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

2020 Spring Math833, Topics in Probability
MWF 12:05-12:55 (VAN VLECK B123)

This is a topics course in stochastic partial differential equations (SPDE). The course will cover theoretical aspects of SPDE, as well as its connections with quantum field theory and statistical mechanics.
Prerequisites are familiarities with probability theory, PDE, and stochastic analysis.
We will suggest many projects during the semester. A `small project' worths 1 point, which is homework level; a `big project' worths 2 points, which is a bit harder. A student should get 2 points in order to get a grade A.
This course will cover topics including: White noise, stochastic heat equation, strong solution, weak solution, regularity, heat equation with multiplicative noise, chaos series solution, renormalization, Da Prato-Debussche argument, Feynman diagrams, introduction to rough paths, introduction to Hairer's theory of regularity structures, stochastic quantization models (Phi4, sine-Gordon, gauge theory), scaling limits of exclusion processes or ferromagnetic models, introduction to KPZ fixed point; if time allows, 2D KPZ equation.

Week 1 (1/22-24). Invitation to SPDE: heat equation adding a white noise. Solution via Fourier transform and via heat kernel
Week 2 (1/27-31). Regularity (Besov space, Holder space and wavelets)
Week 3 (2/3-7). Examples of nonlinear SPDEs. Stochastic heat equation with multiplicative noise (mSHE).
Week 4 (2/10-14). Ito integral wrt space-time white noise. Existence and uniqueness of mild solution for mSHE. Weak solution to SHE.
Week 5 (2/17-21). Phi4 equation in 1D and fixed point. (Formal) perturbation theory and renormalization. Dimensions and (sub)criticality.
Week 6 (2/24-28). Wiener chaos. Da Prato-Debussche argument for Phi4 equation in 2D. Other applications: sine-Gordon, 2D parabolic Anderson
Week 7 (3/2-6). Rough paths. Stochastic integrals for controlled rough paths. Solving SDEs driven by rough paths
Week 8 (3/2-6). Regularity structures.

  • Download lecture notes (updated 3/6)

  • 2020 Spring Math635, Introduction to Brownian motion and stochastic calculus
    MWF 9:55AM - 10:45AM (BIRGE 346)


    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. Large N limit of the O(N) linear sigma model via stochastic quantization. (With Scott Smith, Rongchan Zhu, Xiangchan Zhu)
    2. Scaling limit of a directed polymer among a Poisson field of independent walks. (With Jian Song, Rongfeng Sun, Lihu Xu)
    3. Stochastic Ricci Flow on Compact Surfaces. (With Julien Dubédat)
    4. Some recent progress in singular stochastic PDEs. (With Ivan Corwin) Bulletin of the AMS. Accepted.
    5. 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
    6. The dynamical sine-Gordon model in the full subcritical regime. (With Ajay Chandra and Martin Hairer)
    7. Stochastic Telegraph equation limit for the stochastic six vertex model. (With Li-Cheng Tsai) Proc. Amer. Math. Soc. 147 (2019), 2685-2705
    8. Stochastic PDE Limit of the Six Vertex Model. (With Ivan Corwin, Promit Ghosal and Li-Cheng Tsai) Comm. Math. Phys. accepted
    9. Stochastic quantization of an Abelian gauge theory.
    10. Open ASEP in the weakly asymmetric regime. (With Ivan Corwin) Comm. Pure Appl. Math. 71(10), pp.2065-2128.
    11. 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
    12. 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.
    13. 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.
    14. Weak universality of dynamical Φ4_3: non-Gaussian noise. (With Weijun Xu) Stoch PDE: Anal Comp (2017).
    15. A central limit theorem for the KPZ equation. (With Martin Hairer) Ann. Probab. 45(2017), no. 6B, 4167-4221.
    16. The dynamical sine-Gordon model. (With Martin Hairer) Comm. Math. Phys. 341 (2016), no. 3, 933-989
    17. The strict-weak lattice polymer. (With Ivan Corwin and Timo Seppäläinen) J. Stat. Phys. 160(2015), no. 4, 1027-1053
    18. Exact renormalization group analysis of turbulent transport by the shear flow. (With Weinan E) J. Stat. Phys. 153 (2013), no. 4, 553-571
    19. Mean field limit of a dynamical model of polymer systems. (With Weinan E) Sci. China Math. 56 (2013), no. 12, 2591-2598
    20. A renormalization group method by harmonic extensions and the classical dipole gas. Ann. Henri Poincaré 17 (2016), no. 4, 861-911
    21. 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
    22. PhD Thesis: Renormalization Theory in Statistical Physics and Stochastic Analysis (Advisor: Weinan E)

    Seminars and Conferences

    UW-Madison, Probability Seminar (VV901 Th 2:25PM), Analysis Seminar (VV B139 Tu 4:00PM), PDE Geometric Analysis Seminar (VV901 Mon 3:30pm), Colloquia,

    Random Matrix EurAsia 2020, (4 - 29 May 2020), Singapore

    The 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications June 5-9, 2020, Atlanta, USA

    Bernoulli-IMS 10th World Congress, Seoul, Korea, August 17-21, 2020

    The 10th International Conference on Stochastic Analysis and its Applications, Kyoto University (Japan), 7-11 September 2020