weak convergence of the solutions to stochastic heat equation

Question

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.

$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$

$$u(0,x)=f(x)\in C_C(R^d)$$

If $W$ satisfies some condition, if regarding $\{u(t,\cdot)\}$ as a process in $L^2_\rho$ $(\rho>2d)$ , $(\|u\|^2_{L^2_\rho}=\int_{R^d}u^2(x)(1+|x|)^{-\rho}dx)$, I know the distribution of $\{u(t,\cdot)\}_{t\geq 0}$ is tight. Can one prove that all the limit points are the same one? (You can add some condition on $W$ and $f$. )

Answer 1

In linear random dynamical systems (cocycles) the notion of Lyapunov exponent usually applies. In more compact systems than yours, it is a theorem (consequence of Oseledets' multiplicative ergodic theorem and its infinite-dimensional generalizations) that there is a real number $\lambda$ such that the logarithm of the norm of most solutions is asymptotic to $\lambda t$ for large $t$.

Some form of that statement should apply to your SHE (maybe you even have enough compactness since your norm gives much more weight to the bulk then to the "tails").

Then tightness would mean $\lambda\le 0$. It is unlikely that $\lambda=0$, so one should have $\lambda<0$, and then, of course, you have contraction to zero, and identical zero is a unique limit point.