On low frequency inference for diffusions without the hot spots conjecture

TL;DR

This paper establishes minimax convergence rates for estimating the transition operator of diffusions on convex domains without relying on the hot-spots conjecture, and proves a Lipschitz stability estimate for the inverse problem.

Contribution

It removes the hot-spots conjecture assumption from key theorems, providing new minimax rates and stability results for diffusion inverse problems on convex domains.

Findings

Minimax convergence rates characterized for transition operator estimation.

Lipschitz stability estimate proven for the inverse map from operators to diffusion coefficients.

Results hold on arbitrary convex domains with smooth boundaries.

Abstract

We remove the dependence on the `hot-spots' conjecture in two of the main theorems of the recent paper of Nickl (2024, Annals of Statistics). Specifically, we characterise the minimax convergence rates for estimation of the transition operator $P_{f}$ arising from the Neumann Laplacian with diffusion coefficient $f$ on arbitrary convex domains with smooth boundary, and further show that a general Lipschitz stability estimate holds for the inverse map $P_f\mapsto f$ from $H^2\to H^2$ to $L^1$.