SOME LINEAR SPDEs DRIVEN BY A FRACTIONAL NOISE WITH HURST INDEX GREATER THAN 1/2

作者：Balan Raluca M^*

来源：Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2012, 15(4): 1250023.

DOI：10.1142/S0219025712500233

• Stochastic partial differential equations
• fractional Brownian motion
• spatially homogeneous Gaussian noise
• Levy processes

In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear stochastic partial differential equations (spde%26apos;s) of parabolic and hyperbolic type. These equations rely on a spatial operator L given by the L-2-generator of a d-dimensional Levy process X = (X-t)(t %26gt;= 0), and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index H %26gt; 1/2. As an application, we consider the case when X is a beta-stable process, with beta is an element of (0, 2]. In the parabolic case, we develop a connection with the potential theory of the Markov process (X) over bar (defined as the symmetrization of X), and we show that the existence of the solution is related to the existence of a %26quot;weighted%26quot; intersection local time of two independent copies of (X) over bar.

• 出版日期2012-12