Brownian Motion Subject to Additive Penalizations

Conditioning stochastic processes on rare events is a classic problem, going back at least to Doob’s work in the 1950s. In this talk, we condition a one-dimensional Brownian motion $(B_t)_{t \ge 0}$ on $\int_0^T u(B_t,t) dt$ being unusually small, as $T \to \infty$, for multiple (classes of) penalization functions $u:\mathbb{R}\times[0,\infty) \to [0,\infty)$. We establish the existence of conditioned processes, provide precise descriptions of these processes and discuss some of their properties. We also consider the asymptotic behavior of the underlying conditioning probabilities, as $T \to \infty$. The talk is based on joint works with Frank Aurzada, Martin Kolb, Mikhail Lifshits and Bastien Mallein.