The macroscopic behavior of dissipative stochastic partial differential equations usually can be described by a finite dimensional system. This article proves that a macroscopic reduced model may be constructed for stochastic reaction-diffusion equations with cubic nonlinearity by artificial separating the system into two distinct slow-fast time parts. An averaging method and a deviation estima...

This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over ...

This paper studies the large fluctuations of solutions of scalar and finite–dimensional affine stochastic functional differential equations with finite memory, as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the partial maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploi...

An error occurs in a part of the statement and proof Proposition 2.2 paper “Stochastic explosion non-uniqueness for α-Riccati equation” [J. Math. Anal. Appl. 476 (2019) 53–85] that is corrected this erratum. The revised result reveals new unexpected critical phenomena, having further implications solutions to nonlinear differential equation Riccati type.

We study a free energy computation procedure, introduced in [4, 7], which relies on the long-time behavior of a nonlinear stochastic differential equation. This nonlinearity comes from a conditional expectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions to this equation has been proved in [10], under some existence and regularity assumpti...

In this paper, we are focused upon the global uniqueness results for a stochastic integro-differential equation in Fréchet spaces. The main results are proved by using the resolvent operators combined with a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to Frigon and Granas. As an application, a controllability result with one parameter is given to illustrate the theory.