4 problems

Let $(M,d, u)$ be the metric-measure space and let $V$ and $V{epsilon}(t)$ be as defined above. Assume that $textup{Vol}(V; alpha1, alpha2)$ and…

Let $K\subset\mathbb{R}^2$ be a compact set of positive capacity, and let a Brownian motion be conditioned on returning to the origin at time $t$. For a disc, let the expected area…

Let $(\xi(s))_{s\geq 0}$ be a standard Brownian motion in $\mathbb{R}^d$, let $r>0$, and let $f:\mathbb{R}_+\to\mathbb{R}^d$ be continuous. Write $\mathcal{B}(x,r)$ for the open ba…

Let $W_g(t)$ denote the Wiener sausage generated up to time $t$ by a function $g$ in ${\mathbb R}^d$, and let $W_0(t)$ be the Wiener sausage generated by the zero function. For a s…