Mathematics – Probability
Scientific paper
2007-01-10
Mathematics
Probability
Scientific paper
Let $M$ be a compact Riemannian manifold without boundary isometrically embedded into $\Rnu^m$, $\W^x_{M,t}$ be the distribution of a Brownian bridge starting at $x\in M$ and returning to $M$ at time $t$. Let $Q_t: \C(M) \to \C(M)$, $(Q_t f)(x)=\int_{\C([0,1],\Rnu^m)}f(\om(t)) \W^x_{M,t}(d\om)$, and let $\mc P = \{0=t_0 < t_1 < ... < t_n=t\}$ be a partition of $[0,t]$. It was shown in a paper by O. G. Smolyanov, H. v. Weizsaecker, and O. Wittich that $Q_{t_1-t_0}... Q_{t_n-t_{n-1}} f \to e^{-t\frac{\lap_M}2}f, \text{as} |\mc P|\to 0$ in $\C(M)$. Taking into consideration integral representations: $(Q_{t_1-t_0}... Q_{t_n-t_{n-1}} f)(x)=\int_M q_{_{\mc P}}(x,y)f(y)\la_M(dy)$ and $(e^{-t\frac{\lap_M}2}f)(x)=\int_M h(x,y,t) f(y) \la_M(dy)$, where $\la_M$ is the volume measure on $M$, $h(x,y,t)$ is the heat kernel on $M$, one interprets this relation as a weak convergence in $\C(M)$ of the integral kernels: $q_{\mc P}(x,y)\to h(x,y,t)$. The present paper improves the result by Smolyanov and Weizsaecker, and shows that this convergence is uniform on $M\x M$. Keywords: Gaussian integrals on compact Riemannian manifolds, heat kernel, Smolyanov--Weizsaecker approach, Smolyanov--Weizsaecker surface measures
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