Large deviations of Brownian intersections

In this talk, we consider the upper-tail large deviations of intersections of Brownian motions. Apart from intrinsic interest, these quantities are also closely related to the parabolic Anderson model via the Feynman-Kac formula. 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. The key tools are a compactification of the weak topology and an exponentially good approximation of the Brownian occupation measure.