Distribution dependent SDEs driven by additive fractional Brownian motion

Abstract

We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter Hin (0,1). We establish strong well-posedness under a variety of assumptions on the drift; these include the choice B(·,μ)=(f∗μ)(·)+g(·),f,g∈B∞,∞α,α>1-12H,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B(\\cdot ,\\mu )=(f*\\mu )(\\cdot ) + g(\\cdot ), \\quad f,\\,g\\in B^\\alpha _{\\infty ,\\infty },\\quad \\alpha >1-\\frac{1}{2H}, \\end{aligned}$$\\end{document}thus extending the results by Catellier and Gubinelli (Stochast Process Appl 126(8):2323–2366, 2016) to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances.