Abstract. We consider different sub-Laplacians on a sub-Riemannian manifold M . Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in [7]. This operator is canonical with respect to the horizontal Brownian motion, we are able to define the sub-Laplacian without some a priori choice of measure. The other operator is div ω grad H for some volume form ω on M . We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: SU (2), the Heisenberg group and the affine group.
This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold M . To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators naturally connected to the geometry of the underlining manifold. In the case when M is a Riemannian (non-degenerate) manifold, we recover the Laplace-Beltrami operator. We then construct the corresponding random walk, and under standard assumptions on the sub-Laplacian and M we show that this random walk weakly converges to a process, horizontal Brownian motion, whose infinitesimal generator is the sub-Laplacian. An example of the Heisenberg group equipped with a standard sub-Riemannian metric is considered in detail, in which case the sub-Laplacian we introduced is shown to be the sum of squares (Hörmander's) operator.Contents 1991 Mathematics Subject Classification. Primary 60J65, 58G32; Secondary 58J65.
Abstract. This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. A L 2 Riemannian metric G P is given on the space of piecewise geodesic paths H P (M ) adapted to the partition P of [0, 1], whence a finite-dimensional approximation of Wiener measure is developed. It is proved that, as mesh(P) → 0, the approximate Wiener measure converges in a L 1 sense to the measure exp{− 2+ √ 3 20 √ 3 1 0 Scal(σ(s))ds}dν(σ) on the Wiener space W (M ) with Wiener measure ν. This gives a possible prescription for the path integral representation of the quantized Hamiltonian, as well as yielding such a result for the natural geometric approximation schemes originating in [3] and followed by [34].
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