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40
SubGaussian estimates of heat kernels on infinite graphs
 Duke Math. J
, 2000
"... We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. ..."
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Cited by 32 (10 self)
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We prove that a two sided subGaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay.
Gaussian Upper Bounds For The Heat Kernel On Arbitrary Manifolds
, 1997
"... In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional ..."
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Cited by 23 (1 self)
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In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...
A scheme for simulating onedimensional diffusion processes with discontinuous coefficients
 ANN. APPL. PROBAB
, 2006
"... The aim of this article is to provide a scheme for simulating diffusion processes evolving in onedimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of t ..."
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Cited by 14 (8 self)
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The aim of this article is to provide a scheme for simulating diffusion processes evolving in onedimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion.
Boundary Harnack principle for Brownian motions with measurevalued drifts in bounded Lipschitz domains
 MATHEMATISCHE ANNALEN
, 2007
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Quantitative estimates of unique continuation for parabolic equations, determination of unknown timevarying boundaries and optimal stability estimates
, 2007
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Asymptotic Behaviour Of A NonAutonomous Population Equation With Diffusion In
"... . We prove existence and uniqueness of positive solutions of an agestructured population equation of McKendrick type with spatial diffusion in L 1 . The coefficients may depend on age and position. Moreover, the mortality rate is allowed to be unbounded and the fertility rate is time dependent. I ..."
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Cited by 4 (0 self)
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. We prove existence and uniqueness of positive solutions of an agestructured population equation of McKendrick type with spatial diffusion in L 1 . The coefficients may depend on age and position. Moreover, the mortality rate is allowed to be unbounded and the fertility rate is time dependent. In the time periodic case, we estimate the essential spectral radius of the monodromy operator which gives information on the asymptotic behaviour of solutions. Our work extends previous results in [NR], [Rh1], [Th1], and [Th2] to the nonautonomous situation. We use the theory of evolution semigroups and extrapolation spaces. 1. Introduction The investigation of an agestructured population of McKendrick type with age and space dependent spatial diffusion leads to the mathematical model (P ) 8 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ! ? ? ? ? ? ? ? ? ? ? ? ? ? ? : @ t u(t; a; x) + @ a u(t; a; x) = n X k;l=1 @ k a kl (a; x) @ l u(t; a; x) + n X k=1 b k (a; x) @ k u(t; a; x) +c(a; x)u(t; a; ...
Hitting probabilities for Brownian motion on Riemannian manifolds
"... this paper is to estimate the hitting probability function / K (t; x) := P x (9 s 2 [0; t] ; X s 2 K) where K ae M is a fixed compact set. In words, / K (t; x) is the probability that Brownian motion started at x hits K by time t. Our goal is to obtain precise estimates on / K for all t ? 0 and x ..."
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Cited by 4 (2 self)
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this paper is to estimate the hitting probability function / K (t; x) := P x (9 s 2 [0; t] ; X s 2 K) where K ae M is a fixed compact set. In words, / K (t; x) is the probability that Brownian motion started at x hits K by time t. Our goal is to obtain precise estimates on / K for all t ? 0 and x outside a neighborhood of K, hence avoiding the somewhat different question of the behavior of / K near the boundary of K. In the context of Riemannian manifolds, this natural question has been considered only in a handful of papers including [2], [4]. We were led to study / K in our attempt to develop sharp heat kernel estimates on manifolds with more than one end. Indeed, the proof of the heat kernel estimates announced in [20] depends in a crucial way on the results of the present paper (see [21]). In this context, it turns out to be important to estimate also the time derivative @ t / K (t; x) which is a positive function. We develop a general approach which allows to obtain estimates of / K in terms of the heat kernel p(t; x; y) or closely related objects such as the Dirichlet heat kernel p U (t; x; y) of some open set U . In the case when X t is transient, that is, M is nonparabolic, we show that the behavior of / K (t; x), away from K, is comparable to that of Z t 0 p(s; x; y)ds; where y is a reference point on @K. If (X t ) t?0 is recurrent, that is, M is parabolic, we obtain similar estimates through Z t 0 p U (s; x; y)ds where U is a certain region slightly larger than\Omega := M n K. We also show that @ t / K (t; x) is comparable to p\Omega (t; x; y) where y stays at a certain distance from @K. For precise statements, see Theorems 3.3, 3.5, 3.7 and Corollaries 3.9, 3.10. Using the known results concerning the heat kernel p(t; x; y) and the results of [23...
Exponential separation and principal Floquet bundles for linear parabolic equations on R^N
 INDIANA UNIV. MATH. J
"... We consider linear nonautonomous second order parabolic equations on R^N. Under an instability condition, we prove the existence of two complementary Floquet bundles, one spanned by a positive entire solution the principal Floquet bundle, the other one consisting of signchanging solutions. We esta ..."
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Cited by 4 (0 self)
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We consider linear nonautonomous second order parabolic equations on R^N. Under an instability condition, we prove the existence of two complementary Floquet bundles, one spanned by a positive entire solution the principal Floquet bundle, the other one consisting of signchanging solutions. We establish an exponential separation between the two bundles, showing in particular that a class of signchanging solutions are exponentially dominated by positive solutions.
Doubling Properties For Second Order Parabolic Equations
"... . We prove the doubling property of Lcaloric measure corresponding to the second order parabolic equation in the whole space and in Lipschitz domains. For parabolic equations in the divergence form, a weaker form of the doubling property follows easily from a recent result, the backward Harnack ine ..."
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Cited by 3 (1 self)
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. We prove the doubling property of Lcaloric measure corresponding to the second order parabolic equation in the whole space and in Lipschitz domains. For parabolic equations in the divergence form, a weaker form of the doubling property follows easily from a recent result, the backward Harnack inequality, and known estimates of the Green's function. Our method works for both the divergence and nondivergence cases. Moreover, the backward Harnack inequality and estimates of the Green's function are not needed in the course of proof.