Stochastic Processes Induced by Singular Operators

Stochastic Analysis and Applications

2016

We study kernel functions, and associated reproducing kernel Hilbert spaces H over infinite, discrete, and countable sets V . Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding approximate solutions to boundary value problems; using multiresolution-subdivision schemes in continuous domains. In this article, we turn the tables: Our object of study is realistic infinite discrete models in their own right; and we then use an analysis of suitable continuous counterpart problems, but now serving as a tool for obtaining solutions in the discrete world.

“…Readers are referred to the following papers, and the references cited there, for details regarding this: [25][26][27][28][29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Graph Laplacians and discrete reproducing kernel Hilbert spaces from restrictions

Stochastic Analysis and Applications

2016

“…Such an extension will adapt much better to a host of applications, including problems from probability theory, from stochastic processes [AJ12,JT16a], from mathematical physics [HKL + 14, RAKK05], and measurable dynamics; and for the latter, especially to the context of reversible processes (see, e.g., [TB13,CXY15,Sko13,Her12,DJ11,Rob11], and also [BP17,PSS16]).…”

Section: Introductionmentioning

confidence: 99%

Realizations and Factorizations of Positive Definite Kernels

2018

J Theor Probab

Given a fixed sigma-finite measure space (X, B, ν), we shall study an associated family of positive definite kernels K. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure ν is infinite, but sigmafinite. We introduce such positive definite kernels K (·, ·) with the two variables from the subset of the sigma-algebra B, sets having finite ν measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for K; and we give necessary and sufficient conditions for K to have realizations and factorizations in L 2 (ν). Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals.

“…As for positive definite functions, their use and applications are extensive and includes such areas as stochastic processes, see e.g., [JP13,AJSV13,JP12,AJ12]; harmonic analysis (see [JÓ00]) , and the references there); potential theory [Fug74,KL14]; operators in Hilbert space [Alp92,AD86]; and spectral theory [AH13,Nus75,Dev72,Dev59]. We stress that the literature is vast, and the above list is only a small sample.…”

Section: The Reproducing Kernel Hilbert Space H Fmentioning

confidence: 99%

Von Neumann indices and classes of positive definite functions

Journal of Mathematical Physics

2014

Abstract. With view to applications, we establish a correspondence between two problems: (i) the problem of finding continuous positive definite extensions of functions F which are defined on open bounded domains Ω in R, on the one hand; and (ii) spectral theory for elliptic differential operators acting on Ω, (constant coefficients.) A novelty in our approach is the use of a reproducing kernel Hilbert space H F computed directly from (Ω, F ), as well as algorithms for computing relevant orthonormal bases in H F .