## C O L U M B I A / C O U R A N T

J O I N T P R O B A B I L I T Y S E M I N A R S E R I E S

**Feb. 5th 2016, Math 417 Columbia University**

**This Columbia / Courant joint probability seminar will focus on the recent developments of stochastic partial differential equations, especially the new solution theory of regularity structures developed by Martin Hairer, as well as the alternative theories such as paracontrolled distributions by M.Gubinelli et al.**

**Speakers:**

****

__Ajay Chandra__(University of Warwick)__Hendrik Weber__(University of Warwick)__Weijun Xu__**(University of Warwick)**

**Schedule:**

**9:00 - 9:30 Coffee, tea and light breakfast**

9:30 - 10:30 Ajay Chandra: "An analytic BPHZ theorem for Regularity Structures"

10:45 - 11:45 Hendrik Weber: "Global well-posedness for the dynamic Phi^4_3 model the torus"

12:00 - 1:00 Weijun Xu: "Large scale behaviour of phase coexistence models"

9:30 - 10:30 Ajay Chandra: "An analytic BPHZ theorem for Regularity Structures"

10:45 - 11:45 Hendrik Weber: "Global well-posedness for the dynamic Phi^4_3 model the torus"

12:00 - 1:00 Weijun Xu: "Large scale behaviour of phase coexistence models"

**Practical Information:**

**The talks will take place at Room 417 Math Building of Columbia University on February 5th. No registration is needed. For any questions, please contact**

__Hao Shen__.**Title and abstracts:**

**Ajay Chandra**

(i) the insertion of the counter-term corresponds to a renormalization of the equation and is allowed by the algebraic structure of regularity structures,

(ii) there is a way to choose the value of counterterms which yield the right stochastic estimates.

This verification is difficult when the divergences become numerous and are nested/overlapping.

Recent work by Bruned, Hairer, and Zambotti provides a robust framework to systematically handle the first issue, I will describe how this can be combined with multiscale techniques from constructive field theory in order to handle the second.This is joint work with Martin Hairer.

Hendrik Weber

Hairer and Gubinelli devised methods to give an interpretation and show local well-posedness for a

class of very singular SPDEs from Mathematical Physics.

In this talk I will discuss how to extend their method to get global bounds in a prominent example, the dynamic

Phi^4 model. I will first show how to use a simple PDE argument to show global in time well-posedness for the

dynamic Phi^4 equation on the two-dimensional plane. The emphasis of the talk will be on an extension of this

method which yields global in time solutions for the three dimensional Phi^4 model on the torus.

__Title:__An analytic BPHZ theorem for Regularity Structures__Abstract:__When trying to tame divergences using counterterms within regularity structures there are two key things one has to verify:(i) the insertion of the counter-term corresponds to a renormalization of the equation and is allowed by the algebraic structure of regularity structures,

(ii) there is a way to choose the value of counterterms which yield the right stochastic estimates.

This verification is difficult when the divergences become numerous and are nested/overlapping.

Recent work by Bruned, Hairer, and Zambotti provides a robust framework to systematically handle the first issue, I will describe how this can be combined with multiscale techniques from constructive field theory in order to handle the second.This is joint work with Martin Hairer.

Hendrik Weber

__Title:__Global well-posedness for the dynamic Phi^4_3 model the torus__Abstract:__The theory of non-linear stochastic PDEs has recently witnessed an enormous breakthrough whenHairer and Gubinelli devised methods to give an interpretation and show local well-posedness for a

class of very singular SPDEs from Mathematical Physics.

In this talk I will discuss how to extend their method to get global bounds in a prominent example, the dynamic

Phi^4 model. I will first show how to use a simple PDE argument to show global in time well-posedness for the

dynamic Phi^4 equation on the two-dimensional plane. The emphasis of the talk will be on an extension of this

method which yields global in time solutions for the three dimensional Phi^4 model on the torus.

**This is joint work with Jean-Christophe Mourrat.**

**Weijun Xu**

__Title:__Large scale behaviour of phase coexistence models__Abstract:__The solutions to many interesting stochastic PDEs are often obtained after suitable renormalisations. These renormalisations often change the original equation by a quantity which is infinity, but they do have concrete physical meanings. We will explain the meaning of the infinities in the context of the Phi^4_3 equation. As a consequence, we will see how this equation, interpreted after suitable renormalisations, arises naturally as the universal limit for symmetric phase coexistence models. We will also see how this universality can be lost when asymmetry is present. Based on joint works with Martin Hairer and Hao Shen.