Hao Shen   Associate Professor   Department of Mathematics   University of Wisconsin-Madison   Email: 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). CV.

Recent events

July 1 - 12, 2024. MSRI/SLMath Summer Graduate School on Stochastic quantization (More information is on SLMath website)

Dec 17 - 22, 2023. Oberwolfach Arbeitsgemeinschaft: QFT and Stochastic PDEs

Aug 9 - 11, 2023. Frontiers of Stochastic Analysis (Chicago, Organized by Cheng Ouyang and Jing Wang).

September 10 - 11, 2022. Graduate Student Probability Conference, Madison (Registration closed.)

Teaching

  Here are some of the previous courses I have taught.

Teaching in Spring 2023

Math833, Topics in stochastic PDE.
MWF 12:05PM - 12:55PM
TEXTBOOK: Lecture notes uploaded on Canvas.

Research

A survey:

1. Some recent progress in singular stochastic partial differential equations. (with Ivan Corwin) Bulletin of the AMS. 57 (2020), no. 3, 409-454.

Stochastic quantization of Yang-Mills and other gauge theories

1. Stochastic quantization of an abelian gauge theory. (2019). Comm. Math. Phys. 384, 1445-1512 (2021).
2. Langevin dynamic for the 2D Yang-Mills measure. (with Ajay Chandra, Ilya Chevyrev, Martin Hairer) (2020). Publ. Math. IHÉS. 136, 1-147 (2022).
3. Stochastic quantisation of Yang-Mills-Higgs in 3D. (with Ajay Chandra, Ilya Chevyrev, Martin Hairer) (2022).
4. A new derivation of the finite N master loop equation for lattice Yang-Mills. (with Scott Smith, Rongchan Zhu) (2022).
5. A stochastic analysis approach to lattice Yang-Mills at strong coupling. (with Rongchan Zhu, Xiangchan Zhu) (2022). Comm. Math. Phys.
6. Invariant measure and universality of the 2D Yang-Mills Langevin dynamic. (with Ilya Chevyrev) (2023).

Stochastic PDE limits of statistical physics models

1. ASEP(q,j) converges to the KPZ equation. (with Ivan Corwin and Li-Cheng Tsai) (2016) Ann. Inst. Henri Poincaré (B) Probab. Stat.
2. Glauber dynamics of 2D Kac-Blume-Capel model and their stochastic PDE limits. (with Hendrik Weber) (2016). J. Funct. Anal. 275, Issue 6, 1321-1367 (2018).
3. Open ASEP in the weakly asymmetric regime. (with Ivan Corwin) (2016). Comm. Pure Appl. Math. 71, Issue 10, 2065-2128 (2018).
4. Stochastic PDE limit of the six vertex model. (with Ivan Corwin, Promit Ghosal and Li-Cheng Tsai) (2018). Comm. Math. Phys. 375, 1945-2038 (2020).
5. Stochastic Telegraph equation limit for the stochastic six vertex model. (with Li-Cheng Tsai) (2018). Proc. Amer. Math. Soc.
6. Scaling limit of a directed polymer among a Poisson field of independent walks. (with Jian Song, Rongfeng Sun, Lihu Xu), J. Funct. Anal. 281 (2021), no. 5, 55pp
7. Mean field limit of a dynamical model of polymer systems. (with Weinan E) Sci. China Math. 56 (2013), no. 12, 2591-2598

Stochastic PDE and quantum field theory problems

1. A stochastic PDE approach to large N problems in quantum field theory: a survey. J. Math. Phys. (2022)
2. An SPDE approach to perturbation theory of Φ42: asymptoticity and short distance behavior. (with Rongchan Zhu, Xiangchan Zhu) (2021). Ann. Appl. Prob. (2022)
3. Large N limit of the O(N) linear sigma model via stochastic quantization. (with Scott Smith, Rongchan Zhu, Xiangchan Zhu) (2020), Ann. Probab. 50 (2022), no. 1, 131-202.
4. Large N limit of the O(N) linear sigma model in 3D. (with Rongchan Zhu, Xiangchan Zhu) (2021) Comm. Math. Phys. (2022)
5. Large N limit and 1/N expansion of invariant observables in O(N) linear σ-model via SPDE. (with Rongchan Zhu, Xiangchan Zhu) (2023)
6. Stochastic Ricci Flow on Compact Surfaces. (with Julien Dubédat) Int. Math. Res. Not. (2021).
7. Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equations. (with Weinan E and Arnulf Jentzen) (2013) Nonlinear Analysis.

Other papers on regularity structures (KPZ, stochastic quantization, weak universality etc.)

1. A central limit theorem for the KPZ equation. (With Martin Hairer) Ann. Probab. 45 (2017), no. 6B, 4167-4221.
2. The dynamical sine-Gordon model. (With Martin Hairer) Comm. Math. Phys. 341 (2016), no. 3, 933-989
3. Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem. (With Ajay Chandra) Electron. J. Probab. Vol 22 (2017), 1-32.
4. Weak universality of dynamical Φ43: non-Gaussian noise. (With Weijun Xu) Stoch PDE: Anal Comp. 6 (2018), 211-254.
5. Local solution to the multi-layer KPZ equation. (With Ajay Chandra and Dirk Erhard), J. Stat. Phys. 175 (2019), Issue 6, 1080-1106
6. The dynamical sine-Gordon model in the full subcritical regime. (With Ajay Chandra and Martin Hairer)

Integrable probability

1. The strict-weak lattice polymer. (With Ivan Corwin and Timo Seppäläinen) J. Stat. Phys. 160 (2015), no. 4, 1027-1053

Rigorous renormalization group analysis

1. Exact renormalization group analysis of turbulent transport by the shear flow. (With Weinan E) J. Stat. Phys. 153 (2013), no. 4, 553-571
2. A renormalization group method by harmonic extensions and the classical dipole gas. Ann. Henri Poincaré, 17 (2016), no. 4, 861-911

Other articles

1. Oberwolfach report: Rough Paths, Regularity Structures and Related Topics (2016)
2. Oberwolfach report: A dynamical approach to lattice Yang-Mills. Universality: Random Matrices, Random Geometry and SPDEs (2022)
3. PhD Thesis: Renormalization Theory in Statistical Physics and Stochastic Analysis. (Advisor: Weinan E)

Grants and Awards