Functional Inequalities for doubly weighted Brownian Motion with Sticky-Reflecting Boundary Diffusion

TL;DR

This paper establishes bounds for functional inequalities related to doubly weighted Brownian motion with sticky-reflecting boundaries on manifolds, using curvature assumptions and interpolation techniques.

Contribution

It introduces new bounds for Poincaré and Logarithmic Sobolev constants, and provides eigenvalue estimates for weighted Brownian motion with sticky reflection.

Findings

Upper bounds for Poincaré and Logarithmic Sobolev constants.

Lower bound on the first nontrivial doubly weighted Steklov eigenvalue.

Upper bound on the boundary trace operator norm.

Abstract

We give upper bounds for the Poincar\'e and Logarithmic Sobolev constants for doubly weighted Brownian motion on manifolds with sticky reflecting boundary diffusion under curvature assumptions on the manifold and its boundary. We therefor use an interpolation approach based on energy interactions between the boundary and the interior of the manifold and the weighted Reilly formula. Along the way we also obtain a lower bound on the first nontrivial doubly weighted Steklov eigenvalue and an upper bound on the norm of the doubly weighted boundary trace operator on Sobolev functions. We also consider the case of weighted Brownian motion with pure sticky reflection.