arXiv 1409.73662014-09-25math.PRmath-phmath.APmath.MP

时空分数阶随机偏微分方程

Space-time fractional stochastic partial differential equations

Jebessa B. Mijena, Erkan Nane

备注：: 25 pages
发表信息：: Stochastic Process. Appl. 125, (2015) 3301--3326

摘要：

在$(d+1)$维上考虑非线性时间分数阶随机热型方程$$\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[σ(u)\stackrel{\cdot}{W}(t,x)]$$，其中$ν>0, β\in (0,1)$，$α\in (0,2]$和$d<\min\{2,β^{-1}\}\a$，$\partial^β_t$是Caputo分数阶导数，$-(-Δ)^{α/2} $是各向同性稳定过程的生成元，$I^{1-β}_t$是分数阶积分算子，$\stackrel{\cdot}{W}(t,x)$是时空白噪声，$σ:\RR{R}\to\RR{R}$是Lipschitz连续的.时间分数阶随机热型方程可以用来模拟具有热记忆随机效应的现象。我们证明了该方程温和解的存在性和唯一性，并建立了解连续的条件。我们的结果推广了在\cite{foondun-khoshnevisan-09, walsh}中得到的抛物型随机偏微分方程的结果。与\cite{foondun-khoshnevisan-09, khoshnevisan-cbms, walsh}中研究的随机偏微分方程形成鲜明对比的是，在某些情况下，我们的结果给出了空间维度$d=1,2,3$中随机场解的存在性。在$σ$的增长速度大于线性增长的条件下，证明了时间分数阶随机偏微分方程不存在有限能量解。这推广了Foondun和Parshad \cite{foondun-parshad}在抛物型随机偏微分方程情形下的结果。建立了时间分数阶随机偏微分方程与高阶抛物型随机微分方程的联系。

英文摘要：

We consider non-linear time-fractional stochastic heat type equation $$\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[σ(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $ν>0, β\in (0,1)$, $α\in (0,2]$ and $d<\min\{2,β^{-1}\}\a$, $\partial^β_t$ is the Caputo fractional derivative, $-(-Δ)^{α/2} $ is the generator of an isotropic stable process, $I^{1-β}_t$ is the fractional integral operator, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and $σ:\RR{R}\to\RR{R}$ is Lipschitz continuous. Time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove existence and uniqueness of mild solutions to this equation and establish conditions under which the solution is continuous. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in \cite{foondun-khoshnevisan-09, walsh}. In sharp contrast to the stochastic partial differential equations studied earlier in \cite{foondun-khoshnevisan-09, khoshnevisan-cbms, walsh}, in some cases our results give existence of random field solutions in spatial dimensions $d=1,2,3$. Under faster than linear growth of $σ$, we show that time fractional stochastic partial differential equation has no finite energy solution. This extends the result of Foondun and Parshad \cite{foondun-parshad} in the case of parabolic stochastic partial differential equations. We also establish a connection of the time fractional stochastic partial differential equations to higher order parabolic stochastic differential equations.

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