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Stochastic Invariant Set

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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