# 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 $\|f\|_{L^2(\mathbb{R})}$.

# 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, while $\mathbb{P}(\exists t>t_c: \mu_t \text{ locally finite})=0$. What about the critical value $t=t_c$?

# Coffee-Break Problem, 05/09/2022

Fix finite sets $S, A$, together with probability measures $\mu\in \mathcal{P}(S), \nu_1, \nu_2\in \mathcal{P}(A)$, together with a map $\Xi: S\times A\times A\to S$.

I now take a random variable $X_0\sim \mu$ on $S$ and independent samples $\Theta_1\sim \nu_1, \Theta_2\sim \nu_2$. I will now generate two new random variables by

$X_1:=\Xi(X_0, \Theta_1, \theta_2);\qquad X_1' :=\Xi(X_0, \Theta_1, \Theta_2)$

where $\theta_2 \in A$ is nonrandom. Is it true that the entropies satisfy $H(X_1')\le H(X_1)$? What about for `typical’ $\theta_2\in A$?

# Coffee-Break Problem, 29/08/2022

Suppose that $u_N \in L^1([0,1])$ and $u\in L^1([0,1])$ are such that $u_N \rightarrow u$ in the weak topology of $L^1([0,1])$, that is, for all $f\in L^\infty([0,1])$, it holds that $\int_0^1 f(x)u_N(x) dx \to \int_0^1 f(x)u(x) dx$.

Is it true that, for any $\lambda \in \mathbb{R}$, the functions $u_N^\lambda(x):=\max(u_N(x),\lambda)$ converge weakly to $u^\lambda(x):=\max(u(x), \lambda)$? What about for almost all $\lambda \in \mathbb{R}$? What about if we replaced $L^1([0,1])$ by $\ell^1(\mathbb{N})$ or by $L^p([0,1])$ with $p>1$?

# Coffee-Break Problem, 15/08/2022

It is well-known that, for $\alpha>1$, all functions $f: [0,1]\to \mathbb{R}$ which are Hölder-continuous of exponent $\alpha$ are constant. What about functions $f:C\to \mathbb{R}$ defined on the Cantor set $C$?

# Coffee-Break Problem, 18/07/2022

Let $u\in C^\infty(\mathbb{R}, (0,\infty))$ be a smooth, compactly supported function. Suppose we have estimates, for some $p>1$,

$\int_{\mathbb{R}} |\partial_x(u^p)|^2 dx \le C_1, \qquad \int_{\mathbb{R}} u dx \le C_2$.

Show that these estimates imply an estimate on $\int_\mathbb{R} u^q dx$ for some $q>p$, where the upper bound depends on $C_1, C_2$ and one other quantity, but no other norms of $u$. What other quantity is necessary?

# Coffee-Break Problem, 11/07/2022

Let $X_\lambda$ be a Poisson random variable with mean $\lambda$, and fix $\alpha>0$. What are the asymptotics of

$F_\alpha(\lambda):=\mathbb{E}((X_\lambda!)^{-\alpha})?$

# Coffee-Break Problem, 04/07/2022

Let $b, \sigma: \mathbb{R}\to \mathbb{R}$ be Lipschitz functions. For every $x\in \mathbb{R}$, consider the solution $X^x_t$ to

$dX^x_t=b(X^x_t)dt+ \sigma(X^x_t)dB^x_t, \qquad X^x_0=x$

for a Brownian motion $B^x_t$. Must the two-point process $(X^x_t, Y^x_t)$ be a Markov process, for any $x, y$?

# Coffee-Break Problem, 27/06/2022

Let $X_t$ be a continuous-time Markov chain on a finite state space $S$. Let $f:S\to S'$ be a function to another set; must $Y_t:= f(X_t)$ be a Markov chain? What about the jump process $N^X_t \in \{0,1,2,\dots\}$ which counts the jumps of $X_t$?

# 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?