Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type

作者：Kotelenez Peter M^*; Kurtz Thomas G

来源：Probability Theory and Related Fields, 2010, 146(1-2): 189-222.

DOI：10.1007/s00440-008-0188-0

• Stochastic partial differential equations
• Partial differential equations
• Macroscopic limit
• Particle systems
• Stochastic ordinary differential equations
• Exchangeable sequences

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita.

• 出版日期2010-1