This talk is based on the paper [1]. The main focus is pathwise uniqueness for mild solutions to stochastic PDEs with drift given in differential form. The singularity of the drift perturbation allows to achieve novel pathwise uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models, previously studied only in the context of weak uniqueness, see [2,4]. Finally, the technique introduced here also yields significant improvements over the results already known in the non-singular case, see [3].

References:

[1]D. Addona, D. A. Bignamini, C. Orrieri, L. Scarpa, Pathwise uniqueness by noise for singular stochastic PDEs, e-print arXiv:2512.17736, 2025.

[2] Bertacco F., Orrieri C., Scarpa L., Weak uniqueness by noise for singular stochastic PDES, Transactions of the American Mathematical Society 378, 7977-8023 (2025).

[3]G. Da Prato, F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, Journal of Functional Analysis 259, 243-267 (2010).

[4]E. Priola, An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs, Annals of Probability 49, 1310–1346 (2021).