Zero-Noise Limit for High-Dimensional ODE with Measurable Drift

TL;DR

This paper investigates the zero-noise limit of high-dimensional stochastic differential equations with measurable, bounded drift, revealing that the limit distribution is supported on instantaneous escape Filippov solutions and is singular with respect to Lebesgue measure.

Contribution

It provides a comprehensive analysis combining probabilistic, geometric, and differential inclusion methods to characterize the zero-noise limit in systems with non-unique ODE solutions.

Findings

Support of the limit measure is the closure of instantaneous escape solutions.

The limit measure is singular and supported on a set with Hausdorff dimension less than the ambient space.

Uniform weak convergence holds under small drift perturbations.

Abstract

This paper studies the zero-noise limit of high-dimensional small-noise diffusion processes governed by the stochastic differential equation (SDE): \[ dX_{t}^{\varepsilon }=b(X_{t}^{\varepsilon })\,dt+\varepsilon \,dW_{t}, \quad X_{0}^{\varepsilon }=0, \quad \varepsilon >0, \] where drift $b$ is measurable and bounded. The associated ordinary differential equation (ODE) $\dot{x}_{t}=b(x_{t})$ may have multiple Filippov solutions due to lack of Lipschitz continuity, while non-degenerate additive noise ensures unique strong solutions for each $\varepsilon >0$. Integrating the Stroock-Varadhan support theorem, comparison theorem for diffusion processes, law of the iterated logarithm (LIL) for Brownian motion, and Hausdorff dimension from geometric measure theory, we analyze the weak limit distribution $\mu ^{0}=\lim_{\varepsilon \rightarrow 0}\mathcal{L}(X_{t}^{\varepsilon })$. We find instantaneous escape Filippov solutions dominate the zero-noise limit, with the support of $\mu ^{0}$ being the closure of points reached by these solutions at fixed $t$ (delayed solutions are geometrically negligible). The comparison theorem verifies uniform weak convergence under small drift perturbations; LIL quantifies $X_{t}^{\varepsilon }$ fluctuations as $\varepsilon \rightarrow 0$; Hausdorff dimension analysis shows the support has dimension strictly less than ambient space dimension $d$, making $\mu ^{0}$ singular with respect to the Lebesgue measure. The compact support set's structure depends only on drift dynamics and instantaneous escape solutions, not Brownian motion or $d$. Our work unifies probabilistic limit theory, geometric measure theory, ODE non-uniqueness and differential inclusion theory, providing a comprehensive framework for high-dimensional non-unique systems' zero-noise limit and new insights into singular limit distributions in stochastic analysis.