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

Abstract

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 , \begin{equation*} \frac {\partial }{\partial t}u_t(\boldsymbol {x})= -(-\Delta )^{\frac {\alpha }{2}}u_t(\boldsymbol {x}) +b\left (u_t(\boldsymbol {x})\right )+\sigma \left (u_t(\boldsymbol {x})\right ) \dot F(t, \boldsymbol {x}), \quad t\ge 0,\: \boldsymbol {x}\in \mathbb R^d, \end{equation*} where − ( − Δ ) α 2 -(-\Delta )^{\frac {\alpha }{2}} denotes the fractional Laplacian with power α 2 ∈ ( 1 2 , 1 ] \frac {\alpha }{2}\in (\frac 12,1] , and the driving noise F ˙ \dot 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+{\varepsilon }}(\boldsymbol {x})-u_t(\boldsymbol {x}) at any fixed t > 0 t > 0 and x ∈ R d \boldsymbol {x}\in \mathbb R^d , as ε ↓ 0 {\varepsilon }\downarrow 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(\boldsymbol {x})\}_{ t \ge 0} of the solution, where x ∈ R d \boldsymbol {x}\in \mathbb R^d is fixed.