Feynman-Kac formula for tempered fractional general diffusion equations driven by TFBM

Abstract

In this article, we investigate the following stochastic tempered fractional general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ t β , η u ( t , x ) = L u ( t , x ) + u ( t , x ) B ˙ H , λ ( t ) , ( t , x ) ∈ R + × R d , where ∂ t β , η denotes the Caputo tempered fractional derivative with order β ∈ ( 0 , 1 ) and tempered parameter η > 0, 𝔏 is the infinitesimal generator of some general time-homogeneous symmetric strong Markov process { X t } t ≥ 0 with the corresponding transition semigroup { P t , t ≥ 0 } defined by P t f(x): = 𝔼 x [f(X t )], and { B H , λ ( t ) } t ≥ 0 is a TFBM with Hurst index H ∈ ( 1 2 , 1 ) and tempering parameter λ > 0 which is independent of { X t } t ≥ 0 . First of all, a novel estimate of multiple stochastic integrals with respect to the inverse tempered β-stable subordinator D β, η (t) is presented, which greatly contributes to the stochastic analysis. Then after establishing the Fubini Theorem for the stochastic integrals with respect to TFBM B H, λ (t) and the inverse tempered β-stable subordinator D β, η (t), we show that the Feynman-Kac representation u ( t , x ) = E D [ P D β , η ( t ) f ( x ) e ∫ 0 t ∫ 0 t δ ( s − r ) d D β , η ( s ) d B H , λ ( r ) ] is the weak solution of the stochastic tempered fractional general diffusion equation driven by TFBM with the initial value f by means of the techniques of Malliavin calculus and convergence analysis. From the Feynman-Kac formula, several kinds of continuity of the solutions with respect to time and tempered parameter η of tempered fractional derivative are proved by using the scaling property of the inverse tempered β-stable subordinator D β, η (t) and the novel estimate of stochastic integrals with respect to B H, λ (t) and D β, η (t). In particular for the stochastic fractional general diffusion equation driven by fractional Brownian motion (FBM) but in the time fractional derivative case, we can also obtain the sharp upper and lower bounds for the Hölder regularity and the moments of the solution.