## Estimates of derivatives of the heat kernel on a compact Riemannian manifold.(English)Zbl 0948.58025

The author proves an estimate for the $$N$$-th covariant derivative of the logarithm of the heat kernel $$p(T,x,y)$$ on a compact Riemannian manifold $$M$$: $|\nabla^N\log{p(T,x,y)}|\leq c_N\left\{\frac{d(x,y)}{T}+\frac{1}{\sqrt{T}}\right\}^N$ with the correct order of magnitude. Here $$c_{N}$$ is a constant depending on $$N$$ and $$M$$, $$(T,x,y)\in]0,1]\times M\times M$$ (see also [S.-J. Sheu, Ann. Probab. 19, No. 2, 538-561 (1991; Zbl 0738.60060)], [R. S. Hamilton, Commun. Anal. Geom. 1, No. 1, 113-126 (1993; Zbl 0799.53048); D. W. Stroock and P. Malliavin, J. Differ. Geom. 44, No. 3, 550-570 (1996; Zbl 0873.58063), D. W. Stroock, N. Ikeda (ed.) et al., “Ito’s stochastic calculus and probability theory. Tribute dedicated to Kiyosi Ito on the occasion of his 80th birthday.” Tokyo: Springer, 355-371 (1996; Zbl 0868.58075)] or [D. W. Stroock and O. Zeitouni, Hommage à P. A. Meyer et J. Neveu. Paris: Société Mathématique de France, Astérisque. 236, 291-301 (1996; 863.58065)]. For the proof, the author applies repeatedly Ito’s formula for $$\log{p}(T-t,u_{t},y)$$, where $$u_{t}$$ is the horizontal lift, with initial value $$u_{0}$$, of the Brownian bridge from $$x$$ to $$y$$, and where $$u_{0}$$ is an orthonormal frame over $$x$$. Therefore, $$\nabla^{N}\log{p}(T,x,y)$$ is expressed in terms of lower derivatives and the estimate is obtained by induction.

### MSC:

 58J65 Diffusion processes and stochastic analysis on manifolds 60J60 Diffusion processes
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