On Some Properties of a Class of Fractional Stochastic Heat Equations

Abstract

We consider nonlinear parabolic stochastic equations of the form ∂ t u = Lu + λσ (u)ξ on the ball B(0, R), whereξ denotes some Gaussian noise and σ is Lipschitz continuous. Here L corresponds to a symmetric α-stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in

Moment bounds for some fractional stochastic heat equations on the ball

“…Following Walsh [21], we define the mild solution to equation (1.1) as the random field u = {u t (x)} t>0,x∈D satisfying where p D (t, x, y) denotes the Dirichlet fractional heat kernel on D, and the stochastic integral is understood in an extended Itô sense. Following Dalang [7], it is well-known (see also [19,Appendix] and [8]), that if the spectral measure µ satisfies that R d µ(dξ) 1 + |ξ| α < ∞, (1.4) then there exists a unique random field solution u to equation (1.3). Moreover, for all p ≥ 2 and T > 0, sup…”

Exponential behavior and upper noise excitation index of solutions to evolution equations with unbounded delay and tempered fractional Brownian motions

“…It is worth noticing that, in NMR spectroscopy, intermittency can be strongly associated to nonlinear noise excitation (see, e.g., [1,22]). The effect of noise intensity on stochastic parabolic equations driven by Brownian motion has been discussed in recent years, in particular the relationship between the energy of solutions at time t and the level of the noise was established in [12,25,19,20]. However, there has been little literature about the relationship between the energy of solutions and the level of the noise for stochastic delay evolution equations even in the case of Brownian motion.…”

“…Here we consider stochastic evolution equations with infinite delay and TFBM, the upper bound of the upper excitation index of the solution at time t will be presented. e(t) and e(t), respectively, denote the lower and upper excitation indices of the mild solution at time t [12,25,19,20], where we may use the notation…”

On the distribution and q-variation of the solution to the heat equation with fractional Laplacian

“…[17], [5]). Their stochastic counterparts (3.1) and (4.1) are valuable models for similar phenomena with random effects, including random effects with thermal memory (see [5], [16], [17]). The link between the fractional (stochastic) differential equations and the conservation law has been studied in [1] or [22] among many others.…”