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

TL;DR

This paper investigates the power variations of solutions to a class of parabolic stochastic PDEs with fractional Laplacians in fractional Sobolev spaces, revealing a phase transition at a critical regularity level and explicit formulas involving spectral zeta functions.

Contribution

It introduces a detailed analysis of power variations in fractional Sobolev spaces for stochastic PDEs with fractional Laplacians, identifying a phase transition and explicit spectral zeta function expressions.

Findings

Nontrivial quadratic variation for r < -d/2

Power variations follow a law of large numbers or degenerate limits depending on r

Explicit quadratic variation formula involving spectral zeta functions

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 $\gamma$ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r < \gamma - d/2$, 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 $p$th order variation for $p= 2\gamma/(\gamma-d/2-r)>2$ when $r>-d/2$. More generally, suitably 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.