Some linear SPDEs driven by a fractional noise with Hurst index greater than 1/2

Abstract

In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear s.p.d.e.'s of parabolic and hyperbolic type. These equations rely on a spatial operator $\cL$ given by the $L^2$-generator of a $d$-dimensional Lévy process $X=(X_t{)}_{t\ge 0}$, and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index $H>1/2$. As an application, we consider the case when $X$ is a $\beta$-stable process, with $\beta \in (0,2]$. In the parabolic case, we develop a connection with the potential theory of the Markov process $\bar{X}$ (defined as the symmetrization of $X$), and we show that the existence of the solution is related to the existence of a "weighted" intersection local time of two independent copies of $\bar{X}$.

37 pages

References in corpus (2)

• The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution
• On the Stochastic Heat Equation with Spatially-Colored Random forcing