Nonlinear stochastic time-fractional slow and fast diffusion equations on [formula omitted].

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: ∂ β + ν 2 (− Δ) α ∕ 2 u (t , x) = I t γ ρ (u (t , x)) W ̇ (t , x) , t > 0 , x ∈ R d , where W ̇ is the space–time white noise, α ∈ (0 , 2 ] , β ∈ (0 , 2) , γ ≥ 0 and ν > 0. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang's condition: d < 2 α + α β min (2 γ − 1 , 0). In some cases, the initial data can be measures. When β ∈ (0 , 1 ] , we prove the sample path regularity of the solution. [ABSTRACT FROM AUTHOR]