Spurious length-scale cutoff emerges in propagator defined in Costello's "Renormalization and EFT" In page 9 of the introductory chaper of Renormalization and Effective Field Theory (the introductory chapter is available free here), Kevin Costello defines a propagator $P$ for the Laplace operator $D$ in a Riemannian spacetime $M$. 
In order to define a "length-scale version of the renormalization group flow", he provides a formula for this propagator in terms of the heat kernel $K_\tau$ of the Laplace operator: 
$$P(x,y):=\int_0^\infty d\tau\; K_\tau(x,y).\tag{1}$$
Using the interpretation of the heat kernel as a transition probability, one may then rigorously write (expanding upon equation 1)
$$P(x,y)=\int_0^\infty d\tau \int_{\gamma:[0,\tau]\to M\\\gamma(0)=x\\\gamma(\tau)=y}\exp\left(-\int_0^\tau \|d\gamma\|^2\right).\tag{2}$$
My question regards the interpretation of this "path integral" above involving $\gamma$, as something is confusing to me. Since Costello defines $\tau$ to be the proper time of the worldline $\gamma$ as measured by the Riemannian metric, it occurred to me that the integrand in equation (2) vanishes for $\tau < \text{dist}(x,y)$, as there will not exist paths with lengths smaller than that of the geodesic.
Therefore, we can write, more subtly,
$$P(x,y)=\int_{\text{dist}(x,y)}^\infty d\tau \int_{\gamma:[0,\tau]\to M\\\gamma(0)=x\\\gamma(\tau)=y}\exp\left(-\int_0^\tau \|d\gamma\|^2\right).\tag{2}$$
Look how, naturally, a length-scale cutoff emerges. Clearly, I have made a serious conceptual error. Which is it?
 A: There is no space cutoff because it is $d(x,y)$ which depends on the two points. If you put a constant $c$ in your integral $\int_c^\infty d\tau$ then yes you would have introduced a spurrious cutoff in position space.
If you are in $\mathbb{R}^d$ you can write the free massless propagator as
$$
(-\Delta)^{-1}(x,y)=\int_0^{\infty}\frac{dl}{l} l^{-(d-2)}
u\left(\frac{x-y}{l}\right)
$$
for some nonnegative, positive semidefinite smooth function $u$ with compact support say in the unit Euclidean ball around the origin.
See for example my article "A complete renormalization group trajectory between two fixed points". Comm. Math. Phys. 276 (2007), no. 3, 727-772.
This can be very useful for a rigorous renormalization group analysis (see this MO answer).
Clearly, there is no cutoff above because the integral over length scales $l$ is from $0$ to $\infty$. However, because of the support property of the infinitesimal multiscale slice $u((x-y)/l)$, the previous expression is equal to
$$
\int_{|x-y|}^{\infty}\frac{dl}{l} l^{-(d-2)}
u\left(\frac{x-y}{l}\right)\ .
$$
There is a theory for this type of compactly supported multiscale decompositions developed by Brydges, Guadagni, Mitter, Bauerschmidt, Talarczyk, and others. See the JSP article by Mitter "On a finite range decomposition of the resolvent of a fractional power of the Laplacian" (and its erratum) for a recent account.
