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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.)

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    $\begingroup$ 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/… $\endgroup$ Jun 9, 2022 at 16:15
  • $\begingroup$ 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 $V(x) = |x|^2$. $\endgroup$ Jun 9, 2022 at 16:38

1 Answer 1

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Thanks to Nawaf Bou-Rabee's comments, I can post a first answer. Specifically, Theorem 2.2.1 of [Cerrai '01] seems to state the following - although it is hard to say for certain that I have interpreted all the notations correctly as Google isn't giving me access to the whole book!

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 $$ (b(y)-b(x)) \boldsymbol{\cdot} (y-x) + p\|\sigma(y)-\sigma(x)\|^2 \leq c_p|y-x|^2\text{.} $$

Then the SDE has a stationary probability measure.

Reference:
[Cerrai '01] S. Cerrai, Second Order PDE's in Finite and Infinite Dimension: A Probabilistic Approach, Lecture Notes in Mathematics 1762, Springer-Verlag Berlin Heidelberg, 2001.

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