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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.

We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(x j) on a random set of points x j in the unit cube (the " sampling problem for trigonometric polynomi-als ") and estimate the probability distribution of the condition number for the associated Vandermonde-type and Toeplitz-like… (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. 1. Introduction. There is a huge literature on the subject of transition probabilities of random… (More)

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

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

- Richard F Bass
- 2004

This paper is a survey of uniqueness results for stochastic differential equations with jumps and regularity results for the corresponding harmonic functions. 1 1. Introduction. Researchers have increasingly been studying models from economics and from the natural sciences where the underlying randomness contains jumps. To give an example from financial… (More)

We consider a class of pure jump Markov processes in R d 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 investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.

We study the random sampling of band-limited functions of several variables. If a band-limited function with bandwidth has its essential support on a cube of volume R d , then O(R d log R d) random samples suffice to approximate the function up to a given error with high probability.

- Siva R Athreya, Richard F Bass, Maria Gordina, Edwin A Perkins
- 2006

We consider the operator Lf ðxÞ ¼ 1 2 X 1