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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martingale (probability Theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. History Originally, ''martingale'' referred to a class of betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users due to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman Integral
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are ''coordina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Integral
Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields. In an ordinary integral (in the sense of Lebesgue integration) there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the domain of integration). The process of integration consists of adding up the values of the integrand for each point of the domain of integration. Making this procedure rigorous requires a limiting procedure, where the domain of integration is divided into smaller and smaller regions. For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characterization Theorem
In mathematics, a characterization of an object is a set of conditions that, while different from the definition of the object, is logically equivalent to it. To say that "Property ''P'' characterizes object ''X''" is to say that not only does ''X'' have property ''P'', but that ''X'' is the ''only'' thing that has property ''P'' (i.e., ''P'' is a defining property of ''X''). Similarly, a set of properties ''P'' is said to characterize ''X'', when these properties distinguish ''X'' from all other objects. Even though a characterization identifies an object in a unique way, several characterizations can exist for a single object. Common mathematical expressions for a characterization of ''X'' in terms of ''P'' include "''P'' is necessary and sufficient for ''X''", and "''X'' holds if and only if ''P''". It is also common to find statements such as "Property ''Q'' characterizes ''Y'' up to isomorphism". The first type of statement says in different words that the extension of ''P'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pettis Integral
In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral. Definition Let f : X \to V where (X,\Sigma,\mu) is a measure space and V is a topological vector space (TVS) with a continuous dual space V' that separates points (that is, if x \in Vis nonzero then there is some l \in V' such that l(x) \neq 0), for example, V is a normed space or (more generally) is a Hausdorff locally convex TVS. Evaluation of a functional may be written as a duality pairing: \langle \varphi, x \rangle = \varphi The map f : X \to V is called if for all \varphi \in V', the scalar-valued map \varphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skorokhod Integral
In mathematics, the Skorokhod integral (also named Hitsuda-Skorokhod integral), often denoted \delta, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod and japanese mathematician Masuyuki Hitsuda. Part of its importance is that it unifies several concepts: * \delta is an extension of the Itô integral to non-adapted processes; * \delta is the adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations (Malliavin calculus); * \delta is an infinite-dimensional generalization of the divergence operator from classical vector calculus. The integral was introduced by Hitsuda in 1972 and by Skorokhod in 1975. Definition Preliminaries: the Malliavin derivative Consider a fixed probability space (\Omega, \Sigma, \mathbf) and a Hilbert space H; \mathbf denotes expectation with respect to \mathbf \mathbf := \int_ X(\omega) \, \mathrm \mathbf(\omega). Intuit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace–Beltrami Operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami. For any twice- differentiable real-valued function ''f'' defined on Euclidean space R''n'', the Laplace operator (also known as the ''Laplacian'') takes ''f'' to the divergence of its gradient vector field, which is the sum of the ''n'' pure second derivatives of ''f'' with respect to each vector of an orthonormal basis for R''n''. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Test Functions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function. A function f is normally thought of as on the in the function domain by "sending" a point x in its domain to the point f(x). Instead of acting on points, distribution theory reinterpr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Kondratiev
Yuri Kondratiev ( uk, Кондратьєв Юрій Григорович; born October 23, 1953, in Kyiv, USSR) is a Ukrainian mathematician, professor at the Bielefeld University, Germany. * His research interests include functional analysis, mathematical physics and stochastic calculus. * Y. Kondratiev is a member of the Kyiv school of functional analysis founded by M. Krein and lead, for many years, by Y. Berezansky. * His interests in mathematical physics were inspired by cooperation with the Moscow seminar in statistical Physics ( R. Dobrushin, R. Minlos, Y. Sinai). Yuri's collaboration with A. Skorokhod has influenced most of his subsequent papers on stochastics. Biography * In 1975, Yuri Kondratiev graduated with honours from Kyiv University. * In 1979 he completed his Candidate of Sciences with thesis entitled “Generalized functions in problems of infinite dimensional analysis” at the Kyiv University. * In 1987 Yuri obtained the Degree of Doctor of Sciences in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and more contemporary developments in certain directions are closely related to ideas of Mikio Sato, on what he calls algebraic analysis. Important influences on the subject have been the technical requirements of theories of partial differential equations, and group representation theory. Some early history In the mathematics of the nineteenth century, aspects of generalized function t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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