Asymptotics of Brownian occupation measures with unusually large intersections

TL;DR

This paper establishes the asymptotic behavior of Brownian occupation measures conditioned on large intersections, showing convergence to a measure related to the Gagliardo-Nirenberg inequality and deriving large deviation principles.

Contribution

It introduces a large deviation principle for Brownian occupation measures conditioned on large intersections, generalizing previous work and connecting to optimization problems.

Findings

Weak convergence of conditioned occupation measures to a measure with a Gagliardo-Nirenberg optimizer density

A compact large deviation principle for unconditioned Brownian occupation measures

LDP for intersection measures of independent Brownian motions

Abstract

We prove that the occupation measures of Brownian motions conditioned to have large intersections converge weakly, up to spatial shifts, to the measure whose density is the square of an optimizer of the Gagliardo-Nirenberg inequality. We do so by proving a large deviation principle (LDP) for Brownian occupation measures conditioned either on large self-intersections or large mutual intersections. To this end, we derive a compact LDP for unconditioned Brownian occupation measures, generalizing the work of Mukherjee and Varadhan. We also prove the LDP for Brownian occupation measures tilted by their intersections in the same topology. A key tool of independent interest is an exponentially good approximation of the intersection measure tested against all bounded measurable functions, from which we further get the LDP for the intersection measure of independent Brownian motions.