Generalized fractional kinetic equations: another point of view

Abstract

In this paper we deal with generalized fractional kinetic equations driven by a Gaussian noise, white in time and correlated in space, and where the diffusion operator is the composition of the Bessel and Riesz potentials for any fractional parameters. We give results on the existence and uniqueness of solutions by means of a weak formulation and study the Hölder continuity. Moreover, we prove the existence of a smooth density associated to the solution process and study the asymptotics of this density. Finally, when the diffusion coefficient is constant, we look for its Gaussian index.

Generalized space-time fractional stochastic kinetic equation

“…In this paper it is assumed (20) with ̺ = α+γ 2 . So when β is close to one, the exponent ̺ in (21) is coincides with the exponent studied in Lemma 2.1 in Márquez-Carreras [17]. On the other hand, our assumption ( 20) is weaker when β is close to zero.…”

“…If the parameter β = 1, Eq. (1) reduces to the SPDE (1.1) studied by Márquez-Carreras [17]. In this paper it is assumed (20) with ̺ = α+γ 2 .…”

“…The fractional SPDEs represent a combination of the deterministic fractional equations and the stochastic integration theory developped by Walsh (see [27], see also Dalang's seminal paper [9]). Several types of fractional SPDEs have been considered in Chen, Kim and Kim [7], Chen et al [6], Foondun and Nane [12], Hu and Hu [13], Liu and Yan [16], Márquez-Carreras [17], [18], Mijena and Nane [20], [21] and and references therein.…”

“…We refer, for example, to the classical stochastic heat equation with β = 1, γ = 0 and α = 2 (see e.g. Dalang [9], Khoshnevisan [14]), the fractional stochastic heat equation with β = 1, γ = 0 and α > 0 (see example, Chen and Dalang [4], [5], Foondun and Nane [12], Márquez-Carreras [18], Tudor [24]), the generalized fractional kinetic equation with β = 0, γ ≥ 0 and α > 0 (see example Márquez-Carreras [17]), the space-time fractional stochastic partial differential equation with 0 < β < 1, γ = 0 and 0 < α ≤ 2 (see example, Mijena and Nane [20], [21]).…”

“…To be more precise, we extend the result in [20], [17] and [11] by including in the model the Bessel operator (I − ∆) γ 2 with γ ≥ 0 and by generalizing the stochastic noise, in the sense that we allow a more general structure for the spatial covariance of the Gaussian noise W in (1) (which is taken to be space-time white noise in [20] and colored by a Riesz kernel in space in [11]). The presence of this Bessel operator brings more flexibility to the model, by including for γ = 0 the situation treated in [20], [11] or [17]. From the technical point of view, the appearance of the Bessel operator leads to a new expression of the fundamental solution associated to the equation (1).…”