Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations

Abstract

By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

Existence, uniqueness and exponential ergodicity under Lyapunov conditions for McKean-Vlasov SDEs with Markovian switching

“…Song [23] studied exponential ergodicity for MVSDEs with jumps. In [20], Ren et al proved the existence and uniqueness of solutions in infinite dimension under a Lyapunov condition (different from ours in the present paper).…”

A local-in-time theory for singular SDEs with applications to fluid models with transport noise

“…So, the well-known approximation scheme under a Gelfand triple developed for quasi-linear SPDEs does not work for the present model. Motivated by [51], we will employ a regularization argument to overcome this difficulty. Let us give some explanations on Assumption (A) that makes precise the required regularization procedure.…”

“…For LA SALT the velocity field is randomly transported by white-noise vector fields as well as by its own average over realizations of this noise. For the even more general distribution-path dependent case of transport type equations, we refer to [51]. Generally speaking, the distribution of the solution is a global object on the path space, and it does not exist for explosive stochastic processes whose paths are killed at the life time.…”

Distribution Dependent Stochastic Porous Media Equations

“…The author in [17] studied the strong solutions to DDSDEs in finite as well as infinite dimensional cases with delay. For more recent results on DDS(P)DEs, one can see [7,10,18,20,26,34,35] and references therein. To the best of our knowledge, the references we mentioned above mainly focus on DDSDEs or semilinear DDSPDEs, there are very few results in the literature concerning nonlinear DDSPDEs due to the technical difficulties caused by the nonlinear terms.…”