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I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to f …

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 …

Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel …

By way of introduction: As expressed in some of the comments, I find the "locally compact" assumption possibly a bit too strong. A weaker assumption than having a locally compact second-countable Ha …

Let $\mathbf{\Omega}=(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space satisfying the Usual Conditions. Let $P \colon [0,\infty) \times \mathbb{R} \t …

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" …

I've found the answer - it's NO! The paper I found addressing the question is the following: http://projecteuclid.org/euclid.aop/1175287757 ("0-1 Laws for Regular Conditional Probabilities") A simple …

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 …

I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order: The answer to Q2 is yes; the structure of the proo …

Given a compact metrisable topological space $X$, we write $\mathcal{N}(X)$ for the set of non-empty closed nowhere dense subsets of $X$, which is a Polish space under the topology induced by the Haus …

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 $(M_n)_{n \geq 1}$ be a uniformly bounded martingale over a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Define the probability measure $\mu$ on $\mathbb{R}^\mathbb{N}$ to be the law 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, …

I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before. So I'm wondering if there are any papers/text …