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Notation: We denote by $\mathcal M$ the set of vector valued measures on $\mathbb R^d$ whose divergence is a scalar measure (in the weak sense). Definitions: Consider the Beckmann flow minimisation pr …

An individual, henceforth called the runner starts at the center of an open ball $\Omega_r \subset \mathbb R^2$ of radius $r > 1$. At each turn, a vector $x \in S^1$ is chosen uniformly at random, an …

Consider the deterministic controlled system: $$\dot x(t) = Ax(t) + Bu(t), \ t \in [0, T]$$ $$x(0) = x_0$$ where $x: [0, T] \to \mathbb R^n$ is the controlled state process, $A \in \mathbb R^{n \times …

An individual, henceforth called the runner starts at the center of an open two dimensional square $\Omega$ of side length $r \geq 2$. At each turn, a vector $x \in S^1$ is chosen uniformly at random, …

Consider a drunk, blind man starting in the middle of the two dimensional open unit ball. At each turn, the man chooses a direction to move a step of size $\delta > 0$ in. Unfortunately, he is very dr …

We recall that a coupling of probability distributions $\mu_1, \dots, \mu_n$ on $\mathbb R$ is a set of random variables $X_1, \dots, X_n$ defined on the same probability space such that $X_i$ is dist …

Let $N \gg n \geq 2$ be fixed natural numbers. In the Gacha stamp game, players are given an $N \times N$ square grid, with each point occupied by a unique stamp. On every turn, they may choose a subs …

Recall that a coupling of probability measures $\mu_i$ is a set of random variables $X_i$ defined on the same probability space $\Omega$ such that $X_i \sim \mu_i$. Question: Let $\mu_1, \dots, \mu_n$ …

An invisible target on the integer line starts at $0$. On each round it either stays put, moves to the left or moves to the right by $1$ with probability $\frac{1}{3}$ each. You are then asked to gues …

A number $p$ is drawn uniformly at random from $[0, 1]$. You are then given a biased coin that turns up heads with probability $p$, but the number $p$ is not known to you. You start with a total wealt …

$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or lo …

Consider the following game - I draw a number from $[0, 1]$ uniformly, and show it to you. I tell you I am going to draw another $1000$ numbers in sequence, independently and uniformly. Your task is t …

$N \geq 2$ players play a cooperative game on the integers $\mathbb Z$. All of them start from $0$. At each turn, they are simultanously given the same yes or no question to answer. The questions give …

I’m looking for a nice readable introductory text to stochastic control theory. Background wise, I know some general stochastic analysis and deterministic optimal control theory. Some criterion I’m lo …

Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which. He has \$1 with which to bet with. On ea …