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 Fall 2021

  Math 222, Calculus and Analytic Geometry
(Syllabus, course materials, homework are also posted on Canvas.)

Instructor Email/Preferred Contact hshen3@wisc.edu

REQUIRED TEXTBOOK: Stewart Calculus 8/e and WebAssign.


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. An SPDE approach to perturbation theory of Φ4_2: asymptoticity and short distance behavior. (With Rongchan Zhu, Xiangchan Zhu)
  2. Large N limit of the O(N) linear sigma model in 3D. (With Rongchan Zhu, Xiangchan Zhu)
  3. Langevin dynamic for the 2D Yang-Mills measure. (With Ajay Chandra, Ilya Chevyrev, Martin Hairer)
  4. Large N limit of the O(N) linear sigma model via stochastic quantization. (With Scott Smith, Rongchan Zhu, Xiangchan Zhu) Ann. Probab. accepted.
  5. Scaling limit of a directed polymer among a Poisson field of independent walks. (With Jian Song, Rongfeng Sun, Lihu Xu) J. Funct. Anal. (2021), 281(5).
  6. Stochastic Ricci Flow on Compact Surfaces. (With Julien Dubédat) Int. Math. Res. Not. accepted.
  7. Some recent progress in singular stochastic PDEs. (With Ivan Corwin) Bulletin of the AMS. 57.3 (2020): 409-454.
  8. Local solution to the multi-layer KPZ equation. (With Ajay Chandra and Dirk Erhard) J. Stat. Phys. (2019) Vol 175, Issue 6, pp 1080-1106
  9. The dynamical sine-Gordon model in the full subcritical regime. (With Ajay Chandra and Martin Hairer)
  10. Stochastic Telegraph equation limit for the stochastic six vertex model. (With Li-Cheng Tsai) Proc. Amer. Math. Soc. 147 (2019), 2685-2705
  11. Stochastic PDE limit of the Six Vertex model. (With Ivan Corwin, Promit Ghosal and Li-Cheng Tsai) Comm. Math. Phys. 375 (2020), pp 1945-2038
  12. Stochastic quantization of an Abelian gauge theory. Comm. Math. Phys. 384 (2021), pp 1445-1512.
  13. Open ASEP in the weakly asymmetric regime. (With Ivan Corwin) Comm. Pure Appl. Math. 71(10), pp.2065-2128.
  14. 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
  15. Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem. (With Ajay Chandra) Electron. J. Probab. Vol 22 (2017), paper no. 68.
  16. ASEP(q,j) converges to the KPZ equation. (With Ivan Corwin and Li-Cheng Tsai) Ann. Inst. Henri Poincaré (B) Probab. Stat. (2018), 54, No. 2, 995-1012.
  17. Weak universality of dynamical Φ4_3: non-Gaussian noise. (With Weijun Xu) Stoch PDE: Anal Comp (2017).
  18. A central limit theorem for the KPZ equation. (With Martin Hairer) Ann. Probab. 45(2017), no. 6B, 4167-4221.
  19. The dynamical sine-Gordon model. (With Martin Hairer) Comm. Math. Phys. 341 (2016), no. 3, 933-989
  20. The strict-weak lattice polymer. (With Ivan Corwin and Timo Seppäläinen) J. Stat. Phys. 160(2015), no. 4, 1027-1053
  21. Exact renormalization group analysis of turbulent transport by the shear flow. (With Weinan E) J. Stat. Phys. 153 (2013), no. 4, 553-571
  22. Mean field limit of a dynamical model of polymer systems. (With Weinan E) Sci. China Math. 56 (2013), no. 12, 2591-2598
  23. A renormalization group method by harmonic extensions and the classical dipole gas. Ann. Henri Poincaré 17 (2016), no. 4, 861-911
  24. Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. (With Weinan E and Arnulf Jentzen) Nonlinear Anal. 142 (2016), 152- 193
  25. PhD Thesis: Renormalization Theory in Statistical Physics and Stochastic Analysis (Advisor: Weinan E)

Recent activities

Bernoulli-IMS 10th World Congress, Seoul, Korea, July 19 - 23, 2021

International Congress on Mathematical Physics, Geneva, Switzerland, Aug 2 - 7, 2021

PDE and Randomness symposium, University of Bath, UK, 1st - 10th of September 2021

The 10th International Conference on Stochastic Analysis and its Applications, Kyoto University (Japan), 6-10 September 2021

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)