Title: Large deviations of Brownian occupation measures with large intersections

Abstract: In probability theory, large deviation theory refers to the study of events that occur with exponentially decaying probability. In this talk, we give an introduction to large deviations with a focus on the occupation measures of Brownian motions. Based on this framework, 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.