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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… (More)
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 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 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.
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.
We consider the stochastic differential equation dX t = dW t + dA t , where W t is d-dimensional Brownian motion with d ≥ 2 and the ith component of A t is a process of bounded variation that stands in the same relationship to a measure π i as t 0 f (X s) ds does to the measure f (x) dx. We prove weak existence and uniqueness for the above stochastic… (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 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.
A new technique for proving uniqueness of martingale problems is introduced. The method is illustrated in the context of elliptic diffusions in R d .
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)