Sharp explicit lower bounds of heat kernels. (English) Zbl 0918.58070

Let \(M\) be a complete Riemannian manifold of dimension \(d\) and let \(L={1\over 2}(\Delta+Z)\) where \(Z\) is a \(C^1\)-vector field. Denote by \(\rho_t(x,y)\) the heat kernel of \(L\).
The author uses a probabilistic approach called “logarithmic transformation” to prove explicit lower bounds for \(\rho_t\). The advantage of the probabilistic approach by W. H. Fleming [Lect. Notes Contr. Inf. Sci. 42, 131-141 (1982; Zbl 0502.93076)] is that it can be applied also in the case of unbounded curvature. A major assumption made in this paper is that the Ricci curvature of \(M\) can be bounded from below by a function depending quadratically on the distance function \(\rho_y(x)\), i.e., \(\text{Ric}(X,X)\geq-(k_1+k_2\rho_y(x))^2(d-1) | X|^2, X\in T_xM\), where \(k_i, i=1,2\) are positive constants depending on \(y\) and \(y\in M\) is a fixed point. Further he assumes that \(b_2:=\sup\{\langle\nabla_UZ,U\rangle:U\in TM, | U|=1\}\) is bounded and defines \(b_1:=| Z(y)|\) and \(\beta_i:=\max\{0,(d-1)k_i+ b_i\}\). The main theorem then states that for any \(\sigma>0\) one has \[ \rho_t(x,y)\geq (2\pi t)^{-d/2}\exp\{-f_1(\sigma,t)\rho^2 (x,y)-f_2(\sigma, t)\}, \quad t>0,\;x,y\in M, \] where \(f_i, i=1,2\) are two positive functions given in the paper and depending on terms involving \(\sqrt{t}\) up to fifth order and on \(\sigma,\beta_1, \beta_2\) and \(d\).


58J35 Heat and other parabolic equation methods for PDEs on manifolds
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions


Zbl 0502.93076
Full Text: DOI


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