Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type
Under appropriate Lipschitz conditions, \textit{P. Kotelenez} [Probab. Theory Relat. Fields 102, 159--188 (1995; Zbl 0821.60066)] proved the strong Itô existence and uniqueness of solutions for the system of stochastic ordinary differential equations (SODEs) \[ dr^i(t)=F_{\varepsilon}(r^i(t),X(t),t)dt+\int {\mathcal J}_{\varepsilon}(r^i(t),q,X(t),t)w(dq,dt) \] where \(w_l(dq,dt)\) are i.i.d. Gaussian standard white noise random fields on \({\mathbb R}^d \times {\mathbb R}_+, l=1,\ldots,d\), \(X(t)\) is the empirical distribution of the \(r^i(t), i=1,\dots,N\), and \(\varepsilon >0\) is a correlation parameter. The empirical distributions \(X(t)\) are solutions of quasilinear stochastic partial differential equations (SPDEs). After passing to a continuum limit, the solutions of the SPDEs are proved to converge to the solution of a deterministic quasilinear partial differential equation as the correlation parameter \(\varepsilon\) tends to \(0\). The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. This approach is compared with the one of Kunita about SODEs driven by a Gaussian space-time random field.
stochastic partial differential equations
stochastic ordinary differential equations
scientific article; zbMATH DE number 5659494
scientific article