Stochastic Trace Formula for Closed, Negatively Curved Manifolds

[DRAFT] Last updated by Joe Schaefer on Sun, 30 Mar 2025    source

 

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My 1997 Ph.D. thesis as a blog entry.

There Is Only One n-Dimensional Wiener Measure $\mu$

Piecewise Linear Approximations to Brownian Motion

The Development Map DM

The Cameron-Martin Formula

Heat Kernels as Radon-Nicodym Derivatives of Weiner Measure

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 in general.

$\Omega_t^0$ 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 $\Omega_t^0[\gamma]$.
In horocyclic coordinates, $\det{g(\vec{x})} = 1$:

$$\begin{aligned} ds^2 &= dx\odot dx + h(x,y) dx\odot dy + (1+h^2(x,y)) dy\odot dy \\ g(x,0) &= 0,\\ \sigma_1 &= dx + h(x,y)dy \\ \sigma_2 &= h(x,y)dx + (1+h^2(x,y))dy \\ \sigma_1 \wedge \sigma_2 &= dx \wedge dy\\ d\sigma_1 &= \frac{\partial h}{\partial x}dx\wedge dy \\ d\sigma_2 &= (-\frac{\partial h}{\partial y}+2h\frac{\partial h}{\partial x}) dx\wedge dy \\ \end{aligned}$$

So the connection 1-form $\alpha := Adx + Bdy$ satisfies

$$\begin{aligned} d\sigma_1 &= \alpha \wedge \sigma_1 \\ d\sigma_2 &= \sigma_2 \wedge \alpha \\ \implies \\ A &= \frac{\partial h}{\partial y} - 3h\frac{\partial h}{\partial x} \\ B &= h\frac{\partial h}{\partial y} - (1+3h^2)\frac{\partial h}{\partial x} \\ \\ K &= \frac{\partial B}{\partial x} - \frac{\partial A}{\partial y} \\ &= \frac{\partial h}{\partial x}\frac{\partial h}{\partial y} - 6h\frac{\partial h}{\partial x}^2 - (1+3h^2)\frac{\partial^2 h}{\partial x \partial x} - \frac{\partial^2 h}{\partial y \partial y} + 2h \frac{\partial^2 h}{\partial x \partial y} ,\\ h &= h(y) \implies \\ \alpha &= \frac{\partial }{\partial y} (hdx +\frac{h^2}{2} dy)\\ K(y) &= -\frac{\partial^2 h}{\partial y \partial y}\ \\ \text{has Galilean Symmetry:} \\ h \mapsto h(y,\beta) &= h(y) + \beta y \ .\\ \end{aligned}$$

IMPORTANT Therefore, when $h=h(y)$, the parallel transport equation reduces to $\dot{\vec{c}}(t) = -\alpha(\dot{\gamma}(t))\vec{c}(t) = -\partial_y(h\ dx/dt + h^2/2\ dy/dt)\vec{c}(t)$. Of course, this has closed solution

$$\vec{c}(t) = exp(-((h+\beta y)\ dx/dt + (h^2/2 + (\beta y)^2/2 + \beta y h) \ dy/dt)|_{x_0,y_0}^{x_t,y_t})\vec{c}(0)$$

which is a function of the transport curve $\gamma$’s endpoints alone. This fact implies the Development Map preserves loops.

$Z_{-\Delta/2}(t) := \int_M k_{-t\Delta/2}(x,x) \sqrt{g}dx = \sum_{j=0}^\infty e^{-\lambda_i t/2}$ is the trace of the heat kernel.

Finally let us define the following from their Radon-Nicodym derivatives:

$$\begin{aligned} DM_*\mu(\Omega_t) &:= \int_M DM_*\mu(\Omega_t\vert_x \sqrt{g}dx)\\ DM_*\mu(\Omega^0_t) &:= \int_M DM_*\mu(\Omega^0_t\vert_x \sqrt{g}dx)\\ DM_*\mu(\Omega_t[\gamma]) &:= \int_M DM_*\mu(\Omega_t[\gamma]\vert_x \sqrt{g}dx) \\ \end{aligned}$$

Stochastic Trace Formula

$$\begin{aligned} Z_{-\Delta/2}(t) = DM_*\mu(\Omega_t) &= DM_*\mu(\Omega^0_t) + \sum_{\set{\gamma}} DM_*\mu(\Omega_t[\gamma]) \\ DM_*\mu(\Omega_t^0) &\approx_{t\rightarrow 0} (2\pi t)^{-n/2}(vol(M) + t/6\int_M K(x)\sqrt{g} dx + O(t^2))\space \small\text{by McKean-Singer}\\ DM_*\mu(\Omega_t[\gamma]) &= e^{-\ell(\gamma)^2/2t}\int_M DM_*\mu(e^{\bra{J_BB_t}\ket{B_t}} _t \Omega_t^0[\gamma]\vert_x\sqrt{g}dx)\space\small \text{ by Cameron-Martin}\\ &= e^{-\ell(\gamma)^2/2t}\int_{T_{\gamma_0}M} E(e^{J_B}_{t} | \Omega_t^0[\gamma]\vert_{x(\tau)})dx^1(\tau)\dots dx^n(\tau) d\tau\\ \end{aligned}$$

$$\begin{aligned} \text{Horocyclic coordinates}: \\ h&= h(y) \implies \\ &=\frac{\ell(\gamma_0)}{2\pi t}\int_{\Reals}\frac{e^{- (1+h^2(y))\ell(\gamma)^2/2t}}{2\sinh \sqrt{-K(y)}\ell(\gamma)/2}\ell(\gamma) dy\ ,\\ h(y) = y \implies \\ &= \frac{e^{-\ell(\gamma)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t}\sinh \sqrt {-K}\ell(\gamma)/2}\int_{\Reals}e^{-y^2\ell(\gamma)^2/2t}{\frac{\ell(\gamma)dy}{\sqrt{2\pi t}}} \end{aligned}$$

Approximation and the Selberg Trace Formula

In the $\dim = 2$ constant curvature $K = -\kappa^2$ case,

$$\begin{aligned} \det |I-J_\gamma| &= (e^{\kappa\ell(\gamma)} - 1)(1 - e^{-\kappa\ell(\gamma)}) = 2 \sinh \kappa\ell(\gamma)/2\\ \gamma(t) = \gamma_0(kt)\implies \\ DM_*\mu(\Omega_t[\gamma])&=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t}\sinh k\kappa\ell(\gamma_0)/2}\\ \end{aligned}$$

In the $\dim=3$ hyperbolic manifold case, we use complex coordinates $(z,\bar{z})$ on the normal bundle to write

$$\begin{aligned} J_{DM(\vec{x}+(\tau+\ell(\gamma))\vec{e}^1)} &= \begin{pmatrix} e^{\kappa\ell(\gamma)} && 0 && 0\\ 0 && e^{-\kappa\ell(\gamma)+i\theta(\gamma)} && 0 \\ 0 && 0 && e^{-\kappa\ell(\gamma)-i\theta(\gamma)} \\ \end{pmatrix}\\ \implies& \\ \det I-{\perp_{\gamma_0}}^k &= |1-e^{-k(\kappa\ell(\gamma_0)-i\theta(\gamma_0))}|^2 \end{aligned}$$

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:

$$\begin{aligned} \kappa &= 1 \implies \\ DM_*\mu(\Omega_t[\gamma]) &=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t (1-e^{-k\ell(\gamma_0)})}|e^{k\ell(\gamma_0)/2}-e^{-k(\ell(\gamma_0)/2-i\theta(\gamma_0))}|}\\ \end{aligned}$$