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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 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)
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)