# Stochastic invariant subset

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.

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.

• Just after exiting the compact, is the probability of the state to return to C strictly lower than 1 or is always 1? Mar 26, 2020 at 17:02
• The probability of return is $1$. Mar 30, 2020 at 17:42
• How can I prove that? Mar 31, 2020 at 9:12
• This follows by Birkhoff‘s theorem. Jul 15, 2020 at 19:00
• 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? Jul 19, 2020 at 16:57