Heat equations with fractional white noise potentials

This paper is concerned with the following stochastic heat equations: partial derivativeu(t) (chi)/partial derivativet = 1/2 Deltau(t) (chi) + omega (H) . u(t) (chi), chi is an element of R-d, t > 0, where omega (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 1/2 < 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 1/2 < 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,, p E R, is introduced so that every chaos of an element in S-rho is in L-2. The Lyapunov exponents in S, of the solution are also estimated.