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Abstract Wiener Space
The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Cameron–Martin space. The classical Wiener space is the prototypical example. The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction. Motivation Let H be a real Hilbert space, assumed to be infinite dimensional and separable. In the physics literature, one frequently encounters integrals of the form :\frac\int_H f(v) e^ Dv, where Z is supposed to be a normalization constant and where Dv is supposed to be the non-existent Lebesgue measure on H. Such integrals arise, notably, in the context of the Euclidean path-integral formulation of quantum field theory. At a mathematical level, such an integral cannot be interpreted as integration against a measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonard Gross
Leonard Gross (born February 24, 1931) is an American mathematician and Professor Emeritus of Mathematics at Cornell University. Gross has made fundamental contributions to mathematics and the mathematically rigorous study of quantum field theory. Education and career Leonard Gross graduated from James Madison High School in December 1948. He was awarded an Emil Schweinberg scholarship that enabled him to attend college. He studied at City College of New York for one term and then studied electrical engineering at Cooper Union for two years. He then transferred to the University of Chicago, where he obtained a master's degree in physics and mathematics (1954) and a Ph.D. in mathematics (1958). Gross taught at Yale University and was awarded a National Science Foundation Fellowship in 1959. He joined the faculty of the mathematics department of Cornell University in 1960. Gross was a member of the Institute for Advanced Study in 1959 and in 1983 and has held other visiting posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
White Noise Analysis
In probability theory, a branch of mathematics, white noise analysis, otherwise known as Hida calculus, is a framework for infinite-dimensional and stochastic calculus, based on the Gaussian white noise probability space, to be compared with Malliavin calculus based on the Wiener process. It was initiated by Takeyuki Hida in his 1975 Carleton Mathematical Lecture Notes. The term white noise was first used for signals with a flat spectrum. White noise measure The white noise probability measure \mu on the space S'(\mathbb) of tempered distributions has the characteristic function : C(f)=\int_\exp \left( i\left\langle \omega ,f\right\rangle \right) \, d\mu (\omega )=\exp \left( -\frac\int_ f^2(t) \, dt\right), \quad f\in S(\mathbb). Brownian motion in white noise analysis A version of Wiener's Brownian motion B(t) is obtained by the dual pairing : B(t) = \langle \omega, 1\!\!1_\rangle, where 1\!\!1_ is the indicator function of the interval [0,t) . Informally : B(t)=\int_0^t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Algebra
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel. For a topological space ''X'', the collection of all Borel sets on ''X'' forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on ''X'' is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets). Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory. In some contexts, Borel sets are defined to be generated by the compact sets of the topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Formal definition Let X be a locally compact Hausdorff space, and let \mathfrak(X) be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. A Borel measure is any measure \mu defined on the σ-algebra of Borel sets. A few authors require in addition that \mu is locally finite, meaning that every point has an open neighborhood with finite measure. For Hausdorff spaces, this implies that \mu(C) 0 and ''μ''(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then the Hausdorff dimension dimHaus(''X'') ≥ ''s''. A partial converse is provided by the Frostman lemma: Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Linear Map
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous linear operators Characterizations of continuity Suppose that F : X \to Y is a linear operator between two topological vector spaces (TVSs). The following are equivalent: F is continuous. F is continuous at some point x \in X. F is continuous at the origin in X. If Y is locally convex then this list may be extended to include: for every continuous seminorm q on Y, there exists a continuous seminorm p on X such that q \circ F \leq p. If X and Y are both Hausdorff locally convex spaces then this list may be extended to include: F is weakly continuous and its transpose ^t F : Y^ \to X^ maps equicontinuous subsets of Y^ to equicontinuous subsets of X^. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Integral
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is \int_^\infty e^\,dx = \sqrt. Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809, attributing its discovery to Laplace. The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution. The same integral with finite limits is closely related to both the error function and the cumulative distribution function of the normal distribution. In physics this type of integral appears frequently, for example, in quantum mechanics, to find the probability density of the ground state of the harmonic oscillator. This integral is also used in the path integral formulation, to find the propagator of the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylindrical σ-algebra
In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures or probability measures of random variables on Banach spaces. For a product space, the cylinder σ-algebra is the one that is generated by cylinder sets. In the context of a Banach space X and its dual space of continuous linear functionals X', the cylindrical σ-algebra \mathfrak(X,X') is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on X is a measurable function. In general, \mathfrak(X,X') is ''not'' the same as the Borel σ-algebra on X, which is the coarsest σ-algebra that contains all open subsets of X. Definition Consider two topological vector spaces N and M with dual pairing \langle,\rangle:=\langle,\rangle_, then we can define the so called Borel cylinder sets :C_=\ for som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylinder Set Measure
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a Measure (mathematics), measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be Sigma additivity, countably additive but only Sigma additivity, finitely additive), but can be used to define measures, such as the Classical Wiener space#Classical Wiener measure, classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space. This is done in the construction of the abstract Wiener space where one defines a cylinder set Gaussian measure on a separable Hilbert space and chooses a Banach space in such a way that the cylindrical measure becomes σ-additive on the cylindrical algebra. The terminology is not always consistent in the literature. Some authors call cylinder set measures just cylinder m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Representation Theorem
The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the underlying field is the real numbers, the two are isometrically isomorphic; if the underlying field is the complex numbers, the two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism. Preliminaries and notation Let H be a Hilbert space over a field \mathbb, where \mathbb is either the real numbers \R or the complex numbers \Complex. If \mathbb = \Complex (resp. if \mathbb = \R) then H is called a (resp. a ). Every real Hilbert space can be extended to be a dense subset of a unique (up to bijective isometry) complex Hilbert space, called its complexification, which is why Hilbert spaces are often automatically assumed to be complex. Real and complex Hilbert spaces have in common many, but b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with area or volume. In probability theory, they are used to define events with a well-defined probability. In this way, σ-algebras help to formalize the notion of ''size''. In formal terms, a σ-algebra (also σ-field, where the σ comes from the German "Summe", meaning "sum") on a set ''X'' is a nonempty collection Σ of subsets of ''X'' closed under complement, countable unions, and countable intersections. The ordered pair (X, \Sigma) is called a measurable space. The set ''X'' is understood to be an ambient space (such as the 2D plane or the set of outcomes when rolling a six-sided die ), and the collection Σ is a choice of subsets declared to have a well-defined size. The closure requirements for σ-algebras are designed to cap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |