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Stochastic Partial Differential Equations
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They have relevance to quantum field theory, statistical mechanics, and spatial modeling. Examples One of the most studied SPDEs is the stochastic heat equation, which may formally be written as : \partial_t u = \Delta u + \xi\;, where \Delta is the Laplacian and \xi denotes space-time white noise. Other examples also include stochastic versions of famous linear equations, such as wave equation and Schrödinger equation. Discussion One difficulty is their lack of regularity. In one dimensional space, solutions to the stochastic heat equation are only almost 1/2-Hölder continuous in space and 1/4-Hölder continuous in time. For dimensions two and higher, solutions are not even function-valued, but can be made sense of as random distributi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Distribution (mathematics)
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 reint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Stochastic Differential Equations
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations. Background Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stocha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Zakai Equation
Zakai is a surname. Notable people with the surname include: *Johanan ben Zakai :''See Yohanan for more rabbis by this name''. Yohanan ben Zakkai ( he, יוֹחָנָן בֶּן זַכַּאי, ''Yōḥānān ben Zakkaʾy''; 1st century CE), sometimes abbreviated as Ribaz () for Rabbi Yohanan ben Zakkai, was one of the Tan ..., Mishnah rabbi * Moshe Zakai (born 1926), Israeli scientist * Shafrira Zakai (born 1932), Israeli translator * Shmuel Zakai (born 1963), Israeli general * Yehezkel Zakai (born 1932), Israeli politician {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Wick Product
In probability theory, the Wick product is a particular way of defining an adjusted product of a set of random variables. In the lowest order product the adjustment corresponds to subtracting off the mean value, to leave a result whose mean is zero. For the higher order products the adjustment involves subtracting off lower order (ordinary) products of the random variables, in a symmetric way, again leaving a result whose mean is zero. The Wick product is a polynomial function of the random variables, their expected values, and expected values of their products. The definition of the Wick product immediately leads to the Wick power of a single random variable and this allows analogues of other functions of random variables to be defined on the basis of replacing the ordinary powers in a power-series expansions by the Wick powers. The Wick powers of commonly-seen random variables can be expressed in terms of special functions such as Bernoulli polynomials or Hermite polynomials. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Malliavin Calculus
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, the significant contributors such as S. Kusuoka, D. Stroock, Bismut, S. Watanabe, I. Shigekawa, and so on finally completed the foundations. Malliavin calculus is named after Paul Malliavin whose ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. The calculus has been applied to stochastic partial differential equations as well. The calculus allows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kushner Equation
In filtering theory the Kushner equation (after Harold Kushner) is an equation for the conditional probability density of the state of a stochastic non-linear dynamical system, given noisy measurements of the state. It therefore provides the solution of the nonlinear filtering problem in estimation theory. The equation is sometimes referred to as the Stratonovich–Kushner Stratonovich, R.L. (1960). ''Conditional Markov Processes''. Theory of Probability and Its Applications, 5, pp. 156–178. (or Kushner–Stratonovich) equation. Overview Assume the state of the system evolves according to :dx = f(x,t) \, dt + \sigma dw and a noisy measurement of the system state is available: :dz = h(x,t) \, dt + \eta dv where ''w'', ''v'' are independent Wiener processes. Then the conditional probability density ''p''(''x'', ''t'') of the state at time ''t'' is given by the Kushner equation: :dp(x,t) = L (x,t)dt + p(x,t) (x,t)-E_t h(x,t) \top \eta^\eta^ z-E_t h(x,t) dt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Brownian Surface
A Brownian surface is a fractal surface generated via a fractal elevation function. As with Brownian motion, Brownian surfaces are named after 19th-century biologist Robert Brown. Example For instance, in the three-dimensional case, where two variables ''X'' and ''Y'' are given as coordinates, the elevation function between any two points (''x''1, ''y''1) and (''x''2, ''y''2) can be set to have a mean or expected value that increases as the vector distance between (''x''1, ''y''1) and (''x''2, ''y''2). There are, however, many ways of defining the elevation function. For instance, the fractional Brownian motion variable may be used, or various rotation functions may be used to achieve more natural looking surfaces. Generation of fractional Brownian surfaces Efficient generation of fractional Brownian surfaces poses significant challenges. Since the Brownian surface represents a Gaussian process with a nonstationary covariance function, one can use the Cho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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C0-semigroup
In mathematics, a ''C''0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations. Formally, a strongly continuous semigroup is a representation of the semigroup (R+,+) on some Banach space ''X'' that is continuous in the strong operator topology. Thus, strictly speaking, a strongly continuous semigroup is not a semigroup, but rather a continuous representation of a very particular semigroup. Formal definition A strongly continuous semigroup on a Banach space X is a map T : \mathbb_+ \to L(X) such that # T(0) = I , (identity operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hölder Continuous
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Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |