Regular occupation measures of Volterra processes

In this work, 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 measure and the self-intersection measure. For the particular case of solutions to a stochastic Volterra equation, we also obtain the regularity of the one-dimensional distributions and study the absolute continuity of the finite-dimensional distributions. Finally, based on the previously shown regularity of the self-intersection measure, we prove the existence, uniqueness and stability of Volterra equations with distributional drifts of "self-intersection" type in terms of corresponding nonlinear Young equations.