω) with paths in M define

|u| = (𝔼(|u(·, ω)|^{2}))^{\frac{1}{2}} = (𝔼 ∫_{0}^{∞} |u(t, ω)|^{2} exp(−t)dt)^{\frac{1}{2}} ,

which is simply the norm of L^{2}(Ω, M). Let ρ be the corresponding metric ρ(u, v) = |u − v|.

We will be interested in the laws of solutions viewed as probability distributions on the space of paths M, so we need the following definitions.

Definition 3.4 Let u(t, ω) be a process with paths in M.

(a) law(u) is the probability law on M induced by u; i.e.

law(u)(E) = ℙ(u(·, ω) ∈ E)

for Borel E ⊆ M.

(b) law_{w}(u) = law(u, w), the probability law induced on M × C_{0} by the pair of processes (u(t, ω), w(t, ω)), where C_{0} = C_{0}[0, ∞).

For the space of probability laws M_{1}(S) on a separable metric space S a fundamental metric is the Prohorov metric, which we denote by d_{0}; this makes M_{1}(S) separable. Here we are thinking of S = M and S = M × C_{0}.

There is a natural projection mapping π : M_{1}(M × C_{0}) → M_{1}(M) defined by

π(λ)(E) = λ(E × C_{0}).

In the current situation the laws on the space M that we are interested are laws of L^{2} random variables, so it is appropriate to define a stronger metric to reflect this.

Definition 3.5 (a) M_{1,2}(M) = {μ ∈ M_{1}(M) : 𝔼_{μ}(|u|^{2}) < ∞}.

(b) the metric d on M_{1,2}(M) is defined by

d(μ_{1}, μ_{2}) = d_{0}(μ_{1}, μ_{2}) + |𝔼_{μ_{1}}(|u|^{2}) − 𝔼_{μ_{2}}(|u|^{2})| .

(c) M_{1,2}(M × C_{0}) = {λ ∈ M_{1}(M × C_{0}) : π(λ) ∈ M_{1,2}(M)}.

(d) the metric d on M_{1,2}(M × C_{0}) is defined by

d(λ_{1}, λ_{2}) = d_{0}(λ_{1}, λ_{2}) + |𝔼_{μ_{1}}(|u|^{2}) − 𝔼_{μ_{2}}(|u|^{2})| ,

where μ_{i} = π(λ_{i}) (i = 1, 2).

The following lemma is easily checked.

Lemma 3.6 (a) The function law maps L^{2}(Ω, M) into M_{1,2}(M) and is continuous with respect to the metrics ρ and d.

(b) The function law_{w} maps L^{2}(Ω, M) into M_{1,2}(M × C_{0}) and is continuous with respect to the metrics ρ and d.

(c) The mapping π : M_{1}(M × C_{0}) → M_{1}(M) defined by

π(λ)(E) = λ(E × C_{0})

is continuous with respect to the metric d.

Suppose now that X ⊆ L^{2}(Ω, M) is a class of solutions to the stochastic Navier–Stokes equations on Ω Ω for the given Wiener process w. Then each u ∈ X is a stochastic process such that |u| as defined above is finite. By a bounded subset of X we will mean a subset of X which is bounded in the norm | · |.

Let us assume further that S_{t}X ⊆ X for all t ≥ 0, that is, S_{t} is a semigroup on X. A semigroup Ŝ_{t} : M_{1,2}(M × C_{0}) → M_{1,2}(M × C_{0}) is induced in a natural way: in detail

Ŝ_{t}(λ) = λ ∘ h_{t}^{−1},

where h_{t} : M × C_{0} → M × C_{0} is given by

h_{t}(ξ, w)(s) = (ξ(t + s), w(t + s) − w(t)).

For future reference note that

Ŝ_{t} ∘ law_{w} = law_{w} ∘ S_{t}.  (4)

This allows the following definition of an attractor for a semiflow S_{t} on X.

Definition 3.7 (a) A set of laws A ⊆ law_{w}(X) is a law-attractor for the semiflow S_{t} on X if:

(i) (Invariance) Ŝ_{t}A = A for all t ≥ 0.

(ii) (Attraction) For any open set O ⊃ A and d-bounded set Z ⊆

law_{w}(X),

Ŝ_{t}Z ⊆ O

eventually (i.e. for some t_{0} = t_{0}(O, Z), this holds for all t ≥ t_{0}).

(iii) (Compactness) A is compact in the metric d.

(b) An attractor for the semiflow S_{t} on X is a set of processes A ⊆ X such that:

(i) law_{w}(A) is a law-attractor (in particular law_{w}(A) is compact in the metric d, and so A is bounded).

(ii) (Invariance) S_{t}A = A for all t ≥ 0.

(iii) (Attraction) For any bounded set Z ⊆ X and compact set K ⊆

L^{2}(Ω, M),

\underline{lim}_{t→∞}ρ(S_{t}Z, K) ≥ ρ(A, K).

(iv) A is closed in the space L^{2}(Ω, M).

Remarks on Definition 3.7.

1. Since existence results for the stochastic Navier–Stokes equations require a rather large probability space, it is to be expected that any space carrying a whole class of solutions X as above will be too big to allow an attractor A ⊆ X that is compact in the usual sense. In a later section we will give evidence to support this remark. We will see, however, that on a suitable space there is an attractor A in the above sense. In the sequel [18] to this paper we improve this result, and show that the attractor to be constructed in this paper is neocompact in the sense of [20]. Neocompact sets share many properties with compact sets. Among other things, they are closed and have compact laws.

2. The attraction property 3.7(b)(iii) is equivalent to the following:

S_{t}Z ⊆ O eventually  (5)

for any bounded Z and any open O ⊃ A of the form O = L^{2}(Ω, M) \ K^{≤ε}, with K compact. Property 3.7(b)(i) means that in addition (5) holds for any open set O of the form O = law_{w}^{−1}(O′) where O′ is an open set of laws with law_{w}(A) ⊆ O′ ⊆ M_{1,2}(M × C_{0}).

3. The usual attraction property for attractors, namely that property (5) holds for any bounded Z and any open O ⊃ A, is probably too much to expect. However, in the sequel [18] we will show that the attractor to be constructed in this paper has property (5) for a smaller class of open sets – those that are neoopen in the sense of [20]. Sets O of the form L^{2}(Ω, M)\K^{≤ε} or law_{w}^{−1}(O′) as above are neoopen.

We can now state the main theorem of this paper.

Theorem 3.8 There is a filtered probability space Ω Ω, a class X of adapted weak solutions to the stochastic Navier-Stokes equation in Ω Ω, and a semiflow S_{t} on X, such that

(a) there exists u ∈ X for all L^{2} F_{0}-measurable initial conditions;

(b) there is an attractor for the semiflow S_{t} on X.

Before embarking on the proof of this result we note some further properties of an attractor (if it exists) that can be deduced immediately.

Theorem 3.9 Given a semiflow S_{t} on a set X ⊆ L^{2}(Ω, M), there is at most one attractor A for S_{t} on X. A has the following properties:

(a) law(A) is compact in the metric d.

(b) For any open set of laws O ⊃ law(A), and bounded Z ⊆ X, S_{t}Z ⊆ law^{−1}(O) eventually.

(c) A = ⋂_{n∈ℕ} S_{n}(Z) = ⋂_{t≥0} S_{t}(Z) for any bounded set Z with A ⊆ Z ⊆ X.

In particular

(d) A = ⋂_{n∈ℕ} S_{n}(law^{−1}(law(A)) ∩ X).

Remark The significance of (d) is that although A may not be compact, it may be defined from a compact set of laws. Proof For uniqueness, suppose that A and A′ are attractors with u ∈ A′ \A. Since A is closed, ρ(A, u) = ε > 0 . Since A′ ⊆ X is bounded we have \underline{lim}_{t→∞}ρ(S_{t}A′, u) ≥ ε, and hence ρ(S_{t}A′, v) > ε/2 for some t, contradicting v ∈ A′ = S_{t}A′.

(a) and (b) follow immediately from the continuity of π.

For (c), it is clear that A is contained in the right-hand side, since A = S_{t}A ⊆ S_{t}Z for all t ≥ 0. For the other inclusion, suppose that u ∈ ⋂_{n∈ℕ} S_{n}(Z) \ A. Since A is closed, ε = ρ(A, u) > 0. Since Z is bounded the attraction property gives ρ(S_{n}Z, v) > ε/2 for some n, contradicting v ∈ S_{n}Z.

For (d), apply (c) to Z = law^{−1}(law(A)) ∩ X. Clearly A ⊆ Z ⊆ X and Z has the same bound as A. □

4 SOLUTIONS TO THE STOCHASTIC NAVIER–STOKES EQUATIONS12

4 Solutions to the stochastic Navier–Stokes equations

We define below a special class X of weak solutions to the stochastic Navier– Stokes equations (3). Each element u of X is thus an adapted stochastic process (with u(t, ω) ∈ H for all t, ω). The properties required for membership of X are among those that can be deduced heuristically from (3) using elementary stochastic calculus. (Of course we will later show rigorously that X is non empty!)

Definition 4.1 (a) Given positive real constants k_{1}, k_{2}, k_{3}, α, β, denote by X the class of adapted stochastic processes u : (0, ∞) × Ω → H with the following properties.

(X1) For a.a. ω the path u(·, ω) belongs to the following spaces:

L_{loc}^{∞}(0, ∞; H) ∩ L_{loc}^{2}[0, ∞; H) ∩ L_{loc}^{2}(0, ∞; V) ∩ C(0, ∞; H_{weak}).

(X2) For all t_{1} ≥ t_{0} > 0

u(t_{1}) = u(t_{0}) + ∫_{t_{0}}^{t_{1}}[−νAu(t) − B(u(t)) + f(u(t))]dt + ∫_{t_{0}}^{t_{1}} g(u(t))dw_{t}.

(X3) For a.a. t_{0} > 0 and all t_{1} ≥ t_{0},

𝔼(|u(t_{1})|^{2}) ≤ 𝔼(|u(t_{0})|^{2}) exp(−k_{1}(t_{1} − t_{0})) + k_{2}.  (6)

(X4) For a.a. t_{0} > 0 and all t_{1} ≥ t_{0},

𝔼 (sup_{t_{0}≤s≤t_{1}}|u(s)|^{2} + ∫_{t_{0}}^{t_{1}} ∥u(s)∥^{2}ds) ≤ α𝔼(|u(t_{0})|^{2})+β(t_{1} −t_{0}). (7)

(X5) For a.a. t_{0} > 0 and all t_{1} ≥ t_{0} and n ∈ ℕ,

𝔼(φ_{n}(u(t_{1}))) ≤ 𝔼(φ_{n}(u(t_{0})) exp(−k_{3}(t_{1} − t_{0}))) + n^{− \frac{1}{2}} (α𝔼(|u(t_{0})|^{2}) + β). (8)

(X6) 𝔼 ∫_{0}^{1} |u(t)|^{2}dt < ∞.

(b) Denote by X_{k} the set of u ∈ X with

(X6_{k}) 𝔼 ∫_{0}^{1} |u(t)|^{2}dt ≤ k.

Remarks

1. The class X depends on the constants k_{1}, k_{2}, k_{3}, α, β. We will show later that there is an explicit choice of constants for which X is non-empty.

2. The sets X_{k} obviously increase with k.

3. The above conditions tell us nothing about u(t, ω) at t = 0 and there may be a singularity there. In this sense the class X is a class of generalized weak solutions to the stochastic Navier–Stokes equations (cf. [32], p.12).

4. The meaning of “loc” in the path properties (X1) is as follows: L_{loc}^{p}(0, ∞) means L^{p}[1/n, n] for all n, whereas L_{loc}^{p}[0, ∞) means L^{p}[0, n] for all n.

5. The conditions (X5) follow naturally from the Foias equation for the stochastic Navier-Stokes equations (see [5]), which may be deduced heuristically from the equation (3). The choice of the functions φ_{n} makes (X5) a uniform integrability condition for |u(t)|^{2} on any [t_{0}, ∞).

The following lemma is our motivation for singling out the sets X_{k}; it relates these sets to bounded sets.

Lemma 4.2 (a) X ⊆ M.

(b) If u ∈ X and |u|^{2}e ≤ k then u ∈ X_{k}.

(c) If u ∈ X_{k} then |u|^{2}e ≤ k(1 + e) + k_{2}.

(d) If Z ⊆ X, then Z is bounded if and only if Z ⊆ X_{k} for some k ∈ ℕ.

Proof (a) Let u ∈ X. It follows from (X6) that 𝔼(|u(t))|^{2}) < ∞ for a.a. t ∈ (0, 1). Thus, from (X3) we see that 𝔼(|u(t))|^{2}) is bounded on [\frac{1}{n}, ∞) for all n ∈ ℕ. It follows from u ∈ L_{loc}^{2}[0, ∞) that 𝔼(|u|^{2}) < ∞, so u ∈ M.

(b) For u ∈ M we have

𝔼 ∫_{0}^{1} |u(t)|^{2}dt ≤ e𝔼 ∫_{0}^{1} |u(t)|^{2} exp(−t)dt ≤ e|u|^{2}.

So if |u|^{2}e ≤ k then 𝔼 ∫_{0}^{1} |u(t)|^{2}dt ≤ k.

(c) Let u ∈ X_{k}. Then the set {t ∈ [0, 1] : 𝔼(|u(t)|^{2}) ≤ k} has positive Lebesgue measure. By (X3) there is a t_{0} ∈ (0, 1] such that (6) holds for all t_{1} ≥ t_{0} and 𝔼(|u(t)|^{2}) ≤ k. Then for all t_{1} ≥ t_{0}, 𝔼(|u(t_{1})|^{2}) ≤ k + k_{2}. Thus

|u|^{2} = ∫_{0}^{1} 𝔼|u(t)|^{2}exp(−t)dt + ∫_{1}^{∞} 𝔼|u(t)|^{2}exp(−t)dt

≤ k + ∫_{1}^{∞}(k_{2} + k)exp(−t)dt = k + (k_{2} + k)/e,

and (c) follows.

(d) follows easily from (b) and (c). □

We can now reformulate the main theorem for the particular class of solutions X that has just been introduced.

Theorem 4.3 There is a filtered probability space Ω Ω with a semiflow S_{t} on X, and constants k_{1}, k_{2}, k_{3}, α, β, such that there exist u ∈ X for all L^{2} F_{0}-measurable initial conditions, and there is an attractor for the semiflow S_{t} on X.

In fact, we will show that there exists weak solutions in the following smaller class Y whose members are defined at 0.

Definition 4.4 Denote by Y the class of stochastic processes u : [0, ∞) × Ω → H with u ∈ X (that is, the restriction of u to (0, ∞) lies in X) with the following additional properties:

(Y1) For a.a. ω, the path u(·, ω) is in

L_{loc}^{∞}[0, ∞; H) ∩ L_{loc}^{2}[0, ∞; H) ∩ L_{loc}^{2}[0, ∞; V) ∩ C[0, ∞; H_{weak}).

(Y2) For all t_{1} ≥ t_{0} ≥ 0,

u(t_{1}) = u(t_{0}) + ∫_{t_{0}}^{t_{1}}[−νAu(t) − B(u(t)) + f(u(t))]dt + g(u(t))dw_{t}.

(Y3) 𝔼(|u(t)|^{2}) is bounded on [0, ∞).

Note that (Y1) implies (X1), (Y2) implies (X2), and (Y3) implies (X6).

5 The space Ω Ω and the semiflow

The particular space Ω Ω that we use is a filtered Loeb space similar to that used in [7] for the construction of solutions to the stochastic Navier–Stokes equations. Loeb spaces constitute a special class of probability spaces that are very rich – in a sense that can be made precise (see for example [28]). The richness is needed to be able to solve the general stochastic Navier–Stokes equations in dimension 3, and it will also come into play when showing that the single space Ω Ω has solutions to (3) with the same (prescribed) Wiener process w_{t} for any random initial condition.

From this point on we assume the basics of nonstandard analysis and in particular the Loeb construction. Some details are provided in the appendix; for a full exposition see any of [1, 7, 14, 16, 17, 29, 30].

We set Ω = ^{∗}(C_{0}(ℝ)), the internal space of ^{∗}continuous functions ω : ∗ℝ → ^{∗}ℝ with ω(0) = 0, and let Q be the internal ^{∗}Wiener measure on Ω. Thus the canonical process

W (τ, ω) = ω(τ)

is a two-sided ^{∗}Wiener process under Q. This gives the internal filtered probability space

Ω̄ Ω = (Ω, G, (G_{τ})_{τ∈^{∗}ℝ}, Q),

where G_{τ} = ^{∗}σ({W(τ′) : τ′ ≤ τ}) and G = ⋁_{τ∈^{∗}ℝ} G_{τ}.

A family of internal measure preserving maps Θ_{τ} : Ω → Ω is defined for τ ∈ ^{∗}ℝ by

(Θ_{τ}(ω))(σ) = ω(σ − τ) − ω(−τ).

That is, Θ_{τ} is a shift of the path ω to the right by τ and then adjusted to be 0 at 0.

Now let P = Q_{L} be the Loeb measure obtained from Q with the corresponding Loeb σ-algebra F = L(G), giving the Loeb probability space (Ω, L(G), Q_{L}) = (Ω, F, P), and denote the P -null sets by N.

Definition 5.1

(a) The filtered probability space Ω Ω is

Ω Ω = (Ω, F, (F_{t})_{t≥0}, P),

where the right continuous filtration (F_{t})_{t≥0} is defined by

F_{t} = ⋂_{t<^{∘}τ} σ(G_{τ}) ∨ N .

(b) The Wiener process w(t, ω) on Ω Ω is defined by

w(t, ω) = ^{∘}W (t, ω).  (9)

(c) The family of measure preserving transformations (θ_{t})_{t≥0} is given by

θ_{t} = Θ_{t}.

That is, the restriction of the family (Θ_{τ}) to non-negative standard times.

It is well known that (9) defines an almost surely continuous Wiener process on Ω Ω. It is clear that the family θ_{t} satisfies conditions (θ1), (θ2), (θ3).

With the space Ω Ω and the family (θ_{t})_{t≥0} now fixed, the semiflow of processes S_{t} defined by Definition 3.1 is also fixed for the rest of the paper.

6 Construction of solutions

From now on all discussion is in the context of the fixed space Ω Ω of the previous section, so that in particular the classes of processes X and Y are classes of processes on Ω Ω. It is clear from Proposition 3.2 that both classes of solutions X and Y are closed under S_{t} for all t. In fact S_{t}X ⊆ Y for t > 0.

This section is devoted to a proof that, for an explicit choice of the constants, the class Y (and hence X) is non-empty, and in particular contains solutions with any prescribed F_{0}-measurable L^{2} initial condition. (In this connection, note that the richness of the space Ω Ω means that for any probability law on H there is an F_{0}-measurable u_{0}(ω) on Ω Ω with the same law.) In the course of the proof the constants in the definition of the class X (Definition 4.1) are given explicitly.

Theorem 6.1 There are constants k_{1}, k_{2}, k_{3}, α, β such that for any F_{0}-measurable u_{0}(ω) ∈ L^{2}(Ω, H) there is a process u(t, ω) ∈ Y with u(0, ω) = u_{0}(ω).

Proof

As in [7] a solution u ∈ Y is constructed as the standard part of an internal Galerkin approximate solution on the internal space Ω̄ Ω.

Suppose that u_{0} ∈ L^{2}(Ω, H) is F_{0}−measurable. Fix an infinite natural number N and take an SL^{2} lifting U_{0}(ω) ∈ H_{N} of u_{0} that is G_{δ}-measurable for some positive δ ≈ 0. Consider the following internal SDE on ^{∗}[δ, ∞) in H_{N}, where B, F, G denote the projections of ^{∗}B,^{∗}f,^{∗}g respectively onto H_{N}. (Here and elsewhere we use interchangeably the notation U_{τ}(ω) = U(τ, ω), etc.)

\{ \begin{matrix}dU_{τ} = [−ν^{∗}A U_{τ} − B(U_{τ}) + F (U_{τ})]dτ + G(U_{τ})dW_{τ} \\ U_{δ}(ω) = U_{0}(ω). \end{matrix}

Elementary SDE theory and the usual energy considerations give an internal solution U (not necessarily unique) to this Galerkin approximation. Fix any one such solution U. The task now is to derive some properties of U that will show that its standard part u = ^{∘}U belongs to Y as required.

Using Itô’s lemma and the fact that (B(U), U) = 0 we have

d|U_{τ}|^{2} = (−2ν∥U_{τ}∥^{2} +2(F (U_{τ}), U_{τ})+|G(U_{τ})|^{2})dτ +2(U_{τ}, G(U_{τ}))dW_{τ}. (10)

Put Z_{τ} = 𝔼(|U_{τ}|^{2}). The growth conditions on f and g, together with an application of Young’s inequality give positive constants k_{1}, l such that

\frac{d}{dτ} Z_{τ} + k_{1}Z_{τ} ≤ l.  (11)

(See the appendix to this section for details.) Hence (using Gronwall’s lemma), for all τ_{0} ≤ τ_{1}

Z_{τ_{1}} ≤ Z_{τ_{0}} exp(−k_{1}(τ_{1} − τ_{0})) + \frac{l}{k_{1}} (1 − exp(−k_{1}(τ_{1} − τ_{0}))).  (12)

So

𝔼(|U_{τ_{1}}|^{2}) ≤ 𝔼(|U_{τ_{0}}|^{2}) exp(−k_{1}(τ_{1} − τ_{0})) + k_{2},  (13)

where k_{2} = l/k_{1}.

We will see below that this is almost sufficient to obtain the inequality (X3) in the definition of the class X. Turning now to the condition (X4), we go back to (10).

The growth conditions on f and g, together with an application of Young’s inequality, give

2(F (U), U) + |G(U)|^{2} ≤ c(1 + |U|^{2}),

where c = c_{0}^{2} + c_{0}(1 + d_{1}).

¿From (10), this gives, for δ ≤ τ_{0} ≤ τ_{1},

sup_{τ_{0}≤σ≤τ_{1}} |U_{σ}|^{2} + 2ν ∫_{τ_{0}}^{τ_{1}} ∥U_{σ}∥^{2})dσ

≤ |U_{τ_{0}}|^{2} + c ∫_{τ_{0}}^{τ_{1}}(1 + |U_{σ}|^{2})dσ + sup_{τ_{0}≤σ≤τ_{1}} |M_{σ}|,  (14)

where M_{τ} is the martingale

M_{τ} = 2 ∫_{τ_{0}}^{τ_{1}}(U_{σ}, G(U_{σ}))dW_{σ},

and so

[M]_{τ} = 4 ∫_{τ_{0}}^{τ_{1}}(U_{σ}, G(U_{σ}))^{2}dσ.

Now using the bound on g we obtain

[M]_{τ} ≤ 4c′ sup_{τ_{0}≤σ≤τ} |U_{τ}|^{2} ∫_{τ_{0}}^{τ}(1 + |U_{σ}|^{2})dσ,  (15)

where c′ = c_{0}d_{2} + max(c_{0}, d_{2}^{2}).

Using Young’s inequality again gives

[M]_{τ}^{1/2} ≤ \frac{c′}{γ} sup_{τ_{0}≤σ≤τ} |U_{τ}|^{2} + γ ∫_{τ_{0}}^{τ}(1 + |U_{σ}|^{2})dσ  (16)

for γ to be specified below. The following Burkholder-Davis-Gundy inequality is now applicable (κ is a universal constant independent of the dimension N):

𝔼 ( sup_{τ_{0}≤σ≤τ_{1}} |M_{σ}|) ≤ κ𝔼 ([M]_{τ_{1}}^{1/2}) .  (17)

Use (14) and (16), with γ = 2(κc′)^{−1}, together with the Burkholder-Davis-Gundy inequality (17) to obtain

𝔼 ( sup_{τ_{0}≤σ≤τ_{1}} |U_{σ}|^{2} + ∫_{τ_{0}}^{τ_{1}} ∥U_{σ}∥^{2}dσ) ≤ c′′𝔼 (|U_{τ_{0}}|^{2} + ∫_{τ_{0}}^{τ_{1}}(1 + |U_{σ}|^{2})dσ) ,

(18)

where c′′ = max(1, c + 2/c′)/ min(\frac{1}{2}, 2ν). This, together with (12) gives

𝔼 ( sup_{τ_{0}≤σ≤τ_{1}} |U_{σ}|^{2} + ∫_{τ_{0}}^{τ_{1}} ∥U_{σ}∥^{2}dσ) ≤ α𝔼(|U_{τ_{0}}|^{2}) + β(τ_{1} − τ_{0}) (19)

for δ ≤ τ_{0} ≤ τ_{1}, provided that α ≥ c′′(1 + k_{1}^{−1}), β ≥ c′′(1 + lk_{1}^{−1}).

The construction of the standard process u(t) from U_{τ} now proceeds as in the main existence result Theorem 6.4.1 of [7] using the inequality (19). This shows that for a.a. ω the function |U_{τ}(ω)| is finite for all finite τ ≥ δ and U_{τ} is weakly S-continuous. Thus we can define a standard process u(t, ω) for a.a. ω by

u(t, ω) = ^{∘}U(τ, ω)

(the weak standard part) for any finite τ ≈ t with τ ≥ δ.

The inequality (19) also shows that for a.a. ω, for almost all finite τ we have ∥U_{τ}(ω)∥ finite and so u = ^{∘}U in the strong topology, hence F (U) ≈ f(u) and similarly with g. From the theory developed in [7] it follows that u(t, ω) is a solution; that is, condition (Y2) is satisfied. It is also clear that condition (Y1) holds. From (13) we also have (Y3) since |^{∘}U| ≤ ^{∘}|U| always and 𝔼(|U_{δ}|^{2}) ≈ 𝔼(|u_{0}|^{2}) < ∞.

It follows that conditions (X1), (X2), and (X6) hold. It is now necessary to check conditions (X3)–(X5).

We have the internal energy decay inequality (13), but in order to translate this into condition (X3) we need to know that |U_{τ}(ω)|^{2} is S-integrable for all finite τ ≥ δ. To show this we need to prove an internal analogue of the formula (X5).

Applying Itô’s lemma to the process φ_{n}(U_{τ}(ω)) for any n ∈ ^{∗}ℕ (of course we mean ^{∗}φ_{n} as usual in such contexts) gives:

dφ_{n}(U_{τ}) = [−ν((U_{τ}, φ′_{n}(U_{τ}) )) + (F (U_{τ}), φ′_{n}(U_{τ})) + \frac{1}{2}G(U_{τ})^{T} φ′′_{n}(U_{τ})G(U_{τ})] dτ
+(G(U_{τ}), φ′_{n}(U_{τ}))dW_{τ}

since (B(U), φ′_{n}(U)) = 0.

Now from the explicit form of φ′_{n} and φ′′_{n} (see Definition 2.1(f),(g) ) and the growth conditions on f, g we have:

((U, φ′_{n}(U))) ≥ λ_{1}|U||φ′_{n}(U)|,
(F (U), φ′_{n}(U)) ≤ (c_{0} + d_{1}|U|)|φ′_{n}(U)|,

and

G^{T} (U)φ′′_{n}(U)G(U) ≤ 3d_{2}^{2}|U||φ′_{n}(U)| + 12c_{0}(c_{0} + 2d_{2}|U|)ψ_{n}(U^{2}).

Putting Y_{τ} = 𝔼(φ_{n}(U_{τ})) and Ỹ_{τ} = 𝔼(|U_{τ}||φ′_{n}(U_{τ})|), it follows that for all δ ≤ τ_{0} ≤ τ,

\frac{dY_{τ}}{dτ} + νλ_{1}Ỹ_{τ} ≤

(d_{1} + \frac{3}{2}d_{2}^{2})Ỹ_{τ} + c_{0}𝔼(|φ′_{n}(U_{τ})|) + 6c_{0}𝔼((c_{0} + 2d_{2}|U_{τ}|)ψ_{n}(U_{τ}^{2})).  (20)

Hence, using the estimates in Lemma 2.3 together with 2.2(c)

\frac{dY_{τ}}{dτ} + k_{3}Y_{τ} ≤ n^{− \frac{1}{2}} (β_{0} + α_{0}Z_{τ})
≤ n^{− \frac{1}{2}} (β_{1} + α_{0}Z_{τ_{0}} exp(−k_{1}(τ − τ_{0})))
≤ n^{− \frac{1}{2}} (β_{1} + α_{0}Z_{τ_{0}}),

where k_{3} = 2νλ_{1} − 2d_{1} − 3d_{2}^{2} and Z_{τ} = 𝔼(|U_{τ}|^{2}) as before, and we have used the inequality (13). The constants are given explicitly as follows:

α_{0} = 2c_{0}(2 + 3c_{0} + 6d_{2}),
β_{0} = 3α_{0},
β_{1} = β_{0} + α_{0}k_{2}.

Finally, using Gronwall’s inequality, we get an internal analogue of (X5):

𝔼φ_{n}(U_{τ_{1}}) ≤ 𝔼φ_{n}(U_{τ_{0}}) exp[−k_{3}(τ_{1} − τ_{0}) + n^{− \frac{1}{2}} (α𝔼(|U_{τ_{0}}|^{2}) + β)]  (21)

for all τ_{1} ≥ τ_{0} ≥ δ, provided that β ≥ β_{1}/k_{3} and α ≥ α_{0}/k_{3}.

We can now see that |U_{τ}(ω)|^{2} is S-integrable for all finite τ ≥ δ. We know that |U_{δ}(ω)|^{2} is S-integrable (it was chosen thus). The property of the “truncation” function φ_{n} given by Corollary 13.8 (in the Appendix) means that ^{∘}𝔼φ_{n}(U_{δ}) → 0 as n → ∞. Now use (21) to see that ^{∘}𝔼φ_{n}(U_{τ}) → 0 as n → ∞ for all τ ≥ δ, and so |U_{τ}(ω)|^{2} is S-integrable for all τ ≥ δ.

We can now proceed to verify conditions (X3)–(X5). First note that applying the internal Fubini theorem to (19) shows that for all finite τ_{0} in a Loeb full set T_{0}, say, we have

𝔼(∥U_{τ_{0}}∥^{2}) < ∞.

Thus for any such τ_{0} we have ∥U_{τ_{0}}∥ < ∞ for a.a. ω and hence ^{∘}|U_{τ_{0}}| = |^{∘}U_{τ_{0}}|. Consequently, now that we have S-integrability of |U_{τ}(ω)|^{2} for all τ ≥ δ, Loeb theory gives the following: for all τ_{0} ∈ T_{0},

𝔼(|U_{τ_{0}}|^{2}) ≈ 𝔼(^{∘}|U_{τ_{0}}|^{2})
= 𝔼(|^{∘}U_{τ_{0}}|^{2})
= 𝔼(|u(t_{0})|^{2}),

where t_{0} = ^{∘}τ_{0}. Loeb theory also gives that for all τ and t_{1} = ^{∘}τ,

∘𝔼(|U_{τ}|^{2}) ≥ 𝔼(^{∘}|U_{τ}|^{2})
≥ 𝔼(|^{∘}U(τ)|^{2})
= 𝔼(|u(t_{1})|^{2}).

Putting this together and using the internal energy decay inequality (13), it follows that for all t_{0} = ^{∘}τ_{0} ∈ ^{∘}T_{0} and t_{1} = ^{∘}τ ≥ t_{0},

𝔼(|u(t_{1})|^{2}) ≤ ^{∘}𝔼(|U_{τ}|^{2})
≤ ^{∘}(𝔼(|U_{τ_{0}}|^{2}) exp(−k_{1}(τ − τ_{0})) + k_{2})
= 𝔼(|u(t_{0})|^{2}) exp(−k_{1}(t_{1} − t_{0})) + k_{2}

which is (X3), since ^{∘}T_{0} is a full set.

A similar argument shows that (X4) follows from (19). For (X5) note that since φ_{n}(U) ≤ |U|^{2} then φ_{n}(U_{τ}) is also S-integrable. Thus, for t_{0} = ^{∘}τ_{0} ∈ ^{∘}T_{0} we have, using the continuity of ψ_{n} and the fact that ^{∘}|U_{τ_{0}}(ω)| = |^{∘}U_{τ_{0}}(ω)| for a.a. ω,

𝔼φ_{n}(U_{τ_{0}}) ≈ 𝔼^{∘}φ_{n}(U_{τ_{0}})
= 𝔼φ_{n}(^{∘}U_{τ_{0}})
= 𝔼φ_{n}(u(t_{0})).

For all other t = ^{∘}τ we have

𝔼φ_{n}(u(t)) ≤ ^{∘}𝔼φ_{n}(U_{τ}),

using |^{∘}U(τ, ω)| ≤ ^{∘}|U_{τ}| and the continuity of ψ_{n}. These facts, together with (21), give (X5), and we are done. □

Appendix. Calculation of constants

Starting from (10), use ∥U∥^{2} ≥ λ_{1}|U|^{2} and the growth conditions on f, g to give

\frac{d}{dτ} Z_{τ} + 2νλ_{1}𝔼(|U|^{2}) ≤ 𝔼(2|U|(c_{0} + d_{1}|U|) + (c_{0} + d_{2}|U|)^{2})
= 𝔼(c_{0}^{2} + 2c_{0}(1 + d_{2})|U| + (2d_{1} + d_{2}^{2})|U|^{2})
≤ c_{0}^{2} + K/ε + 𝔼((εK + 2d_{1} + d_{2}^{2})|U|^{2})

for any given ε > 0, where K = c_{0}(1 + d_{2}). Here we have used Young’s inequality 2|U| ≤ ε^{−1} + ε|U|^{2}.

Now let δ = 2νλ_{1} − 2d_{1} − d_{2}^{2} > 0 and set ε = δ/2K. Then, putting k_{1} = δ/2 and l = c_{0}^{2} + K/ε = c_{0}^{2} + 2K^{2}/δ, we have

\frac{d}{dτ} Z_{τ} + k_{2}𝔼(|U|^{2}) ≤ l,

which is (11). For the constants α, β we simply set

α = max{c′′(1 + k_{1}^{−1}), α_{0}/k_{3}}

and

β = max{c′′(1 + lk_{1}^{−1}), β_{1}/k_{3}}

to ensure that (19) and (21) hold.

Stipulation 6.2 From now on we fix a set X of solutions as given by Definition 4.1, corresponding to a given choice of the constants k_{1,}k_{2}, k_{3}, α, β such that X ≠ Ø.

7 Internal approximate solutions

In order to construct an attractor we need to obtain an internal representation of all solutions u ∈ X. There may be many more than those that are obtained by means of Theorem 6.1, and to represent these we need the notion of an internal approximate solution. The following definition explains this. Here we work with the internal filtered probability space Ω̄ Ω = (Ω, G, (G_{τ})_{τ≥0}, Q). For any element V ∈ H_{N}, we let V_{(n)} = (V,^{∗}e_{n}).

Definition 7.1 (a) For each k ∈ ℕ and n ∈ ^{∗}ℕ denote by X_{k,n} the internal class of ^{∗}-adapted (with respect to (G_{τ})_{τ≥0}) processes

U : ^{∗}[0, ∞) × Ω → H_{N}

with the following properties (numbered to match with the corresponding properties in the Definition 4.1 of X_{k}).

(X 1) U_{τ}(ω) has paths ^{∗}a.s. in ^{∗}M and U ∈ ^{∗}L^{2}(Ω,^{∗}M), i.e.

𝔼 (∫_{0}^{^{∗}∞} |U_{τ}(ω)|^{2} exp(−τ)dτ) < ^{∗}∞.

(X 2_{n}) With Q-probability ≥ 1 − \frac{1}{n} on Ω, for all τ_{1} ∈ ^{∗}[\frac{1}{n}, n] and all m ≤ n,

|U(τ_{1})_{(m)} − U(1/n)_{(m)} − ∫ τ_{1}_{\frac{1}{n}}[−(νAU_{τ})_{(m)} − B(U_{τ})_{(m)} + F (U_{τ})_{(m)}]dτ−

∫ τ_{1}_{\frac{1}{n}} G(U_{τ})_{(m)}dW_{τ} | ≤ 2^{−n}.  (22)

(X 3_{n}) For all τ_{0} ∈ ^{∗}[0, ∞) except for a set of ^{∗}Lebesgue measure \frac{1}{n}, for all τ_{1} ≥ τ_{0},

𝔼(|U_{τ_{1}}|^{2}) ≤ 𝔼(|U_{τ_{0}}|^{2}) exp(−k_{1}(τ_{1} − τ_{0})) + k_{2} + \frac{1}{n}. (23)

(X 4_{n}) For all τ_{0} ∈ ^{∗}[0, ∞) except for a set of ^{∗}Lebesgue measure \frac{1}{n}, for all τ_{1} ≥ τ_{0},

𝔼 (sup_{τ_{0}≤σ≤τ_{1}}|U_{σ}| + ∫_{τ_{0}}^{τ_{1}} ∥U_{σ}∥^{2}dσ) ≤ α𝔼(|U_{τ_{0}}|^{2}) + β(τ_{1} − τ_{0}) + \frac{1}{n}.

(24)

(X 5_{n}) For all τ_{0} ∈ ^{∗}[0, ∞) except for a set of ^{∗}Lebesgue measure \frac{1}{n}, for all τ_{1} ≥ τ_{0} and all m ≤ n,

𝔼(φ_{m}(U_{τ_{1}})) ≤ 𝔼(φ_{m}(U_{τ_{0}})) exp(−k_{3}(τ_{1}−τ_{0}))+m^{− \frac{1}{2}} (α𝔼(|U_{τ_{0}}|)^{2}+β)+\frac{1}{n}. (25)

(X 6_{k,n}) 𝔼 ∫_{0}^{1} |U_{τ}|^{2}dτ ≤ k + \frac{1}{n}.

(b) For each k ∈ ℕ, define

X_{k} = ⋂_{n∈ℕ}X_{k,n}.

(c) Define X = ⋃_{k∈ℕ} X_{k}.

Remark 7.2 1. The sets X_{k} and X_{k,n} obviously increase with k.

2. The choice of the bound 2^{−n} on the right of (X 2_{n}) ensures that (X 2_{n+1}) ⇒ (X 2_{n}) for all n. Hence the sets X_{k,n} decrease with n.

3. Each of the sets X_{k,n} is internal.

Note that although X_{k,n} is defined for all n ∈ ^{∗}ℕ, the sets X_{k} only involve X_{k,n} for finite n. However, it important to note that for each fixed k the whole family (X_{k,n})_{n∈^{∗}ℕ} is defined and is internal, and we will make use of certain X_{k,J} for infinite J ∈ ^{∗}ℕ.

To help explain the next definition note that if U ∈ X then for a.a. ω, |U(τ, ω)| < ∞, and so U(τ, ω) is weakly nearstandard, for all finite τ ≉ 0 (see Lemma 13.3(a) in the Appendix); the weak standard part has coordinates ∘(U(τ, ω)_{(m)}) for m ∈ ℕ.

Definition 7.3 Given U ∈ X, a weak standard part of U is a process

u : (0, ∞) × Ω → H

such that for a.a. ω, whenever t ∈ (0, ∞) and ^{∘}τ = t,

u(t, ω)_{(m)} = ^{∘}(U(τ, ω)_{(m)})

for each m ∈ ℕ. If U has a weak standard part, it is a.s. unique and is denoted by ^{∘}U.

For a set Z ⊆ X, ^{∘}Z = {^{∘}U : U ∈ Z and ^{∘}U exists}.

Proposition 7.4 Let U ∈ X.

(a) U has a weak standard part if and only if for each m ∈ ℕ, for P - almost all ω, U(·, ω)_{(m)} is S-continuous on (0, ∞), that is, whenever σ ≈ τ and ^{∘}τ ∈ (0, ∞), U(σ, ω)_{(m)} ≈ U(τ, ω)_{(m)} and U(τ, ω)_{(m)} is finite.

(b) If the weak standard part u = ^{∘}U exists, then for P -almost all ω, u(·, ω) is weakly continuous on (0, ∞), that is, for each m ∈ ℕ, u(·, ω)_{(m)} is continuous on (0, ∞).

Proof This follows from Theorem 2.5 in [26], which is the analogous result for processes V : [0, 1] × Ω → ℝ. For (b) note that weak continuity for a norm bounded function v(t) with values in H is equivalent to continuity of the coordinate functions v(t)_{(m)}.

□

Definition 7.5 We denote by SL^{2}[a, b] the set of internal processes

U(τ, ω) : ^{∗}[0, ∞) → H_{N}

such that the restriction of |U(τ, ω)|^{2} to^{∗}[a, b]×Ω is S-integrable with respect to the product measure on ^{∗}[a, b] × Ω.

The importance of X lies in the following result.

Theorem 7.6 (a) For each k ∈ ℕ, if U ∈ X_{k} ∩ SL^{2}[0, 1] then the weak standard part u = ^{∘}U exists and is in X_{k}.

(b)

∘(X_{k} ∩ SL^{2}[0, 1]) = X_{k},

and hence

∘(X ∩ SL^{2}[0, 1]) = X.

We will prove this in two halves. First we have: Proof of Theorem 7.6(a) Let U ∈ X_{k}. By overspill, U ∈ X_{k,J} for some infinite J. From (X 6_{k,J}) there is a Loeb-full subset of τ_{0} ≤ 1 such that 𝔼(|U_{τ_{0}}|^{2}) < ∞, so by (23), 𝔼(|U_{τ_{0}}|^{2}) < ∞ for all τ_{0} ≉ 0. By (X 4_{J}), there is a Loeb-full set of finite τ_{0} such that (24) holds for all finite τ_{1}. Taking arbitrarily small such τ_{0} gives

𝔼 (sup_{\frac{1}{n} ≤σ≤τ_{1}}|U_{σ}|^{2} + ∫ τ_{1}_{\frac{1}{n}} ∥U_{σ}∥^{2}dσ) < ∞  (26)

for all n ∈ ℕ and all finite τ_{1} ≥ \frac{1}{n}. So |U_{τ}(ω)| is finite for all finite τ ≉ 0, for a.a. ω.

Let m ∈ ℕ. Fixing n for the moment and putting

V (τ)_{(m)} = ∫ τ_{\frac{1}{n}}[−(νAU_{σ})_{(m)} − B(U_{σ})_{(m)} + F (U_{σ})_{(m)}]dσ − ∫ τ_{\frac{1}{n}} G(U_{σ})_{(m)}dW_{σ},

the theory of [7] shows that for a.a. ω the process V (τ)_{(m)} is S-continuous on (0, ∞). Then (X 2_{J}) gives that for a.a. ω the process U(τ)_{(m)} is S-continuous on [\frac{1}{n}, ∞). Since this holds for all n we have established the existence and almost sure continuity of the weak standard part u = ^{∘}U.

We must now verify the properties of u that are needed to place it in X_{k}. First we have from (X 2_{J}) that for m ∈ ℕ,

u(t, ω)_{(m)} = u(\frac{1}{n}, ω)_{(m)} +^{∘}V (t, ω)_{(m)}.

The book [7] shows that

∘V (t)_{(m)} = (∫ t_{\frac{1}{n}}[−νAu(t) − B(u(t)) + f(u(t))]dt + ∫ t_{\frac{1}{n}} g(u(t))dw_{t})_{(m)}

almost surely, and this is sufficient to establish (X2).

The inequalities (X3)–(X5) follow from (X 3_{J}) − (X 5_{J}) provided that, as in the proof of Theorem 6.1, it can be shown that |U_{τ}(ω)|^{2} is S-integrable for all finite τ ≉ 0. The condition that U ∈ SL^{2}[0, 1] ensures that |U_{τ}(ω)|^{2} is S-integrable for a.a. τ ∈ ^{∗}[0, 1]. The formula (21) which was established in the proof of Theorem 6.1 is the same as condition (X 5_{J}). As in the proof of Theorem 6.1, this condition together with Corollary 13.8 imply that |U_{τ}(ω)|^{2} is S-integrable for all τ ≉ 0. Then (X3)–(X5) follow routinely from (X 3_{J}) − (X 5_{J}), using (X 4_{J}) to give that for almost all τ_{0} ≉ 0 we have 𝔼(∥U(τ_{0})∥^{2}) < ∞. Hence, putting t_{0} = ^{∘}τ_{0},

𝔼|u(t_{0})|^{2} = 𝔼^{∘}|U(τ_{0})|^{2} = ^{∘}𝔼|U(τ_{0})|^{2},

and for all t = ^{∘}τ we have

𝔼|u(t)|^{2} ≤ 𝔼^{∘}|U_{τ}|^{2} ≤ ^{∘}𝔼|U(τ)|^{2}.

We have

𝔼 ∫_{0}^{1} |u(t)|^{2}dt ≤ 𝔼 ∫_{0}^{1} ^{∘}|U_{τ}|^{2}d_{L}τ = 𝔼 ∫ 1_{\frac{1}{J}} ^{∘}|U_{τ}|^{2}d_{L}τ ≤ ^{∘}𝔼 ∫_{0}^{1} |U_{τ}|^{2}dτ ≤ k,

using basic Loeb theory and (X 6_{k,J}), so that (X6_{k}) holds.

For the path properties (X1), we have u(·, ω) ∈ L_{loc}^{∞}(0, ∞; H)∩L_{loc}^{2}[0, ∞; H) from (X 3_{J}) combined with 𝔼 ∫_{0}^{1} |u(t)|^{2}dt ≤ k. The weak continuity of u(·, ω) has already been established. Finally, the condition u(·, ω) ∈ L_{loc}^{2}(0, ∞; V) is immediate from (26). □

Implicit in the proof of Theorem 7.6 (a) is the following.

Theorem 7.7

X ∩ SL^{2}[0, 1] = X ∩ NS,

where NS = ns^{2}(Ω, M) as defined in the Appendix. Hence for U ∈ X ∩ SL^{2}[0, 1] the weak standard part ^{∘}U as defined in Definition 7.3 is the same as the standard part ^{∘}U of U ∈ NS in L^{2}(Ω, M) as defined in the Appendix.

Proof Take U = U_{τ}(ω) ∈ X ∩ SL^{2}[0, 1] and let u = ^{∘}U be its weak standard part. Regarded as random functions we have U ∈ ^{∗}L^{2}(Ω,^{∗}M) and u ∈ L^{2}(Ω, M). To see that U ∈ NS we first show that ^{∘}U(ω) = u(ω) in M for a.a. ω.

For a.a. ω, the proof of Theorem 7.6 and basic Loeb theory gives for V = U(ω) and v = u(ω):

(a) for a.a. finite τ, V (τ) ≈ v(^{∘}τ) and (by Anderson’s Lusin Theorem) ∗v(τ) ≈ v(^{∘}τ) (both strongly in H), so ^{∘}|V (τ) −^{∗}v(τ)| = 0.

(b) |V (τ)| ∈ SL^{2}[0, 1] and is bounded for τ ≥ 1, so |V (τ)| ∈ SL^{2}[0, n] for each n ∈ ℕ.

(c) |^{∗}v(τ)| ∈ SL^{2}[0, n] for each n ∈ ℕ.

Hence, for each n ∈ ℕ,

∫_{0}^{n} |V (τ) −^{∗}v(τ)|^{2} exp(−τ)dτ ≈ ∫_{0}^{n} ^{∘}|V (τ) −^{∗}v(τ)|^{2} exp(−τ)d_{L}τ = 0.

To get V ≈ v in M it is now sufficient to show that

∘(∫_{n}^{^{∗}∞} |V (τ) −^{∗}v(τ)|^{2} exp(−τ)dτ) → 0 as n → ∞.

We have ∫_{n}^{∞} |^{∗}v(τ)|^{2} exp(−τ)dτ = ∫_{n}^{∞} |v(t)|^{2} exp(−t)dt → 0 as n → ∞. It is enough then to show that for a.a. ω, I_{n} = ^{∘}( ∫_{n}^{^{∗}∞} |V (τ)|^{2} exp(−τ)dτ) → 0. Now from (X 3) and (X 6) we deduce that there is k ∈ ℕ such that

𝔼I_{n} ≤ k exp(−n),

and so P (I_{n} ≥ exp(−\frac{n}{2})) ≤ k exp(−\frac{n}{2}) by Chebychev. Borel-Cantelli gives that P (I_{n} ≥ exp(−\frac{n}{2}) i.o.\} = 0 so that I_{n} → 0 a.s. as required.

To obtain U ∈ NS it remains to show that |U(ω)|^{2} is S-integrable; that is

𝔼|U(ω)|^{2} ≈ 𝔼(^{∘}|U(ω)|^{2}) = 𝔼|u(ω)|^{2}.

Now

𝔼(|U(ω)|^{2}) = 𝔼 ∫_{0}^{^{∗}∞} |U_{τ}(ω)|^{2} exp(−τ)dτ
= 𝔼 ∫_{0}^{n} |U_{τ}(ω)|^{2} exp(−τ)dτ + 𝔼 ∫_{n}^{^{∗}∞} |U_{τ}(ω)|^{2} exp(−τ)dτ

for each n ∈ ℕ, and the second term becomes infinitesimal as n → ∞. The same is true for 𝔼(|u(ω)|^{2}), so it suffices to show that for each n ∈ ℕ,

𝔼 ∫_{0}^{n} |U_{τ}(ω)|^{2} exp(−τ)dτ ≈ 𝔼 ∫_{0}^{n} |u(t, ω)|^{2} exp(−t)dt.

Since |U_{τ}(ω)|^{2} is assumed to be S-integrable on [0, 1] × Ω, we have

𝔼 ∫_{0}^{1} |U_{τ}(ω)|^{2} exp(−τ)dτ ≈ 𝔼 ∫_{0}^{1} |u(t, ω)|^{2} exp(−t)dt.

Now 𝔼|U_{τ}(ω)|^{2} is bounded for τ ≥ 1 and, from the proof of the previous theorem, |U_{τ}(ω)|^{2} is S-integrable for a.a. τ ≉ 0. Putting this together we have

𝔼 ∫_{1}^{n} |U_{τ}(ω)|^{2} exp(−τ)dτ = ∫_{1}^{n} 𝔼|U_{τ}(ω)|^{2} exp(−τ)dτ
≈ ∫_{1}^{n} ^{∘}𝔼|U_{τ}(ω)|^{2} exp(−τ)d_{L}τ
= ∫_{1}^{n} 𝔼^{∘}|U_{τ}(ω)|^{2} exp(−τ)d_{L}τ
= 𝔼 ∫_{1}^{n} ^{∘}|U_{τ}(ω)|^{2} exp(−τ)d_{L}τ
= 𝔼 ∫_{0}^{n} |(u(t, ω)|^{2} exp(−t)dt

from the fact that ^{∘}U(ω) = u(ω) in M for a.a. ω.

For the other direction, we show that NS ⊆ SL^{2}[0, 1]. This is routine: if U ∈ NS with ^{∘}U = u then from 𝔼(|U|^{2}) ≈ 𝔼(|u|^{2}) and U_{τ}(ω) ≈ u(^{∘}t, ω) in H a.s. in [0, 1]×Ω, we have that |U_{τ}(ω)|^{2} exp(−τ) is S-integrable over [0, 1]×Ω. □

Now we turn to part (b) of Theorem 7.6. In view of part (a) and Theorem 7.7, it suffices to prove the following.

Theorem 7.8 If u ∈ X_{k} then there is U ∈ X_{k} ∩NS with weak standard part u = ^{∘}U.

Proof Take u ∈ X_{k}. By Theorem 7.7, it suffices to find a U ∈ X_{k} ∩ SL^{2}[0, 1] with u = ^{∘}U. Proposition 13.10 in the Appendix gives a positive δ ≈ 0, and for each i ∈ ℕ an internal G_{τ}-adapted process Ū(τ, ω)_{(i)} such that for a.a. ω and for all i ∈ ℕ, Ū(τ, ω)_{(i)} = 0 for all τ < δ, and

Ū(τ, ω)_{(i)} ≈ u(^{∘}τ, ω)_{(i)}

for all finite τ ≥ δ. By ℵ_{1}-saturation the sequence of co-ordinate processes extends to an internal family (Ū_{(i)})_{i≤N} of G_{τ}-adapted processes, giving a G_{τ}- adapted process Ū(τ, ω) ∈ H_{N}. This process must now be suitably truncated in order meet the conditions for membership of X_{k} ∩ NS.

For each m ≤ N define a process Ū^{(m)}(τ, ω) ∈ H_{N} by

Ū_{(i)}^{(m)} = \{ \begin{matrix}(Ū(τ, ω)_{(i)} ∧ m) ∨ −m & if i ≤ m \\ 0 & otherwise \end{matrix}

This process is again G_{τ}-adapted. For each m ∈ ℕ and all n ≤ m we have

𝔼 (∫_{0}^{n} |Ū_{τ}^{(m)}|^{2}dτ) ≈ 𝔼 (∫_{0}^{n} |^{∘}Ū_{τ}^{(m)}|^{2}d_{L}τ)
< 𝔼 (∫_{0}^{n} |u(t)|^{2}dt) + \frac{1}{m}

and

𝔼 (∫ n_{\frac{1}{n}} ∥U_{τ}^{(m)}∥^{2}dτ) ≈ 𝔼 (∫ n_{\frac{1}{n}} ∥^{∘}Ū_{τ}^{(m)}∥^{2}d_{L}τ)

< 𝔼 (∫ n_{\frac{1}{n}} ∥u(t)∥^{2}dt) + \frac{1}{m}.

By overspill there is an infinite J ≤ N such that for all n ∈ ℕ,

𝔼 (∫_{0}^{n} |Ū_{τ}^{(J)}|^{2}dτ) < 𝔼 (∫_{0}^{n} |u(t)|^{2}dt) + \frac{1}{J}

and

𝔼 (∫ n_{\frac{1}{n}} ∥Ū_{τ}^{(J)}∥^{2}dτ) < 𝔼 (∫ n_{\frac{1}{n}} ∥u(t)∥^{2}dt) + \frac{1}{J} .

Let V = Ū^{(J)}. Then for i ∈ ℕ, for a.a. ω we have

V (τ, ω)_{(i)} ≈ u(^{∘}τ, ω)_{(i)}

for all finite τ ≥ δ, and so |u(^{∘}τ, ω)|^{2} ≤ ^{∘}|V (τ, ω)|^{2} for all finite τ ≥ δ. Hence for n ∈ ℕ,

∘𝔼 ∫_{0}^{n} |V (τ, ω)|^{2}dτ ≤ 𝔼 (∫_{0}^{n} |u(t, ω)|^{2}dt)
= 𝔼 (∫_{0}^{n} |u(^{∘}τ, ω)|^{2}d_{L}τ)
≤ 𝔼 (∫_{0}^{n} ^{∘}|V (τ, ω)|^{2}d_{L}τ)
≤ ^{∘}𝔼 ∫_{0}^{n} |V (τ, ω)|^{2}dτ.

So V (τ, ω) is an adapted lifting of u(t, ω) which is in SL^{2}[0, n] for each n. Therefore V ∈ SL^{2}[0, 1]. Moreover, for each n ∈ ℕ and a.a. (τ, ω) ∈ ^{∗}[0, n] × Ω we have V (τ, ω) ≈ u(τ, ω) in H (strongly). Note also that for each n ∈ ℕ and a.a. τ ∈ ^{∗}[0, n] we have

𝔼(|V (τ)|^{2}) ≈ 𝔼(|u(^{∘}τ)|^{2}).  (27)

Similar reasoning shows that for all n ∈ ℕ, ∥V (τ, ω)∥ ∈ SL^{2}[\frac{1}{n}, n] and ∥V (τ, ω)∥ ≈ ∥u(τ, ω)∥ for a.a. (τ, ω) ∈ ^{∗}[0, n] × Ω. Thus, for a.a. (τ, ω) ∈ ∗[0, n] × Ω we also have V (τ, ω) ≈ u(τ, ω) in V (strongly).

It is now necessary to truncate further (i.e. take V′ = Ū^{(J′)} for some infinite J′ ≤ J) in order to ensure that the approximate inequalities (X 3_{n}) − (X 5_{n}) are all satisfied. Note that a fortiori all the properties of V noted above are valid for any such V′ also.

Let T_{0} be the Loeb-full set of finite τ_{0} such that

(i) (27) holds for τ = τ_{0}.

(ii) t_{0} = ^{∘}τ_{0} is a value for which (6), (7) and (8) hold for all t_{1} ≥ t_{0}.

Choose an increasing chain of internal sets A_{m} ⊆ ^{∗}[0, m] ∩ T_{0} with ∗Leb(A_{m}) ≥ m − \frac{1}{m}. Then for all m ∈ ℕ, τ_{0} ∈ A_{m}, and τ_{0} ≤ τ_{1} ≤ m we have 𝔼(|V (τ_{0})|^{2}) ≈ 𝔼(|u(t_{0})|^{2}) and ^{∘}𝔼(|Ū^{(m)}(τ_{1})|^{2}) ≤ 𝔼(|u(t_{1})|^{2}) (where t_{0} = ^{∘}τ_{0} and t_{1} = ^{∘}τ_{1}). Hence, using (6), for m ∈ ℕ, τ_{0} ∈ A_{m}, and all τ_{0} ≤ τ_{1} ≤ m,

𝔼(|Ū^{(m)}(τ_{1})|^{2}) ≤ 𝔼(|V (τ_{0})|^{2}) exp(−k_{1}(τ_{1} − τ_{0})) + k_{2} + \frac{1}{m}.  (28)

Similar reasoning gives

𝔼 (sup_{τ_{0}≤σ≤τ_{1}}|Ū^{(m)}(σ)|^{2} + ∫_{τ_{0}}^{τ_{1}} ∥Ū^{(m)}(σ)∥^{2}dσ) ≤ α𝔼(|V (τ_{0})|^{2})+β(τ_{1}−τ_{0})+\frac{1}{m},

(29)

and for all n ≤ m,

𝔼(φ_{n}(Ū_{τ_{1}}^{(m)})) ≤ 𝔼(φ_{n}(V_{τ_{0}})) exp(−k_{3}(τ_{1}−τ_{0}))+n^{− \frac{1}{2}} (α𝔼(|V_{τ_{0}}|^{2})+β)+ \frac{1}{m}. (30)

By ℵ_{1}-saturation we extend A_{m}, m ∈ ℕ to an internal increasing chain of sets A_{m}, m ∈ ^{∗}ℕ such that A_{m} ⊆ ^{∗}[0, m] and A_{m} has ^{∗}Lebesgue measure ≥ m − 1/m. Overspill gives an infinite J′ ≤ J such that (28)–(30) all hold with m = J′ for τ_{0} ∈ A_{J′} and τ_{0} ≤ τ_{1} ≤ J′.

Writing V′ = Ū^{(J′)}, for all m ∈ ℕ and τ_{0} ∈ A_{m} we have 𝔼(|V (τ_{0})|^{2}) ≈ 𝔼(|u(t_{0})|^{2}) ≈ 𝔼(|V′(τ_{0})|^{2}), and so for m ∈ ℕ, τ_{0} ∈ A_{m}, and τ_{0} ≤ τ_{1} ≤ m,

𝔼(|V′(τ_{1})|^{2}) ≤ 𝔼(|V′(τ_{0})|^{2}) exp(−k_{1}(τ_{1} − τ_{0})) + k_{2} + \frac{1}{m}.  (31)

Similar reasoning gives

𝔼 (sup_{τ_{0}≤σ≤τ_{1}}|V′(σ)|^{2} + ∫_{τ_{0}}^{τ_{1}} ∥V′(σ)∥^{2}dσ) ≤ α𝔼(|V′(τ_{0})|^{2}) + β(τ_{1} − τ_{0}) + \frac{1}{m},

(32)

and for all n ≤ m,

𝔼(φ_{n}(V′(τ_{1})) ≤ 𝔼(φ_{n}(V′_{τ_{0}})) exp(−k_{3}(τ_{1}−τ_{0}))+n^{− \frac{1}{2}} (α𝔼(|V′_{τ_{0}}|^{2})+β)+ \frac{1}{m}. (33)

By overspill, there is some infinite J′′ ≤ J′ such that (31)–(33) all hold with m = J′′ and for all τ_{0} ∈ A_{J′′} and τ_{0} ≤ τ_{1} ≤ J′′. Put

U = \{ \begin{matrix}V′(τ, ω) & for τ ≤ J′′ \\ 0 & for τ > J′′ \end{matrix}

Since V′ ∈ SL^{2}[0, 1], we have U ∈ SL^{2}[0, 1]. It follows from (27) with τ = τ_{0} and (28) that U satisfies the condition (X 1). Using (31)–(33), it is routine to check that (X 3_{n}) − (X 5_{n}) hold for this U for each n ∈ ℕ, the exceptional set in each case being ^{∗}[0, J′′] \ A_{J′′}.

As noted earlier, V′ (like V) is a lifting of u(t, ω) which is in SL^{2}[0, 1], so

∘𝔼 ∫_{0}^{1} |U_{τ}|^{2}dτ = 𝔼 ∫_{0}^{1} |u_{t}|^{2}dt,

giving (X 6_{k,n}) for each n.

To see that U ∈ X_{k} as required we now have only to check that (X 2_{n}) holds.

For this, fix n and note that U is an adapted lifting of u in SL^{2}[0, 1] with the property (24) for τ ≥ δ. Thus the theory of [7] shows that for a.a. ω, for all τ_{1} ∈ [\frac{1}{n}, n] and m ≤ n,

U(τ_{1})_{(m)} − U(1/n)_{(m)} ≈ u(^{∘}τ_{1})_{(m)} − u(1/n)_{(m)}

= ∫ ^{∘}τ_{1}_{\frac{1}{n}}[−νAu(t)_{(m)} − B(u(t))_{(m)} + f(u(t))_{(m)}]dt − ∫ ^{∘}τ_{1}_{\frac{1}{n}} g(u(t))_{(m)}dw_{t}

≈ ∫ τ_{1}_{\frac{1}{n}}[−(νAU_{τ})_{(m)} − B(U(τ))_{(m)} + F (U(τ))_{(m)}]dτ − ∫ τ_{1}_{\frac{1}{n}} G(U(τ))_{(m)}dW_{τ}.

□

Corollary 7.9 X ∩ NS is nonempty.

Proof By Theorem 6.1, there is a k ∈ ℕ such that X_{k} is nonempty. Then by Theorem 7.8, X_{k} ∩ SL^{2}[0, 1] is nonempty, and by Theorem 7.7, X_{k} ∩ NS is nonempty. □

8 The internal semiflow

Let H be the internal set consisting of all internal stochastic processes U : ^{∗}[0, ∞) × Ω → H_{N}. There is a natural internal semiflow T_{τ} : H → H defined for all 0 ≤ τ ∈ ^{∗}ℝ, as follows.

Definition 8.1 (Semiflow of internal processes) Suppose that U ∈ H. Then for any τ ≥ 0 the process V = T_{τ}U ∈ H is defined by

V (σ, ω) = U(σ + τ, Θ_{τ}ω).

This semiflow has internal properties corresponding to those for S_{t} as given in Proposition 3.2 Proposition 8.2

(a) T_{τ} is an internal semigroup on the class H.

(b) If U is adapted to the filtration (G_{τ})_{τ≥0}, then so is V = T_{τ}U.

(c) If U is adapted and V = T_{τ}U, then for τ_{1} ≥ 0: (i) For appropriate internal ^{∗}continuous F,

∫_{0}^{τ_{1}} F (V (σ, ω))dσ = ∫_{τ}^{τ+τ_{1}} F (U(σ, Θ_{τ}ω))dσ.

(By appropriate we mean that the integrals are defined.) (ii) For appropriate internal ^{∗}continuous G,

∫_{0}^{τ_{1}} G(V (σ, ω))dW (σ, ω) = ∫_{τ}^{τ+τ_{1}} G(V (σ, Θ_{τ}ω))dW (σ, Θ_{τ}ω)

(meaning that I(ω) = J(Θ_{τ}ω) as random variables, where I(ω) is the left-hand integral and J(ω) = ∫_{τ}^{τ+τ_{1}} G(u(σ, ω))dW (σ, ω)).

(d) If U is adapted and V = T_{τ}U and τ_{1} ≥ τ_{0}, then

|U(τ_{1} + τ) − U(τ_{0} + τ) − ∫_{τ_{0}+τ}^{τ_{1}+τ} F (U(σ))dσ − ∫_{τ_{0}+τ}^{τ_{1}+τ} G(U(σ))dW (σ)|

and

|V (τ_{1}) − V (τ_{0}) − ∫_{τ_{0}}^{τ_{1}} F (V (σ))dσ − ∫_{τ_{0}}^{τ_{1}} G(V (σ))dW (σ)|

have the same internal probability distribution. Proposition 8.3 For finite τ > 0:

(a) T_{τ}X ⊆ X. (b)

T_{τ}(X ∩ SL^{2}[0, 1]) ⊆ X ∩ SL^{2}[0, 1],

and for U ∈ SL^{2}[0, 1] we have

∘(T_{τ}U) = S_{^{∘}τ}^{∘}U.  (34)

Proof (a) Clauses (X 3_{n})–(X 5_{n}) are clearly invariant under the operation of T_{τ}. For (X 2), it is easy to check using Proposition 8.2 that T_{τ}U has property (X 2_{n}) provided that U has the property (X 2_{n′}) for some n′ ≥ n+τ.

If 𝔼 ∫_{0}^{1} |U_{σ}|^{2}dσ < ∞ then it is clear from (X 3_{n}) that 𝔼 ∫_{0}^{1} |(T_{τ}U)_{σ}|^{2}dσ < ∞ also.

(b) Suppose that U ∈ X ∩SL^{2}[0, 1]. From the proof of Theorem 7.6 (a) we see that |U_{σ}(ω)|^{2} is S-integrable for all σ ≉ 0. Further, from (X 3) we know that 𝔼(|U_{σ}|^{2}) is bounded on ^{∗}[σ, ∞) for any s ≉ 0. Thus |U_{σ}(ω)| ∈ SL^{2}[1, n] for any n ≥ 1, and hence on ^{∗}[τ, τ + 1] for any τ > 0. This means that T_{τ}U ∈ SL^{2}[0, 1] as required. □

We next show that for a suitable ρ ∈ ℕ the set X_{ρ} is S-absorbing.

Lemma 8.4 There is a ρ ∈ ℕ such that X_{ρ} is S-absorbing. That is, for each k ∈ ℕ there is an r(k) ∈ ℕ such that

T_{τ}X_{k} ⊆ X_{ρ}

for all finite τ ≥ r(k).

Proof This follows from the fact that U ∈ X_{k} has the property (X 3_{n}) for all n and 𝔼 ∫_{0}^{1} |U_{τ}|^{2}dτ ≤ k + \frac{1}{n} for all n. From these properties we can deduce that for any τ ≉ 0

𝔼 ∫_{τ}^{τ+1} |U_{σ}|^{2}dσ ≤ k exp(−k_{1}τ) + k_{2} + \frac{1}{n}

for any n. To see this, let Z_{σ} = 𝔼|U_{σ}|^{2} and note that from (X 6) and (X 3), Z_{σ} is bounded and hence S-integrable on [τ, τ + 1]. Moreover, for a.a. σ in this interval, σ − τ has the property of τ_{0} in (X 3_{n}) for all n. Thus

∫_{0}^{1} 𝔼|(T_{τ}U)(σ)|^{2}dσ = ∫_{τ}^{τ+1} Z_{σ}dσ ≈ ∫_{τ}^{τ+1} ^{∘}Z_{σ}d_{L}σ
≤ ∫_{τ}^{τ+1} ^{∘}(Z_{σ−τ} exp(−k_{1}τ) + k_{2})d_{L}σ
= exp(−k_{1}^{∘}τ) ∫_{0}^{1} ^{∘}Z_{σ}d_{L}σ + k_{2}
≤ exp(−k_{1}^{∘}τ)^{∘}(∫_{0}^{1} Z_{σ}dσ) + k_{2}
< k exp(−k_{1}τ) + k_{2} + \frac{1}{n}

as required. To verify the lemma we may take any natural number ρ > k_{2} and then r(k) = r such that

k exp(−k_{1}r) + k_{2} = ρ.

□

Now write B = X_{ρ} and B_{n} = X_{ρ,n}. Thus B is the intersection of a decreasing chain of internal sets, B = ⋂_{n∈ℕ} B_{n}.

Corollary 8.5 For each τ,

T_{τ}B = ⋂_{n∈ℕ}T_{τ}B_{n}.

Proof This follows from B = ⋂_{n∈ℕ} B_{n} by ℵ_{1}-saturation. □

Corollary 8.6 B ∩ NS is nonempty.

Proof By Corollary 7.9, X ∩ NS ≠ Ø, so there is a k ∈ ℕ such that X_{k} ∩ NS ≠ Ø. Take finite τ such that T_{τ}X_{k} ⊆ B and then

B ∩ NS ⊇ T_{τ}(X_{k} ∩ NS) ≠ Ø

using Proposition 8.3(b) and Theorem 7.7. □

Definition 8.7 Define r_{0} = r(ρ); then r_{0} is finite and T_{τ}B ⊆ B for finite τ ≥ r_{0}.

The following will be useful in several situations below.

Proposition 8.8 For any n, m ∈ ^{∗}ℕ, if r_{0} ≤ σ ≤ n − m then

T_{σ}B_{n} ⊆ B_{m}.

Proof Let U ∈ B_{n}. It is routine to check from the definitions that for any σ the properties (X 1) and (X 3_{n}) − (X 5_{n}) are preserved under T_{σ}. If σ ≥ r_{0} then the calculation in the proof of Lemma 8.4 shows that T_{σ}U has property (X 6_{ρ,n}) also. Finally, as noted in the proof of Proposition 8.3, T_{σ}U has the property (X 2_{m}) provided that σ + m ≤ n. Then, since m ≤ n, T_{σ}U belongs to X_{ρ,m} = B_{m}. □

9 Construction of global attractors

The main theorem of this section shows the existence of a global attractor A for the semiflow S_{t} on X. The idea for constructing A is similar to that used in the earlier papers [6, 8, 9] of the first author and Capiński: A = ^{∘}C for a set C ⊆ X that is an S-attractor for the internal semiflow T_{τ}.

From the S-absorbing set B we define the following set C, which we will call the S-attractor for the internal semiflow T_{τ} on X.

Definition 9.1 Define sets C, C_{n} and Ĉ_{n} (for n ∈ ℕ) as follows.

(a) C = ⋂_{τ finite^{0≤τ}} T_{τ}B.

(b) C_{n} = ⋂_{0≤τ≤n} T_{τ}B, so that C = ⋂_{n∈ℕ} C_{n}.

(c)  Ĉ_{n} = ⋂_{0≤τ≤n} T_{τ}B_{n}.

We first show that C is nonempty.

Proposition 9.2 (a) The sets Ĉ_{n} are internal and decreasing.

(b)  Ĉ_{n} ⊇ C_{n} ⊇ T_{r_{0}+n}B, hence Ĉ_{n} ≠ Ø.

(c) C = ⋂_{n∈ℕ} Ĉ_{n}, and so C ≠ Ø.

Proof (a) is obvious.

(b) Since B_{n} ⊇ B, we have C_{n} ⊇ Ĉ_{n}. The second inclusion follows from the choice of r_{0}: for V ∈ B we have T_{r_{0}+n}V = T_{τ}T_{r_{0}+n−τ}V ∈ T_{τ}B for all τ ≤ n. Since B is nonempty, it follows that Ĉ_{n} is nonempty.

(c)

⋂_{n∈ℕ} Ĉ_{n} = ⋂_{n∈ℕ} ⋂_{0≤τ≤n} T_{τ}B_{n} = ⋂_{τ finite^{0≤τ}} ⋂_{n∈ℕ}T_{τ}B_{n} = ⋂_{τ finite^{0≤τ}} T_{τ}B = C

(the third equality following from Corollary 8.5). The fact that C ≠ Ø now follows from (a) and ℵ_{1}-saturation. □

In preparation for the next theorem, which gives some of the key properties of C, we have:

Proposition 9.3

T_{τ}C = ⋂_{n∈ℕ}T_{τ}C_{n} = ⋂_{n∈ℕ}T_{τ} Ĉ_{n}.

Proof The equality of the outer sets is an elementary application of ℵ_{1}- saturation, since C = ⋂_{n∈ℕ} Ĉ_{n} and each Ĉ_{n} is internal. The middle set is squeezed between the outer ones since C ⊆ C_{n} ⊆ Ĉ_{n} for all n. □

Some crucial properties of the set C are now gathered together.

Theorem 9.4

(a) C ⊆ B(= X_{ρ}).

(b) C is a countable intersection of internal sets.

(c) (Invariance of C) For finite τ,

T_{τ}C = C.

(d) (C is nearstandard)

C ⊆ NS.

Proof (a) C ⊆ T_{0}B = B.

(b) Obvious from Proposition 9.2.

(c) For any finite τ and σ we have

C ⊆ T_{σ+r_{0}}B and T_{τ+r_{0}}B ⊆ B,

so

T_{τ}C ⊆ T_{τ}T_{σ+r_{0}}B = T_{σ}T_{τ+r_{0}}B ⊆ T_{σ}B.

Since this holds for all finite σ ≥ 0, it follows that T_{τ}C ⊆ C.

For the opposite inclusion, from Proposition 9.3 it is sufficient to show that C ⊆ T_{τ}C_{n} for each n. Take any U ∈ C; then U = T_{τ+r_{0}+n}V = T_{τ}T_{r_{0}+n}V for some V ∈ B. Now for each σ ≤ n we have T_{r_{0}+n}V = T_{σ}T_{r_{0}+n−σ}V ∈ T_{σ}B, so T_{r_{0}+n}V ∈ C_{n} and U ∈ T_{τ}C_{n}.

(d) Let U ∈ C. By Proposition 9.2 (b), U ∈ ⋂_{n∈ℕ} Ĉ_{n}. Recall that the sets B_{n} and hence Ĉ_{n} are already defined for all n ∈ ^{∗}ℕ. By overflow, there is an infinite J with U ∈ Ĉ_{J}. By the definition of Ĉ_{J}, U = T_{J}V for some V ∈ B_{J}. Using (X 6), there is a τ_{0} ∈ ^{∗}[0, 1] such that 𝔼(|V (τ_{0})|^{2}) is finite, and (25) holds for V with n = J. That is,

𝔼(φ_{m}(V_{τ_{1}})) ≤ 𝔼(φ_{m}(V_{τ_{0}})) exp(−k_{3}(τ_{1} −τ_{0}))+m^{− \frac{1}{2}} (α𝔼(|V_{τ_{0}}|^{2})+β)+ \frac{1}{J} (35)

for all τ_{1} ∈ ^{∗}[τ_{0}, ∞) and each m ≤ J. Since ψ(x) ≤ 1 for all x, φ_{m}(V_{τ_{0}}) ≤ |V_{τ_{0}}|^{2}, so 𝔼(φ_{m}(𝕍_{τ_{0}})) is also finite for all m.

Consider any infinite K ≤ J. For each σ ∈ ^{∗}[0, 1] using (35) with τ_{1} = J + σ we have

𝔼(φ_{K}(U(σ))) = 𝔼(φ_{K}(V_{J+σ})) ≤

𝔼(φ_{K}(V_{τ_{0}})) exp(−k_{3}(J + σ − τ_{0})) + K^{− \frac{1}{2}} (α𝔼(|V_{τ_{0}}|^{2}) + β) + \frac{1}{J} ,

so 𝔼(φ_{K}(U(σ))) ≈ 0. Therefore, by Corollary 13.8 in the Appendix, |U_{σ}(ω)|^{2} is S-integrable for all σ ∈ ^{∗}[0, 1]. Then by Proposition 13.6 in the Appendix, |U|^{2} is S-integrable in the product ^{∗}[0, 1] × Ω, that is, U ∈ SL^{2}[0, 1]. Since U ∈ X, we have U ∈ NS by Theorem 7.7. □

It is of interest to note the following seemingly weaker characterization of C 9 CONSTRUCTION OF GLOBAL ATTRACTORS 35

Proposition 9.5 For any sequence (n_{k}) from ℕ such that n_{k} → ∞, we have

C = ⋂_{k∈ℕ} T_{n_{k}}B.

Proof One inclusion is obvious. For the other, if τ ≥ 0 take n_{k} with n_{k} ≥ r_{0} + τ. Then

T_{n_{k}}B = T_{τ}T_{n_{k}−τ}B ⊆ T_{τ}B,

so that ⋂_{k} T_{n_{k}}B ⊆ T_{τ}B for all τ. □

We also need the following property of C. Proposition 9.6 (C attracts) For every n, k ∈ ℕ,

T_{τ}X_{k} ⊆ C_{n}

for all finite τ ≥ n + r_{0} + r(k). In particular, taking k = ρ so that r(k) = r_{0}

T_{τ}B ⊆ C_{n}

for all finite τ ≥ n + 2r_{0}. Proof We have τ = n + σ + r(k) for some finite σ ≥ r_{0}. Then

T_{τ}X_{k} = T_{n}T_{σ}T_{r(k)}X_{k} ⊆ T_{n}T_{σ}B ⊆ T_{n}B ⊆ C_{n}.

□

Remark 9.7 The above attraction property together with the invariance and the fact that C ⊆ NS (Theorem 9.4) is the reason for calling C the S-attractor for the internal set of processes X.

The attractor for the semiflow S_{t} can now be defined. Definition 9.8 Define the sets A_{n}, A by

A_{n} = ^{∘}(C_{n} ∩ NS), A = ^{∘}C.

Immediate properties of A are as follows (where we write B = X_{ρ}). Theorem 9.9 (a) S_{t}A = A for all t ≥ 0.

(b) A ⊆ B, where B = X_{ρ}.

(c) A ⊆ Y.

(d) A = ⋂_{n∈ℕ} A_{n}.

(e) A = ⋂_{t≥0} S_{t}B = ⋂_{k} S_{t_{k}}B for any sequence t_{k} → ∞.

(f) A is closed in the space L^{2}(Ω, M).

Proof (a) follows from Theorem 9.4 and Proposition 8.3(b).

(b) follows from the fact that C ⊆ B.

(c) By part (b), A = S_{1}A ⊆ S_{1}X ⊆ Y.

For (d), it is clear that A ⊆ ⋂ A_{n}. Consider an element u in the righthand set. For each n we have u = ^{∘}U_{n} for some U_{n} ∈ C_{n} ∩ NS. Then U_{n} ∈ Ĉ_{n} for all n ∈ ℕ. Let | · |_{k} denotes the norm | · | restricted to the time interval [0, k], that is,

|U|_{k} = (𝔼 ∫_{0}^{k} |U_{τ}|^{2}μ(dτ))^{\frac{1}{2}} .

Then for all n and m ≤ n we have

|U_{m} − U_{n}|_{n} ≤ \frac{1}{n}.

By ℵ_{1}-saturation there is an infinite J with U_{J} ∈ Ĉ_{J} ⊆ C and

|U_{m} − U_{J}|_{J} ≈ 0

for all m ∈ ℕ. Thus u = ^{∘}U_{J} ∈ A.

(e) Since C ⊆ T_{τ}B we have A ⊆ S_{t}B for all t ≥ 0, so it is enough to prove that the third set is contained in A. For any n we have

S_{t}B ⊆ ^{∘}(T_{t}B ∩ NS) ⊆ ^{∘}(C_{n} ∩ NS) = A_{n}

for all t ≥ n + 2r_{0}, by Proposition 9.6. The result follows.

(f) follows from Proposition 13.5 in the Appendix. □

In order to show that A has the required attracting properties, we first show:

Theorem 9.10 (a) For every n, k ∈ ℕ,

S_{t}X_{k} ⊆ A_{n}

for all t ≥ n + r_{0} + r(k).

(b) Let Z be a bounded subset of X (with the norm of L^{2}(Ω, M)). Then for every n ∈ ℕ there is finite t_{0}(n, Z) such that

S_{t}Z ⊆ A_{n}

for all t ≥ t_{0}(n, Z).

Proof (a) is immediate from Proposition 9.6 after observing that

S_{t}X_{k} ⊆ ^{∘}(T_{t}X_{k}).

(b) By Lemma 4.2, Z ⊆ X_{k} for some k ∈ ℕ. The result now follows from (a). In fact, Lemma 4.2 (a) shows that one can take t_{0}(n, Z) = n + r_{0} + r(k) where k ≥ (sup{|u| : u ∈ Z})^{2}e. □

Lemma 9.11 A has the attracting property of Definition 3.7(b)(iii)

Proof Let K be compact with ρ(A, K) = ε > 0 (otherwise there is nothing to prove). Then K^{≤ε} ∩ A = Ø. By Theorem 9.10 it is sufficient to show that K^{≤ε} ∩ A_{n} = Ø for some n ∈ ℕ.

For each n there is a finite set {v_{n,i} : 1 ≤ i ≤ k_{n}} ⊆ K such that K is covered by the open balls \{v_{n,i}\}^{< \frac{1}{n}} , 1 ≤ i ≤ k_{n}. ( \{v\}^{<r} denotes the open ball of radius r with centre v.) Taking v_{n,i} = ^{∘}V_{n,i} with V_{n,i} ∈ NS we have

K^{≤ε} = ^{∘} ( ⋂_{n∈ℕ} ⋃_{i=1}^{k_{n}}{U : |U − V_{n,i}| ≤ ε + 1/n}) = ^{∘} ( ⋂_{n∈ℕ}K_{n})

where the sets K_{n} are all internal. Similar reasoning gives K^{≤ε} = ^{∘} (⋂_{n∈ℕ} K_{n}^{≤1/n}). ℵ_{1}-saturation shows that there is n ∈ ℕ with

⋂_{m≤n} K_{m}^{≤ \frac{1}{m}} ∩ Ĉ_{n} = Ø  (36)

(for otherwise there is U ∈ ⋂_{n∈ℕ}(K_{n}^{≤ \frac{1}{n}} ∩ Ĉ_{n}), and such a U belongs to C and is thus nearstandard with ^{∘}U ∈ K^{≤ε} ∩ A).

It follows that K^{≤ε} ∩ A_{n} = Ø, for if not there is u = ^{∘}U = ^{∘}V with U ∈ ⋂_{m≤n} K_{m} and V ∈ Ĉ_{n}. But then V ∈ ⋂_{m≤n} K_{m}^{≤ \frac{1}{m}} contradicting (36).

□

Now we can prove the main theorem of the paper:

Theorem 9.12 The set A is an attractor for the set of solutions X.

Proof We have seen in Theorem 9.9 and Lemma 9.11 that A is closed and invariant and has the required attracting property, so it remains to prove that law_{w}(A) is a law-attractor. Write A = law_{w}(A).

For invariance, using (4) gives

Ŝ_{t}A = law_{w}(S_{t}A) = law_{w}(A) = A

.

Using Proposition 13.9(b) in the Appendix), we have

A = law_{w}(A) = law_{w}(^{∘}C) = ^{∘}^{∗}law_{w}(C) = ^{∘ ⋂}_{n∈ℕ}^{∗}law_{w}(Ĉ_{n}).

Since ^{∗}law_{w}(C) ⊆ ns(M_{1,2}(M × C_{0})), the compactness of A follows from Proposition 13.2 in the Appendix with E_{n} = ^{∗}law_{w}(Ĉ_{n}).

For the attraction property, suppose that Z ⊆ law_{w}(X) is d-bounded, and O ⊃ A is open. We will prove first that there is some n ∈ ℕ with

A_{n} = law_{w}(A_{n}) ⊆ O  (37)

and then that Ŝ_{t}Z ⊆ A_{n} eventually.

For (37), let F = M_{1,2}(M ×C_{0})\O, so the hypothesis is that A∩F = Ø. This means that

⋂_{n∈ℕ}^{∗}law_{w}(Ĉ_{n}) ∩ ⋂_{n}^{∗}(F^{≤1/n}) = Ø,

since any member of this set has the form Λ = law_{w}(U) with U ∈ C and so Λ is nearstandard in M_{1,2}(M × C_{0}) with ^{∘}Λ ∈ A ∩ F. Saturation gives an m ∈ ℕ with

⋂_{n≤m}^{∗}law_{w}(Ĉ_{n}) ∩ ⋂_{n≤m}^{∗}(F^{≤1/n}) = Ø,

and so A_{n} ∩ F = Ø, which is (37).

Since Z is d-bounded, Z ⊆ law_{w}(X_{k}) for some k. By Theorem 9.10, S_{t}X_{k} ⊆ A_{n} eventually. Hence, eventually

Ŝ_{t}Z ⊆ Ŝ_{t}(law_{w}(X_{k})) = law_{w}(S_{t}X_{k}) ⊆ law_{w}(A_{n}) = A_{n},

as required. □

The attraction property in Lemma 9.11 and its proof generalizes easily to the following, from which the law-attraction property of A can also be deduced.

Theorem 9.13 If O ⊇ A is an open set whose complement has the form O^{c} = ^{∘}( ⋂_{n∈ℕ} E_{n}) with each E_{n} internal, then A_{n} ⊆ O eventually, and so O attracts bounded subsets of X.

Remark This result will be generalized and explored further in the sequel to this paper, [18].

10 Two-sided solutions

In this section we formulate the notion of a two-sided solution to the Navier-Stokes equation (3) – that is, a solution defined for all time, negative and positive. It will be shown that the set X̄ of two-sided solutions is non-empty and that the attractor A identified in the previous section is simply the set of restrictions to non-negative times of the solutions in X̄. The corresponding fact for the deterministic equations is noted by Sell in [32].

We must first modify the space Ω Ω so that it can accommodate two-sided processes u(t, ω) with t ∈ ℝ. Recall that the underlying internal filtered probability space is

Ω̄ Ω = (Ω, G, (G_{τ})_{τ∈^{∗}ℝ}, Q),

where G_{τ} = ^{∗}σ({W(τ′) : τ′ ≤ τ}) and G = ⋁_{τ∈^{∗}ℝ} G_{τ}. So we simply extend the filtration F_{t} to negative times using the same recipe as before: for any t ∈ ℝ define

F_{t} = ⋂_{t<^{∘}τ} σ(G_{τ}) ∨ N .

Thus, the space we now work with is

Ω Ω = (Ω, F, (F_{t})_{t∈ℝ}, P).

The Wiener process w(t, ω) is extended to negative times in the same way:

w(t, ω) = ^{∘}W (t, ω)  (38)

for all real times.

Definition 10.1 Denote by X̄ the set of bounded two-sided solutions to the stochastic Navier-Stokes equations as follows. The members of X̄ are adapted stochastic processes u : ℝ × Ω → H with the following properties.

(X̄1) For a.a. ω the path u(·, ω) belongs to the following spaces:

L_{loc}^{∞}(−∞, ∞; H) ∩ L_{loc}^{2}( − ∞,∞;V) ∩ C( − ∞,∞;H_{weak}).

(X̄2) For all t_{1} ≥ t_{0} ∈ ℝ,

u(t_{1}) = u(t_{0}) + ∫_{t_{0}}^{t_{1}}[−νAu(t) − B(u(t)) + f(u(t))]dt + ∫_{t_{0}}^{t_{1}} g(u(t))dw_{t}.

(X̄3) For a.a. t_{0} ∈ ℝ and all t_{1} ≥ t_{0},

𝔼(|u(t_{1})|^{2}) ≤ 𝔼(|u(t_{0})|^{2}) exp(−k_{1}(t_{1} − t_{0})) + k_{2}.  (39)

(X̄4) For a.a. t_{0} ∈ ℝ and all t_{1} ≥ t_{0},

𝔼 (sup_{t_{0}≤s≤t_{1}}|u(s)|^{2} + 2ν ∫_{t_{0}}^{t_{1}} ∥u(s)∥^{2}ds) ≤ α𝔼(|u(t_{0})|^{2}) + β(t_{1} − t_{0}).

(40)

(X̄5) For a.a. t_{0} ∈ ℝ and all t_{1} ≥ t_{0} and n ∈ ℕ,

𝔼(φ_{n}(u(t_{1}))) ≤ 𝔼(φ_{n}(u(t_{0})) exp(−k_{3}(t_{1} − t_{0})) + n^{− \frac{1}{2}} (α𝔼(|u(t_{0})^{2}|) + β). (41)

(X̄6) 𝔼(|u(t)|^{2}) ≤ k_{2} for all t ∈ ℝ.

Remark It is immediate from the definitions that

X̄ ↾ [0, ∞) ⊆ Y ∩ X_{k_{2}} ⊆ X.

Corresponding to the space M of paths for one-sided solutions, the natural space that contains paths of two-sided solutions to the stochastic Navier– Stokes equations is the space M̄ defined as follows.

Definition 10.2 (a) For a measurable (deterministic) function ξ : ℝ → H define a norm

|ξ| = (∫_{−∞}^{∞} ξ(t)^{2} exp(−|t|)dt)^{\frac{1}{2}} = (∫_{−∞}^{∞} ξ(t)^{2}μ(dt))^{\frac{1}{2}} ,

where μ(dt) = exp(−|t|)dt, and write

M̄ = {ξ : |ξ| < ∞}

for this space of paths, which is a separable Hilbert space.

(b) For a process u(t, ω) with paths in M̄ define

|u| = (𝔼(|u(·, ω)|^{2}))^{\frac{1}{2}} = (𝔼 ∫_{−∞}^{∞} |u(t, ω)|^{2} exp(−|t|)dt)^{\frac{1}{2}} ,

which is simply the norm of L^{2}(Ω, M̄).

Remark It is clear from (X̄6) that X̄ ⊆ L^{2}(Ω, M̄). In fact, |u|^{2} ≤ 2k_{2} for all u ∈ X̄.

It would be possible to proceed by adapting the basic existence result Theorem 6 to show that X̄ ≠ Ø. However, this follows from the more detailed analysis of X̄ to come, and in particular from the main theorem of this section:

Theorem 10.3

A = X̄ ↾ [0, ∞).

The proof of this requires some additional analysis involving approximate two-sided solutions, which we now define (the counterpart of Definition 7.1 for one-sided solutions).

Definition 10.4 (a) For each n ∈ ℕ denote by X̄_{n} the internal class of ∗-adapted (with respect to (G_{τ})_{τ∈^{∗}ℝ}) processes U : ^{∗}ℝ × Ω → H_{N} with the following properties:

(X̄1) U_{τ}(ω) has paths ^{∗}a.s. in ^{∗}M̄ and U ∈ ^{∗}L^{2}(Ω,^{∗}M̄), i.e.

𝔼 (∫_{−^{∗}∞}^{^{∗}∞} |U_{τ}(ω)|^{2} exp(−|τ|)dτ) < ^{∗}∞.

(X̄2_{n}) With Q-probability ≥ 1 − \frac{1}{n} on Ω, for all τ_{1} ∈ ^{∗}[ − n, n] and all m ≤ n,

|U(τ_{1})_{(m)} − U(−n)_{(m)} − ∫_{−n}^{τ_{1}}[−(νAU_{τ})_{(m)} − B(U_{τ})_{(m)} + F (U_{τ})_{(m)}]dτ−

∫_{−n}^{τ_{1}} G(U_{τ})_{(m)}dW_{τ} | ≤ 2^{−n}.  (42)

(X̄3_{n}) For all τ_{0} ∈ ^{∗}[ − n, ∞) except for a set of ^{∗}Lebesgue measure \frac{1}{n}, and for all τ_{1} ≥ τ_{0},

𝔼(|U_{τ_{1}}|^{2}) ≤ 𝔼(|U_{τ_{0}}|^{2}) exp(−k_{1}(τ_{1} − τ_{0})) + k_{2} + \frac{1}{n}. (43)

(X̄4_{n}) For all τ_{0} ∈ ^{∗}[−n, ∞) except for a set of ^{∗}Lebesgue measure \frac{1}{n}, for all τ_{1} ≥ τ_{0},

𝔼 (sup_{τ_{0}≤σ≤τ_{1}}|U_{σ}| + ∫_{τ_{0}}^{τ_{1}} ∥U_{σ}∥^{2}dσ) ≤ α𝔼(|U_{τ_{0}}|^{2}) + β(τ_{1} − τ_{0}) + \frac{1}{n}.

(44)

(X̄5_{n}) For all τ_{0} ∈ ^{∗}[−n, ∞) except for a set of ^{∗}Lebesgue measure \frac{1}{n}, for all τ_{1} ≥ τ_{0} and all m ≤ n,

𝔼(φ_{m}(U_{τ_{1}})) ≤ 𝔼(φ_{m}(U_{τ_{0}})) exp(−k_{3}(τ_{1}−τ_{0}))+m^{− \frac{1}{2}} (α𝔼(|U_{τ_{0}}|^{2})+β)+\frac{1}{n}. (45)

(X̄6_{n}) 𝔼(|U_{τ}|^{2}) ≤ k_{2} + \frac{1}{n} for τ ∈ ^{∗}[ − n, ∞).

(b) Define

X̄ = ⋂_{n∈ℕ} X̄_{n}.

Remark It is immediate from the definitions that

X̄_{n} ↾ ^{∗}[0, ∞) ⊆ X_{k_{2},n} ⊆ X_{n}, X̄ ↾ ^{∗}[0, ∞) ⊆ X_{k_{2}} ⊆ X .

The notion of the standard part of an internal process from earlier sections extends naturally to two-sided internal processes. Corresponding to Theorem 7.6 (a) we have:

Theorem 10.5 If U ∈ X̄ then

(a) For P -almost all ω the path U(·, ω) has |U(τ, ω)| finite and U(τ, ω)_{(m)} S-continuous for m ∈ ℕ and all finite τ (positive and negative).

(b) The process u = ^{∘}U defined by

u(t, ω) = ^{∘}U(τ, ω)

for τ ≈ t (where ^{∘}U denotes the weak standard part in H) belongs to X̄.

Proof We first show that, unlike Theorem 7.6, it is not necessary to restrict to internal processes U that are S-integrable in some sense.

Take U ∈ X̄; then U ∈ X̄_{J} for some infinite J. From (X̄6) we have that 𝔼(|U_{τ}|^{2}) < k_{2} + \frac{1}{J} for all finite τ. Then (X̄5_{J}) ensures that |U_{τ}(ω)|^{2} is S-integrable for all finite τ, using the criterion of Corollary 13.8(b) of the Appendix.

Now simply follow the proof of Theorem 7.6 (a) with the origin t = 0 moved to t = −n to construct u = ^{∘}U with u fulfilling the requirements of a two-sided solution on each time interval [−n, ∞). Consequently u ∈ X̄. □

We also have the converse of Theorem 10.5:

Theorem 10.6

X̄ = ^{∘}X̄.

Proof Let u ∈ X̄. Simply adapt the proof of Theorem 7.8 to obtain U^{n} ∈ X̄_{n} for each n with ^{∘}U^{n} ↾ ^{∗}[ − n, ∞) = u ↾ [−n, ∞). Then for all sufficiently small infinite J ∈ ^{∗}ℕ we have U = U^{J} ∈ X̄ and ^{∘}U = u. □

To continue we must introduce a group of shift operators on ^{∗}L^{2}(Ω, M̄) as follows:

Definition 10.7 For V ∈ ^{∗}L^{2}(Ω, M̄) define the left shift operator L_{τ} for τ ∈ ^{∗}ℝ by

(L_{τ}V)(σ, ω) = V (τ + σ, Θ_{τ}ω).

Clearly L_{τ} ∘ L_{−τ} = identity for all τ, and L_{σ} ∘ L_{τ} = L_{σ+τ}, so the left shift operators L_{τ} form an internal group of mappings on ^{∗}L^{2}(Ω, M̄) under composition.

It is also clear from the definitions that L_{τ} X̄ = X̄ for every finite τ. The connection with the internal semigroup T_{τ} is given by:

Lemma 10.8 For any V ∈ ^{∗}L^{2}(Ω, M̄),

T_{τ}(V ↾ ^{∗}[0, ∞)) = (L_{τ}V) ↾ ^{∗}[0, ∞)).  (46)

Proof From the definitions. □

To proceed it is necessary to make one further definition, the counterpart of C for two-sided solutions. Recall the set B = X_{ρ} of one-sided approximate solutions.

Definition 10.9 (a) Define B̄ by

B̄ = {U ∈ ^{∗}L^{2}(Ω, M̄) : U ↾ ^{∗}[0, ∞) ∈ B},

(i.e. the processes in B are extended to negative times in an arbitrary manner) and in the same way for each k ∈ ℕ define

B̄_{k} = {U ∈ ^{∗}L^{2}(Ω, M̄) : U ↾ ^{∗}[0, ∞) ∈ B_{k}},

(b) Define

D = ⋂{L_{τ} B̄: τ ≥ 0, τ finite}.

Lemma 10.10 X̄ ⊆ B̄.

Proof Let U ∈ X̄ and let r = r(k_{2}). Since X̄ = L_{r}(X̄), U = L_{r}V for some V ∈ X̄. Then V ↾ ^{∗}[0, ∞) ∈ X_{k_{2}}, and by (46) we have

U ↾ ^{∗}[0, ∞) = (L_{r}V) ↾ ^{∗}[0, ∞) = T_{r}(V ↾ ^{∗}[0, ∞)) ∈ T_{r}X_{k_{2}} ⊆ B.

Therefore U ∈ B̄. □

The main result will follow from its counterpart at the level of approximate solutions:

Theorem 10.11 (a) D = X̄.

(b) C = D ↾ ^{∗}[0, ∞).

Proof (a) We have X̄ ⊆ B̄ and L_{τ} X̄ = X̄ for finite τ, so

X̄ = L_{τ} X̄ ⊆ L_{τ} B̄

for finite τ ≥ 0. Hence X̄ ⊆ D.

Conversely, let V ∈ D. Then for each n ∈ ℕ we have V = L_{n}V_{n} for some V_{n} ∈ B̄_{2n}. Overflow gives an infinite J ∈ ^{∗}ℕ with V = L_{J}V_{J} and V_{J} ∈ B̄_{2J}. Thus V_{J} ↾ ^{∗}[0, ∞) = U_{J} is in B_{2J}. From the definition of B_{2J} we can read off the properties of V on the interval ^{∗}[ − J, ∞) and check that V ∈ X̄. For example, the property (X 2_{2J}) for U_{J} gives the approximate equation (22) on ∗[1/(2J), 2J], and so V satisfies the same equation on ^{∗}[ − J + 1/(2J), J], which ensures that V satisfies condition (X̄2_{n}) for all n ∈ ℕ.

The rest of the properties needed for V ∈ X̄ are easily checked.

(b) Writing q(V) = V ↾ ^{∗}[0, ∞) for a two-sided process V we have by definition

B̄ = q^{−1}(B) and D = ⋂_{0≤τ<∞} L_{τ} B̄.

Thus

D ↾ ^{∗}[0, ∞) = q(D)
= q ( ⋂_{0≤τ<∞} L_{τ}(q^{−1}(B)))
⊆ ⋂_{0≤τ<∞} q(L_{τ}(q^{−1}(B)))
= ⋂_{0≤τ<∞} T_{τ}(q(q^{−1}(B)))
= ⋂_{0≤τ<∞} T_{τ}B
= C,

giving one inclusion. For the other direction, let U ∈ C. So U = T_{n}U_{n} with U_{n} ∈ B_{2n} for each n ∈ ℕ. By overflow we have U = T_{J}U_{J} for some infinite J ∈ ^{∗}ℕ, with U_{J} ∈ B_{2J}. Take any V_{J} such that V_{J} ↾ ^{∗}[0, ∞) = U_{J}, so that (L_{J}V_{J}) ↾ ^{∗}[0, ∞) = T_{J}U_{J} = U, using (46). It now suffices to show that L_{J}V_{J} ∈ D.

For any finite τ ≥ 0,

L_{J}V_{J} = L_{τ} ∘ L_{J−τ}V_{J}

Now

(L_{J−τ}V_{J}) ↾ ^{∗}[0, ∞) = T_{J−τ}(V_{J} ↾ ^{∗}[0, ∞)) = T_{J−τ}U_{J} ∈ B_{J},

using (46) again, and Proposition 8.8. This gives L_{J−τ}V_{J} ∈ B̄_{J} ⊆ B̄, and so L_{J}V_{J} ∈ L_{τ} B̄. □

As a simple corollary we now have the main result of this section.

Theorem 10.12 (=Theorem 10.3)

A = X̄ ↾ [0, ∞).

Proof

A = ^{∘}C = ^{∘}(D ↾ ^{∗}[0, ∞)) = ^{∘}(X̄ ↾ ^{∗}[0, ∞)) = X̄ ↾ [0, ∞)

using Theorems 10.6 and 10.11. □

The next theorem is proved in the same way as the corresponding results for A:

Theorem 10.13 law_{w}(X̄) is compact.

Proof Modify the proof that law_{w}(A) is compact in Theorem 9.12. □

11 Noncompactness Results

In this section we prove results showing that one usually cannot expect the attractor for the stochastic Navier-Stokes equation to be a compact set.

Theorem 11.1 Suppose that Ω Ω = (Ω, F, (F_{t})_{t∈ℝ}, P) is an arbitrary filtered probability space carrying a two-sided Brownian motion and suppose there is an attractor A for the stochastic Navier-Stokes equations satisfying Theorem 10.12. Suppose further that A has at least two distinct elements u, v, and there are sets Z_{n}, n ∈ ℕ in F_{−∞} of measure 1/2 that are independent of each other and of w, u, v. Then A is not compact in the topology of convergence in probability (a fortiori, A is not compact in L^{2}(Ω, M)).

Proof We have P [|u − v| ≥ r] ≥ r for some r > 0. By definition of A, there exist two-sided solutions u′, v′ ∈ X such that u_{t} = u′_{t} and v_{t} = v′_{t} for all t ≥ 0. For each n, define the process x_{n} on Ω Ω by

x_{n}(ω) = \{ \begin{matrix}u(ω) & if ω ∈ Z_{n} \\ v(ω) & otherwise. \end{matrix}

and define x′_{n} analogously. Since Z_{n} is F_{−∞}-measurable, it follows that each x′_{n} is a two-sided solution of the Navier-Stokes equation belonging to X̄, and therefore each x_{n} belongs to A. For each m ≠ n,

P [|x_{m} − x_{n}| ≥ r] ≥ r · P [Z_{m}ΔZ_{n}] = r/2,

since {|x_{m} − x_{n}| ≥ r} ⊇ {|u − v| ≥ r]} ∩ (Z_{m}ΔZ_{n}). Therefore A is not compact in the topology of convergence in probability. □

Remark It is well-known that when the filtration (F_{t})_{t∈ℝ} is generated by the Wiener process w_{t} (t ∈ ℝ), F_{−∞} is the trivial algebra (e.g. see [11], page 583), so the above theorem does not apply. In all known proofs of existence for solutions to the stochastic Navier-Stokes equations in 3-dimensions, however, the space required is richer than the Wiener space - for example the Loeb space that we have used in the present paper. Here the non-compactness of A can be verified, due to the following consequence of the richness of the space.

Lemma 11.2 In the filtered Loeb space Ω Ω of Section 10, for any countable collection of Loeb measurable sets y_{0}, y_{1}, … there exists an F_{−∞}-measurable set Z of measure 1/2 which is independent of each y_{n}.

Proof Let Q be the internal measure on Ω which generates the Loeb space Ω Ω. For each n ∈ ℕ, let Y_{n} be an internal approximation of y_{n}. For each h ∈ ℕ there is a finite internal partition of Ω into G_{−h}-measurable sets of Q-measure 1/h. By saturation, there is a hyperfinite partition X_{i}, i ≤ H of Ω into G_{−H}-measurable sets of Q-measure 1/H. Let P_{H} be the internal counting measure on 2^{H}. For each γ ∈ 2^{H} let Z(γ) be the internal F_{−∞}-measurable set

Z(γ) = ⋃{X_{i} : i ≤ H, γ(i) = 1}.

Fix n ∈ ℕ for the moment and let Y = Y_{n}. It is sufficient to show that for a.a. γ we have Q(Z(γ)) ≈ \frac{1}{2} and Z(γ) is independent of Y with respect to the Loeb measure, – i.e. Q(Z(γ) ∩ Y) ≈ \frac{1}{2}Q(Y).

For each 1 ≤ i ≤ H and γ ∈ 2^{H}, define the random variable η_{i}(γ) by

η_{i}(γ) = (Q(Y ∩ Z(γ) ∩ X_{i}) − \frac{1}{2}Q(Y ∩ X_{i}))
= \{ \begin{matrix}\frac{1}{2}Q(Y ∩ X_{i}) & if γ(i) = 1 \\ −\frac{1}{2}Q(Y ∩ X_{i}) & if γ(i) = 0. \end{matrix}

With respect to P_{H}, the random variables η_{i}, i ≤ H, are mutually independent and have expected values 𝔼η_{i} = 0. Moreover, |η_{i}(γ)| ≤ (2H)^{−1}, so η_{i} has variance ≤ (2H)^{−2} with respect to P_{H}.

Define a random variable

ξ(γ) = ∑_{i=1}^{H} η_{i}(γ).

Then with respect to P_{H},

𝔼(ξ^{2}) = ∑_{i=1}^{H} 𝔼(η_{i}^{2}) ≤ \frac{1}{4H} ≈ 0.

Since the sets X_{i} partition Ω, we have

∑_{i=1}^{H} Q(Y ∩ Z(γ) ∩ X_{i}) = Q (⋃_{i=1}^{H}(Y ∩ Z(γ) ∩ X_{i})) = Q(Y ∩ Z(γ))

and

∑_{i=1}^{H} Q(Y ∩ X_{i}) = Q (⋃_{i=1}^{H}(Y ∩ X_{i})) = Q(Y).

Therefore

ξ(γ) = Q(Y ∩ Z(γ)) − \frac{1}{2}Q(Y).

Thus, for (P_{H})_{L}-a.a. γ we have Q(Y ∩ Z(γ)) − \frac{1}{2}Q(Y) ≈ 0, and so Q(Z(γ) ∩ Y) ≈ \frac{1}{2}Q(Y) as required. Applying this to Y = Ω also gives that Q(Z(γ)) ≈ \frac{1}{2} for a.a. γ. □

Corollary 11.3 In the filtered Loeb space Ω Ω of Section 10, if the Navier-Stokes attractor has more than one element, then it is not compact in the topology of convergence in probability.

To see how, in a general setting, simply enlarging the space eliminates compactness of an attractor, we have the following.

Let Ω Ω = (Ω, F, (F_{t})_{t∈ℝ}, P) be a filtered probability space equipped with a family of measure preserving maps (θ_{t})_{t≥0} satisfying properties (θ1, θ2, θ3) of Section 3. Let (Ω′, F′, P′) be another probability space. The product Ω̄ Ω = Ω Ω×Ω Ω′ is the filtered probability space (Ω×Ω′, F×F′, (F_{t}×F′)_{t∈ℝ}, P ×P′) with the measure preserving maps (θ̄)_{t}(ω, γ) = (θ_{t}(ω), γ).

For a stochastic process u on Ω Ω, let ū be the process defined on Ω̄ by ū_{t}(ω, γ) = u_{t}(ω).

If w is a two-sided Brownian motion on Ω Ω, then w̄ is a two-sided Brownian motion on Ω̄ Ω.

Note that for each set D ∈ F′, Ω × D is F_{−∞} × F′-measurable.

With this notation we have the following non-compactness result.

Corollary 11.4 Let w be a two-sided Brownian motion in Ω Ω, and suppose the stochastic Navier-Stokes equations have an attractor A satisfying Theorem 10.12, and A has at least two distinct elements. Let Ω Ω′ be an atomless probability space, and let Ā be the stochastic Navier-Stokes attractor for (Ω̄ Ω, w̄). Then Ā is not compact in the topology of convergence in probability.

12 Final Remarks

On any filtered probability space Ω Ω with a family of measure preserving maps θ_{t} as in section 3 we can define the sets X_{k}, X and X̄ of solutions and a set A by

A = ⋂ S_{n}B

with B = X_{ρ}. A natural question is to isolate the properties of Ω Ω that are needed to give the main results above – namely that A = X̄ ↾ [0, ∞) and that A is an attractor in some sense (and the above results about compactness of families of laws hold).

A further natural question is to isolate the particular properties of the Navier-Stokes equations and/or the families of solutions X that are needed to develop a theory such as this.

We will present one approach to these questions in the sequel [18] to this paper.

13 Appendix: Nonstandard preliminaries

We work in an ℵ_{1}-saturated nonstandard universe that contains a nonstandard extension ^{∗}J for every mathematical object J involved in our theory. In particular we have ^{∗}H,^{∗}M,^{∗}C_{0}(ℝ),^{∗}Wiener measure, etc.

Given a standard Hausdorff space S, we identify each point x ∈ S with ∗x, so that S ⊆ ^{∗}S. If x ∈ S and X ∈ ^{∗}S, we say that x is the standard part of X, in symbols x = ^{∘}X, if X ∈ ^{∗}O for every open neighborhood O of x. Since S is Hausdorff, each X ∈ ^{∗}S has at most one standard part. An element X ∈ ^{∗}S is said to be near-standard, in symbols X ∈ ns(S), if X has a standard part. Thus the standard part function maps ns(S) onto S and is the identity on S. The standard part of a set B ⊆ ns(S) is the set ^{∘}B = {^{∘}X : X ∈ B}. Here is a useful immediate consequence of the definition of standard part.

Remark 13.1 Suppose x = ^{∘}X in a Hausdorff space S.

(a) If O is open in S, x ∈ O implies X ∈ ^{∗}O.

(b) If C is closed in S, X ∈ ^{∗}C implies x ∈ C.

In the particular case of a standard metric space (S, ρ), x = ^{∘}X if and only if ^{∗}ρ(X, x) ≈ 0, and two points X, Y ∈ ^{∗}S are said to be infinitely close, in symbols X ≈ Y, if ^{∗}ρ(X, Y) ≈ 0.

We need the following compactness criterion ([20], Lemma 4.9).

Proposition 13.2 Let S be a standard metric space, let E_{n} be an internal subset of ^{∗}S for each n, and let E = ⋂_{n} E_{n}.

(a) ^{∘}(E ∩ ns(S)) is closed.

(b) If E ⊆ ns(S), then ^{∘}E is compact.

Proof We give a proof here for the sake of completeness, and so we can refer to the proof later on. We may assume without loss of generality that E_{1} ⊇ E_{2} ⊇ · · ·.

(a) Suppose x_{n} ∈ ^{∘}(E ∩ ns(S)) and x_{n} → x ∈ S. Take a subsequence x_{m} such that ρ(x_{m}, x) < 1/m for each m ∈ ℕ. Choose X_{m} ∈ E ⊆ E_{m} with ∘X_{m} = x_{m}, and choose X ∈ ^{∗}S with ^{∘}X = x. Then ^{∗}ρ(X_{m}, X) < 1/m for each m. By the triangle inequality, ^{∗}ρ(X_{m}, X_{n}) < 2/m whenever m ≤ n. By ℵ_{1}-saturation, there is a Y ∈ ^{∗}S such that Y ∈ E_{n} for each n, and ∗ρ(X_{m}, Y) < 2/m for each m. Then Y ∈ E and Y ≈ X, so x = ^{∘}Y and Y ∈ E ∩ ns(S). Thus x ∈ ^{∘}(E ∩ ns(S)), so ^{∘}(E ∩ ns(S)) is closed.

(b) Since S is a metric space, to show that ^{∘}E is compact it suffices to show that every countable open cover {O_{n}} of ^{∘}E has a finite subcover. Let A_{mn} = \{x ∈ S : ρ(x, S \ O_{m}) > \frac{1}{n}\}. Then A_{mn} is open, \overline{A_{mn}} ⊆ O_{m}, and {A_{mn}} covers ^{∘}E. By 13.1 (a), {^{∗}A_{mn}} covers E. By ℵ_{1}-saturation, there is a k such that \{^{∗}A_{mn} : m, n ≤ k\} covers E_{k}. Then by 13.1 (b), \{\overline{A_{mn}} : m, n ≤ k\} covers ^{∘}E. Therefore {O_{m} : m ≤ k} covers ^{∘}E. □

The book [7] gives information about the standard part mapping for various topologies on the standard set H. The most important are as follows. Here, ^{∗}H has an internal ^{∗}basis {^{∗}e_{n}}_{n∈^{∗}ℕ}, and we write E_{n} = ^{∗}e_{n}. Thus for each N ∈ ^{∗}ℕ, H_{N} = ^{∗}span{E_{1}, …, E_{N}} ⊆ ^{∗}H. We also write u_{(n)} = (u, e_{n}) and U_{(n)} = (U, E_{n}).

Lemma 13.3 Let U ∈ H_{N}. Then:

(a) If |U| < ∞ (i.e. |U| is finite) then U is weakly nearstandard in H, and the weak standard part u = st_{weak}(U) is defined by

u_{(n)} = ^{∘}(U_{(n)}), n ∈ ℕ.

(b) If U is nearstandard in the strong topology of H then |U| < ∞ and

st_{weak}(U) = st(U).

(c) If ∥U∥ < ∞ then U is (strongly) nearstandard in H.

In view of the consistency (b) above we use ^{∘}U to denote the standard part of U whenever |U| is finite.

Near-standard points and standard parts also appear in the setting of Loeb spaces. For an internal probability space Ω Ω = (Ω, G, Q) we write Q_{L} for the corresponding Loeb measure. The theory of Loeb measure and Loeb integration is assumed (see [7, 16, 17] for example). For convenience, we assume here that Ω Ω is ^{∗}countably additive, although most of the general theory carries over to the ^{∗}finitely additive case. Recall that a ^{∗}measurable function U : Ω → ^{∗}ℝ is S-integrable if 𝔼(|U|) is finite and ∫_{|U(ω)|≥J} |U|dQ(ω) ≈ 0 for all infinite J ∈ ^{∗}ℕ. The following notation, taken from [19, 20], is used.

Definition 13.4 Suppose that (S, ρ) is a separable metric space (we will mainly have S = M and S = M × C_{0}).

(a) L^{0}(Ω, S) is the space of all random variables on v : Ω → S with the topology of convergence in probability, and for 0 < p ≤ ∞ the spaces L^{p}(Ω, S) are defined as usual.

(b) SL^{0}(Ω, S) = ^{∗}L^{0}(Ω,^{∗}S) = {V : Ω → ^{∗}S; V is G-measurable}.

(c) ns^{0}(Ω, S) = {V ∈ SL^{0}(Ω, S) : V (ω) ∈ ns(S) for Q_{L} a.a. ω ∈ Ω}. If V : Ω → ^{∗}S is in ns^{0}(Ω, S) then we write v = ^{∘}V for the member of L^{0}(Ω, S) given by

(^{∘}V)(ω) = ^{∘}(V (ω)),

and we say that V is a lifting of v and that v is the standard part of V.

(d) For p ∈ [1, ∞), SL^{p}(Ω, S) is the set of all V ∈ ^{∗}L^{0}(Ω, S) such that ρ(V (ω), z)^{p} is S-integrable for some (equivalently, any) z ∈ S. If V ∈ SL^{p}(Ω, S), we say that V is SL^{p} on Ω.

(e) For p ∈ [1, ∞),

ns^{p}(Ω, S) = ns^{0}(Ω, S) ∩ SL^{p}(Ω, S).

(Note that ns^{p}(Ω, S) ⊆ ^{∗}L^{p}(Ω,^{∗}S), and if V ∈ ns^{p}(Ω, S) then ^{∘}V ∈ L^{p}(Ω, S))

(f) Since p = 2 occurs frequently, we write NS = ns^{2}(Ω, S) with the particular Ω and S being clear from the context.

This notation is actually a generalization of the standard part in stars of Hausdorff spaces. In the case that Ω = {ω} is a one-point space, the points of SL^{0}({ω}, S) can be identified with the points of ^{∗}S in the obvious way, so that

SL^{0}({ω}, S) = SL^{p}({ω}, S) = ^{∗}S, ns^{0}({ω}, S) = ns^{p}({ω}, S) = ns(S),

and the standard part mapping for ns^{0}({ω}, S) is the same as for ns(S).

Given a set B ⊆ SL^{p}(Ω, S), where p ∈ {0} ∪ [1, ∞), the standard part of B is the set ^{∘}B = {^{∘}V : V ∈ B}. The following result is needed:

Proposition 13.5 Let E = ⋂_{n∈ℕ} E_{n} where each set E_{n} ⊆ SL^{0}(Ω, S) is internal. Then for each p ∈ {0} ∪ [1, ∞), the set

∘(E ∩ ns^{p}(Ω, S))

is closed in L^{p}(Ω, S).

Proof Exactly the same as the proof of Proposition 13.2 (a). □

We need the following fact about S-integrability on a product space.

Proposition 13.6 Suppose V : ^{∗}[0, 1] × Ω → ^{∗}[0, ∞) is ^{∗}-measurable on the product, and V (τ, ·) is S-integrable on Ω for each τ. Then V is S-integrable on the product.

Proof This result is well-known, but we include a proof for completeness. Take any infinite J ∈ ^{∗}ℕ. For each τ ∈ ^{∗}[0, 1], ∫ V (τ, ω)dQ(ω) is finite, and by overspill, ∫ V (τ, ω)dQ(ω) has a uniform finite bound. Therefore ∫_{0}^{1} ∫ V (τ, ω)dQ(ω)dτ is finite. Moreover, for each infinite J ∈ ℕ,

∫_{V (τ,ω)≥J} V (τ, ω)dQ(ω) ≈ 0.

Therefore

∫_{0}^{1} ∫_{V (τ,ω)≥J} V (τ, ω)dQ(ω)dτ ≈ 0.

By the transfer of the Fubini Theorem, it follows that V is S-integrable on the product. □

The following proposition and corollary explain how the “truncation” functions ψ_{n} and φ_{n} are used to characterize S-integrability.

Proposition 13.7 Let V : Ω → ^{∗}[0, ∞) be an internal random variable. Then the following are equivalent.

(a) V is S-integrable.

(b) 𝔼(V ψ_{J}(V)) ≈ 0 for all sufficiently small infinite J.

(c) ^{∘}𝔼(V ψ_{n}(V)) → 0 as n → ∞, (n ∈ ℕ).

Proof Assume (a), so that ∫_{V ≥K} V ≈ 0 for all infinite K ∈ ^{∗}ℝ. Writing

𝔼(V ψ_{J}(V)) = ∫_{V <K} V ψ_{J}(V) + ∫_{V ≥K} V ψ_{J}(V)

for infinite K we see that the second term on the right is infinitesimal since 0 ≤ ψ_{J} ≤ 1. For the first term, if we now set K = J^{\frac{1}{2}} we have

∫_{V <K} V ψ_{J}(V) ≤ Kψ_{J}(K) = K ((\frac{K}{J^{2}} − 1)^{3} + 1) ≈ 0,

which establishes (b). The converse implication (assuming (b)) is trivial since 𝔼(V ψ_{J}(V)) ≥ ∫_{V ≥J} V. The equivalence of (b) and (c) is routine. □

Recalling that φ_{n}(u) = |u|^{2}ψ_{n}(|u|^{2}) in any Hilbert space, we have

Corollary 13.8 Let U : Ω → ^{∗}S be an internal random vector where S is a Hilbert space. The following are equivalent.

(a) |U|^{2} is S-integrable, that is, U ∈ SL^{2}(Ω, S).

(b) 𝔼(φ_{J}(U)) ≈ 0 for all sufficiently small infinite J.

(c) ^{∘}𝔼(φ_{n}(U)) → 0 as n → ∞, (n ∈ ℕ).

As indicated in Section 3.2, the space of all Borel probability measures on a separable metric space S is denoted by M_{1}(S), and for v ∈ L^{0}(Ω, S), law(v) ∈ M_{1}(S) is the probability induced by v. The Prohorov metric on M_{1}(S) gives weak convergence of measures, and part (a) of the following is well-known (see [20] Prop 5.7 for example).

Proposition 13.9 Let V ∈ SL^{0}(Ω, S).

(a)

V ∈ ns^{0}(Ω, S) ⇔ ^{∗}law(V) ∈ ns(M_{1}(S)),

and if either side holds then law(^{∘}V) = ^{∘}(^{∗}law(V)).

(b) For the particular case S = M

V ∈ NS = ns^{2}(Ω, M) ⇔ ^{∗}law(V) ∈ ns(M_{1,2}(M)),

and if either side holds then law(^{∘}V) = ^{∘}(^{∗}law(V)).

Proof (b) follows from (a) together with the observation that for Λ = ∗law(V) ∈ ns(M_{1,2}(M)) there is the additional requirement that

𝔼_{Λ}|U|^{2} = ∫ |V (ω)|^{2}dP (ω) ≈ ∫ |^{∘}V (ω)|^{2}dP_{L} = 𝔼_{λ}|u|^{2},

where λ = law(^{∘}V). This condition is precisely that V (ω) is SL^{2} on Ω, which ensures that V ∈ NS. □

Finally, we need the following result from [26], Theorem 8.1, on liftings of continuous adapted stochastic processes. (The equivalence of (a) and (b) is proved there, and it is an easy exercise to show that (b) is equivalent to (c)).

Proposition 13.10 Let x : [0, ∞) × Ω → ℝ be a stochastic process in the filtered Loeb space Ω Ω given in Definition 5.1. The following are equivalent.

(a) x is adapted and almost surely continuous.

(b) There is an internal process X : ^{∗}[0, ∞) × Ω → ^{∗}ℝ and a positive δ ≈ 0 such that X(·, τ) is G_{τ}-measurable for all τ ≥ δ, and for almost all ω, ∘X(τ, ω) = x(^{∘}τ, ω) for all finite τ.

(c) There is an internal G_{τ}-adapted process X : ^{∗}[0, ∞) × Ω → ^{∗}ℝ and a positive δ ≈ 0 such that X(ω, τ) = 0 for τ < δ, and for almost all ω, ∘X(τ, ω) = x(^{∘}τ, ω) for all finite τ ≥ δ.

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