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 Mytnik, Perkins and Sturm (2006) that relate the Hölder coefficient with the singularity of the correlation kernel of the noise. For this we use new ideas of Mytnik and Perkins (2011) 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.
"New results on pathwise uniqueness for the heat equation with colored noise." Electron. J. Probab. 18 1 - 46, 2013. https://doi.org/10.1214/EJP.v18-2506