Feng Tian

Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive definite kernel is also the covariance kernel of a Gaussian process. Given a fixed sigma-finite measure $\mu$, we consider positive definite kernels defined on the subset of the sigma algebra having finite $\mu$ measure. We show that then the corresponding Hilbert factorizations consist of signed measures, finitely additive, but not automatically sigma-additive. We give a necessary and sufficient condition for when the measures in the RKHS, and the Hilbert factorizations, are sigma-additive. Our emphasis is the case when $\mu$ is assumed non-atomic. By contrast, when $\mu$ is known to be atomic, our setting is shown to generalize that of Shannon-interpolation. Our RKHS-approach further leads to new insight into the associated Gaussian processes, their It\^{o} calculus and diffusion. Examples include fractional Brownian motion, and time-change processes.

We study positive transfer operators $R$ in the setting of general measure spaces $\left(X,\mathscr{B}\right)$. For each $R$, we compute associated path-space probability spaces $\left(Ω,\mathbb{P}\right)$. When the transfer operator $R$ is compatible with an endomorphism in $\left(X,\mathscr{B}\right)$, we get associated multiresolutions for the Hilbert spaces $L^{2}\left(Ω,\mathbb{P}\right)$ where the path-space $Ω$ may then be taken to be a solenoid. Our multiresolutions include both orthogonality relations and self-similarity algorithms for standard wavelets and for generalized wavelet-resolutions. Applications are given to topological dynamics, ergodic theory, and spectral theory, in general; to iterated function systems (IFSs), and to Markov chains in particular.

We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.

Given a fixed sigma-finite measure space $\left(X,\mathscr{B},\nu\right)$, 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 $\nu$ is infinite, but sigma-finite. We introduce such positive definite kernels $K\left(\cdot,\cdot\right)$ with the two variables from the subset of the sigma-algebra $\mathscr{B}$, sets having finite $\nu$ 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 $L^{2}\left(\nu\right)$. 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.

In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum probability, there has also been much research which seeks to study fractal and multiresolution structures with the use of certain systems of non-commutative operators; non-commutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these non-commutative algebras are the Cuntz algebras; denoted $\mathcal{O}_{N}$, $N$ for the number of isometry generators. $N$ is at least 2. Our focus is on the representations of $\mathcal{O}_{N}$. We aim to develop new non-commutative tools, involving both representation theory and stochastic processes. They serve to connect these parallel developments. In outline, boundaries, Poisson, or Martin, are certain measure spaces (often associated to random walk models), designed to encode the asymptotic behavior, e.g., how trajectories diverge when the number of steps goes to infinity. We stress that our present boundaries (commutative or non-commutative) are purely measure-theoretical objects. Although, as we show, in some cases our boundaries may be compared with more familiar topological boundaries.

We give explicit transforms for Hilbert spaces associated with positive definite functions on $\mathbb{R}$, and positive definite tempered distributions, incl., generalizations to non-abelian locally compact groups. Applications to the theory of extensions of p.d. functions/distributions are included. We obtain explicit representation formulas for positive definite tempered distributions in the sense of L. Schwartz, and we give applications to Dirac combs and to diffraction. As further applications, we give parallels between Bochner's theorem (for continuous p.d. functions) on the one hand, and the generalization to Bochner/Schwartz representations for positive definite tempered distributions on the other; in the latter case, via tempered positive measures. Via our transforms, we make precise the respective reproducing kernel Hilbert spaces (RKHSs), that of N. Aronszajn and that of L. Schwartz. Further applications are given to stationary-increment Gaussian processes.