Comparison principles for stochastic reaction-diffusion equations on metric measure spaces

We study parabolic stochastic partial differential equations on metric measure spaces $(\mathbb{X}, d,m)$ of the form $$ \partial_t u(t,x) = \mathcal{L}^* u(t,x) + b(t,x,u(t,x)) + σ(t,x,u(t,x)) \dot{W}(t,x),\quad t>0,\, x \in \mathbb X, $$ where $\mathcal{L}$ is the generator of a Markov pro