Regularising Flows and Singular Fractional SDEs with Superlinear Dissipation

Source abstract

We study stochastic differential equations driven by the fractional Riemann-Liouville process with Hurst parameter $H\in(0,1)$, whose drift is the sum of a possibly distributional vector field $b$ and a smooth, one-sided Lipschitz, dissipative vector field $F$ with coercivity of order $k+1$ and polynomially growing derivatives. If $b \in \mathcal{C}^γ(\mathbb{R}^d;\mathbb{R}^d)$ for some $γ \max\{1-1/(2H),2-1/H\}$, and if $H+1/(k-1)>(1-γ)H/2$, we construct a solution by smooth approximation and prove uniqueness in a class defined through local comparison with the flow generated by the dissipative drift. We also establish quantitative temporal regularity and stability estimates. The proof combines a conditional decomposition of fractional Brownian motion, Malliavin integration by parts, uniform non-degeneracy estimates for the regularising flow, and the stochastic sewing lemma.