## Hao Shen

### Assistant Professor

### Department of Mathematics, Wisconsin - Madison

**Address:** Van Vleck 619, Department of Mathematics, University of Wisconsin - Madison

**Email**: *hshen3 at wisc dot edu or pkushenhao at gmail dot edu*

**Short bio: **PhD 2013 (Princeton); Postdoc 2014-2015 (Warwick); Ritt assistant prof. 2015-2018 (Columbia)

**Teaching Fall 2018: Stochastic analysis (Math 735)**

Office Hours: Tuesday and Thursday 1:10 pm - 2:20 pm (or by appointment via email)

Lecture notes: by Timo Seppäläinen

Week 1 (9/6): Introduciton to stochastic analysis.

Week 2 (9/11, 13): Stochastic processes, filtrations, stopping times, quadratic variation.

Week 3 (9/18, 20): Path spaces, (Strong) Markov.

Week 4 (9/25, 27): Brownian motion, Poisson point process and Poisson process, Levy Processes

(Fun stuff: You may find this paper by Peres and Virag interesting. On page 10 the picture compares Poisson point process with a more nontrivial point processes - namely, zeroes of random power series)

Week 5 (10/2, 4): Blumenthal 0-1 law and running maximum of Brownian motion. Theorey of L2 martinglaes.

Week 6 (10/9, 11): Applications of Martingale convergence theorem: 1) Kahane's construction of Gaussian multiplicative chaos 2) Bolthausen's proof of diffusion of directed polymers in d>2. Predictable quadratic variation.

(Reference paper: a review of Gaussian multiplicative chaos ; Bolthausen's paper on directed polymer)

Week 7 (10/16): Stochastic integral w.r.t. Brownian motion. Ito formula. No class on Thursday (Midwest Probability Seminar)

Week 8 (10/23, 25): Stochastic integral w.r.t. Brownian motion and Ito formula continued. Application of Ito formula: deriving Dyson Brownian motion

Week 9 (10/30, 11/1): Stochastic integral: L2 martingale integrators. Midterm quiz.

Week 10 (11/6, 8): Stochastic integral: local martingale and semi-martingale integrators. Quadratic variation, integration by parts, change of variable formula.

Week 11 (11/13, 15): Ito formula, Applications.

Week 12 (11/20): Solving linear SDEs.

Week 13 (11/27, 29): Existence and uniqueness of strong solution. Weak solution.

**Teaching Spring 2019: **431/001, Intro-Theory of Probability 09:55 - 10:45 MWF

**Publications and Preprints - in reversed chronological order**

- A. Chandra, M. Hairer and H. Shen, The dynamical sine-Gordon model in the full subcritical regime, arXiv:1808.02594.
- H. Shen and L.-C. Tsai, Stochastic Telegraph equation limit for the stochastic six vertex model, arXiv:1807.04678,
**Proc. Amer. Math. Soc.**Accepted. - I. Corwin, P. Ghosal, H. Shen and L.-C. Tsai, Stochastic PDE Limit of the Six Vertex Model, arXiv:1803.08120.
- H. Shen, Stochastic quantization of an Abelian gauge theory, arXiv:1801.04596.
- I. Corwin and H. Shen, Open ASEP in the weakly asymmetric regime,
**Comm. Pure Appl.Math.**(2018) - H. Shen and H. Weber, Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits,
**J. Funct. Anal.**Vol 275, Issue 6, (2018), 1321-1367 - A. Chandra and H. Shen, Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem,
**Electron. J. Probab.**Vol 22 (2017), paper no. 68. - I. Corwin, H. Shen and L-C. Tsai, ASEP(q,j) converges to the KPZ equation,
**Ann. Inst. Henri Poincaré (B) Probab. Stat.**(2018), 54, No. 2, 995-1012. - H. Shen and W. Xu, Weak universality of dynamical Φ
^{4}_{3}: non-Gaussian noise,**Stoch PDE: Anal Comp**(2017). - M. Hairer and H. Shen, A central limit theorem for the KPZ equation,
**Ann. Probab.**45(2017), no. 6B, 4167-4221. - M. Hairer and H. Shen, The dynamical sine-Gordon model,
**Comm. Math. Phys.**341 (2016), no. 3, 933-989 - I. Corwin, T. Seppäläinen and H. Shen, The strict-weak lattice polymer,
**J. Stat. Phys.**160(2015), no. 4, 1027-1053 - W. E and H. Shen, Exact renormalization group analysis of turbulent transport by the shear flow,
**J. Stat. Phys.**153 (2013), no. 4, 553-571 - W. E and H. Shen, Mean field limit of a dynamical model of polymer systems,
**Sci. China Math.**56 (2013), no. 12, 2591-2598 - H. Shen, A renormalization group method by harmonic extensions and the classical dipole gas,
**Ann. Henri Poincaré**17 (2016), no. 4, 861-911 - W. E, A. Jentzen and H. Shen, Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations,
**Nonlinear Anal.**142 (2016), 152- 193 - PhD Thesis: Renormalization Theory in Statistical Physics and Stochastic Analysis, available online at: http://dataspace.princeton.edu/jspui/handle/88435/dsp01b2773v77m