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).
Hao Shen

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.)


  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.


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).
  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).

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

Stochastic PDE and 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.

Other 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)

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:
NSF CAREER DMS-2044415 (2021-2026);
NSF DMS-1954091 (2020-2023);
NSF DMS-1712684 / DMS-1909525 (2017 - 2020)