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We consider a class of pure jump Markov processes in R 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.

We consider bounded entire solutions of the non-linear PDE ∆u+u−u 3 = 0 in 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 non-uniformly elliptic divergence form operator.

- Richard F. Bass, Karlheinz Gröchenig
- SIAM J. Math. Analysis
- 2005

We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(xj) on a random set of points xj in the unit cube (the “sampling problem for trigonometric polynomials”) and estimate the probability distribution of the condition number for the associated Vandermonde-type and Toeplitz-like matrices.… (More)

We consider a class of fractal subsets of Rd 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… (More)

- Richard F. Bass, Xia Chen, φεduds, Xia Chen
- 2004

BY RICHARD F. BASS AND XIA CHEN University of Connecticut and University of Tennessee If βt is renormalized self-intersection local time for planar Brownian motion, we characterize when Eeγβ1 is finite or infinite in terms of the best constant of a Gagliardo–Nirenberg inequality. We prove large deviation estimates for β1 and −β1. We establish lim sup and… (More)

Let (G,E) be a graph with weights {axy} for which a parabolic Harnack inequality holds with space-time scaling exponent β ≥ 2. Suppose {axy} is another set of weights that are comparable to {axy}. We prove that this parabolic Harnack inequality also holds for (G,E) with the weights {axy}. We also give stable necessary and sufficient conditions for this… (More)

- Richard F. Bass, MORITZ KASSMANN
- 2004

We consider harmonic functions with respect to the operator Lu(x) = ∫ [u(x+ h)− u(x)− 1(|h|≤1)h · ∇u(x)]n(x, h) dh. Under suitable conditions on n(x, h) we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator L is allowed to be anisotropic and of variable order.

Let L be a second-order partial differential operator in R e. Let R e be the finite union of disjoint polyhedra. Suppose that the diffusion matrix is everywhere non singular and constant on each polyhedron, and that the drift coefficient is bounded and measurable. We show that the martingale problem associated with L is well-posed.

dXt = dWt + dAt , where Wt is d-dimensional Brownian motion with d ≥ 2 and the ith component of At is a process of bounded variation that stands in the same relationship to a measure πi as ∫ t 0 f (Xs)ds does to the measure f (x)dx. We prove weak existence and uniqueness for the above stochastic differential equation when the measures πi are members of the… (More)

We consider symmetric Markov chains on the integer lattice in d dimensions, where α ∈ (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.