Richard F. Bass

Harnack Inequalities for Jump Processes

We consider a class of pure jump Markov processes in Rd whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.

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.

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 Probabilities for Symmetric Jump Processes

We consider symmetric Markov chains on the integer lattice in d dimensions, where a ∈ (0, 2) and the conductance between x and y is comparable to |x-y| -(d+α) . We establish upper and lower bounds for the transition probabilities that are sharp up to constants.

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.

Stochastic differential equations with jumps

This paper is a survey of uniqueness results for stochastic differential equations with jumps and regularity results for the corresponding harmonic functions.

Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order

ABSTRACT We consider a class of integrodifferential operators and their corresponding harmonic functions. Under mild assumptions on the family of jump measures we prove a priori estimates and establish Hölder continuity of bounded functions that are harmonic in a domain.

Stability of parabolic Harnack inequalities on metric measure spaces

Let

Uniqueness of Brownian motion on Sierpinski carpets

(X,d,\mu)

Uniqueness for diffusions with piecewise constant coefficients

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