Stochastic differential equations with coefficients in Sobolev spaces

2010/01/18 by Shizan Fang, Fang, Shizan, Dejun Luo +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #34F05 (Secondary) #37C10 #60H10 (Primary) #60J60 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1001.3007

openalex publication_date 2010/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider Itô SDE \d X_t=∑_j=1^m A_j(X_t) \d w_t^j + A₀(X_t) \d t on \R^d. The diffusion coefficients A₁,..., A_m are supposed to be in the Sobolev space W_loc^1,p (\R^d) with p>d, and to have linear growth; for the drift coefficient A₀, we consider two cases: (i) A₀ is continuous whose distributional divergence δ(A₀) w.r.t. the Gaussian measure γ_d exists, (ii) A₀ has the Sobolev regularity W_loc^1,p' for some p'>1. Assume ∫_\R^d exp[λ₀(|δ(A₀)| + ∑_j=1^m (|δ(A_j)|² +|∇ A_j|²))] \dγ_d0, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X_t)_# γ_d admits a density with respect to γ_d. In particular, if the coefficients are bounded Lipschitz continuous, then X_t leaves the Lebesgue measure \Leb_d quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps.

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