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Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each …

$\newcommand\sdiff{\mathbin\triangle}$Starting from literally just the measurable space $(X,\Sigma)$, I wouldn't think there's any particularly nice and natural $\sigma$-algebra on $\Sigma$; but with …

Preliminary notations: For a compact metrisable space $X$, $\mathcal{B}(X)$ is the Borel $\sigma$-algebra on $X$. $\overline{\mathcal{B}}(X)$ is the universal completion of $\mathcal{B}(X)$. $\mathca …

Do you not also want that $\mathbb{P}_Y$ is $\phi$-invariant? In any case, yes there are extremely many continuous-time continuous-path real-valued stochastic processes whose law is ergodic under the …

Let $X$ be a compact metric space, and fix an arbitrary point $x_\ast \in X$. By the Kantorovich-Rubinstein duality theorem, the $1$-Wasserstein metric $W_1$ on the set of Borel probability measures o …

Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff convergence" …

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 …

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 …

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 …

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 …

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, …

Let $(X_n)_{n \geq 0}$ be an i.i.d. sequence of $\{0,1\}$-valued random variables $X_n \sim \mathrm{Bernoulli}(\frac{1}{2})$, i.e. a sequence of independent tosses of a fair coin. Does there exist a …

Let $(X,d)$ be a compact metric space, and let $\{\mu_n\}_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures on $X$. Fix $L \geq 1$. I will say that $\mu_n$ converges in op …

I have a question that seems very basic, and yet I have not managed to find an answer after probably several hours of Google-searching. Fix $0<a<b<\infty$, and let $\mathcal{P}_{[a,b]}$ be the set of …

It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it wo …