Upper Bounds for Symmetric Markov Transition Functions.

Cárlen, Eric A.; Kusuoka, Shigeo; Stroock, Daniel W.

doi:10.21236/ada170010

Cited by 272 publications

(346 citation statements)

References 6 publications

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“…In particular, during the past 30 years many authors attacked the problem of describing the global behavior of the heat diffusion kernel p (t, x, y) on various Euclidean domains and manifolds. See for instance [3,7,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,26,30,31,33,32,34,48,52,56,55,60,61,62,63,64].…”

Section: Motivationmentioning

confidence: 99%

Heat kernel on manifolds with ends

Grigor’yan¹,

Saloff‐Coste²

2009

Ann. inst. Fourier

103

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Section: Motivationmentioning

confidence: 99%

Heat kernel on manifolds with ends

Grigor’yan¹,

Saloff‐Coste²

2009

Ann. inst. Fourier

103

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“…2). In two-dimensions, the walk becomes the weighted nearest neighbor walk on C(n, 2) with loops added and weighted.…”

Section: The Metropolis Algorithm In a Boxmentioning

confidence: 99%

Nash inequalities for finite Markov chains

1996

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This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) 2 steps are necessary and suffice to reach stationarity. We consider local Poincar6 inequalities and use them to prove Nash inequalities. These are bounds on (,_-norms in terms of Dirichlet forms and l~-norms which yield decay rates for iterates of the kernel. This method is adapted from arguments developed by a number of authors in the context of partial differential equations and, later, in the study of random walks on infinite graphs. The main results do not require reversibility.

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“…[21], [9], [12], [4], [18], [5]). Since the work of Nash [19] and Aronson [1], many methods have been discovered for deriving Gaussian upper and lower bounds of H(x, y, t), see e.g., [7], [18], [10], [13], [11], [17].…”

Section: Introductionmentioning

confidence: 99%

Davies type estimate and the heat kernel bound under the Ricci flow

Zhu

2015

Trans. Amer. Math. Soc.

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Abstract. We prove a Davies type double integral estimate for the heat kernel H(y, t; x, l) under the Ricci flow. As a result, we give an affirmative answer to a question proposed in [8]. Moreover, we apply the Davies type estimate to provide a new proof of the Gaussian upper and lower bounds of H(y, t; x, l) which were first shown in [6]. IntroductionOn a complete Riemannian manifold (M n , g ij ), the heat kernel H(x, y, t), is the smallest positive fundamental solution to the heat equation ∂u ∂t = ∆u.( is the distance between U 1 and U 2 .In this paper, we consider the heat kernel of the time-evolving heat equation under the Ricci flow on a complete manifold M n , i.e., where ∆ t is the Laplacian with respect to a complete solution g ij (t), t ∈ [0, T ) and T < ∞, of the following Ricci flow equationThe existence and uniqueness of the heat kernel H(y, t; x, l) to (1.3) were proved in Our main goal is to give an affirmative answer to the question above. More specifically, we prove Theorem 1.2. Let (M n , g ij (t)) be a complete solution to (1.4) on [0, T ) and T < ∞. Suppose that H(y, t; x, l) is the heat kernel of (1.3), and Rc ≥ −K 1 on [0, T ) for some nonnegative constant K 1 . Then for any open sets, U 1 and U 2 , with compact closure in M, and 0 ≤ l < t < T , we havewhere C 0 is a constant depends only on n.Using Theorem 1.2, we are able to provide a new proof of the Gaussian upper bound and lower bounds of H(y, t; x, l) which was first shown by . For the Gaussian upper bound, we have where constants C 1 , C 2 ,, C 3 and C 4 depend only on n.Next, following a method of Li- Tam H(y, t; x, l) ≥ C 5 e −C 6 e C 7 Λ+C 8 K 1 T exp − 4d 2 t (x, y) (t − l)Vol t (B t (x, t−l 8 ))

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Upper Bounds for Symmetric Markov Transition Functions.

Cited by 272 publications

References 6 publications

Heat kernel on manifolds with ends

Heat kernel on manifolds with ends

Nash inequalities for finite Markov chains

Davies type estimate and the heat kernel bound under the Ricci flow

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