We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on ℝd for d≥1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Hölder continuity of the solutions near zero is established.
"On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients." Ann. Probab. 34 (5) 1910 - 1959, September 2006. https://doi.org/10.1214/009117906000000331