Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

Junfeng Liu () and Litan Yan ()
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Junfeng Liu: Nanjing Audit University
Litan Yan: Donghua University

Journal of Theoretical Probability, 2016, vol. 29, issue 1, 307-347

Abstract: Abstract In this article, we will prove the existence, uniqueness and Hölder regularity of the solution to the fractional stochastic partial differential equation of the form $$\begin{aligned} \frac{\partial }{\partial t}u(t,x)=\mathfrak {D}(x,D)u(t,x)+\frac{\partial f}{\partial x}(t,x,u(t,x))+\frac{\partial ^2 W^H}{\partial t\partial x}(t,x), \end{aligned}$$ ∂ ∂ t u ( t , x ) = D ( x , D ) u ( t , x ) + ∂ f ∂ x ( t , x , u ( t , x ) ) + ∂ 2 W H ∂ t ∂ x ( t , x ) , where $$\mathfrak {D}(x,D)$$ D ( x , D ) denotes the Markovian generator of stable-like Feller process, $$f:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$$ f : [ 0 , T ] × R × R → R is a measurable function, and $$\frac{\partial ^2 W^H}{\partial t\partial x}(t,x)$$ ∂ 2 W H ∂ t ∂ x ( t , x ) is a double-parameter fractional noise. In addition, we establish lower and upper Gaussian bounds for the probability density of the mild solution via Malliavin calculus and the new tool developed by Nourdin and Viens (Electron J Probab 14:2287–2309, 2009).

Keywords: Stable-like generator of variable order; Green function; Fractional noise; Hölder regularity; Malliavin calculus; 60G15; 60H05; 60H07 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-014-0578-4

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