Stochastic Calculus for the Theta Process

Abstract: The Theta process, $X(t)$, is a complex valued stochastic process of number theoretical origin arising as a scaling limit of quadratic Weyl sums $$\sum_{n=1}^N e^{2\pi i \left(\frac{1}{2}(n^2+\beta)x+\alpha n\right)},$$ where $(\alpha,\beta)\in \mathbb R^2 \setminus \mathbb Q^2$ and $x\in \mathbb R$ is chosen at random according to any law absolutely continuous with respect to Lebesgue measure. The Theta process can be explicitly represented as $X(t)=\sqrt{t} \Theta(\Gamma g \Phi^{2 \log t})$ where $\Theta$ is an automorphic function defined on Lie group $G$, invariant under left multiplication under lattice $\Gamma$. Additionally, $g\in \Gamma \setminus G$ is chosen Haar uniformly at random and $\Phi$ is the geodesic flow on $\Gamma \setminus G$. The Theta process shares several similar properties with the Brownian motion. In particular, both lack differentiability and have the same $p$ variation and H\”older properties. Similarly to Brownian motion, standard calculus and even Young/Riemann-Stieltjes calculus techniques do not work. However, Brownian motion is what is known as a martingale allowing for a classical theory of It\^o calculus which makes use of cancellations “on average”. The It\^o calculus can be used to prove several properties of Brownian motion such as its conformal invariance, bounds on its running maximum in terms of its quadratic variation, absolutely continuous changes in measure and much more. Unfortunately, we show that the Theta process $X$ is not a (semi)martingale, therefore It\^o techniques don’t work. However, a new theory introduced in 1998 by Terry Lyons called rough paths theory handles processes with the same analytic regularity as $X$. The key idea in rough paths theory is that constructing stochastic calculus for a signal can be reduced to constructing the “iterated integrals” of the signal. In this talk, we will show the construction of the iterated integrals – the “rough path” – above the process $X$. Joint with Francesco Cellarosi.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar