Temporal properties of the stochastic fractional heat equation with spatially-colored noise

Consider the stochastic partial differential equation ∂ ∂ t u t ( x ) = − ( − Δ ) α 2 u t ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ ( t , x ) , t ≥ 0 , x ∈ R d , {equation*} ∂ /∂ tu_t(x)= -(-Δ )α /2u_t(x) +b (u_t(x) )+σ (u_t(x) ) F(t, x), t≥ 0,\: x∈ R^d, {equation*} where − ( − Δ ) α 2 -(-Δ )α /2 denotes the fractional Laplacian with power α 2 ∈ ( 1 2 , 1 ] α /2∈ ( 12,1] , and the driving noise F ˙ F is a centered Gaussian field which is white in time and has a spatial homogeneous covariance given by the Riesz kernel. We study the detailed behavior of the approximation of the temporal gradient u t + ε ( x ) − u t ( x ) u_{t+{ε }}(x)-u_t(x) at any fixed t > 0 t > 0 and x ∈ R d x∈ R^d , as ε ↓ 0 {ε }↓ 0 . As applications, we deduce Khintchin’s law of iterated logarithm, Chung’s law of iterated logarithm, and a result on the q q -variations of the temporal process { u t ( x ) } t ≥ 0 \{u_t(x)\}t ≥ 0 of the solution, where x ∈ R d x∈ R^d is fixed.