# Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise

## TL;DR

This paper extends the theory of stochastic differential equations with singular drifts to infinite-dimensional spaces by employing fractal noise and Malliavin calculus, establishing well-posedness without classical PDE or semimartingale methods.

## Contribution

It introduces a novel approach using fractal noise and Malliavin calculus to prove well-posedness of infinite-dimensional SDEs with singular drifts, bypassing traditional techniques.

## Key findings

- Established existence of unique strong solutions in infinite dimensions.
- Demonstrated regularization effect of fractal noise.
- Provided a new framework for analyzing infinite-dimensional SDEs.

## Abstract

In this paper we aim at generalizing the results of A. K. Zvonkin and A. Y. Veretennikov 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-H\"{o}lder 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.