Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs

Abstract

We consider a class of parabolic stochastic PDEs on bounded domains $D\subseteq {\mathbb{R}}^d$ that includes the stochastic heat equation but with a fractional power γ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r<\mathit{\gamma }-d∕2$, we study its power variations in $H_r$ along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at $r=-d∕2$: the solutions have a nontrivial quadratic variation when $r<-d∕2$ and a nontrivial pth order variation for $p=2\mathit{\gamma }∕(\mathit{\gamma }-d∕2-r)>2$ when $r>-d∕2$. More generally, normalized power variations of any order satisfy a genuine law of large numbers in the first case and a degenerate limit theorem in the second case. When $r<-d∕2$, the quadratic variation is given explicitly via an expression that involves the spectral zeta function, which reduces to the Riemann zeta function when $d=1$ and D is an interval.