Hao Shen
Hao Shen 


Short Bio:
PhD 2013 (Princeton, Advisor:Weinan E);
Postdoc 20142015 (Warwick, Mentor: Martin Hairer);
Ritt Assistant Professor 20152018 (Columbia, Mentor:Ivan Corwin).



Teaching
Here are some of the previous courses I have taught
2020 Spring Math833, Topics in Probability
MWF 12:0512:55 (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 PratoDebussche argument, Feynman diagrams, introduction to rough paths, introduction to Hairer's theory of regularity structures, stochastic quantization models (Phi4, sineGordon, gauge theory), scaling limits of exclusion processes or ferromagnetic models, introduction to KPZ fixed point; if time allows, 2D KPZ equation.
Week 1 (1/2224). Invitation to SPDE: heat equation adding a white noise. Solution via Fourier transform and via heat kernel
Week 2 (1/2731). Regularity (Besov space, Holder space and wavelets)
Week 3 (2/37). Examples of nonlinear SPDEs. Stochastic heat equation with multiplicative noise (mSHE).
Week 4 (2/1014). Ito integral wrt spacetime white noise. Existence and uniqueness of mild solution for mSHE. Weak solution to SHE.
Week 5 (2/1721). Phi4 equation in 1D and fixed point. (Formal) perturbation theory and renormalization. Dimensions and (sub)criticality.
Week 6 (2/2428). Wiener chaos. Da PratoDebussche argument for Phi4 equation in 2D. Other applications: sineGordon, 2D parabolic Anderson
Week 7 (3/26). Rough paths. Stochastic integrals for controlled rough paths. Solving SDEs driven by rough paths
Week 8 (3/26). 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)
Research
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

Large N limit of the O(N) linear sigma model via stochastic quantization.
(With Scott Smith, Rongchan Zhu, Xiangchan Zhu)

Scaling limit of a directed polymer among a Poisson field of independent walks.
(With Jian Song, Rongfeng Sun, Lihu Xu)

Stochastic Ricci Flow on Compact Surfaces.
(With Julien Dubédat)

Some recent progress in singular stochastic PDEs.
(With Ivan Corwin) Bulletin of the AMS. Accepted.

Local solution to the multilayer KPZ equation.
(With Ajay Chandra and Dirk Erhard)
J. Stat. Phys. (2019) Vol 175, Issue 6, pp 10801106

The dynamical sineGordon model in the full subcritical regime.
(With Ajay Chandra and Martin Hairer)

Stochastic Telegraph equation limit for the stochastic six vertex model.
(With LiCheng Tsai)
Proc. Amer. Math. Soc. 147 (2019), 26852705

Stochastic PDE Limit of the Six Vertex Model.
(With Ivan Corwin, Promit Ghosal and LiCheng Tsai)
Comm. Math. Phys. accepted

Stochastic quantization of an Abelian gauge theory.

Open ASEP in the weakly asymmetric regime.
(With Ivan Corwin)
Comm. Pure Appl. Math. 71(10), pp.20652128.

Glauber dynamics of 2D KacBlumeCapel model and their stochastic PDE limits.
(With Hendrik Weber)
J. Funct. Anal. Vol 275, Issue 6, (2018), 13211367

Moment bounds for SPDEs with nonGaussian fields and application to the WongZakai problem.
(With Ajay Chandra)
Electron. J. Probab. Vol 22 (2017), paper no. 68.

ASEP(q,j) converges to the KPZ equation.
(With Ivan Corwin and LiCheng Tsai)
Ann. Inst. Henri Poincaré (B) Probab. Stat. (2018), 54, No. 2, 9951012.

Weak universality of dynamical Φ4_3: nonGaussian noise.
(With Weijun Xu)
Stoch PDE: Anal Comp (2017).

A central limit theorem for the KPZ equation.
(With Martin Hairer)
Ann. Probab. 45(2017), no. 6B, 41674221.

The dynamical sineGordon model.
(With Martin Hairer)
Comm. Math. Phys. 341 (2016), no. 3, 933989

The strictweak lattice polymer.
(With Ivan Corwin and Timo Seppäläinen)
J. Stat. Phys. 160(2015), no. 4, 10271053

Exact renormalization group analysis of turbulent transport by the shear flow.
(With Weinan E)
J. Stat. Phys. 153 (2013), no. 4, 553571

Mean field limit of a dynamical model of polymer systems.
(With Weinan E)
Sci. China Math. 56 (2013), no. 12, 25912598

A renormalization group method by harmonic extensions and the classical dipole gas.
Ann. Henri Poincaré 17 (2016), no. 4, 861911

Renormalized powers of OrnsteinUhlenbeck processes and wellposedness of stochastic GinzburgLandau equations.
(With Weinan E and Arnulf Jentzen)
Nonlinear Anal. 142 (2016), 152 193

PhD Thesis: Renormalization Theory in Statistical Physics and Stochastic Analysis (Advisor: Weinan E)
Seminars and Conferences
UWMadison,
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 59, 2020, Atlanta, USA
BernoulliIMS 10th World Congress, Seoul, Korea, August 1721, 2020
The 10th International Conference on Stochastic Analysis and its Applications, Kyoto University (Japan), 711 September 2020