A functional Law of the Iterated Logarithm for weakly hypoelliptic diffusions at time zero

TL;DR

This paper establishes a functional law of the iterated logarithm for weakly hypoelliptic diffusions at time zero, revealing almost sure behaviors even in degenerate noise settings using large deviations and control theory.

Contribution

It introduces a new functional LIL for SDEs with weak Hörmander conditions, extending understanding of their small-time behavior and regularity criteria.

Findings

Derived the functional LIL for weakly hypoelliptic diffusions.

Identified proper rescaling techniques for concrete examples.

Provided control-theoretic criteria for regular points.

Abstract

We study the almost sure behavior of solutions of stochastic differential equations (SDEs) as time goes to zero. Our main general result establishes a functional law of the iterated logarithm (LIL) that applies in the setting of SDEs with degenerate noise satisfying the weak Hormander condition but not the strong Hormander condition}. That is, SDEs in which the drift terms must be used in order to conclude hypoellipticity. As a corollary of this result, we obtain the almost sure behavior as time goes to zero of a given direction in the equation, even if noise is not present explicitly in that direction. The techniques used to prove the main results are based on large deviations applied to a non-trivial rescaling of the original system. In concrete examples, we show how to find the proper rescaling to obtain the functional LIL. Furthermore, we apply the main results to the problem of identifying regular points for hypoelliptic diffusions. Consequently, we obtain a control-theoretic criteria for a given point to be regular for the process.