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

##### 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
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