Particle representations for stochastic partial differential equations with boundary conditions
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Published version
Author(s)
Crisan, DO
Kurtz, Thomas
Janjigian, Christopher
Type
Journal Article
Abstract
In this article, we study weighted particle representations for a class of stochastic partial differential equations (SPDE) with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations. The locations are given by independent, stationary reflecting diffusions in a
bounded domain, and the weights evolve according to an infinite system of stochastic differential equations driven by a common Gaussian white noise W which is the stochastic input for the SPDE. The weights interact through V, the associated weighted empirical measure, which gives the solution of the SPDE. When a particle hits the boundary its weight jumps to a value given by a function of the location of the particle on the
boundary. This function determines the boundary condition for the SPDE. We show existence and uniqueness of a solution of the infinite system of stochastic differential equations giving the locations and weights of the particles and derive two weak forms for the corresponding SPDE depending on the choice of test functions. The weighted empirical measure V is the unique solution for each of the nonlinear stochastic par-
tial differential equations. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong in [14, 15]
bounded domain, and the weights evolve according to an infinite system of stochastic differential equations driven by a common Gaussian white noise W which is the stochastic input for the SPDE. The weights interact through V, the associated weighted empirical measure, which gives the solution of the SPDE. When a particle hits the boundary its weight jumps to a value given by a function of the location of the particle on the
boundary. This function determines the boundary condition for the SPDE. We show existence and uniqueness of a solution of the infinite system of stochastic differential equations giving the locations and weights of the particles and derive two weak forms for the corresponding SPDE depending on the choice of test functions. The weighted empirical measure V is the unique solution for each of the nonlinear stochastic par-
tial differential equations. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong in [14, 15]
Date Issued
2018-07-26
Date Acceptance
2018-06-07
Citation
Electronic Journal of Probability, 2018, 23 (65), pp.1-29
ISSN
1083-6489
Publisher
Institute of Mathematical Statistics
Start Page
1
End Page
29
Journal / Book Title
Electronic Journal of Probability
Volume
23
Issue
65
Copyright Statement
Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/
Identifier
https://projecteuclid.org/euclid.ejp/1532570593#abstract
Subjects
0104 Statistics
Statistics & Probability
Publication Status
Published
Date Publish Online
2018-07-26