Martin T. Barlow

Brownian motion on the Sierpinski gasket

SummaryWe construct a “Brownian motion” taking values in the Sierpinski gasket, a fractal subset of ℝ2, and study its properties. This is a diffusion process characterized by local isotropy and homogeneity properties. We show, for example, that the process has a continuous symmetric transition density, pt(x,y), with respect to an appropriate Hausdorff measure and obtain estimates on pt(x,y).

Brownian Motion and Harmonic Analysis on Sierpinski Carpets

We consider a class of fractal subsets of R d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincar´ e inequalities to this setting.

Non-local Dirichlet forms and symmetric jump processes

We consider the non-local symmetric Dirichlet form (E,F) given by with F the closure with respect to E 1 of the set of C 1 functions on R d with compact support, where E 1 (f, f):= E(f, f) + f Rd f(x) 2 dx, and where the jump kernel J satisfies for 0 < α < β < 2, |x - y| < 1. This assumption allows the corresponding jump process to have jump intensities whose sizes depend 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 non-negative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.

Random walks on supercritical percolation clusters

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 C1 in the Euclidean lattice. The bounds, analogous to Aronsens bounds for uniformly elliptic divergence form diusions, 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(!), where the constant Sx(!) depends on the percolation congura- tion !.

The Liouville Property and a Conjecture of De Giorgi

We consider bounded entire solutions of the nonlinear PDE Δu + u u 3 = 0i n R d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes. The proof uses a Liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator. c 2000 John Wiley & Sons, Inc.

Transition densities for Brownian motion on the Sierpinski carpet

SummaryUpper and lower bounds are obtained for the transition densitiesp(t, x, y) of Brownian motion on the Sierpinski carpet. These are of the same form as those which hold for the Sierpinski gasket. In addition, the joint continuity ofp(t, x, y) is proved, the existence of the spectral dimension is established, and the Einstein relation, connecting the spectral dimension, the Hausdorff dimension and the resistance exponent, is shown to hold.

Stability of parabolic Harnack inequalities on metric measure spaces

Let

Invariance principle for the random conductance model with unbounded conductances

(X,d,\mu)

Uniqueness of Brownian motion on Sierpinski carpets

be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global space-time scaling exponent

Heat kernel upper bounds for jump processes and the first exit time

\beta\ge 2