On the supercritical fractional diffusion equation with Hardy-type drift

Abstract

We study the heat kernel of the supercritical fractional diffusion equation with the drift in the critical Hölder space. We show that such a drift can have point irregularities strong enough to make the heat kernel vanish at a point for all t > 0.

Stochastic equations with time-dependent singular drift

“…appropriate Hölder continuity of drift b in order to have Hölder continuous solutions [39]. In fact, if 0 < α ≤ 1 and b is not Hölder continuous enough even at a single point, then the heat kernel of (−∆) α 2 + b • ∇ can vanish in the second variable [26]. In the case 1 < α < 2 one observes the same effect for locally unbounded repulsing drift b(x) = −κ|x| −α x [28].…”