Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds
By O. Smolyanov, H. Weizsäcker, Olaf Wittich. 2004
111 citations recorded by the source corpus.
Let $(S(t))_{t \ge 0}$ be a one-parameter family of positive integral operators on a locally compact space $L$. For a possibly non-uniform partition of $[0,1]$ define a finite measure on the path space $C_L[0,1]$ by using a) $S(\Delta t)$ for the transition between any two consecutive partition times of distance $\Delta t$ and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize the result to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tightness results which yield convergence in law of such measures as the partition gets finer. In particular let $L$ be a closed smooth submanifold of a 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 density with respect to Brownian motion on $L$ which contains scalar, mean and sectional curvatures terms. Various approximation schemes for Brownian motion on $L$ are also given.
Subfields: Finance, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5b0ce1cd63d33ecad2f2f289f8256b2756b3c2a2
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Citations in the corpus
- From Graphs to Manifolds - Weak and Strong Pointwise Consistency of Graph Laplacians (2005, methodology): “Approximations of the Laplace-Beltrami operator based on averaging with the Gaussian kernel have been studied in the special case of the uniform measure on a compact submanifold without boundary in Smolyanov et al.[9, 10 ] and Belkin [1].”
- Towards a theoretical foundation for Laplacian-based manifold methods (2005, methodology): “Now we make use of the fact (see [35,36] and references therein) that the metric tensor has an asymptotic expansion in exponential coordinates given by det(gij) = 1 − 1 6 x Rx + O(‖x‖) where R is the Ricci curvature tensor.”