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 <1 for i=1, 2,..., d and if |H|>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 <1 for i=0, 1,..., d and if |H|>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.