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Asymptotic properties of some space-time fractional stochastic equations

Research paper by Mohammud Foondun, Erkan Nane

Indexed on: 19 Jan '17Published on: 03 Jan '17Published in: Mathematische Zeitschrift



Abstract

Consider non-linear time-fractional stochastic heat type equations of the following type, $$\begin{aligned} \partial ^\beta _tu_t(x)=-\nu (-\Delta )^{\alpha /2} u_t(x)+I^{1-\beta }_t[\lambda \sigma (u)\mathop {F}\limits ^{\cdot }(t,x)] \end{aligned}$$ in \((d+1)\) dimensions, where \(\nu >0, \beta \in (0,1), \alpha \in (0,2]\) . The operator \(\partial ^\beta _t\) is the Caputo fractional derivative while \(-(-\Delta )^{\alpha /2} \) is the generator of an isotropic stable process and \(I^{1-\beta }_t\) is the Riesz fractional integral operator. The forcing noise denoted by \(\mathop {F}\limits ^{\cdot }(t,x)\) is a Gaussian noise. And the multiplicative non-linearity \(\sigma :\mathbb {R}\rightarrow \mathbb {R}\) is assumed to be globally Lipschitz continuous. Mijena and Nane (Stochastic Process Appl 125(9):3301–3326, 2015) have introduced these time fractional SPDEs. These types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter \(\lambda \) . In particular, our results are significant extensions of those in Ann Probab (to appear), Foondun and Khoshnevisan (Electron J Probab 14(21): 548–568, 2009), Mijena and Nane (2015) and Mijena and Nane (Potential Anal 44:295–312, 2016). Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation. Consider non-linear time-fractional stochastic heat type equations of the following type, $$\begin{aligned} \partial ^\beta _tu_t(x)=-\nu (-\Delta )^{\alpha /2} u_t(x)+I^{1-\beta }_t[\lambda \sigma (u)\mathop {F}\limits ^{\cdot }(t,x)] \end{aligned}$$ in \((d+1)\) dimensions, where \(\nu >0, \beta \in (0,1), \alpha \in (0,2]\) . The operator \(\partial ^\beta _t\) is the Caputo fractional derivative while \(-(-\Delta )^{\alpha /2} \) is the generator of an isotropic stable process and \(I^{1-\beta }_t\) is the Riesz fractional integral operator. The forcing noise denoted by \(\mathop {F}\limits ^{\cdot }(t,x)\) is a Gaussian noise. And the multiplicative non-linearity \(\sigma :\mathbb {R}\rightarrow \mathbb {R}\) is assumed to be globally Lipschitz continuous. Mijena and Nane (Stochastic Process Appl 125(9):3301–3326, 2015) have introduced these time fractional SPDEs. These types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter \(\lambda \) . In particular, our results are significant extensions of those in Ann Probab (to appear), Foondun and Khoshnevisan (Electron J Probab 14(21): 548–568, 2009), Mijena and Nane (2015) and Mijena and Nane (Potential Anal 44:295–312, 2016). Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation. $$\begin{aligned} \partial ^\beta _tu_t(x)=-\nu (-\Delta )^{\alpha /2} u_t(x)+I^{1-\beta }_t[\lambda \sigma (u)\mathop {F}\limits ^{\cdot }(t,x)] \end{aligned}$$ $$\begin{aligned} \partial ^\beta _tu_t(x)=-\nu (-\Delta )^{\alpha /2} u_t(x)+I^{1-\beta }_t[\lambda \sigma (u)\mathop {F}\limits ^{\cdot }(t,x)] \end{aligned}$$ \((d+1)\) \((d+1)\) \(\nu >0, \beta \in (0,1), \alpha \in (0,2]\) \(\nu >0, \beta \in (0,1), \alpha \in (0,2]\) \(\partial ^\beta _t\) \(\partial ^\beta _t\) \(-(-\Delta )^{\alpha /2} \) \(-(-\Delta )^{\alpha /2} \) \(I^{1-\beta }_t\) \(I^{1-\beta }_t\) \(\mathop {F}\limits ^{\cdot }(t,x)\) \(\mathop {F}\limits ^{\cdot }(t,x)\) \(\sigma :\mathbb {R}\rightarrow \mathbb {R}\) \(\sigma :\mathbb {R}\rightarrow \mathbb {R}\)2015 \(\lambda \) \(\lambda \)200920152016