Spatial asymptotics and strong comparison principle for some fractional stochastic heat equations

Consider the following stochastic heat equation, \begin{align*} \frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in R^d. \end{align*} Here $-\nu(-\Delta)^{\alpha/2}$ is the fractional Laplacian with $\nu>0$ and $\alpha \in (0,2]$, $\sigma: R\rightarrow R$ is a globally Lipschitz function, and $\dot{F}(t,\,x)$ is a Gaussian noise which is white in time and colored in space. Under some suitable additional conditions, we prove a strong comparison theorem and explore the effect of the initial data on the spatial asymptotic properties of the solution. This constitutes an important extension over a series of recent works.