Concentration for multidimensional diffusions and their boundary local times

TL;DR

This paper establishes concentration inequalities for multidimensional diffusions and their boundary local times, providing explicit applications to Brownian motions with rank-based drifts, under quadratic transportation cost inequalities.

Contribution

It proves that certain multidimensional semimartingales satisfy Quadratic Transportation Cost Inequalities, leading to new concentration results for process functionals and boundary local times.

Findings

Proves quadratic transportation cost inequalities for multidimensional diffusions.

Derives concentration bounds for Lipschitz functions of process paths.

Provides explicit applications to Brownian motion with rank-based drifts.

Abstract

We prove that probability laws of certain multidimensional semimartingales which includes time-inhomogenous diffusions, under suitable assumptions, satisfy Quadratic Transportation Cost Inequality under the uniform metric. From this we derive concentration properties of Lipschitz functions of process paths that depend on the entire history. In particular, we estimate concentration of boundary local time of reflected Brownian motions on a polyhedral domain. We work out explicit applications of consequences of measure concentration for the case of Brownian motion with rank-based drifts.