First, we will consider the SDE dX_t = b(t,X_t) dt + d B_t, where b is a singular drift (e.g. a distribution) and B is a fractional Brownian motion. We will review some recent results on existence and uniqueness for this equation, providing criteria linking the regularity of b and the Hurst parameter H of the fractional Brownian motion. Next, we will study the time-space regularity of the conditional density of the solution in Lebesgue-Besov spaces, and also provide Gaussian bounds. Then by exploiting this regularity, we will demonstrate the existence of solutions for McKean-Vlasov equations of the form dY_t = μ_t ∗ b(t, Y_t) + dB_t, where μt is the law of the solution Y_t, for a drift b that can be more singular than in the linear case, and chosen in the full sub-critical regime of such SDEs. Finally, we discuss uniqueness for this singular McKean-Vlasov equation.
Joint work with L. Anzeletti, L. Galeati and E. Tanré. Abstract
Regularity of the density of SDEs driven by fractional noise and application to McKean-Vlasov equations
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