Regular occupation measures of Volterra processes

Abstract: We give a local non-determinism condition applicable to general Volterra Ito processes that allow us to obtain the space-time regularity of the occupation and self-intersection measures. For the particular case of solutions to a stochastic Volterra equation, we also obtain the one-dimensional distributions' regularity and study the finite-dimensional distributions' absolute continuity. Finally, based on the previously shown regularity of the self-intersection measure, we prove the existence, uniqueness and stability of stochastic equations with distributional drifts of "self-intersection" type in terms of corresponding two-parameter nonlinear Young equations.