Particle representations for stochastic partial differential equations with boundary conditions

Abstract

E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractIn 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].

Well-posedness of a coupled system of Skorohod-like stochastic differential equations

“…Instead of a measure-valued equation for the smoke dynamics, one could also use a stochastically perturbed diffusion-transport equation. In this case, the approach from [10] is potentially applicable, provided the coupling between the SDEs for the crowd dynamics and the SPDE for the smoke evolution is done in a well-posed manner. However, in both cases, it is not yet clear cut how to couple correctly the model equations.…”

Particle representation for the solution of the filtering problem. Application to the error expansion of filtering discretizations

“…Such results would be very hard to obtain (under the same general assumptions) by other methods such as PDE methods (Sobolev embedding theorems) or Malliavin calculus. Separately, in [7], a similar particle representation is used to study a class of semilinear stochastic partial differential equations with Dirichlet boundary conditions that includes the stochastic Allen-Cahn equation and the Φ 4 d equation of Euclidean quantum field theory. Particle representations arise naturally in the study of McKean-Vlasov type models, for example, [23,14,6] where the representations are used to prove limit theorems.…”

Weak existence and uniqueness for McKean–Vlasov SDEs with common noise

“…For models motivated by application to finance and neuroscience, see Hambly and Søjmark [18] and Ledger and Søjmark [39]. Crisan, Janjigian and Kurtz [14] study a class of SPDEs that includes the Stochastic Allen-Cahn equation, extending the earlier work of Kurtz and Xiong [34] where strong solutions to an infinite system of mean-field interacting particles driven by correlated noises are connected to strong solutions to a non-linear stochastic partial differential equation (SPDE) via the empirical distribution of the particles. Another approach to studying the types of SPDEs associated to particle systems driven by correlated noises is that of Dawson and Vaillancourt [15] who obtain measurevalued solutions of the aforementioned SPDE by studying the limit of empirical distributions to interacting systems of finitely many particles as the particle number increases to infinity.…”