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Full-Text Articles in Other Mathematics
Spectral Theory For Gaussian Processes: Reproducing Kernels, Random Functions, Boundaries, And L2-Wavelet Generators With Fractional Scales, Daniel Alpay
Mathematics, Physics, and Computer Science Faculty Articles and Research
A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by a host of applications, e.g., in mathematical physics, and in stochastic differential equations, and their use in financial models. In this paper, we show that, for three classes of cases in the correspondence, it is possible to obtain explicit formulas which are amenable to computations of the respective Gaussian stochastic processes. For achieving this, we first develop two functional analytic tools. They are: …
Stochastic Processes Induced By Singular Operators, Daniel Alpay, Palle Jorgensen
Stochastic Processes Induced By Singular Operators, Daniel Alpay, Palle Jorgensen
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure σ on Rn. The case when σ is assumed absolutely continuous with respect to Lebesgue measure was stud- ied earlier in the literature, when n = 1. Our focus here is on showing how different equivalence classes (defined from relative absolute continuity for pairs of measures) translate into concrete spectral decompositions of the corresponding stochastic processes under study. The measures σ we consider are typically purely singular. Our proofs rely on the theory of (singular) unbounded operators in …