Fokker-Planck equation for McKean-Vlasov SPDEs driven by time-space Brownian sheet
Nacira Agram, Bernt Oksendal, Frank Proske, Olena Tymoshenko
TL;DR
This paper investigates McKean-Vlasov SPDEs driven by a Brownian sheet, establishing existence, uniqueness, and a Fokker-Planck equation for their solutions, with applications to space-time Ornstein-Uhlenbeck processes.
Contribution
It introduces a Fokker-Planck framework for McKean-Vlasov SPDEs driven by time-space Brownian sheets, including existence and uniqueness results.
Findings
Proved existence and uniqueness of solutions.
Derived a Fokker-Planck equation for the law of solutions.
Provided examples illustrating theoretical results.
Abstract
In this paper, we consider a McKean-Vlasov (mean-field) stochastic partial differential equations (SPDEs) driven by a Brownian sheet. We study the propagation of chaos for a space-time Ornstein-Uhlenbeck SPDE type. Subsequently, we prove the existence and uniqueness of a nonlinear McKean-Vlasov SPDE. Finally, we establish a Fokker-Planck equation for the law of the solution of the McKean-Vlasov type SPDE driven by a time-space Brownian sheet, and we provide some examples to illustrate the results obtained.
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Taxonomy
TopicsStatistical Mechanics and Entropy · Stochastic processes and financial applications · Advanced Thermodynamics and Statistical Mechanics