Abstract
In this paper, we consider a stochastic heat equation with multiplicative bi-fractional Brownian sheet. Using the technique of Feynman-Kac formula and Malliavin calculus, we give an explicit formula of the weak solution and study the regularity.
Highlights
In recent years, there has been considerable interest in studying fractional Brownian motion due to its interesting properties and wide applications in various scientific areas such as turbulence, telecommunications, finance, and image processing
Some surveys and complete literatures for fractional Brownian motion (fBm) can be found in Alós et al [ ], Biagini et al [ ], Decreusefond and Üstünel [ ], Gradinaru et al [ ], Hu [ ], Mishura [ ], Nourdin [ ], Nualart [ ], Tudor [ ], and the references therein
The main reason is the complexity of dependence structures for self-similar Gaussian processes that do not have stationary increments
Summary
There has been considerable interest in studying fractional Brownian motion (fBm) due to its interesting properties and wide applications in various scientific areas such as turbulence, telecommunications, finance, and image processing. Many authors have proposed to use more general self-similar Gaussian processes and random fields as stochastic models. Such applications have raised many interesting theoretical questions about selfsimilar Gaussian processes and fields in general. It seems interesting to study some extensions of fBm. The bi-fractional Brownian motion BH,K with indices H ∈ ( , ) and K ∈ ( , ] is an extension of fBm with Hurst index H ∈ ( , ), which was first introduced by Houdré and Villa [ ]. We extend the Feynman-Kac formula to the stochastic heat equation driven by a bi-fractional noise: u(t, x). The stochastic integral is the Stratonovich integral, and φ is a bounded measurable function
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