Hao Shen

Hao Shen

  Assistant Professor
  Department of Mathematics
  University of Wisconsin-Madison
  Email: hshen3 at wisc or 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).


  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.

Publications and Preprints - in reversed chronological order

  1. Invariant measure and universality of the 2D Yang-Mills Langevin dynamic. (With Ilya Chevyrev).
  2. A stochastic PDE approach to large N problems in quantum field theory: a survey. J. Math. Phys. (2022)
  3. A stochastic analysis approach to lattice Yang-Mills at strong coupling. (With Rongchan Zhu, Xiangchan Zhu). Comm. Math. Phys. (2022)
  4. A new derivation of the finite N master loop equation for lattice Yang-Mills. (With Scott Smith, Rongchan Zhu).
  5. Stochastic quantisation of Yang-Mills-Higgs in 3D. (With Ajay Chandra, Ilya Chevyrev, Martin Hairer).
  6. An SPDE approach to perturbation theory of Φ42: asymptoticity and short distance behavior. (With Rongchan Zhu, Xiangchan Zhu). Ann. Appl. Prob. (2022)
  7. Large N limit of the O(N) linear sigma model in 3D. (With Rongchan Zhu, Xiangchan Zhu) Comm. Math. Phys. (2022)
  8. Langevin dynamic for the 2D Yang-Mills measure. (With Ajay Chandra, Ilya Chevyrev, Martin Hairer) Publ. Math. IHÉS. 136, 1-147 (2022)
  9. Large N limit of the O(N) linear sigma model via stochastic quantization. (With Scott Smith, Rongchan Zhu, Xiangchan Zhu), Ann. Probab. 50 (2022), no. 1, 131-202.
  10. 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
  11. Stochastic Ricci Flow on Compact Surfaces. (With Julien Dubédat) Int. Math. Res. Not. (2021).
  12. Some recent progress in singular stochastic partial differential equations. (With Ivan Corwin) Bulletin of the AMS. 57 (2020), no. 3, 409-454.
  13. Local solution to the multi-layer KPZ equation. (With Ajay Chandra and Dirk Erhard), J. Stat. Phys. 175 (2019), Issue 6, 1080-1106
  14. The dynamical sine-Gordon model in the full subcritical regime. (With Ajay Chandra and Martin Hairer)
  15. Stochastic Telegraph equation limit for the stochastic six vertex model. (With Li-Cheng Tsai), Proc. Amer. Math. Soc. 147 (2019), 2685-2705
  16. Stochastic PDE limit of the Six Vertex model. (With Ivan Corwin, Promit Ghosal and Li-Cheng Tsai), Comm. Math. Phys. 375 (2020), 1945-2038
  17. Stochastic quantization of an Abelian gauge theory. Comm. Math. Phys. 384 (2021), 1445-1512.
  18. Open ASEP in the weakly asymmetric regime. (With Ivan Corwin) Comm. Pure Appl. Math. 71 (2018), Issue 10, 2065-2128.
  19. Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits. (With Hendrik Weber) J. Funct. Anal. Vol 275, Issue 6, (2018), 1321-1367
  20. 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.
  21. ASEP(q,j) converges to the KPZ equation. (With Ivan Corwin and Li-Cheng Tsai) Ann. Inst. Henri Poincaré (B) Probab. Stat. 54 (2018), No. 2, 995-1012.
  22. Weak universality of dynamical Φ43: non-Gaussian noise. (With Weijun Xu) Stoch PDE: Anal Comp. 6 (2018), 211-254.
  23. A central limit theorem for the KPZ equation. (With Martin Hairer) Ann. Probab. 45 (2017), no. 6B, 4167-4221.
  24. The dynamical sine-Gordon model. (With Martin Hairer) Comm. Math. Phys. 341 (2016), no. 3, 933-989
  25. The strict-weak lattice polymer. (With Ivan Corwin and Timo Seppäläinen) J. Stat. Phys. 160 (2015), no. 4, 1027-1053
  26. Exact renormalization group analysis of turbulent transport by the shear flow. (With Weinan E) J. Stat. Phys. 153 (2013), no. 4, 553-571
  27. Mean field limit of a dynamical model of polymer systems. (With Weinan E) Sci. China Math. 56 (2013), no. 12, 2591-2598
  28. A renormalization group method by harmonic extensions and the classical dipole gas. Ann. Henri Poincaré, 17 (2016), no. 4, 861-911
  29. Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. (With Weinan E and Arnulf Jentzen), Nonlinear Analysis, 142 (2016), 152-193
  30. PhD Thesis: Renormalization Theory in Statistical Physics and Stochastic Analysis (Advisor: Weinan E)

Recent news and activities

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).

July 24-28, 2023. 43rd Conference on Stochastic Processes and their Applications (Lisboa, Portugal).

June 12-14, 2023. Stochastic analysis meets QFT - critical theory (Muenster, Germany, Organized by A.Chandra, H.Weber and R.Wulkenhaar).

May 31 - June 4, 2023. The 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Wilmington, NC).

Nov 14 - 16, 2022. Stochastic PDEs and Related Topics, Maryland

September 10 - 11, 2022. Graduate Student Probability Conference, Madison (Registration closed.)

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)