Small-time asymptotics for hypoelliptic diffusions

TL;DR

This paper develops an inductive method to analyze the small-time behavior of hypoelliptic diffusions with polynomial drift and degenerate noise, linking asymptotics to control theory and boundary regularity criteria.

Contribution

It introduces a novel inductive procedure for asymptotic analysis of hypoelliptic diffusions, connecting geometric control problems to boundary regularity conditions.

Findings

Derived two rescaling limits: law of the iterated logarithm and distributional.

Solved related control problems in nontrivial examples.

Provided practical criteria for boundary regularity in hypoelliptic diffusions.

Abstract

An inductive procedure is developed to calculate the asymptotic behavior at time zero of a diffusion with polynomial drift and degenerate, additive noise. The procedure gives rise to two different rescalings of the process; namely, a functional law of the iterated logarithm rescaling and a distributional rescaling. The limiting behavior of these rescalings is studied, resulting in two related control problems which are solved in nontrivial examples using methods from geometric control theory. The control information from these problems gives rise to a practical criteria for points to be regular on the boundary of a domain in $\mathbf{R}^n$ for such diffusions.