2012/02/19 by Liangquan Zhang, Zhang, Liangquan
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1202.4131
openalex publication_date 2012/02/19 · openalex created_date 2024/03/30 · openalex updated_date 2026/07/28
This paper studies the zero-noise limit of high-dimensional small-noise diffusion processes governed by the stochastic differential equation (SDE): dXtε =b(Xtε ) dt+ε dWt, X0ε =0, ε gt;0, where drift b is measurable and bounded. The associated ordinary differential equation (ODE) xt=b(xt) may have multiple Filippov solutions due to lack of Lipschitz continuity, while non-degenerate additive noise ensures unique strong solutions for each ε >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 μ0=limε → 0L(Xtε ). We find instantaneous escape Filippov solutions dominate the zero-noise limit, with the support of μ0 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 Xtε fluctuations as ε → 0; Hausdorff dimension analysis shows the support has dimension strictly less than ambient space dimension d, making μ0 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.