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.