[Paper Review] Trace asymptotics for subordinate semigroups

Rodrigo Bañuelos, Fabrice Baudoin|arXiv (Cornell University)|Aug 22, 2013

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper establishes small-time trace asymptotics for subordinate semigroups generated by subordinated Laplacians on compact Riemannian manifolds. By leveraging Weyl's law and the Karamata tauberian theorem, it proves that the trace of the heat semigroup associated with $ψ(-\Delta)$, where $\psi$ is a regularly varying Bernstein function, behaves asymptotically as $\mathbf{Tr}(\mathbf{Q}_t) \sim C \cdot \varphi(1/t)^{n/2}$ as $t \to 0^+$, extending prior results on $α$-stable processes to a broad class of subordinators.

ABSTRACT

We address a conjecture of D. Applebaum on small time trace asymptotics for subordinate Brownian motion on compact manifolds.

Motivation & Objective

• To resolve a conjecture by D. Applebaum on small-time trace asymptotics for $α$-stable processes on compact Lie groups.
• To generalize trace asymptotics beyond the Cauchy process ($\alpha=1$) to a wide class of subordinated semigroups on compact Riemannian manifolds.
• To establish a general asymptotic formula for the trace of the heat semigroup generated by $\psi(-\Delta)$, where $\psi$ is a Bernstein function with regularly varying Laplace exponent.
• To demonstrate that the asymptotic behavior of the trace is determined by the inverse of the Laplace exponent $\psi$ and the volume of the manifold, via spectral counting function analysis.

Proposed method

• Use of the Bochner subordination principle to define the semigroup $\mathbf{Q}_t$ via subordination of Brownian motion on a compact manifold $\mathbb{M}$.
• Expression of the heat kernel $q(t,x,y)$ as an integral of the standard heat kernel $p(s,x,y)$ against the subordinator's distribution $\eta_t$.
• Derivation of the trace $\mathbf{Tr}(\mathbf{Q}_t)$ as a sum over eigenvalues: $\sum_{k=0}^\infty e^{-\psi(\lambda_k)t}$, using spectral decomposition.
• Application of Weyl's law for the eigenvalue counting function $N(\lambda) \sim C \lambda^{n/2}$ to relate the asymptotics of $\mathbf{Tr}(\mathbf{Q}_t)$ to the behavior of $\psi^{-1}$.
• Use of the Karamata tauberian theorem to connect the large-$\lambda$ behavior of the spectral counting function $N^\psi(\lambda)$ to the small-$t$ behavior of $\mathbf{Tr}(\mathbf{Q}_t)$.
• Analysis of the case where $\psi$ is regularly varying at infinity with index $r > 0$, leading to the asymptotic formula involving the inverse function $\varphi = \psi^{-1}$.

Experimental results

Research questions

• RQ1 Does the small-time trace asymptotic behavior observed for the Cauchy process on $SU(2)$ and $SO(3)$ extend to all $\alpha$-stable processes on compact Lie groups?
• RQ2 Can the trace asymptotics for subordinate semigroups on compact manifolds be generalized beyond $\psi(\lambda) = \lambda^{\alpha/2}$ to a broader class of Bernstein functions?
• RQ3 What is the precise asymptotic form of $\mathbf{Tr}(\mathbf{Q}_t)$ as $t \to 0^+$ for subordinated semigroups generated by $\psi(-\Delta)$, where $\psi$ is a general increasing Bernstein function?
• RQ4 How does the inverse of the Laplace exponent $\psi$ influence the small-time trace behavior, and what role does its regular variation play?

Key findings

• The trace of the subordinate semigroup $\mathbf{Q}_t$ satisfies $\mathbf{Tr}(\mathbf{Q}_t) \sim \frac{\mathrm{Vol}(\mathbb{M})\Gamma\left(\frac{n}{2r}+1\right)}{\Gamma\left(\frac{n}{2}+1\right)(4\pi)^{n/2}} \cdot \varphi\left(\frac{1}{t}\right)^{n/2}$ as $t \to 0^+$, where $\varphi = \psi^{-1}$ and $\psi$ is regularly varying with index $r > 0$.
• For the case $\psi(\lambda) = \lambda^{\alpha/2}$, the asymptotic trace is $\mathbf{Tr}(\mathbf{Q}_t) \sim \frac{\mathrm{Vol}(\mathbb{M})\Gamma\left(\frac{n}{\alpha}+1\right)}{\Gamma\left(\frac{n}{2}+1\right)(4\pi)^{n/2}} \cdot t^{-n/\alpha}$, recovering known results for $\alpha$-stable processes.
• When $\psi$ is slowly varying ($r=0$), the asymptotic trace behaves as $\mathbf{Tr}(\mathbf{Q}_t) \sim \frac{\mathrm{Vol}(\mathbb{M})}{\Gamma\left(\frac{n}{2}+1\right)(4\pi)^{n/2}} \cdot \varphi\left(\frac{1}{t}\right)^{n/2}$, with $\Gamma(\cdot)$ replaced by 1.
• The on-diagonal heat kernel asymptotics are derived as a byproduct: $q_t(x,x) \sim \frac{\Gamma\left(\frac{n}{2r}+1\right)}{\Gamma\left(\frac{n}{2}+1\right)(4\pi)^{n/2}} \cdot \varphi\left(\frac{1}{t}\right)^{n/2}$ as $t \to 0^+$.
• The results hold for all compact Riemannian manifolds, including compact Lie groups, under the assumption that $\psi$ is increasing and regularly varying at infinity with index $r > 0$.
• The derivation relies on the spectral counting function $N^\psi(\lambda) = N(\varphi(\lambda))$, and the asymptotic equivalence $N^\psi(\lambda) \sim \frac{\mathrm{Vol}(\mathbb{M})}{\Gamma\left(\frac{n}{2}+1\right)(4\pi)^{n/2}} \cdot \varphi(\lambda)^{n/2}$ as $\lambda \to \infty$.