Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

Abstract

In this paper we aim at generalizing the results of A. K. Zvonkin (That removes the drift, 22, 129, 41) and A. Y. Veretennikov (Theory Probab. Appl., 24, 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.