Particle representations for stochastic partial differential equations with boundary conditions

Thomas G. Kurtz,Dan Crisan,Christopher Janjigian

doi:10.1214/18-ejp186

Abstract

In this article, we study a weighted particle representation for a class of stochastic partial differential equations with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations (SDEs). The evolution of the particles is modeled by an infinite system of stochastic differential equations with reflecting boundary condition and driven by independent finite dimensional Brownian motions. The weights of the particles evolve according to an infinite system of stochastic differential equations driven by a common cylindrical noise $W$ and interact through $V$, the associated weighted empirical measure. When the particles hit the boundary their corresponding weights are assigned a pre-specified value. We show the existence and uniqueness of a solution of the infinite dimensional system of stochastic differential equations modeling the location and the weights of the particles. We also prove that the associated weighted empirical measure $V$ is the unique solution of a nonlinear stochastic partial differential equation driven by $W$ with Dirichlet boundary condition. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong.

Highlights

We study particle representations for a class of nonlinear stochastic partial differential equations that includes the stochastic version of the Allen-Cahn equation [1, 2] and that of the equation governing the stochastic quantization of Φ4d Euclidean quantum field theory with quartic interaction [19], that is, the equation dv = ∆v + G(v)v + W, (1.1)
The primary goal in this setting is to prove that the sequence of empirical measures V n converges in distribution and to characterize the limit V as a measure valued process which solves the following nonlinear partial differential equation, written in weak form,3 t φ, V (t) = φ, V (0) + L(V (s))φ, V (s) ds, (1.3)
We show how Theorem 4.3 leads to a weak formulation of the stochastic partial differential equation with a broader class of test functions than those considered above

Summary

Introduction

Where G is a (possibly) nonlinear function and W is a space-time noise2 These particle representations lead naturally to the solution of a weak version of a stochastic partial differential equation similar to (1.1). The primary goal in this setting is to prove that the sequence of empirical measures V n converges in distribution and to characterize the limit V as a measure valued process which solves the following nonlinear partial differential equation, written in weak form, t φ, V (t) = φ, V (0) + L(V (s))φ, V (s) ds,. (B(E) denotes the Borel subsets of a metric space E.) The existence of a measurable version of the process of densities follows by a monotone class argument.

Basic conditions and statement of main theorems

Existence and uniqueness of the weighted particle system

Boundary behavior

Weak equations with boundary terms

Gaussian white noise

Measurability of density process