On Cherny’s results in infinite dimensions: a theorem dual to Yamada–Watanabe

Abstract

We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of the formand show that for such equations uniqueness in law is equivalent to joint uniqueness in law. Here W is a cylindrical Wiener process in a separable Hilbert space U and the equation is considered in a Gelfand triple V ⊆ H ⊆ E, where H is some separable (infinite-dimensional) Hilbert space. This generalizes the corresponding results of A. Cherny for the case of finite-dimensional equations.

A dual Yamada–Watanabe theorem for Lévy driven stochastic differential equations

“…More recently, the dual theorem has been generalized to various infinite-dimensional Brownian frameworks, see [19,20,21]. A version for SDEs with Lévy drivers seems to be missing in the literature.…”

On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

“…More recently, Cherny's result and the dual Yamada-Watanabe theorem have been generalized to several infinite dimensional frameworks. In [15,21] the theorems were established for mild solutions to semilinear stochastic partial differential equations (SPDEs) and in [17,18] for the variational framework.…”

“…The basic strategy of our proof, which is borrowed from the finite dimensional case and also used in [15,17,18] 1 , is to construct an infinite dimensional Brownian motion V , independent of X , such that the noise W can be recovered from the solution process X and V . The technical challenge in this argument is the proof for the independence of X and V .…”

“…Cherny's proof for this used additional randomness, an enlargement of filtration and a conditioning argument. In [15,17,18] these ideas have been adapted to the respective infinite dimensional frameworks. Our approach is different and appears to us more straightforward and less technical.…”

“…The proof in[21] is indirect in the sense that it uses the method of the moving frame to transfer results from[18] for infinite dimensional SDEs to SPDEs.…”