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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 $$ dX_t = b(X_t) dt + \sum_{i=1}^k \sigma_i(X_t) \ast dW_t^i $$ w …

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 interpret …

Preliminaries. (See [1] for further details.) Let $M$ be a compact connected $C^\infty$ Riemannian manifold. We say that a list $\sigma_1,\ldots,\sigma_n$ ($n \in \mathbb{N}$) of $C^\infty$ vector fie …

Suppose we have a $C^\infty$ manifold $M$ and $C^\infty$ vector fields $b,\sigma_1,\ldots,\sigma_k$ on $M$, and for convenience define the set of vector fields $$ \mathcal{S} = \{b,\sigma_1,-\sigma_1, …

By adapting the arguments in Sec. 3.3.6.1 of the Michel & Pardoux notes linked to by Nawaf Bou-Rabee, I think I can prove the result. (I will assume for simplicity that the SDE has global existence of …

I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before. Let $(G,\ast)$ be an abelian $C^1$ Lie grou …

Setting: Let $M$ be a compact connected $C^\infty$ Riemannian manifold of dimension $D \geq 2$, with $\lambda$ the normalised Riemannian volume measure. Write $T_{\neq 0}M \subset TM$ for the non-ze …

Motivation. I’m not an expert on stochastic calculus and stochastic differential equations; I often see the Fokker-Planck equations and Hörmander's theorem formulated as addressing “transition probabi …

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a compact metric space. Let $F \colon \Omega \times X \to X$ and $\bar{F} \colon \Omega \times X \to X$ be measurable functi …

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a compact connected $C^\infty$-smooth manifold. Let $F \colon \Omega \times X \to X$ and $\bar{F} \colon \Omega \times X \to …