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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.

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  • $\begingroup$ 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 $\Omega.$ $\endgroup$
    – Will M.
    Nov 15 '19 at 19:46
  • $\begingroup$ $x_t$, the trajectories of the SDE are continuous almost surely, why $x_t$ may not have a limit? $\endgroup$
    – Ali N777
    Nov 15 '19 at 20:26
  • $\begingroup$ Why should it? ${}$ $\endgroup$
    – Will M.
    Nov 16 '19 at 0:29
  • $\begingroup$ 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) $\endgroup$
    – Ali N777
    Nov 16 '19 at 9:03
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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.

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