2025/09/16 by Claudio Landim, Landim, Claudio, Jungkyoung Lee +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F10 #60J45 #60J60 #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Probability and Risk Models #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2509.13222
openalex publication_date 2025/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix a smooth Morse function U\colon ℝd→ℝ with finitely many critical points, and consider the solution of the stochastic differential equation d\boldsymbolxε(t)=-∇ U(\boldsymbolxε(t)) dt + √(2ε) d\boldsymbolwt , where (\boldsymbolwt)t≥0 represents a d-dimensional Brownian motion, and ε>0 a small parameter. Denote by P(ℝd) the space of probability measures on ℝd, and by Iε \colon P(ℝd)→[0, ∞] the Donsker--Varadhan level two large deviations rate functional. We express Iε as Iε= ε-1 J(-1) + J(0) + ∑_1≤ p≤ \mathfrakq (1/θ(p)ε) J(p), where J(p)\colon P(ℝd) → [0,+∞] stand for rate functionals independent of ε and θ(p)ε for sequences such that θ(1)ε→∞, θ(p)ε/ θ(p+1)ε→ 0 for 1≤ p< \mathfrakq. The speeds θ(p)ε correspond to the time-scales at which the diffusion \boldsymbolxε(⋅) exhibits a metastable behaviour, while the functional J(p) represent the level two, large deviations rate functionals of the finite-state, continuous-time Markov chains which describe the evolution of the diffusion \boldsymbolxε(⋅) among the wells in the time-scale θ(p)ε.