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 $${\mathrm{\partial }}_{t}u(t,x)={\mathcal{L}}^{\ast }u(t,x)+b(t,x,u(t,x))+\sigma (t,x,u(t,x))\dot{W}(t,x),t>0,x\in \mathbb{X},$$ where $\mathcal{L}$ is the generator of a Markov process which possesses transition densities, and $\dot{W}$ is a Gaussian noise that is white in time and possibly with spatial correlation. We assume the coefficients $b$ and $\sigma$ are Lipschitz and satisfy the linear growth condition. We formulate general and checkable assumptions on $(\mathbb{X},\mathcal{L},\dot{W})$ that ensure existence and uniqueness of probabilistically strong, continuous, tempered mild solutions. We then prove comparison principles (including a strong comparison principle) and strict positivity, relative to initial conditions. Our framework allows non-symmetric heat kernels and includes diffusion-type and stable-type examples, such as metric graphs and fractal spaces with sub-Gaussian heat kernel estimates.