## Electronic Journal of Probability

### Particle representations for stochastic partial differential equations with boundary conditions

#### 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 partial 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].

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 65, 29 pp.

Dates
Accepted: 7 June 2018
First available in Project Euclid: 26 July 2018

https://projecteuclid.org/euclid.ejp/1532570593

Digital Object Identifier
doi:10.1214/18-EJP186

Mathematical Reviews number (MathSciNet)
MR3835471

Zentralblatt MATH identifier
06924677

#### Citation

Crisan, Dan; Janjigian, Christopher; Kurtz, Thomas G. Particle representations for stochastic partial differential equations with boundary conditions. Electron. J. Probab. 23 (2018), paper no. 65, 29 pp. doi:10.1214/18-EJP186. https://projecteuclid.org/euclid.ejp/1532570593

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