Regular Occupation Measures of Volterra Processes

Abstract We introduce a local non-determinism condition for Volterra Itô processes that captures smoothing properties of possibly degenerate noise. By combining the stochastic sewing lemma with one-step Euler approximations, we first prove the joint space-time regularity for their occupation measure, self-intersection measure, and time marginals for such Volterra Itô processes. As an application, we obtain the space-time regularity of local times and self-intersection times for rough perturbations of Gaussian Volterra processes, and construct a class of non-Gaussian Volterra Iô processes that are $$C^{\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> -regularising. Secondly, for the particular class of stochastic Volterra equations with Hölder continuous coefficients, using disintegration of measures for their Markovian lifts, we further establish the absolute continuity of finite-dimensional distributions. Finally, we prove the existence, uniqueness, and stability for self-interacting stochastic equations with distributional drifts.