Large Deviations and the Peano Phenomenon in Stochastic Differential Equations with Homogeneous Drift

TL;DR

This paper investigates large deviations in stochastic differential equations with non-Lipschitz homogeneous drift, revealing how the process converges to extreme solutions of the associated deterministic system, and explicitly characterizing the rate function.

Contribution

It extends large deviation analysis to systems with non-Lipschitz potentials exhibiting the Peano phenomenon, providing explicit rate functions and convergence results.

Findings

First-order large deviations match Freidlin-Wentzell rate function.

Second-order deviations depend on the ground state of a Schrödinger operator.

Diffusions converge to extreme solutions of the deterministic system.

Abstract

We consider a diffusion equation in $\mathbb{R}^d$ with drift equal to the gradient of a homogeneous potential of degree $1+\gamma$, with $0<\gamma<1$, and local variance equal to $\varepsilon^2$ with $\varepsilon\to 0$. The associated deterministic system for $\varepsilon=0$ has a potential that is not a Lipschitz function at the origin. Therefore, an infinite number of solutions exist, known as the Peano phenomenon. In this work, we study first- and second-order large deviations for a noisy system, generalizing previous results for the specific potential $b(x)=x |x|^{\gamma-1}$. For the first-order large deviations, we recover the rate function from the well-known Freidlin-Wentzell work. For the second-order large deviation, we use a refinement of Carmona-Simon bounds for the eigenfunctions of a Schr\"odinger operator and prove that the exponential behavior of the process depends only on the ground state of such an operator. Moreover, a refined study of the ground state allows us to obtain the large deviation rate function explicitly and to deduce that the family of diffusions converges to the set of extreme solutions of the deterministic system.