## 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

Zbl 0502.93076
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### References:

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