Random fields on Hilbert spaces with their equivalent Gaussian measures

In this paper, we study the parametrization of conservative extensions of minimal dynamic semigroups used in the quantum theory of open systems for describing irreversible processes. The particular case of extensions parametrized by normal states is discussed in [1, 2]. The construction used below is similar to that of self-ajoint extensions for densely-defined, bounded below, symmetric operators in a Hilbert space. The dynamic semigroup, unlike the unitary operator group, acts not on elements of a Hilbert space 7/but on elements of a topological operator algebra in a Hilbert space B(H); moreover, this semigroup is completely positive (see [3]). For such semigroups, direct methods of describing domains of generating mappings are absent; this complicates the parametrization of extensions of the minimal generating mapping. It turns out that the range of the resolvent of the minimal dynamic semigroup belongs to the complement of the zero set (the kernel) of the limit mapping QOO(.) associated with the formal generating mapping of the dynamic semigroup. For this reason, we first consider general properties of the zero set for a completely positive mapping (see Sec. 1). In what follows, the C*-algebra generated by positive roots of a mapping Q(-) plays the central role. In Sec. 2, we prove that this algebra has the following property: it is maximal in the class of algebras that contain an element X if and only if X* X and X X* are elements of the algebra. In Sec. 3, the mentioned property is extended to the set of positive operators that are zeros of the limit mapping QOO(.) _- s-lira Qn (.). We prove that the domain of the infinitesimal mapping of the minimal dynamic semigroup is contained in the maximal C*-Mgebra generated by positive roots of the mapping Q~(-) associated with the semigroup. In Sec. 4, we give a construction of conservative extensions of the minimal dynamic semigroup, which is based on limit properties of the resolvent mapping. In Sec. 5, we prove the characteristic theorem for completely positive mappings with given ranges, which enables us to consider the set of conservative extensions as a *-weakly compact convex manifold in/3+(/2(7-~)), whose points admit an expansion in the set of extreme elements. The barycentric expansion of elements T in extreme points of E C T establishes a one-to-one correspondence between classes of equivalent normed measures on E and conservative extensions of the minimal dynamic semigroup; in this case, E plays the role of boundary and the class of equivalent measures plays the role of boundary condition for a conservative extension.

We generalize the Gross' method of the construction of a Gaussian probability measure μ in the sense of Bernstein on infinite dimensional Banach spaces to an arbitrary separable Fréchet space M. Next we describe basic properties of the constructed measure and discuss analogies between Gaussian measures in Banach and Fréchet spaces. We investigate, among other things, the Hilbert space H of admissible shifts for the measure μ and show that the closed unit ball in H is a compact subset of M. Finally, we prove a generalized version of the Brunn-Minkowski inequality for Gaussian measures in Fréchet spaces, which is the main tool in the proof of strong limit theorems for sums of Gaussian random elements in Fréchet spaces.

We show that one can smoothly renorm the Hilbert space $H$ such that the rate of convergence in the central limit theorem becomes very bad. More precisely, let us fix a sequence $\xi_n \rightarrow 0$ and $\varepsilon > 0$. We can then construct a norm $N(\cdot)$ on the Hilbert space, and a bounded random variable $X$ on $H$ with the following properties: (a) The norm $N(\cdot)$ is $(1 + \varepsilon)$ equivalent to the usual norm. It is infinitely many times differentiable, and each differential is bounded on the unit sphere. (b) If $(X_i)$ denotes independent copies of $X$, and if $\gamma$ is the Gaussian measure with the same covariance as $X$, then the inequality $\operatorname{Sup}_{t>0}|P\{N(n^{-1/2} \sum^n_{i=1} X_i) \leq t\} - \gamma\{x; N(x) \leq t\}| \geq \xi_n$ occurs for infinitely many $n$.

Let P be a zero mean Gaussian measure in a Hilbert space. The asymptotic behavior of $P\{ {\left \| {x - b} \right \|^2} > \varepsilon \}$ as $\varepsilon \to \infty$ is studied in this note.

We consider a random field X(t) as a function taking values in the Hilbert space of all centred finite-dimensional random vectors with finite expectation of the square of the norm. Examples of inversion formulae, which reconstruct the random measures Z when X is known, are given in Sect. 3.1. Linear differential equations whose right-hand side is a random field, are considered in Sect. 3.2. In Sect. 3.3 we introduce rigged Hilbert spaces of ordinary, smooth, and generalised functions, linear functionals, and spectral characteristics connected to the random field X(t). Using Paley–Wiener theorems, we study how smooth our “smooth functions” are. Bibliographical remarks conclude.

Many problems arising in applications result in the need to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the un- known function. We describe an approach to modifying a whole range of MCMC methods, applicable whenever the target measure has density with respect to a Gaussian process or Gaussian random field reference measure, which ensures that their speed of convergence is robust under mesh refinement.\nGaussian processes or random fields are fields whose marginal distri- butions, when evaluated at any finite set of N points, are RN-valued Gaussians. The algorithmic approach that we describe is applicable not only when the desired probability measure has density with respect to a Gaussian process or Gaussian random field reference measure, but also to some useful non-Gaussian reference measures constructed through random truncation. In the applications of interest the data is often sparse and the prior specification is an essential part of the over- all modelling strategy. These Gaussian-based reference measures are a very flexible modelling tool, finding wide-ranging application. Examples are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration.\nThe key design principle is to formulate the MCMC method so that it is, in principle, applicable for functions; this may be achieved by use of proposals based on carefully chosen time-discretizations of stochas- tic dynamical systems which exactly preserve the Gaussian reference measure. Taking this approach leads to many new algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems.

CONTENTS Introduction § 1. Basic facts from the theory of stochastic processes § 2. Characteristic functionals of stochastic processes § 3. General theorems on the absolute continuity of probability measures in function spaces § 4. Absolute continuity of Gaussian measures in Hilbert space § 5. Equivalence and orthogonality of measures corresponding to stationary Gaussian processes § 6. Absolute continuity of infinitely divisible distributions in Hilbert space § 7. Measures corresponding to processes with independent increments § 8. Absolute continuity of measures corresponding to general Markov processes § 9. Absolute continuity of measures under transformations of the space References

We show that for every n-dimensional lattice [Formula: see text] the torus [Formula: see text] can be embedded with distortion [Formula: see text] into a Hilbert space. This improves the exponential upper bound of O(n3n/2) due to Khot and Naor (FOCS 2005, Math. Ann. 2006) and gets close to their lower bound of [Formula: see text]. We also obtain tight bounds for certain families of lattices. Our main new ingredient is an embedding that maps any point [Formula: see text] to a Gaussian function centered at u in the Hilbert space [Formula: see text]. The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine–Zolotarev bases.

We show that for every n-dimensional lattice \({\mathcal L}\) the torus \({\mathbb{R}}^n / {\mathcal L}\) can be embedded with distortion \(O(n \cdot \sqrt{\log{n}})\) into a Hilbert space. This improves the exponential upper bound of O(n 3n/2) due to Khot and Naor (FOCS 2005, Math. Annal. 2006) and gets close to their lower bound of \(\Omega(\sqrt{n})\). We also obtain tight bounds for certain families of lattices.Our main new ingredient is an embedding that maps any point \(u \in {\mathbb{R}}^n/{\mathcal L}\) to a Gaussian function centered at u in the Hilbert space \(L_2({\mathbb{R}}^n/{\mathcal L})\). The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine-Zolotarev bases.

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).

We study measures on a real separable Hilbert space E that are invariant with respect to both shifts by arbitrary vectors of the space and orthogonal transformations. In particular, our first concern is a finitely additive analog of the Lebesgue measure. We present such an analog; namely, we construct a nonnegative finitely additive measure that is invariant with respect to shifts and rotations and is defined on the minimal ring of subsets of E that contains all infinite-dimensional rectangles such that the products of their side lengths converge absolutely. We also define a Hilbert space $${\mathcal H}$$ of complex-valued functions on E that are square integrable with respect to a shift- and rotation-invariant measure. For random vectors whose distributions are given by families of Gaussian measures on E that form semigroups with respect to convolution, we define expectations of the corresponding shift operators. We establish that such expectations form a semigroup of self-adjoint contractions in $${\mathcal H}$$ that is not strongly continuous, and find invariant subspaces of strong continuity for this semigroup. We examine the structure of an arbitrary semigroup of self-adjoint contractions of the Hilbert space, which may not be strongly continuous. Finally, we show that the method of Feynman averaging of strongly continuous semigroups based on the notion of Chernoff equivalence of operator-valued functions is also applicable to discontinuous semigroups.

We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be (Hilbert spaces) of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different. We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory. A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum. In a final section, we present a stochastic theory for inverse problems (System Identification) in the Hilbert space setting. The basic theoretical problem is the calculation of R-N derivatives for finitely additive measures. A fundamental result concerns Identifiability; in particular the identifiability of diffusion coefficients from boundary data is treated here for the first time.

Gaussian measures in Hilbert spaces

This paper provides a quantitative analysis of the optimal accuracy and resolution in electrical impedance tomography (EIT) based on the Cramér–Rao lower bound. The imaging problem is characterized by the forward operator and its Jacobian. The Fisher information operator is defined for a deterministic parameter in a real Hilbert space and a stochastic measurement in a finite-dimensional complex Hilbert space with a Gaussian measure. The connection between the Fisher information and the singular value decomposition (SVD) based on the maximum likelihood (ML) criterion (the ML-based SVD) is established. It is shown that the eigenspaces of the Fisher information provide a suitable basis to quantify the trade-off between the accuracy and the resolution of the (nonlinear) inverse problem. It is also shown that the truncated ML-based pseudo-inverse is a suitable regularization strategy for a linearized problem, which exploits sufficient statistics for estimation within these subspaces. The statistical-based Cramér–Rao lower bound provides a complement to the deterministic upper bounds and the L-curve techniques that are employed with linearized inversion. To this end, electrical impedance tomography provides an interesting example where the eigenvalues of the SVD usually do not exhibit a very sharp cut-off, and a trade-off between the accuracy and the resolution may be of practical importance. A numerical study of a hypothetical EIT problem is described, including a statistical analysis of the model errors due to the linearization.

In this article we investigate unnormalized samples of Gaussian random elements in a separable Fréchet space 𝕄. First we describe a connection between shifts of a Gaussian measure μ in a separable Fréchet space and the infinite product of standard normal distributions in ℝ∞, and on the basis of this result we derive the so‐called self‐sufficient expansion for Gaussian random elements in a Fréchet space. Moreover, we find lower bounds for the Gaussian measure μ of shifted balls in 𝕄 and estimate the metric entropy of balls in the Hilbert space ℍ ⊂ 𝕄 which generates μ. Finally, applying the Brunn–Minkowski inequality we prove a kind of the logarithmic law of large numbers. The last result is an extension of the analogous theorem obtained by Goodman (Characteristics of normal samples. Ann. Probab. 1988, 16, 1281–1290), for a sequence of Gaussian random elements in a separable Banach space.