Resolvent Estimates for equations with distributional drift

Abstract: It is well known that ordinary differential equations (ODEs) regularize when an additional forcing by Brownian motion is added. Indeed, an ODE driven by a function, \(b\), might have multiple solutions or no solutions when \(b\) is a bounded measurable function. However, once the random forcing by Brownian motion \((B_t)_{t\ge 0}\) is added, the corresponding stochastic differential equation will have a unique strong solution even for bounded measurable \(b\) without any additional assumptions on continuity. This phenomenon is called in the literature ``regularization by noise''. For more general forcing and for equations driven by distributions one needs precise regularity estimates on the associated resolvents to make the above program precise.