Pathwise uniqueness of the stochastic heat equation with spatially inhomogeneous white noise

We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is Hölder continuous of index γ>1−η2(η+1). Here η∈(0,1) is a constant that defines the spatial regularity of the noise