Uniform approximation of the heat kernel on a manifold

We approximate the heat kernel $h(x,y,t)$ on a compact connected Riemannian manifold $M$ without boundary uniformly in $(x,y,t)\in M\times M\times [a,b]$, $a>0$, by $n$-fold integrals over $M^n$ of the densities of Brownian bridges. Moreover, we provide an estimate for the uniform convergence rate. As an immediate corollary, we get a uniform approximation of solutions of the Cauchy problem for the heat equation on $M$.

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