Estimates for transition probabilities on a compact manifold (Q859650)

The purpose of this note is to describe a procedure for transferring familiar estimates for transition probabilities on \(\mathbb{R}^N\) to transition probabilities on compact manifolds, maybe filling a gap in the literature. Thus, the author considers, on a compact Riemannian manifold \((M^N, d)\), a uniformly elliptic order two operator \(L\), having Hölder continuous coefficients, and deduces from the well known Euclidean case, under some mild technical conditions, that \(L\) gives rise to a heat equation kernel \(p_r(m,m')\) satisfying Gaussian estimates: \(\exists c>0\), \(\forall t>0\), \(\forall m,m'\in M\), \[ \frac{1}{c\min \{t^{N/2}, 1\}}\exp\left[-\frac ct d(m,m')^2\right]\leq p_t(m,m')\leq\frac{c}{\min \{t^{N/2},1\}}\exp\left[-\frac{d(m,m')^2}{ct}\right]. \]