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We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM) ∂ u (t , x) ∂ t = L u (t , x) + u (t , x) d B H , λ (t) d t , (t , x) ∈ R + × R d , where L is the infinitesimal generator of some time-homogeneous Markov process { X t } t ≥ 0 and { B H , λ (t) } t ∈ R is a tempered fractional Brownian motion with Hurst index H ∈ (0 , 1 2) ∪ (1 2 , 1) and tempering parameter λ > 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 u (t , x) = E x [ f (X t) ] e B H , λ (t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f , Hurst index H ∈ (0 , 1 2) ∪ (1 2 , 1) and tempering parameter λ > 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 Hölder regularity. We further show that lim t → ∞ ⁡ ln ⁡ ‖ u (t , ⋅) ‖ p g (t) = 0 in the almost surely sense where p ∈ 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. [ABSTRACT FROM AUTHOR]

In this paper, we consider stochastic evolution equations with finite delay driven by fractional Brownian motion (fBm) with Hurst index $ H\in(0,1) $. First, the global existence and uniqueness of mild solutions are established by using a temporally weighted norm, the semigroup theory, the Banach fixed point theorem, and a new estimate of the stochastic integral with respect to fBm. Then after extending the integral inequality to the general case, we obtain the mean-square exponential stability of mild solutions. In particular, the convolution method and the Young convolution inequality are utilized to deal with the difficulty of the noise term caused by fBm with Hurst index $ H\in(0,1) $. Finally, the stochastic evolution equation with multiple time-varying delays and the reaction diffusion neural networks system with time-varying delay are considered, and the global existence, uniqueness and exponential stability of mild solutions are proved by using the abstract results. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]

This paper is concerned with stochastic delay evolution equations driven by tempered fractional Brownian motion (tfBm) $ B_Q^{\sigma, \lambda}(t) $ with time fractional operator of order $ \alpha\in (1/2+\sigma, 1) $, where $ \sigma\in (-1/2, 0) $ and $ \lambda>0 $. First, we establish the global existence and uniqueness of mild solutions by using the new established estimation of stochastic integrals with respect to tfBm. Moreover, based on the relations between the stochastic integrals with respect to tfBm and fBm, we show the continuity of mild solutions for stochastic delay evolution equations when tempered fractional noise is reduced to fractional noise. Finally, we analyze the stability with general decay rate (including exponential, polynomial and logarithmic stability) of mild solutions for stochastic delay evolution equations with tfBm and time tempered fractional operator. [ABSTRACT FROM AUTHOR]

We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order p ≥ 2 driven by a tempered fractional Brownian motion (TFBM) B σ , λ (t) with − 1 / 2 < σ < 0 and λ > 0. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of p th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of λ → 0 , σ ∈ (− 1 / 2 , 0) or λ > 0 , σ → σ 0 ∈ (− 1 / 2 , 0). In particular, we obtain p th moment Hölder regularity in time and p th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise. [ABSTRACT FROM AUTHOR]

This article is devoted to study time fractional stochastic evolution inclusions with infinite delays driven by a nonlinear multiplicative noise and a fractional Brownian motion with Hurst parameter. First of all, we investigate the local and global existence of mild solutions to such evolution inclusions by using the fractional resolvent operator theory and some new results on the measure of noncompactness for the stochastic integral term. Further, we prove that every mild solution decays exponentially to zero in the sense of mean-square topology. [ABSTRACT FROM AUTHOR]