# Seminar za analizu, 26. maj 2022.

Naredni sastanak Seminara biće održan onlajn u četvrtak, 26. maja 2022, sa početkom u 15 časova.

Predavač: Fanuel Mariano

Naslov predavanja: SPECTRAL BOUNDS FOR EXIT TIME MOMENTS OF BROWNIAN MOTION AND RELATED RESULTS

Apstrakt: Consider a drumhead $D\subset\mathbb{R}^{d}$. The first Dirichlet eigenvalue for the negative Laplacian acting on $L^{2}\left(D\right)$ has a property of domain monotonicity that roughly says that ''larger drums have lower fundamental frequencies''. While if we consider Brownian motion started inside the domain $D$, domain monotonicity roughly says that ''larger domains have larger mean exit times''. We characterize this inverse behavior by finding sharp uniform bounds on a domain functional $G_{p,d}\left(D\right)$ involving the bottom of the spectrum of $D$ and the $p-$moments of exit times of Brownian motion. We prove a sharp lower bound and an asymptotically sharp upper bound that depends only on the dimension $d$. We also provide the existence of a maximizer for this functional in the class of convex domains using a probabilistic proof. We also develop new techniques to prove similar results on the general setting of metric measure Dirichlet spaces. Moreover, these results have had applications to many interesting problems such as the Hot Spots constant in the Hot  Spots Conjecture.

This talk is based on joint work with Rodrigo Bañuelos and Jing Wang.

https://us02web.zoom.us/j/6636215428?pwd=bllJSTUvZ0E5MjhkMkxYa3RYYzZpQT09
Meeting ID: 663 621 5428
Passcode: 7h6KR1

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