We consider a special class of mean field SDEs with common noise which depends on the image solution (i.e., conditional distribution given noise). The strong well-posedness is derived under monotone condition weaker than those used in literature games; Feynman–Kac formula established to solve Schrördinegr type PDEs $$\mathscr {P}_2$$ , and ergodicity proved for measure-valued diffusion processes.

We consider the continuous parabolic Anderson model with Gaussian fields under measure-valued initial conditions, covariances of which are homogeneous or nonhomogeneous in time and fractional rough space. mainly study spatial behaviors for Feynman–Kac formulas Stratonovich sense. Benefited from application formula based on Brownian bridge, precise asymptotics can be obtained more general condit...

Abstract We consider the spin boson model with external magnetic field. prove a path integral formula for heat kernel, known as Feynman–Kac–Nelson (FKN) formula. use this representation to express ground state energy stochastic integral. Based on connection, we determine expansion coefficients of respect field strength and them in terms correlation functions continuous Ising model. From recentl...

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) stochastic (SDEs) wider class processes, including Brownian motions {BtH,t≥0} sub-fractional {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection fPDE SDE via Feynman–Kac Theorem, which provides representation general Cauchy problem. In hindsig...

In the first part of this article, we present the main tools and definitions of Markov processes’ theory: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov’s backward and forward equations and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward SDEs (BSDEs). The second part of this article is devote...