Hao Shen
Hao Shen 


Short Bio:
PhD 2013 (Princeton, Advisor:Weinan E);
Postdoc 20142015 (Warwick, Mentor: Martin Hairer);
Ritt Assistant Professor 20152018 (Columbia, Mentor:Ivan Corwin).
CV.



Recent events
July 1  12, 2024.
MSRI/SLMath Summer Graduate School on Stochastic quantization
(More information is on
SLMath website)
Dec 17  22, 2023.
Oberwolfach Arbeitsgemeinschaft: QFT and Stochastic PDEs
Aug 9  11, 2023.
Frontiers of Stochastic Analysis (Chicago, Organized by Cheng Ouyang and Jing Wang).
September 10  11, 2022.
Graduate Student Probability Conference, Madison (Registration closed.)
Teaching
Here are some of the previous courses I have taught.
Teaching in Spring 2023
Math833, Topics in stochastic PDE.
MWF 12:05PM  12:55PM
TEXTBOOK: Lecture notes uploaded on Canvas.
Research
I am interested in stochastic partial differential equations,
and its interaction with quantum field theory, statistical mechanics, interacting particle systems and geometric flows.
A survey:

Some recent progress in singular stochastic partial differential equations.
(with Ivan Corwin) Bulletin of the AMS. 57 (2020), no. 3, 409454.
Stochastic quantization of YangMills and other gauge theories

Stochastic quantization of an abelian gauge theory.
(2019).
Comm. Math. Phys. 384, 14451512 (2021).

Langevin dynamic for the 2D YangMills measure.
(with Ajay Chandra, Ilya Chevyrev, Martin Hairer) (2020). Publ. Math. IHÉS. 136, 1147 (2022).

Stochastic quantisation of YangMillsHiggs in 3D.
(with Ajay Chandra, Ilya Chevyrev, Martin Hairer) (2022).

A new derivation of the finite N master loop equation for lattice YangMills.
(with Scott Smith, Rongchan Zhu) (2022).

A stochastic analysis approach to lattice YangMills at strong coupling.
(with Rongchan Zhu, Xiangchan Zhu) (2022). Comm. Math. Phys.

Invariant measure and universality of the 2D YangMills Langevin dynamic. (with Ilya Chevyrev) (2023).
Stochastic PDE limits of statistical physics models

ASEP(q,j) converges to the KPZ equation.
(with Ivan Corwin and LiCheng Tsai) (2016)
Ann. Inst. Henri Poincaré (B) Probab. Stat.

Glauber dynamics of 2D KacBlumeCapel model and their stochastic PDE limits.
(with Hendrik Weber) (2016).
J. Funct. Anal. 275, Issue 6, 13211367 (2018).

Open ASEP in the weakly asymmetric regime.
(with Ivan Corwin) (2016).
Comm. Pure Appl. Math. 71, Issue 10, 20652128 (2018).

Stochastic PDE limit of the six vertex model.
(with Ivan Corwin, Promit Ghosal and LiCheng Tsai) (2018).
Comm. Math. Phys. 375, 19452038 (2020).

Stochastic Telegraph equation limit for the stochastic six vertex model.
(with LiCheng Tsai) (2018).
Proc. Amer. Math. Soc.

Scaling limit of a directed polymer among a Poisson field of independent walks.
(with Jian Song, Rongfeng Sun, Lihu Xu), J. Funct. Anal. 281 (2021), no. 5, 55pp

Mean field limit of a dynamical model of polymer systems.
(with Weinan E)
Sci. China Math. 56 (2013), no. 12, 25912598
Stochastic PDE and quantum field theory problems

A stochastic PDE approach to large N problems in quantum field theory: a survey.
J. Math. Phys. (2022)

An SPDE approach to perturbation theory of Φ42: asymptoticity and short distance behavior.
(with Rongchan Zhu, Xiangchan Zhu) (2021). Ann. Appl. Prob. (2022)

Large N limit of the O(N) linear sigma model via stochastic quantization.
(with Scott Smith, Rongchan Zhu, Xiangchan Zhu) (2020), Ann. Probab. 50 (2022), no. 1, 131202.

Large N limit of the O(N) linear sigma model in 3D.
(with Rongchan Zhu, Xiangchan Zhu) (2021) Comm. Math. Phys. (2022)

Large N limit and 1/N expansion of invariant observables in O(N) linear σmodel via SPDE.
(with Rongchan Zhu, Xiangchan Zhu) (2023)

Stochastic Ricci Flow on Compact Surfaces.
(with Julien Dubédat) Int. Math. Res. Not. (2021).

Renormalized powers of OrnsteinUhlenbeck processes and wellposedness of stochastic GinzburgLandau equations.
(with Weinan E and Arnulf Jentzen) (2013)
Nonlinear Analysis.
Other papers on regularity structures (KPZ, stochastic quantization, weak universality etc.)

A central limit theorem for the KPZ equation.
(With Martin Hairer)
Ann. Probab. 45 (2017), no. 6B, 41674221.

The dynamical sineGordon model.
(With Martin Hairer)
Comm. Math. Phys. 341 (2016), no. 3, 933989

Moment bounds for SPDEs with nonGaussian fields and application to the WongZakai problem.
(With Ajay Chandra)
Electron. J. Probab. Vol 22 (2017), 132.

Weak universality of dynamical Φ43: nonGaussian noise.
(With Weijun Xu)
Stoch PDE: Anal Comp. 6 (2018), 211254.

Local solution to the multilayer KPZ equation.
(With Ajay Chandra and Dirk Erhard),
J. Stat. Phys. 175 (2019), Issue 6, 10801106

The dynamical sineGordon model in the full subcritical regime.
(With Ajay Chandra and Martin Hairer)
Integrable probability

The strictweak lattice polymer.
(With Ivan Corwin and Timo Seppäläinen)
J. Stat. Phys. 160 (2015), no. 4, 10271053
Rigorous renormalization group analysis

Exact renormalization group analysis of turbulent transport by the shear flow.
(With Weinan E)
J. Stat. Phys. 153 (2013), no. 4, 553571

A renormalization group method by harmonic extensions and the classical dipole gas.
Ann. Henri Poincaré, 17 (2016), no. 4, 861911
Other articles

Oberwolfach report:
Rough Paths, Regularity Structures and Related Topics (2016)

Oberwolfach report: A dynamical approach to lattice YangMills.
Universality: Random Matrices, Random Geometry and SPDEs (2022)

PhD Thesis:
Renormalization Theory in Statistical Physics and Stochastic Analysis.
(Advisor: Weinan E)
Grants and Awards
My research is mainly supported by the following grants:
NSF CAREER DMS2044415 (20212026);
NSF DMS1954091 (20202023);
NSF DMS1712684 / DMS1909525 (2017  2020)