For SPDEs with cylindrical Wiener noise in Banach spaces, weak existence plus pathwise uniqueness is equivalent to strong existence plus joint weak uniqueness under flexible path-space and integrability assumptions.
An extension of Skorohod’s almost sure representation theorem
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The Yamada-Watanabe-Engelbert theorem for SPDEs in Banach spaces
For SPDEs with cylindrical Wiener noise in Banach spaces, weak existence plus pathwise uniqueness is equivalent to strong existence plus joint weak uniqueness under flexible path-space and integrability assumptions.