Feynman-Kac formula for general diffusion equations driven by TFBM with Hurst index H ∈(0,1)

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM) partial derivative u(t, x)/partial derivative t = Lu(t, x) + u(t, x) dB(H,lambda)(t)/dt, (t, x) is an element of R+ x R-d, where L is the infinitesimal generator of some time-homogeneous Markov process {X-t}(t >= 0) and{B-H,B-lambda(t)}(t is an element of R) is a tempered fractional Brownian motion with Hurst index H is an element of(0, 1/2) boolean OR (1/2, 1) and tempering parameter lambda > 0 which is independent of {X-t}(t >= 0). Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formula is established for the general stochastic diffusion equation driven by TFBM with the initial value f, Hurst index H is an element of(0, 1/2) boolean OR (1/2, 1) and tempering parameter lambda > 0. The difficulty is that the Hurst parameter H can be allowed to be less than 1/2. The idea is to explore the above simplest form, to utilize the techniques of Malliavin calculus. By using the properties of TFBM and especially that of the modified Bessel function of the second kind, we prove that the process defined by the Feynman-Kac formula is the mild and weak solutions of the general diffusion equation driven by TFBM. From the Feynman-Kac formula, we exhibit the smoothness of the density of the solution and the sharp Holder regularity. We further show that limt ->infinity ln||u(t,center dot)||(p)/g(t) = 0 in the almost surely sense where p is an element of R and the function g: R+ -> R+ grows at least as fast as a linear function. The Feynman-Kac formula for the stochastic general diffusion equation in the Skorohod sense is also achieved by exploring the relationship between the Stratonovich and Skorohod integrals. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.