# Is it possible to interchange the limit and the probability measure for trajectories of stochastic differential equation?

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.

• As stated, obviously not. For example, $x_t$ not having limit or when it does, if $\Omega$ is open and $x(t)$ has limit in the boundary of $\Omega.$ Nov 15 '19 at 19:46
• $x_t$, the trajectories of the SDE are continuous almost surely, why $x_t$ may not have a limit? Nov 15 '19 at 20:26
• Why should it? ${}$ Nov 16 '19 at 0:29
• you mean sometimes $\mathop {\lim }\limits_{t \to \infty } x(t) = \infty$ and the limit does not exists? In fact I consider $\mathop {\lim }\limits_{x \to \infty } x(t) \in \Omega$ as an event. this event might happen with some level of probability, on the other hand, I think also, with some level of probability this event might not happen (as an example when $x_t$ does not converge to $\omega$ or when the limit does not exists) Nov 16 '19 at 9:03