Particle representations for a class of nonlinear SPDEs
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 \[ \begin{aligned} \langle \psi ,V(t)\rangle = & \langle \psi ,V(0)\rangle + \int ^{t}_{0} \langle d(\cdot ,V(s))\psi + L(V(s))\psi , V(s) \rangle ds \\ & + \int _{U\times [0,t]} \langle \beta (\cdot ,V(s),u)\psi +\alpha ^{\text{T}}(\cdot ,V(s),u)\nabla \psi ,V(s)\rangle W(du,ds)\end{aligned} \] for all \(\psi \in C^2_{b} (\mathbb R^{d})\) where \( L(v)\psi (x) = \frac 12\sum ^{d}_{i,j=1} a_{ij}(x,v) \partial _{x_{i}}\partial _{x_{j}}\psi (x) + \sum ^{d} _{i=1} b_{i}(x,v)\partial _{x_{i}}\psi (x)\) with \[ \begin{aligned} a(x,v) &= \sigma (x,v)\sigma ^{\text{T}}(x,v) + \int _{U} \alpha (x,v,u)\alpha ^{\text{T}}(x,v,u) d\mu (u), \\ b(x,v) &= c(x,v) + \sigma (x,v)\gamma (x,v) + \int _{U} \beta (x,v,u)\alpha (x,v,u) d\mu (u). \end{aligned} \]
stochastic partial differential equations
scientific article; zbMATH DE number 1792532