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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 \(d\).

MSC:

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

Citations:

Zbl 0502.93076
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References:

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