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Journal of Theoretical Probability

, Volume 29, Issue 1, pp 307–347

# Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

Article

## 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}
where $$\mathfrak {D}(x,D)$$ denotes the Markovian generator of stable-like Feller process, $$f:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$$ is a measurable function, and $$\frac{\partial ^2 W^H}{\partial t\partial x}(t,x)$$ 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

## Mathematics Subject Classification (2010)

60G15 60H05 60H07

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## Copyright information

© Springer Science+Business Media New York 2014

## Authors and Affiliations

1. 1.Department of StatisticsNanjing Audit UniversityNanjingPeople’s Republic of China
2. 2.Department of MathematicsDonghua UniversityShanghaiPeople’s Republic of China

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