Heat Equations with Fractional White Noise Potentials

Abstract

This paper is concerned with the following stochastic heat equations: $$\frac{{\partial u_t (x)}} {{\partial t}} = \frac{1} {2}\Delta u_t (x) + w^H \cdot u_t (x), x \in \mathbb{R}^d , t > 0,$$ where w H is a time independent fractional white noise with Hurst parameter H=(h 1 , h 2 ,..., h d ) , or a time dependent fractional white noise with Hurst parameter H=(h 0 , h 1 ,..., h d ) . Denote |H|=h 1 +h 2 +...+h d . When the noise is time independent, it is shown that if ½ <h i d-1 , then the solution is in L 2 and the L 2 -Lyapunov exponent of the solution is estimated. When the noise is time dependent, it is shown that if ½ <h i d- 2 /( 2h 0 -1 ) , the solution is in L 2 and the L 2 -Lyapunov exponent of the solution is also estimated. A family of distribution spaces S ρ , ρ∈ \RR , is introduced so that every chaos of an element in S ρ is in L 2 . The Lyapunov exponents in S ρ of the solution are also estimated.