Pathwise Uniqueness for the Stochastic Heat Equation with H\"older Continuous Drift and Noise Coefficients

We study the solutions of the stochastic heat equation with multiplicative space-time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is H"{o}lder continuous of index $\gamma >3\mathrm{/}4$, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are H"{o}lder continuous.