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

David Criens () and Moritz Ritter ()
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David Criens: Albert–Ludwigs–University of Freiburg
Moritz Ritter: Albert–Ludwigs–University of Freiburg

Journal of Theoretical Probability, 2022, vol. 35, issue 3, 2052-2067

Abstract: Abstract We consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by a cylindrical Brownian motion. The solutions are allowed to take values in Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny’s and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada–Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.

Keywords: Stochastic partial differential equation; Martingale problem; Weak solution; Mild solution; Dual Yamada–Watanabe theorem; Weak uniqueness; Joint weak uniqueness; Pathwise uniqueness; Strong uniqueness; 60H15; 60G44; 60H05 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01107-3

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