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 Fall Math 735: Stochastic Analysis
Course description
This Stochastic Analysis course will cover discussions on stochastic processes, stochastic integration, and stochastic differential equations. In particular the topics we expect to cover are:
Review of probability theory and conditional expectation;
Generalities about stochastic processes, Brownian motion, Poisson process;
Martingale theory;
Stochastic integral with respect to Brownian motion;
Stochastic integral with respect to cadlag martingales and semimartingales;
Ito's formula;
Stochastic differential equations;
Local time for Brownian motion, Girsanov's theorem;
White noise integrals and an introduction to Stochastic Partial Differential Equation
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

Langevin dynamic for the 2D YangMills measure.
(With Ajay Chandra, Ilya Chevyrev, Martin Hairer)

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. 57.3 (2020): 409454.

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. (2020), pp.194

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)