[Paper Review] Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

O. G. Smolyanov, H. von Weizsaecker+1 more|ArXiv.org|Sep 9, 2004

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TL;DR

This paper establishes a convergence result for discrete-time approximations of Brownian motion on a Riemannian submanifold $L$ embedded in a larger manifold $M$, using Chernoff's theorem and path-space conditioning. It proves that the law of $M$-Brownian motion conditioned to stay on $L$ at discrete times converges weakly to a measure on $L$-valued paths, whose density with respect to $L$-Brownian motion explicitly involves scalar, mean, and sectional curvatures of $L$.

ABSTRACT

Let (S(t)) be a one-parameter family S = (S(t)) of positive integral operators on a locally compact space L. For a possibly non-uniform partition of [0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the transition between cosecutive partition times of distance dt, and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tighness results which together yield convergence in law of such measures as the partition gets finer. In particular let L be a closed smooth submanifold of a Riemannian manifold M. We prove convergence of Brownian motion on M, conditioned to visit L at all partition times, to a process on L whose law has a Radon-Nikodym density with repect to Brownian motion on L which contains scalar, mean and sectional curvature terms. Various approximation schemes for Brownian motion are also given. These results substantially extend earlier work by the authors and by Andersson and Driver.

Motivation & Objective

• To extend Chernoff's theorem to non-uniform partitions and positive integral operators on manifolds.
• To establish weak convergence of discrete-time approximations of Brownian motion on submanifolds via conditioning and interpolation.
• To derive explicit expressions for the Radon-Nikodym density of the limiting measure relative to Wiener measure on the submanifold.
• To connect the limiting measure to geometric invariants such as scalar, mean, and sectional curvatures through asymptotic analysis of heat kernels.
• To provide a rigorous framework for approximating surface measures and conditional processes on Riemannian manifolds using finite-dimensional kernels.

Proposed method

• Uses a generalized version of Chernoff's theorem for non-uniform partitions and positive integral operators on a locally compact space.
• Constructs finite measures on path spaces via iterated application of operators $S(t)$, followed by continuous interpolation using geodesics or Brownian bridges.
• Applies tightness results for convergence in law on $C_L[0,1]$ or $D_L[0,1]$, relying on large deviation estimates and semigroup theory.
• Employs Wick’s formula and asymptotic analysis of Gaussian integrals to compute short-time behavior of pseudo-Gaussian kernels.
• Derives curvature-dependent normalization coefficients by Taylor expansion of normalization constants in local coordinates.
• Uses the restriction of the heat kernel on $M$ to $L$ as a kernel $q(t,x,y)$, and computes the conditional law via limiting surface measure procedures.

Experimental results

Research questions

• RQ1 Under what conditions does a family of positive integral operators on a manifold generate a Feller process via discrete-time iteration and path interpolation?
• RQ2 How does the law of Brownian motion on a larger manifold $M$, conditioned to remain on a submanifold $L$ at discrete times, converge as the partition becomes finer?
• RQ3 What is the explicit Radon-Nikodym density of the limiting measure on $L$-valued paths relative to standard Brownian motion on $L$?
• RQ4 How do scalar, mean, and sectional curvatures of $L$ enter the density of the limiting measure in the case of heat kernel restrictions?
• RQ5 Can the convergence of conditional measures be established without requiring normalization of the initial kernels, and how does global normalization affect the limit?

Key findings

• The discrete-time approximation of Brownian motion on a submanifold $L$, constructed via iterated application of a positive integral operator $S(t)$ and geodesic interpolation, converges in law to a measure on $C_L[0,1]$ as the partition mesh tends to zero.
• The limiting measure is equivalent to the law of $L$-valued Brownian motion, with a Radon-Nikodym density given by $\exp\left(\int_0^1 D(\omega(s))\,ds\right)$, where $D$ is a function combining scalar, mean, and sectional curvatures of $L$.
• When the kernel $q(t,x,y)$ is the restriction of the heat kernel on $M$ to $L$, the conditional law of $M$-Brownian motion pinned to $L$ at partition times converges weakly to a measure with curvature-dependent density on $L$.
• The convergence holds under both global normalization (renormalizing the measure after construction) and local normalization (normalizing each kernel), with equivalent limiting laws in the latter case.
• The method allows explicit computation of the density in terms of curvature invariants, providing a new proof of a result on surface measures induced by Brownian motion in the ambient manifold.
• Tightness of the approximating measures is established via large deviation estimates and continuity of normalization, ensuring weak convergence on path space.