Coffee-Break Problem, 28/11/2022

Consider the maximisation problem

\lambda = \max \left(\int_{\mathbb{R}\times\mathbb{R}} |v-w| \mu(dv)\mu(dw): \mu \text{ is a probability measure on }\mathbb{R}, \text{ with }\int_{\mathbb{R}} v\mu(dv)=0\text{ and } \int_{\mathbb{R}}|v|^2 \mu(dv)=1\right).

Explain why the maximum is obtained. Using the Ornstein-Uhlenbeck semigroup, show that any maximisers \mu admit a density \mu(dx)=f(x)dx, f\in L^2(\mathbb{R}) and enjoy a bound on \|f\|_{L^2(\mathbb{R})} .

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Coffee-Break Problem, 19/09/2022

Let (B^n_t, t\ge 0), n\in \mathbb{N} be a sequence of independent, one-dimensional Brownian motions, starting from deterministic starting points B^n_0=x_n \in \mathbb{R} .

Suppose that x_n \to \infty and consider the (unnormalised) empirical measure \mu_t:= \sum_{n\in \mathbb{N}} \delta_{B^n_t} . Give an example to show that \mu_t need not be a locally finite measure.

Harder: prove the existence of a nonrandom t_\text{c}\in [0, \infty] such that \mu_t is, with probability 1, locally finite when t<t_c , while \mathbb{P}(\exists t>t_c: \mu_t \text{ locally finite})=0 . What about the critical value t=t_c ?

Coffee-Break Problem, 20/06/2022

Let \mu_n be a sequence of (Borel) probability measures on a compact metric space X, which converge weakly to a point mass \mu=\delta_{x_0} for some x_0 \in X .

Suppose further that f_n: X\to [0,\infty) are probability density functions with respect to \mu_n , i.e. that \int_X f_n d\mu_n=1 for every n , and are such that f_n(x_0)=\sup_x f_n(x) for every n .

Must it be true that the probability measures \nu_n:=f_n \mu_n converge weakly to \delta_{x_0} ? What about if we replace the hypothesis of weak convergence of \mu_n by total variation convergence?