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61
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 30 (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 co ..."
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Cited by 22 (2 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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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.
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...
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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. 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; ...
A HARMONIC MAP FLOW ASSOCIATED WITH THE STANDARD SOLUTION OF RICCI FLOW
, 2007
"... Let (Rn, g(t)), 0 ≤ t ≤ T, n ≥ 3, be a standard solution of the Ricci flow with radially symmetric initial data g0. We will extend a recent existence result of P. Lu and G. Tian and prove that for any t0 ∈ [0, T) there exists a solution of the corresponding harmonic map flow φt: (Rn, g(t)) → (Rn, ..."
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Cited by 3 (3 self)
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Let (Rn, g(t)), 0 ≤ t ≤ T, n ≥ 3, be a standard solution of the Ricci flow with radially symmetric initial data g0. We will extend a recent existence result of P. Lu and G. Tian and prove that for any t0 ∈ [0, T) there exists a solution of the corresponding harmonic map flow φt: (Rn, g(t)) → (Rn, g(t0)) satisfying ∂φt/∂t = ∆g(t),g(t0)φt of the form φt(r, θ) = (ρ(r, t), θ) in polar coordinates in Rn × (t0, T0), φt0 (r, θ) = (r, θ), where r = r(t) is the radial coordinate with respect to g(t) and T0 = sup{t1 ∈ (t0, T] : ‖eρ(·, t)‖L ∞ (R+) + ‖ ∂ eρ/∂r(·, t)‖L ∞ (R+) < ∞ ∀t0 < t ≤ t1} with eρ(r, t) = log(ρ(r, t)/r). We will also prove the uniqueness of solution of the harmonic map flow within the class of functions of the form φt(r, θ) = (ρ(r, t), θ), ρ(r, t) = re eρ(r,t) , for some function eρ(r, t). We will also use the same technique to prove that the solution u of the heat equation in (Ω \{0}) ×(0, T) has removable singularities at {0} × (0, T), Ω ⊂ R m, m ≥ 3, if and only if u(x, t)  = O(x  2−m) locally uniformly on every compact subset of (0, T).