Asymptotic properties of some space-time fractional stochastic equations

2015/05/18 by Mohammud Foondun, Foondun, Mohammud, Erkan Nane +1 · 3 citations
Mathematics · Economics, Econometrics and Finance · #Fractional Differential Equations Solutions #Stochastic processes and financial applications #Nonlinear Differential Equations Analysis

paper · doi:10.48550/arxiv.1505.04615

Abstract

Consider non-linear time-fractional stochastic heat type equations of the following type, ∂^β_tu_t(x)=-ν(-Δ)^α/2 u_t(x)+I^1-β_t[λσ(u)\stackrel⋅F(t,x)] in (d+1) dimensions, where ν>0, β∈ (0,1), α∈ (0,2]. The operator ∂^β_t is the Caputo fractional derivative while -(-Δ)^α/2 is the generator of an isotropic stable process and I^1-β_t is the fractional integral operator. The forcing noise denoted by \stackrel⋅F(t,x) is a Gaussian noise. And the multiplicative non-linearity σ is assumed to be globally Lipschitz continuous. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter λ. In particular, our results are significant extensions of existing results. Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

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