Hao Shen

Hao Shen

  Associate Professor
  Department of Mathematics
  University of Wisconsin-Madison
  Email: pkushenhao at gmail
  Office: Van Vleck Hall 619

Short Bio: PhD 2013 (Princeton, Advisor:Weinan E);
Postdoc 2014-2015 (Warwick, Mentor: Martin Hairer);
Ritt Assistant Professor 2015-2018 (Columbia, Mentor:Ivan Corwin).
CV.
Hao Shen

Recent events

August 18th - December 19th, 2025. SLMath semester program "Recent Trends in Stochastic Partial Differential Equations"

September 1 - 2, 2024. Graduate Student Probability Conference 2.0, Madison

July 1 - 12, 2024. MSRI/SLMath Summer Graduate School on "Stochastic quantization" (More information is on SLMath website)

July 22 - August 2, 2024. Cornell Probability Summer School

June 17 - 21, 2024. The Ninth Pacific Rim Conference in Mathematics Darwin

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 2024


Math635, Introduction to Brownian motion and stochastic calculus
MWF 9:55AM - 10:45AM (SOC SCI 6240)

TEXTBOOK: A first course in stochastic calculus (by Louis-Pierre Arguin)


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:

  1. Some recent progress in singular stochastic partial differential equations. (with Ivan Corwin) Bulletin of the AMS. 57 (2020), no. 3, 409-454.

Stochastic quantization of Yang-Mills and other gauge theories

  1. Stochastic quantization of an abelian gauge theory. (2019). Comm. Math. Phys. 384, 1445-1512 (2021).
  2. Langevin dynamic for the 2D Yang-Mills measure. (with Ajay Chandra, Ilya Chevyrev, Martin Hairer) (2020). Publ. Math. IHÉS. 136, 1-147 (2022).
  3. Stochastic quantisation of Yang-Mills-Higgs in 3D. (with Ajay Chandra, Ilya Chevyrev, Martin Hairer) (2022).
  4. A new derivation of the finite N master loop equation for lattice Yang-Mills. (with Scott Smith, Rongchan Zhu) (2022). EJP (2024)
  5. A stochastic analysis approach to lattice Yang-Mills at strong coupling. (with Rongchan Zhu, Xiangchan Zhu) (2022). Comm. Math. Phys.
  6. Invariant measure and universality of the 2D Yang-Mills Langevin dynamic. (with Ilya Chevyrev) (2023).
  7. Langevin dynamics of lattice Yang-Mills-Higgs and applications. (with Rongchan Zhu, Xiangchan Zhu) (2024).

Stochastic PDE related with other quantum field theory problems

  1. A stochastic PDE approach to large N problems in quantum field theory: a survey. J. Math. Phys. (2022)
  2. An SPDE approach to perturbation theory of Φ42: asymptoticity and short distance behavior. (with Rongchan Zhu, Xiangchan Zhu) (2021). Ann. Appl. Prob. (2022)
  3. 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, 131-202.
  4. Large N limit of the O(N) linear sigma model in 3D. (with Rongchan Zhu, Xiangchan Zhu) (2021) Comm. Math. Phys. (2022)
  5. Large N limit and 1/N expansion of invariant observables in O(N) linear σ-model via SPDE. (with Rongchan Zhu, Xiangchan Zhu) (2023)
  6. Stochastic Ricci Flow on Compact Surfaces. (with Julien Dubédat) Int. Math. Res. Not. (2021).
  7. Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. (with Weinan E and Arnulf Jentzen) (2013) Nonlinear Analysis.

More papers on regularity structures (KPZ, stochastic quantization, weak universality etc.)

  1. A central limit theorem for the KPZ equation. (With Martin Hairer) Ann. Probab. 45 (2017), no. 6B, 4167-4221.
  2. The dynamical sine-Gordon model. (With Martin Hairer) Comm. Math. Phys. 341 (2016), no. 3, 933-989
  3. Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem. (With Ajay Chandra) Electron. J. Probab. Vol 22 (2017), 1-32.
  4. Weak universality of dynamical Φ43: non-Gaussian noise. (With Weijun Xu) Stoch PDE: Anal Comp. 6 (2018), 211-254.
  5. Local solution to the multi-layer KPZ equation. (With Ajay Chandra and Dirk Erhard), J. Stat. Phys. 175 (2019), Issue 6, 1080-1106
  6. The dynamical sine-Gordon model in the full subcritical regime. (With Ajay Chandra and Martin Hairer) (2018)
  7. Global well-posedness for 2D generalized Parabolic Anderson Model via paracontrolled calculus. (with Rongchan Zhu, Xiangchan Zhu) (2024)

Stochastic PDE limits of statistical physics models

  1. ASEP(q,j) converges to the KPZ equation. (with Ivan Corwin and Li-Cheng Tsai) (2016) Ann. Inst. Henri Poincaré (B) Probab. Stat.
  2. Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits. (with Hendrik Weber) (2016). J. Funct. Anal. 275, Issue 6, 1321-1367 (2018).
  3. Open ASEP in the weakly asymmetric regime. (with Ivan Corwin) (2016). Comm. Pure Appl. Math. 71, Issue 10, 2065-2128 (2018).
  4. Stochastic PDE limit of the six vertex model. (with Ivan Corwin, Promit Ghosal and Li-Cheng Tsai) (2018). Comm. Math. Phys. 375, 1945-2038 (2020).
  5. Stochastic Telegraph equation limit for the stochastic six vertex model. (with Li-Cheng Tsai) (2018). Proc. Amer. Math. Soc.
  6. 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
  7. Mean field limit of a dynamical model of polymer systems. (with Weinan E) Sci. China Math. 56 (2013), no. 12, 2591-2598

Integrable probability

  1. The strict-weak lattice polymer. (With Ivan Corwin and Timo Seppäläinen) J. Stat. Phys. 160 (2015), no. 4, 1027-1053

Rigorous renormalization group analysis

  1. Exact renormalization group analysis of turbulent transport by the shear flow. (With Weinan E) J. Stat. Phys. 153 (2013), no. 4, 553-571
  2. A renormalization group method by harmonic extensions and the classical dipole gas. Ann. Henri Poincaré, 17 (2016), no. 4, 861-911

Other articles

  1. Oberwolfach report: Rough Paths, Regularity Structures and Related Topics (2016)
  2. Oberwolfach report: A dynamical approach to lattice Yang-Mills. Universality: Random Matrices, Random Geometry and SPDEs (2022)
  3. 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:
Simons Fellow in Mathematics (2024-2025);
NSF CAREER DMS-2044415 (2021-2026);
NSF DMS-1954091 (2020-2023);
NSF DMS-1712684 / DMS-1909525 (2017 - 2020)