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I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of generalisation of expr …

I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these. If I understand correctly, mathematicians tend to be quite happy working with …

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 …

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, …

Apologies if this question is too basic for MO. I think it should be the case that for any decreasing $f \colon [A,\infty) \to [0,\infty)$ and $k \geq 0$, if $\int_A^\infty f(x) e^{kx} \, dx < \infty …

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 …

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a diffeomorphism, and let $\mu$ be a probability measure on $M$ with compact support. As stated in the questio …

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 …

Let $V$ be a normed vector space, let $l \in V$, and let $(a_n)$ be a sequence in $V$. We say that $a_n$ is Cesàro-convergent to $l$ if $\frac{1}{n}\sum_{i=1}^n a_i \to l$ as $n\to\infty$. Now I will …

All the notions of convergence of measures that I know of are either in the purely measure-theoretic category (e.g. strong convergence, total variation), or in the topological category (e.g. weak conv …

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support. Definition. The …

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 …

Given any $g \in L^\infty(\mathbb{R})$, we define the associated multiplier operator $T_g \colon L^2(\mathbb{R}) \to L^2(\mathbb{R})$ by $$ \mathcal{F}(T_g f) \ = \ g.\mathcal{F}f $$ where $\mathcal{F …