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Let us consider a stochastic differential equation (SDE),

$$ dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}% $$

and a compact set $C\subset\mathbb{R}^{n}$.

Given a stochastic Lyapunov function $\Phi\left( x_{t}\right) $ for this SDE with respect to $C$, i.e.

(i) $\Phi$ is positive definite.

(ii) $L\Phi\left( x\right) $ is not necessary to be nonpositive in $C$ but $L\Phi\left( x\right) <0$ for all $x\notin C$, where $L$ is the infinitesimal generator of the SDE.

How can I prove that $C$ is an invariant set with respect to the solutions of the SDE? In this I work with convergence in probability.

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This seems wrong to me. Consider $\sigma(x)=\sqrt{2}$ and $f(x)=-x$. Then $L=\triangle-x\cdot\nabla$. $\Phi(x)=|x|^2$ is a Lyapunov function with $C=\overline{B}_1(0)$. But $C$ is certainly not invariant.

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  • $\begingroup$ Just after exiting the compact, is the probability of the state to return to C strictly lower than 1 or is always 1? $\endgroup$
    – UnclePetros
    Commented Mar 26, 2020 at 17:02
  • $\begingroup$ The probability of return is $1$. $\endgroup$
    – julian
    Commented Mar 30, 2020 at 17:42
  • $\begingroup$ How can I prove that? $\endgroup$
    – UnclePetros
    Commented Mar 31, 2020 at 9:12
  • $\begingroup$ This follows by Birkhoff‘s theorem. $\endgroup$
    – julian
    Commented Jul 15, 2020 at 19:00
  • $\begingroup$ I am not a specialist in this theorem, but it seems to mention Cosmology and Theory of relativity. How can it be applied to the stochastic problem that I pose? $\endgroup$
    – UnclePetros
    Commented Jul 19, 2020 at 16:57

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