# Sufficient conditions for a SDE to have a stationary probability measure

Apologies if this question is too basic for MathOverflow.

For a smooth Wiener-driven SDE on a non-compact manifold $$M$$ taking the form $$dX_t = b(X_t) dt + \sum_{i=1}^k \sigma_i(X_t) \ast dW_t^i$$ where either $$\ast$$ is Stratonovich or in the case that $$M$$ is an open subset of $$\mathbb{R}^d$$, $$\ast$$ is Itô,

what reasonably verifiable conditions exist [under any particular extra assumptions if necessary, e.g. $$M=\mathbb{R}^d$$] guaranteeing the existence of (at least one) stationary probability measure?

(If it helps, you can assume a priori that the SDE has global existence of strong solutions.)

• One needs the "existence of invariant measures" theorem or Krylov-Bogolyubox theorem (see Theorem 1.10 of hairer.org/notes/Convergence.pdf). In practice the assumptions of this theorem are verified by using a Foster-Lyapunov condition: see, e.g., Section 4.2 of jstor.org/stable/pdf/… Jun 9, 2022 at 16:15
• For SDEs with multiplicative noise specifically, simple/general existence criteria are presented in Theorem 2.2.1 of link.springer.com/book/10.1007/b80743 The proof boils down to developing a Foster-Lyapunov condition with respect to a quadratic Lyapunov function $V(x) = |x|^2$. Jun 9, 2022 at 16:38

Take $$M=\mathbb{R}^d$$, take $$\ast$$ to be Itô, take $$k \leq d$$ [I guess this probably isn't necessary, but it looks like the book is taking $$k=d$$, which automatically generalises to $$k \leq d$$], and let $$\sigma(x) \in \mathbb{R}^{d \times k}$$ be the matrix formed by horizontally concatenating the vectors $$\sigma_1(x),\ldots,\sigma_k(x)$$. Let $$\|\boldsymbol{\cdot}\|$$ denote the operator norm where Euclidean spaces $$\mathbb{R}^d$$ and $$\mathbb{R}^k$$ are equipped with the standard Euclidean norm. Suppose that:
1. There exist values $$r_b \geq r_\sigma \geq 0$$ such that \begin{align*} \sup_{x \in \mathbb{R}^d} \frac{|b(x)|}{1+|x|^{2r_b + 1}} &< \infty \\ \sup_{x \in \mathbb{R}^d} \frac{\|\sigma(x)\|}{1+|x|^{r_\sigma}} &< \infty \end{align*} and for some $$a,\gamma>0$$ and $$c \in \mathbb{R}$$, every $$x,y \in \mathbb{R}^d$$ has $$(b(y)-b(x)) \boldsymbol{\cdot} (y-x) \leq -a|y-x|^{2r_b+2}+c(|x|^\gamma+1)\text{.}$$
2. For each $$p \geq 1$$ there exists $$c_p \in \mathbb{R}$$ such that every $$x,y \in \mathbb{R}^d$$ has $$(b(y)-b(x)) \boldsymbol{\cdot} (y-x) + p\|\sigma(y)-\sigma(x)\|^2 \leq c_p|y-x|^2\text{.}$$