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.
Nonlinear parabolic stochastic evolution equations in critical spaces part I. Stochastic maximal regularity and local existence
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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.