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New results on pathwise uniqueness for the heat equation with colored noise. (English) Zbl 1411.60100

Summary: We consider strong uniqueness and thus also existence of strong solutions for the stochastic heat equation with a multiplicative colored noise term. Here, the noise is white in time and colored in \(q\) dimensional space (\(q \geq 1\)) with a singular correlation kernel. The noise coefficient is Hölder continuous in the solution. We discuss improvements of the sufficient conditions obtained in [L. Mytnik et al., Ann. Probab. 34, No. 5, 1910–1959 (2006; Zbl 1108.60057)] that relate the Hölder coefficient with the singularity of the correlation kernel of the noise. For this we use new ideas of [L. Mytnik and E. Perkins, Probab. Theory Relat. Fields 149, No. 1–2, 1–96 (2011; Zbl 1233.60039)] who treat the case of strong uniqueness for the stochastic heat equation with multiplicative white noise in one dimension. Our main result on pathwise uniqueness confirms a conjecture that was put forward in their paper.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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