Results 1  10
of
16
Random walk on supercritical percolation clusters
 ANN. PROBAB
, 2003
"... We obtain Gaussian upper and lower bounds on the transition density qt(x, y) of the continuous time simple random walk on a supercritical percolation cluster C ∞ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants ..."
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Cited by 44 (2 self)
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We obtain Gaussian upper and lower bounds on the transition density qt(x, y) of the continuous time simple random walk on a supercritical percolation cluster C ∞ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for qt(x, ·) only holds for t ≥ Sx(ω), where the constant Sx(ω) depends on the percolation configuration ω.
Gaussian estimates for Markov chains and random walks on groups
 Ann. Probab
, 1993
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
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Cited by 37 (2 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
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Cited by 33 (2 self)
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Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 1.
Nonlocal Dirichlet forms and symmetric jump processes
 Transactions of the American Mathematical Society
, 1999
"... We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x ..."
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Cited by 31 (16 self)
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We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x  −d−α ≤ J(x, y) ≤ κ2y − x  −d−β for 0 < α < β < 2, x − y  < 1. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F). We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 27 (3 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
The Moser's Iterative Method for a Class of Ultraparabolic Equations
 Commun. Contemp. Math
, 2004
"... We adapt the iterative scheme by Moser... ..."
Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps
, 2008
"... We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the c ..."
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Cited by 8 (6 self)
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We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.
On uniformly subelliptic operators and stochastic area
, 2006
"... We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by SaloffCoste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then observed by Sturm that many proofs extend naturally to the setting ..."
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Cited by 5 (4 self)
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We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by SaloffCoste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then observed by Sturm that many proofs extend naturally to the setting of locally compact Dirichlet spaces. We relate these results to what is known as rough path theory by showing that they provide a natural and powerful analytic machinery for construction and study of (random) geometric Hölder rough paths. (In particular, we obtain a simple construction of the LyonsStoica stochastic area for a diffusion process with uniformly elliptic generator in divergence form.) Our approach then enables us to establish a number of farreaching generalizations of classical theorems in diffusion theory including WongZakai approximations, FreidlinWentzell sample path large deviations and the StroockVaradhan support theorem. The latter was conjectured by T. Lyons in his recent St. Flour lecture. 1
Secondorder subelliptic operators on Lie groups II: real measurable principal coefficients
, 1999
"... Let G be a connected Lie group with Lie algebra g and a 1 ; : : : ; a d 0 an algebraic basis of g. Further let A i denote the generators of left translations, acting on the L p  spaces L p (G ; dg) formed with left Haar measure dg, in the directions a i . We consider secondorder operators H = \G ..."
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Cited by 3 (2 self)
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Let G be a connected Lie group with Lie algebra g and a 1 ; : : : ; a d 0 an algebraic basis of g. Further let A i denote the generators of left translations, acting on the L p  spaces L p (G ; dg) formed with left Haar measure dg, in the directions a i . We consider secondorder operators H = \Gamma d 0 X i;j=1 A i c ij A j + d 0 X i=1 (c i A i + A i c 0 i ) + c 0 I corresponding to a quadratic form with real measurable coefficients c ij and complex c i , c 0 i , c 0 2 L1 . The matrix C = (c ij ) of principal coefficients, which is not necessarily symmetric, is assumed to satisfy the subellipticity condition !C = 2 \Gamma1 i C + C j I ? 0 uniformly over G. We prove that H generates a strongly continuous holomorphic semigroup S on L 2 with a kernel K which satisfies Gaussian bounds jK z (g ; h)j a jzj \GammaD 0 =2 e !jzj e \Gammab(jgh \Gamma1 j 0 ) 2 jzj \Gamma1 for g; h 2 G and z in a subsector (`) of the sector of holomorphy. Moreover, the...
Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators
 Trans. Amer. Math. Soc
"... Abstract. We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators. 1. ..."
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Cited by 3 (1 self)
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Abstract. We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators. 1.