FEYNMAN-KAC FORMULA FOR TEMPERED FRACTIONAL GENERAL DIFFUSION EQUATIONS WITH NONAUTONOMOUS EXTERNAL POTENTIAL

In this paper, we first establish a version of the Feynman-Kac formula for the tempered fractional general diffusion equation partial derivative(beta,eta)(t)u(t,x) = Sigma u(t,x) + b(t)u(t,x), x epsilon chi , t >= 0, with initial value f belonging to a Banach space (B,parallel to center dot parallel to), where partial derivative(beta,eta)(t) denotes the Caputo tempered fractional derivative with order beta epsilon (0; 1) and tempered parameter eta > 0, b(t) is a bounded and continuous external potential on [0,infinity), pound is the infinitesimal generator of a general time-homogeneous strong Markov process {X-t}t >= 0, and X denotes a Lusin space that is a topological space being homeomorphic to a Borel subset of a compact metric space. By using the properties of the tempered beta-stable subordinator S-beta,S-eta(t) and the inverse tempered fi-stable subordinator D-beta,D-eta(t), and the stochastic calculus for the stochastic integral driven by D-beta,D-eta(t), we show that the Feynman-Kac representation u(t,x) defined by u(t,x) = E-x [f(XD beta,eta(t))e integral(t) (0) b(r)dD(beta,eta)(r)] is the unique mild and weak solutions to the tempered fractional general diffusion equation. From the Feynman-Kac formula, we further show the continuity of the solution with respect to time based on the integral properties of the Mittag-Leffler function and differential formula of covariance for D-beta,D-eta(t). By exploring the scaling property of D-beta,D-eta(t), the explicit order is also presented for the continuity of the solution with respect to tempered parameter eta.