Heat equations with fractional white noise potentials. (English) Zbl 0993.60065

Hu, Y.

Appl. Math. Optimization 43, No. 3, 221-243 (2001).

The author considers the following stochastic partial differential equation on \(\mathbb{R}^d:{\partial u\over\partial t}= {1\over 2}\Delta u+ w^H\cdot u\). Here \(w^H\) is a fractional white noise with Hurst parameter \(H\). The fractional white noise can be either time dependent or time independent. Under suitable assumptions on \(H\), it is shown that the solution to the above stochastic partial differential equation is in \({\mathcal L}_2\) and the author provides an estimate of the \({\mathcal L}_2\)-Lyapunov exponent. For \(\rho\in \mathbb{R}\), a family \(S_\rho\) of distribution spaces is introduced. These spaces have the property that every chaos expansion coefficient of an element in \(S_\rho\) is in \({\mathcal L}_2\). The author also estimates the Lyapunov exponents of the solution of the SPDE in the spaces \(S_\rho\).
The paper is well written and the author develops the relevant notions of stochastic calculus for the fractional Brownian fields considered.

Reviewer: Jeffrey J.Mitchell (Waco)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H05	Stochastic integrals
60G60	Random fields
35K05	Heat equation
35R60	PDEs with randomness, stochastic partial differential equations
60F25	\(L^p\)-limit theorems

Keywords:

Lyapunov exponent; Hurst parameter; chaos expansion; fractional Brownian fields