# weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ 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 $f$. )

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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 $t$.
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.