The Bethe ansatz for sticky Brownian motions

Abstract

We consider a multi-dimensional diffusion whose coordinates behave as one-dimensional Brownian motions, evolving independently when apart, but with a sticky interaction when they coincide. We derive the Kolmogorov backwards equation and show that for a specific choice of interaction it can be solved exactly with the Bethe ansatz. The diffusion in Rn can be viewed as the n-point motions of a stochastic flow of kernels. We use our formulae to study the flow of kernels and show that atoms in the flow are asymptotically exponentially distributed in size at large times.