A Sobolev Space Theory for Time-Fractional Stochastic Partial Differential Equations Driven by Lévy Processes

We present an $$L_{p}$$ L p -theory ( $$p\ge 2$$ p ≥ 2 ) for semi-linear time-fractional stochastic partial differential equations driven by Lévy processes of the type $$\begin{aligned} \partial ^{\alpha }_{t}u=\sum _{i,j=1}^d a^{ij}u_{x^{i}x^{j}} +f(u)+\sum _{k=1}^{\infty }\partial ^{\beta }_{t}\int _{0}^{t} \left( \sum _{i=1}^d\mu ^{ik} u_{x^i} +g^k(u)\right) \textrm{d}Z^k_{s} \end{aligned}$$ ∂ t α u = ∑ i , j = 1 d a ij u x i x j + f ( u ) + ∑ k = 1 ∞ ∂ t β ∫ 0 t ∑ i = 1 d μ ik u x i + g k ( u ) d Z s k given with nonzero initial data. Here, $$\partial ^{\alpha }_t$$ ∂ t α and $$\partial ^{\beta }_t$$ ∂ t β are the Caputo fractional derivatives, $$\begin{aligned} 0