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

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.