Results 1  10
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28
Heat kernel estimates for Dirichlet fractional Laplacian
 J. European Math. Soc
"... In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally ..."
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Cited by 36 (19 self)
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In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C 1,1 open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a nonlocal operator on open sets.
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.
The entropy formula for linear heat equation
 J. Geom. Anal
, 2004
"... ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the wellknown Li–Yau’s gradient estimate. As a byproduct we obtain the sharp ..."
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Cited by 28 (11 self)
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ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the wellknown Li–Yau’s gradient estimate. As a byproduct we obtain the sharp estimates on ‘Nash’s entropy ’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li–Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n. In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman’s entropy in the case of Riemann surfaces. 1. The relation with Li–Yau’s gradient estimates In this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li–Yau’s gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash’s ‘entropy quantity ’ − � M H log Hdvin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9]. Let u(x, t) be a positive solution to � ∂ ∂t − � � u(x, t) = 0 with �
Twosided heat kernel estimates for censored stablelike processes
, 2008
"... In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrins ..."
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Cited by 24 (17 self)
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In this paper we study the precise behavior of the transition density functions of censored (resurrected) αstablelike processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored αstablelike process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp twosided estimates for the transition density functions of a large class of censored αstablelike processes in C 1,1 open sets. We further obtain sharp twosided estimates for the Green functions of these censored αstablelike processes in bounded C 1,1 open sets.
Optimal Smoothing and Decay Estimates for Viscously Damped Conservation Laws, with Application to the 2D Navier Stokes Equation
, 1994
"... Optimal bounds on the Lp (IR n) smoothing and decay are established for certain viscously damped conservations laws, of which the vorticity formulation of the Navier–Stokes equation on IR 2 is a basic example. From the smoothing bounds, we obtain pointwise bounds that provide optimal control on th ..."
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Cited by 13 (1 self)
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Optimal bounds on the Lp (IR n) smoothing and decay are established for certain viscously damped conservations laws, of which the vorticity formulation of the Navier–Stokes equation on IR 2 is a basic example. From the smoothing bounds, we obtain pointwise bounds that provide optimal control on the spatial decay of solutions. We apply this in a study of the physically important case in which the integral of the initial data (i.e., the total vorticity in the example) vanishes. We show in this case that as the time t increases, the L 1 (IR n) norm of the solution decays to zero in two stages: for large initial data, there is a slow decay period during which the L 1 (IR n) norm falls off with an inverse power of the logarithm of t. Then, once the norm has fallen below a critical value, it decays away to zero with t −1/2. Again, this is optimal, and all of the constants in these estimates are explicitly computable in terms of the initial data.
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
QUENCHED NONEQUILIBRIUM CENTRAL LIMIT THEOREM FOR A TAGGED PARTICLE IN THE EXCLUSION PROCESS WITH BOND DISORDER
, 2006
"... Abstract. For a sequence of i.i.d. random variables {ξx: x ∈ Z} bounded above and below by strictly positive finite constants, consider the nearestneighbor onedimensional simple exclusion process in which a particle at x (resp. x + 1) jumps to x + 1 (resp. x) at rate ξx. We examine a quenched noneq ..."
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Cited by 7 (4 self)
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Abstract. For a sequence of i.i.d. random variables {ξx: x ∈ Z} bounded above and below by strictly positive finite constants, consider the nearestneighbor onedimensional simple exclusion process in which a particle at x (resp. x + 1) jumps to x + 1 (resp. x) at rate ξx. We examine a quenched nonequilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder {ξx: x ∈ Z}. We prove that the position of the tagged particle converges under diffusive scaling to a Gaussian process if the other particles are initially distributed according to a Bernoulli product measure associated to a smooth profile ρ0: R → [0, 1]. 1.
A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations
, 2003
"... We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations. ..."
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Cited by 5 (5 self)
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We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations.
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