We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph…

We consider a continuous time Markov process $(X,L)$, where $X$ jumps between a finite number of states and $L$ is a piecewise linear process with state space $\mathbb{R}^d$. The process $L$ represents an "inert drift" or "reinforcement." We find sufficient…

We construct martingales whose 1-dimensional marginals are those of a centered self-decomposable variable multiplied by some power of time $t$. Many examples involving quadratic functionals of Bessel processes are discussed