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We study the solutions of the stochastic heat equation driven by spatiallyinhomogeneous multiplicative white noise based on a fractal measure. We provepathwise uniqueness for solutions of this equation when the noise coefficientis H\"{o}lder continuous of index $\gamma>1-\frac{\eta}{2(\eta+1)}$. Here$\eta\in(0,1)$ is a constant that defines the spatial regularity of the noise.

Pathwise Uniqueness of the Stochastic Heat Equations with Spatially Inhomogeneous White Noise