Related papers: Stochastic processes induced by singular operators
One of the broadest concepts of measurement in quantum theory is the generalized measurement. Another paradigm of measurement--arising naturally in quantum optics, among other fields--is that of continuous-time measurements, which can be…
The paper addresses the question whether a random functional, a map from a set $E$ into the space of real-valued measurable functions on a probability space, has a measurable version with values in ${\mathbb R}^E$. Similarly, one may ask…
This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of…
We address the problem of continual learning in multi-task Gaussian process (GP) models for handling sequential input-output observations. Our approach extends the existing prior-posterior recursion of online Bayesian inference, i.e.\ past…
In this paper we present a new extension of the theory of well-bounded operators to cover operators with complex spectrum. In previous work a new concept of the class of absolutely continuous functions on a nonempty compact subset $\sigma$…
Stochastic processes are a flexible and widely used family of models for statistical modeling. While stochastic processes offer attractive properties such as inclusion of uncertainty properties, their inference is typically intractable,…
Motivated by practical applications, I present a novel and comprehensive framework for operator-valued positive definite kernels. This framework is applied to both operator theory and stochastic processes. The first application focuses on…
The sharp asymptotics for the L^2-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.
We consider the multiple products of relevant and marginal scalar composite operators at the Gaussian fixed-point in $D=4$ dimensions. This amounts to perturbative construction of the $\phi^4$ theory where the parameters of the theory are…
We provide a version of the stochastic Fubini's theorem which does not depend on the particular stochastic integrator chosen as far as the stochastic integration is built as a continuous linear operator from an $L^p$ space of Banach…
We develop a Bayesian approach to learning from sequential data by using Gaussian processes (GPs) with so-called signature kernels as covariance functions. This allows to make sequences of different length comparable and to rely on strong…
Gaussian processes provide a compact representation for modeling and estimating an unknown function, that can be updated as new measurements of the function are obtained. This paper extends this powerful framework to the case where the…
The required set of operations for universal continuous-variable quantum computation can be divided into two primary categories: Gaussian and non-Gaussian operations. Furthermore, any Gaussian operation can be decomposed as a sequence of…
The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $\mathbb R^m$ this notion is near to the separate continuity for which it is required only…
The behaviour of the generalized Hilbert operator associated with a positive finite Borel measure $\mu$ on $[0,1)$ is investigated when it acts on weighted Banach spaces of holomorphic functions on the unit disc defined by sup-norms and on…
The relationship between discontinuous and continuous stochastic processes in Hilbert space is investigated. It is shown that for any continuos process there is a parent discontinuous process, that becomes the continuous one in the proper…
Conditions are established under which the optimal control of processes having both absolutely continuous and singular (with respect to time) controls are equivalent to linear programs over a space of measures on the state and control…
Gaussian states, operations, and measurements are central building blocks for continuous-variable quantum information processing which paves the way for abundant applications, especially including network-based quantum computation and…
We study a family of free stochastic processes whose covariance kernels $K$ may be derived as a transform of a tempered measure $\sigma$. These processes arise, for example, in consideration non-commutative analysis involving free…
We provide a full characterization in terms of the six parameters involved the boundedness of all standard weighted integral operators induced by harmonic Bergman-Besov kernels acting between different Lebesgue classes with standard weights…