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

We consider a class of parabolic stochastic PDEs on bounded domains D c Rd that includes the stochastic heat equation but with a fractional power gamma of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces Hr, with r < gamma - d/2, we study its power variations in Hr 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 gamma/(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.