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

TL;DR

The paper studies diffusion processes with drift given by a gradient of a homogeneous potential of degree $1+\gamma$ ($0<\gamma<1$), where the deterministic system exhibits Peano non-uniqueness at the origin. It first establishes a Freidlin–Wentzell-type large deviation principle with rate $\varepsilon^{-2}$ for small noise, then develops a second-order large deviation principle at the slower scale $\varepsilon_\gamma^{-1}$ by connecting the density of the process to a Schrödinger semigroup with potential $V=\tfrac{1}{2}(|\nabla U|^2+\Delta U)$. A refined Carmona–Simon analysis shows the exponential behavior is governed by the ground state $(\lambda_1,\psi_1)$ of the operator $-\mathcal{L}=-\tfrac{1}{2}\Delta+V$, yielding the key limit $\lim_{\varepsilon\to0}\varepsilon_\gamma \log p^\varepsilon(t,x)=-\lambda_1 t - g(x)$ where $g$ solves $\langle\nabla U(x),\nabla g(x)\rangle = -\lambda_1$ with $g(0)=0$. Consequently, the second-order rate function is $I_2(\varphi)=\lambda_1 T+g(\varphi(T))$ for trajectories solving the deterministic ODE, and $\infty$ otherwise; this implies the fluid limit concentrates on the set of extremal deterministic solutions. The work extends prior 1D results to higher dimensions, clarifies the spectral mechanism driving large deviations in Peano-type systems, and links probabilistic selection under vanishing noise to the principal eigenpair of a Schrödinger operator.

Abstract

We consider a diffusion equation in $\mathbb{R}^d$ with drift equal to the gradient of a homogeneous potential of degree $1+γ$, with $0<γ<1$, 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 large deviations of first and second order for the system with noise, generalizing previous results for the particular potential $b(x)=x |x|^{γ-1}$. 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 Schrödinger 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.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 2.2: First-order LDP
• Lemma 2.3: Theorem 2.14 from Pappalettera
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof