In this article, we investigate the existence and uniqueness of random-field solutions to the elliptic SPDE $-\mathcal{ L}u=\dot{ \xi}$ on a bounded domain $D$ with Dirichlet boundary conditions $u=0$ on $\partial D$, driven by symmetric L\'evy noise $\dot{ \xi}$. Under general sufficient conditions on the coefficients of the second-order operator $\mathcal{ L}$, we prove the existence of a mild solution via the corresponding Green's function and show that the same framework applies to the spectral fractional Laplacian of power $\gamma \in (0,\infty)$. In particular, whenever $\gamma>\tfrac{ d}{ 2}$, the solution admits a continuous modification.
Categories: math.PR