2024/05/22 by Oleg Butkovsky, Butkovsky, Oleg, Leonid Mytnik +1 · 3 citations
Economics, Econometrics and Finance · Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2405.13780
openalex publication_date 2024/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We put forward a new method for proving weak uniqueness of stochastic equations with singular drifts driven by a non-Markov or infinite-dimensional noise. We apply our method to study stochastic heat equation (SHE) driven by Gaussian space-time white noise (∂)/(∂ t) ut(x)=\frac12 (∂2)/(∂ x2)ut(x)+b(ut(x))+Wt(x), tgt;0, x∈ D⊂ℝ, and multidimensional stochastic differential equation (SDE) driven by fractional Brownian motion with the Hurst index H∈(0,1/2) d Xt=b(Xt) dt +d BtH, tgt;0. In both cases b is a generalized function in the Besov space Bα∞,∞, α<0. Well-known pathwise uniqueness results for these equations do not cover the entire range of the parameter α, for which weak existence holds. What happens in the range where weak existence holds but pathwise uniqueness is unknown has been an open problem. We settle this problem and show that for SHE weak uniqueness holds for α>-3/2, and for SDE it holds for α>1/2-1/(2H); thus, in both cases, it holds in the entire desired range of values of α. This extends seminal results of Catellier and Gubinelli (2016) and Gyöngy and Pardoux (1993) to the weak well-posedness setting. To establish these results, we develop a new strategy, combining ideas from ergodic theory (generalized couplings of Hairer-Mattingly-Kulik-Scheutzow) with stochastic sewing of Lê.