Fractional Diffusion in Gaussian Noisy Environment

Abstract

Abstract:We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form:is the Caputo fractional derivative of order α ∈ (0, 1) with respect to the time variable t, B is a second order elliptic operator with respect to the space variable x ∈ R d andẆ H a time homogeneous fractional Gaussian noise of Hurst parameter. We obtain conditions satisfied by α and H, so that the square integrable solution u exists uniquely.

Quasi-Linear (Stochastic) Partial Differential Equations with Time-Fractional Derivatives

“…The case γ = β is investigated in various papers about stochastic heat type or wave type equations, see e.g. [15,17,24,43,44], and the case γ = 1 is studied in [16,26].…”

“…In [43,44], the authors consider a space-time fractional stochastic heat type equation to model phenomenon with random effects with thermal memory, and they prove existence and uniqueness of mild solutions and also some intermittency property (see also [24]). For a linear stochastic partial differential equation of fractional order both in the time and space variables with a different type of noise term we refer to [16,26]. In [15,17] the authors investigate linear stochastic time-fractional partial differential equations for the more general case 0 < β ≤ 2.…”

Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd

“…The spde (1.1) for β ∈ (0, 1] and ν = 1 − β has been recently studied in [25,26]. When the noise does not depend on time, a similar model with a general elliptic operator has been studied in [18]. Another related equation is the stochastic fractional heat equation (SFHE) on R:…”

“…When the initial data are spatially homogeneous (i.e., the initial data are constants), so is the solution u(t, x), and then the Lyapunov exponents m p := lim sup 18) do not depend on the spatial variable. In this case, a solution is called fully intermittent if m 1 = 0 and m 2 > 0 (see [2, Definition III.1.1, on p. 55]).…”

Gradient and stability estimates of heat kernels for fractional powers of elliptic operator

“…In this paper we are concerned with the similar estimates for the semigroup generated by the fractional powers of H, namely, Q t = e −t(−H) α , where α ∈ (0, 1) will be fixed throughout this paper. Our motivation is recent works on fractional diffusion in random environment (see [1,6] and references therein) arisen from super and sub diffusion in random environment. However, we shall deal with this problem in separate project.…”