Electronic Journal of Probability

Particle representations for stochastic partial differential equations with boundary conditions

Dan Crisan, Christopher Janjigian, and Thomas G. Kurtz

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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

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

Received: 29 July 2016
Accepted: 7 June 2018
First available in Project Euclid: 26 July 2018

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F17: Functional limit theorems; invariance principles 60F25: $L^p$-limit theorems 60H10: Stochastic ordinary differential equations [See also 34F05] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 93E11: Filtering [See also 60G35]

Stochastic partial differential equations interacting particle systems diffusions with reflecting boundary stochastic Allen-Cahn equation Euclidean quantum field theory equation with quartic interaction

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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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