consider a typical stochastic differential equation as follows:
$dx = f(x)dt + \sigma (x)dw(t)$
where $w(t)$ is Weiner process. $x(t)$ is the solution of the SDE.
are these two limits equivalent? if they are, how can we prove it?
$\mathop {\lim }\limits_{t \to \infty } \,P\{ x(t) \in \Omega \} = P\{ \mathop {\lim }\limits_{t \to \infty } x(t) \in \Omega \,\}$
where $x \in {\mathbb{R}^n},\Omega \subset {\mathbb{R}^n}$
it should be noted that $\mathop {\lim }\limits_{t \to \infty } \,P\{ x(t) \in \Omega \}$=$P_{st}(x)$ exists.
where $P_{st}(x)$ is the stationary probability distribution. also $P(x,t)$ is the probability measure of the trajectories of the SDE and can be calculated by solving the Fokker-Planck equation.
The answer depends on the equation but, being the additive noise the most typical case, in the additive noise case the limit of trajectories does not exist, in particular it is not in a given set, and thus the right-hand-side is zero. On the contrary, the law at time t (the left hand side before taking the limit) very often converges in law to an invariant measure, hence for certain sets the probability converges.