Particle representations for a class of nonlinear SPDEs
type: Item
instance of: scholarly article
title: Particle representations for a class of nonlinear SPDEs
zbMATH Open document ID: 0996.60071
DOI: 10.1016/S0304-4149(99)00024-1
author: Thomas G. Kurtz
author: Jie Xiong
published in: Stochastic Processes and their Applications
publication date: 2002-08-29T00:00:00Z
review text: The authors study an infinite system \[ \begin{aligned} X_{i}(t) &= X_{i}(0) + \int ^{t}_{0}\sigma (X_{i}(s),V(s)) dB_{i}(s) + \int ^{t}_{0} c(X_{i}(s),V(s)) ds \\ &\quad + \int _{U\times [0,t]}\alpha (X_{i}(s),V(s),u)W( du, ds), \tag{1} \\ A_{i}(t) &= A_{i}(0) + \int ^{t}_{0} A_{i}(s)\gamma ^{\text{T}} (X_{i},V(s)) dB_{i}(s) \\ &\quad + \int ^{t}_{0} A_{i}(s) d(X_{i}(s),V(s)) ds + \int ^{t}_{0} A_{i}(s) \beta (X_{i}(s),V(s),u) W( du,ds), \tag{2} \end{aligned} \] \(i\in \mathbb N\), of stochastic differential equations for the locations and weights of a collection of particles in \(\mathbb R^{d}\). In the equations (1), (2), \(V\) denotes the weighted empirical measure \[ V(t) = \lim _{n\to \infty } \frac 1{n}\sum ^{n}_{j=1} A_{j}(t) \delta _{X_{j}(t)}, \tag{3} \] the limit being taken in the weak star topology of the space \(\mathfrak M\) of finite signed Borel measures on \(\mathbb R^{d}\), \(\{B_{i}\}\) are independent standard \(d\)-dimensional Wiener processes, \(W\) is a Gaussian space-time white noise on \(U\times \mathbb R_{+}\), \(E(W(A,t)W(B,t)) = \mu (A\cap B)t\) with a Borel measure \(\mu \) on a Polish space \(U\). The coefficients \(\sigma :\mathbb R^{d}\times \mathfrak M\to \mathbb R^{d\times d}\), \(c: \mathbb R^{d}\times \mathfrak M\to \mathbb R^{d}\), \(\alpha : \mathbb R^{d}\times \mathfrak M\times U\to \mathbb R^{d}\), \(\gamma : \mathbb R^{d}\times \mathfrak M\to \mathbb R^{d}\), \(d:\mathbb R^{d} \times \mathfrak M\to \mathbb R\), and \(\beta : \mathbb R^{d}\times \mathfrak M \times U\to \mathbb R\) are supposed to satisfy linear growth and Lipschitz type conditions, the sequence \(\{X_{i}(0), A_{i}(0)\}\) of initial data is exchangeable and independent of \(B_{i}\), \(W\). It is shown that there exists a unique solution to the problem (1)--(3) and the weighted empirical measure \(V\) solves a stochastic partial differential equation \[ \begin{aligned} \langle \psi ,V(t)\rangle = & \langle \psi ,V(0)\rangle + \int ^{t}_{0} \langle d(\cdot ,V(s)…
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1792532
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: McKean-Vlasov equations
zbMATH Keywords: particle representations
MaRDI profile type: Publication
cites work: Rate of Convegence of a Stochastic Particle Method for the Kolmogorov Equation with Variable Coefficients
cites work: Propagation of chaos and the McKean-Vlasov equation in duals of nuclear spaces
cites work: Nonlinear filtering and measure-valued processes
cites work: Convergence of a Branching Particle Method to the Solution of the Zakai Equation
cites work: Stochastic McKean-Vlasov equations
cites work: Q4889030
cites work: Q3721531
cites work: McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets
cites work: Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions
cites work: Asymptotic behavior of a system of interacting nuclear-space-valued stochastic differential equations driven by Poisson random measures
cites work: Q4882938
cites work: A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation
cites work: Q4895011
cites work: Q5571997
cites work: Q4895007
cites work: Propagation of chaos and fluction for a system of weakly interacting white noise driven parabolic spde's
cites work: Intermittency-type estimates for some nondegenerate SPDE's
cites work: On the uniqueness of solutions of stochastic differential equations
full work available at URL: s0304-4149(99)00024-1
OpenAlex ID: W1979038769
description: scientific article; zbMATH DE number 1792532