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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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