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 [41] and A. Y. Veretennikov [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.

McKean-Vlasov equations on infinite-dimensional Hilbert spaces with irregular drift and additive fractional noise

“…Since X is as a weak solution to MKV equation ( 2) also a weak solution of SDE (3), it is sufficient to show that every weak solution Y of SDE (3) is a strong solution. Furthermore, if MKV equation ( 2) has a weakly unique solution, the associated SDE (3) is uniquely determined and consequently, pathwise uniqueness of the solution Y of SDE (3) implies pathwise uniqueness of the solution X of MKV equation (2). Thus, applying existence results on SDEs as for example stated in [2], [23], [26], and [29], yields existence of a (pathwisely unique) strong solution of MKV equation (2).…”

“…Note that Hurst parameters in the entire range (0, 1) are admitted, and we introduce the following partition: I − := {k : H k ∈ (0, 1/2)}, I 0 := {k : H k = 1/2}, and I + := {k : H k ∈ (1/2, 1)}. The main objective of this paper is to study existence and uniqueness of a solution to the infinite-dimensional MKV equation (2) for irregular drift coefficients b.…”

“…) is an arbitrary measure process continuous with respect to time. Afterwards Schauder's fixed point theorem [28] is applied to the function ϕ(µ) = P X µ t to show the existence of a fixed point and in particular, to conclude existence of a weak solution to MKV equation (2).…”

“…where b P X (t, y) := b (t, y, P Xt ) and X is a weak solution of MKV equation (2). In order to show that (2) has a strong solution, it suffices to show that there exists a weak solution that is measurable with respect to the filtration generated by the driving noise B.…”

“…Furthermore, if MKV equation ( 2) has a weakly unique solution, the associated SDE (3) is uniquely determined and consequently, pathwise uniqueness of the solution Y of SDE (3) implies pathwise uniqueness of the solution X of MKV equation (2). Thus, applying existence results on SDEs as for example stated in [2], [23], [26], and [29], yields existence of a (pathwisely unique) strong solution of MKV equation (2). Analogously, Malliavin differentiability of the solution to MKV equation ( 2) is deduced from results on SDEs, cf.…”