\documentclass{article} \usepackage[hidelinks]{hyperref} \usepackage[type={CC},modifier={by-nc-nd},version={4.0}]{doclicense} \usepackage[margin=1in]{geometry} \newenvironment{subproof}[1][\proofname]{% \renewcommand{\qedsymbol}{$\blacksquare$}% \begin{proof}[#1]% }{% \end{proof}% } % AMS Packages \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb} % modern, utf8 friendly options (wholly incompatible with amscd): \PassOptionsToPackage{partial=upright}{unicode-math} \usepackage{fontsetup} \usepackage{tikz-cd} \usepackage[style=alphabetic]{biblatex} \addbibresource{/content/joe/stochastic-trace-formula.bib.en} % useful math-physics macros \usepackage{braket} \usepackage{physics} % Unicode \usepackage[utf8]{inputenc} % Metadata \title{Stochastic trace formula for closed, negatively curved manifolds} \date{2024\\May} \author{Joe Schaefer\\\href{mailto://Joe\%20Schaefer\%20}{joe@sunstarsys.com}} % Theorem, Lemma, etc \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{claim}{Claim}[theorem] \newtheorem{axiom}[theorem]{Axiom} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{fact}[theorem]{Fact} \newtheorem{hypothesis}[theorem]{Hypothesis} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{criterion}[theorem]{Criterion} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{problem}[theorem]{Problem} \newtheorem{principle}[theorem]{Principle} \begin{document} \maketitle \tableofcontents My \textit{1997 Ph.D. thesis} as a blog entry. \section{There Is Only One n-Dimensional Wiener Measure} \section{Piecewise Linear Approximations to Brownian Motion} \section{The Development Map DM} \section{The Cameron-Martin Formula} \section{Heat Kernels as Radon-Nicodym Derivatives of Weiner Measure} \section{Notation} $M$ is a negatively curved $\dim=n$ closed Riemannian manifold with metric $g$, metric connection $\nabla$, and (nonnegative) Laplace-Beltrami Operator $\Delta_M$. Let $k_{-t\Delta/2}(x,y)$ represent the heat kernel on $M$. Hence $k_{-t\Delta/2}(x,x) = dDM_*\mu/\sqrt{g}dx$ is the Radon-Nicodym derivative of n-dimensional Wiener Measure $\mu$, restricted to the pull-back of continuous loop space $\Omega_t(M)\vert_x$, via the inverse of the Weiner measure-preserving development map $DM$. *Note:* $DM^{-1}\Omega_t\vert_x$ is not a loop space. $\Omega_t^0$ is the space of continuous contractible loops on $M$. $\Omega_t[\gamma]$ is the space of continuous loops on $M$ homotopic to the closed geodesic $\gamma$. Let $\gamma_0$ be its primitive loop. $DM^{-1}\Omega_t^0[\gamma]$ is the preimage of continuous contractible loops on $M$ written as offsets homotopic to $\gamma(s) = DM(\frac{s\ell(\gamma)}{t}\vec{e}^1), 0\leq s \leq t$. Think Horocyclic Coordinates -- each fiber as the geometric limit of periodic geodesic spheres $S_{\gamma_0(s)}^{n-1}(k\ell(\gamma_0)), 0\leq s \leq t, k\rightarrow\infty$, vectorized in the Normal Bundle over $\gamma_0$. Our curvature constraints imply Horocyclic Coordinates for every $\gamma_0$ exist as a smooth, $DM$-compatible coordinate map for $\Omega_t^0[\gamma]$. Now $\vec{x}(\tau)+\ell(\gamma)\vec{e}^1$ is the \textit{undeveloped} endpoint of the "offset" \textit{kinked geodesic} homotopic to $\gamma: DM(\vec{x}(\tau) + \frac{s\ell(\gamma)}{t}\vec{e}^1), 0\leq s \leq t$. The curve is periodic with period $\ell(\gamma_0)$, and it revisits its kinked starting point $DM(\vec{x}(\tau))$ at time $t$, making the computation of its forward derivative $J=\lim_{s\uparrow t}DM^\prime\vert_{DM(\vec{x}(\tau) + \frac{\ell(\gamma)s}{t}\vec{e}^1)}$ tractible as a linear automorphism of $T_{DM(\vec{x}(\tau))}M$. Importantly, $J_{DM(\vec{x}(\tau)+\ell(\gamma)\vec{e}^1)}$ may be constructed using \textbf{Jacobi Fields}, since $DM$ \textbf{is} the (iterated) exponential map along any series of connected straight lines in $\mathbb{R}^n$. We will study $ 1/2 \int_0^t \bra{dX}\ket{dX}_s$, with the solution \begin{equation} \begin{aligned} X_t &= X_0 + \int_0^t \sqrt{J}_{X_t} dB_t \\ \end{aligned} \end{equation} $Z_{-\Delta/2}(t) := \int_M k_{-t\Delta/2}(x,x) \sqrt{g}dx = \sum_{j=0}^\infty e^{-\lambda_i t/2}$ is the trace of the heat kernel. Finally let us define the following from their Radon-Nicodym derivatives: \begin{equation} \begin{aligned} DM_*\mu(\Omega_t) &:= \int_M DM_*\mu(\Omega_t\vert_x \sqrt{g}dx)\\ DM_*\mu(\Omega^0_t) &:= \int_M DM_*\mu(\Omega^0_t\vert_x \sqrt{g}dx)\\ DM_*\mu(\Omega_t[\gamma]) &:= \int_M DM_*\mu(\Omega_t[\gamma]\vert_x \sqrt{g}dx) \\ \end{aligned} \end{equation} \section{Stochastic Trace Formula} \begin{equation} \begin{aligned} Z_{-\Delta/2}(t) = DM_*\mu(\Omega_t) &= DM_*\mu(\Omega^0_t) + \sum_{\set{\gamma}} DM_*\mu(\Omega_t[\gamma]) \\ DM_*\mu(\Omega_t^0) &\approx_{t\rightarrow 0} (2\pi t)^{-n/2}(vol(M) + t/6\int_M K(x)\sqrt{g} dx + O(t^2))\space \small\text{by McKean-Singer}\\ DM_*\mu(\Omega_t[\gamma]) &= e^{-\ell(\gamma)^2/2t}\int_M DM_*\mu(e^{\bra{J_BB_t}\ket{B_t}} _t \Omega_t^0[\gamma]\vert_x\sqrt{g}dx)\space\small \text{ by Cameron-Martin}\\ &= e^{-\ell(\gamma)^2/2t}\int_{T_{\gamma_0}M} E(e^{J_B}_{t} | \Omega_t^0[\gamma]\vert_{x(\tau)})dx^1(\tau)\dots dx^n(\tau) d\tau\\ \frac{dDM_*\mu(e^{-\ell(\gamma)x^1(t)}\Omega^0_t[\gamma])}{dx^1(\tau)\dots dx^n(\tau)d\tau}\vert_{\vec{y(\tau)}}&\approx_{t\rightarrow 0} \frac{e^{-\bra{|I-J_{DM(\vec{x}(\tau),\vec{y}(\tau))}\vec{x}(\tau)}\ket{\vec{x}(\tau)}/2t}}{(2 \pi t)^{(n+1)/2}}(1+O(t^2))\small \text{ semi-classical limit}\\ \text{Horocyclic coordinates}: z(\tau) - x(\tau) &= x + \ell(\gamma)\vec{e}^1\implies\\ \int_{M/S^1\oplus S^1}k_t(x,z) dx &=\lim_{j\rightarrow\infty}\frac{e^{-\ell(\gamma)^2/2t}}{\sqrt{2\pi t}}E(e^{\bra{J_{X^j_t}\vec{x}}\ket{\vec{x}}})\\ &=\lim_{j\rightarrow\infty}\frac{e^{-\ell(\gamma)^2/2t}}{\sqrt{2\pi t}}\int_{M^j/S^1\oplus S^1}\frac{1}{\sqrt{2\pi t}^{jn}\det|I-J_{X^j}|}e^{-\ell(X^j)^2/2t}X^{j}\\ \end{aligned} \end{equation} \section{Approximation and the Selberg Trace Formula} In the $\dim = 2$ constant curvature $-\kappa^2$ surface case, \begin{equation} \begin{aligned} \sqrt{J_{\vec{x}, \vec{y}}}dRB&= \begin{pmatrix} e^{\kappa d(\vec{x},\vec{y})/2} && 0\\ 0 && e^{-\kappa d(\vec{x},\vec{y})/2}\\ \end{pmatrix} \implies&\\ \bra{\sqrt{J}dRB}\ket{\sqrt{J}dRB} &= e^{\kappa \ell(B)}dRB_1^2 - e^{-\kappa \ell(B)}dRB_2^2\\ \int_0^t \bra{\sqrt{ J}dB}\ket{\sqrt{ J}dB} &= e^{\kappa\ell(\gamma)} - e^{-\kappa\ell(\gamma)}\\ \det I-J_{\gamma} &= (e^{\kappa\ell(\gamma)/2}- e^{-\kappa\ell(\gamma)/2})^2 \end{aligned} \end{equation} which is constant over $(\vec{x},\tau)$, so the approximation $\approx_{t\rightarrow 0}$ line in Equation (2) becomes \textit{exact}: \begin{equation} \begin{aligned} DM_*\mu(\Omega_t[\gamma]) &= \frac{e^{-\ell(\gamma)^2/2t}\ell(\gamma_0)}{\sqrt{2 \pi t}(e^{\kappa\ell(\gamma)/2} -e^{-\kappa\ell(\gamma)/2})}\\ \gamma(t) = \gamma_0(kt)\implies \\ &=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t}\sinh k\kappa\ell(\gamma_0)/2}\\ \end{aligned} \end{equation} In the $\dim=3$ hyperbolic manifold case, we use complex coordinates $(z,\bar{z})$ on the normal bundle to write \begin{equation} \begin{aligned} J_{DM(\vec{x}+(\tau+\ell(\gamma))\vec{e}^1)} &= \begin{pmatrix} e^{\kappa\ell(\gamma)} && 0 && 0\\ 0 && e^{-\kappa\ell(\gamma)+i\theta(\gamma)} && 0 \\ 0 && 0 && e^{-\kappa\ell(\gamma)-i\theta(\gamma)} \\ \end{pmatrix}\\ \implies& \\ \det I-{\perp_{\gamma_0}}^k &= |1-e^{-k(\kappa\ell(\gamma_0)-i\theta(\gamma_0))}|^2 \end{aligned} \end{equation} and since $z=x^2+ix^3 \implies d\bar{z}\wedge dz= (dx^2-idx^3)\wedge(dx^2+idx^3) = 2idx^2\wedge dx^3$, the approximation in Equation (2) again becomes exact: \begin{equation} \begin{aligned} \kappa &= 1 \implies \\ DM_*\mu(\Omega_t[\gamma]) &=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t (1-e^{-k\ell(\gamma_0)})}|e^{k\ell(\gamma_0)/2}-e^{-k(\ell(\gamma_0)/2-i\theta(\gamma_0))}|}\\ \end{aligned} \end{equation} \doclicenseThis \pagebreak \printbibliography \end{document}