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Journal of Theoretical Probability

, Volume 29, Issue 1, pp 307–347 | Cite as

Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

  • Junfeng Liu
  • Litan YanEmail author
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 

Notes

Acknowledgments

We want to thank the Editor and anonymous referees whose valuable remarks and suggestions greatly improved the presentation of this paper.

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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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