TEMPORAL PROPERTIES OF THE STOCHASTIC FRACTIONAL HEAT EQUATION WITH SPATIALLY-COLORED NOISE

Consider the stochastic partial differential equation partial derivative/partial derivative t ut(x) = -(-Delta)(alpha/a) u(t)(x) + b (u(t)(x)) + sigma (u(t)(x)) (F)over dot (t, x), t >= 0, x is an element of R-d, where -(-Delta)(alpha/2) denotes the fractional Laplacian with power (alpha/2) is an element of (1/2, 1], and the driving noise (F) over dot 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+epsilon)(x)- u(t)(x) at any fixed t > 0 and x is an element of R-d, as epsilon down arrow 0. As applications, we deduce Khintchin's law of iterated logarithm, Chung's law of iterated logarithm, and a result on the q-variations of the temporal process {u(t)(x)}(t >= 0) of the solution, where x is an element of R-d is fixed.