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It is true that there is a universal constant $c$ such that $\mathbb E[f(Z)]\le c\mathbb E[f(Y)]$. The best (smallest) value for $c$ that I can prove now is $c=2.187...$, but expect that it holds for …

No. Suppose that $\Omega$ consists of four points arranged in a square, where adjacent points have distance 1 between them and opposite points have distance 2. Specifically, if the points are labeled …

Here's a proof of the statement for $f=0$, so that $X=W$ is a Wiener process. (The proof with general $f$ is a bit more involved, and I give this further below). I'll base the proof on the following s …

Letting $\mu_{a,\Sigma}$ be the Gaussian measure with covariance matrix $\Sigma$ and mean $a$. Then (double) the variation distance can be written as $$ \left\lVert\mu_{a_1,\Sigma_1}-\mu_{a_2,\Sigma_2 …

The following is a construction of an analytic set $A\subseteq\omega^\omega\times\omega^\omega$ whose vertical sections exclude at most a single point, and which does not admit a Borel uniformization. …

Letting $\mu_n$ be the distribution of a randomly chosen root of a random polynomial $f=c_0+c_1X+\cdots+c_nX^n$ in $\mathbb{C}[X]$ for IID random variables $c_i\in\mathbb{C}$, each chosen with some pr …

This problem reduces quickly to Holder continuity of the operator square root. That is, there exists a $C > 0$ such that $$ \begin{align} \lVert\sqrt{A}-\sqrt{B}\rVert\le C\lVert A-B\rVert^{1/2}&&{\rm …

You could alternatively try defining Gaussian measures as $2$-stable distributions. This does remove any reliance on finite dimensional projections, and even removes reference to topology. Let $V$ be …

Yes, the Bochner integral does agree with the Lebesgue integral of the sample paths of the process. We can prove this in a slightly more general situation than that asked for in the question. For a p …

If $\mathbb{E}\left[\lVert[ X\rVert^d\right]$ is finite then the integral in the question is necessarily finite. As mentioned, this holds whenever $\pi$ is radially decreasing. However, in the general …

As suggested in my comment, here's a simple fact which applies to any probability measure $\mu$ on (the Borel σ-algebra of) a second countable topological space $X$. There is a unique minimal closed s …

The set $Z_{\mathcal{P^\ast}}$ is never compact except in the case where it is a singleton (or empty). This is for the general case with $S=(S^1,S^2,\ldots,S^d)$ being an $\mathbb{R}^d$-valued semimar …

As suggested in the question, $P_t$ need not be a well defined operator on $L^\infty(W,\mu)$. That is, $F$ can be zero $\mu$-almost everywhere, but $P_tF$ is nonzero on a set of positive $\mu$-measure …

We can show that $$ \mathbb{P}\left(\sum_ia_iX_i^2\le\epsilon\sum_ia_i\right)\le\sqrt{e\epsilon} $$ so that the inequality holds with $c=1/2$ and $C=\sqrt{e}$. For $\epsilon\ge1$ the right hand side …

Yes, in this case it is true that $p$ is a proper martingale! Note that your integrand $\exp\left(-\int_0^tr_u du\right)$ is an adapted, continuous, and decreasing process bounded by 1. So, the follow …