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 well-known Li–Yau’s gradient estimate. As a by-product 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 �

We consider the symmetric non-local 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 κ1|y − x | −d−α ≤ J(x, y) ≤ κ2|y − 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.

In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp two-sided 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 two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

In this paper we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like 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 α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C 1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C 1,1 open sets.

2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

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.

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.

this paper is to obtain precise quantitative bounds on the constants c 2

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 one-dimensional 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.

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