Chaos expansion of heat equations with white noise potentials

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The asymptotic behavior as t→∞ of the solution to the following stochastic heat equations $$\frac{{\partial u_t }}{{\partial t}} = \frac{1}{2}\sum\limits_{i = 1}^d {\frac{{\partial ^2 u_t }}{{\partial x_i^2 }} + w\;\diamondsuit \;u_t ,{\text{ 0}} < t < \infty ,\;x \in \mathbb{R}^d ,{\text{ }}u_0 (x) = 1} $$ is investigated, where w is a space-time white noise or a space white noise. The use of ⋄ means that the stochastic integral of Ito (Skorohod) type is considered. When d=1, the exact ℒ2 Lyapunov exponents of the solution are studied. When the noise is space white and when d<4 it is shown that the solution is in some “flat” ℒ2 distribution spaces. The Lyapunov exponents of the solution in these spaces are also estimated. The exact rate of convergence of the solution by its first finite chaos terms are also obtained.