Green measure

Last updated April 05, 2019

In mathematics — specifically, in stochastic analysis — the Green measure is a measure associated to an Itō diffusion. There is an associated Green formula representing suitably smooth functions in terms of the Green measure and first exit times of the diffusion. The concepts are named after the British mathematician George Green and are generalizations of the classical Green's function and Green formula to the stochastic case using Dynkin's formula.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the $n$ -dimensional Euclidean space $R n$ . For instance, the Lebesgue measure of the interval $[0, 1]$ in the real numbers is its length in the everyday sense of the word, specifically, 1.

The United Kingdom, officially the United Kingdom of Great Britain and Northern Ireland but more commonly known as the UK or Britain, is a sovereign country lying off the north-western coast of the European mainland. The United Kingdom includes the island of Great Britain, the north-eastern part of the island of Ireland and many smaller islands. Northern Ireland is the only part of the United Kingdom that shares a land border with another sovereign state, the Republic of Ireland. Apart from this land border, the United Kingdom is surrounded by the Atlantic Ocean, with the North Sea to the east, the English Channel to the south and the Celtic Sea to the south-west, giving it the 12th-longest coastline in the world. The Irish Sea lies between Great Britain and Ireland. With an area of 242,500 square kilometres (93,600 sq mi), the United Kingdom is the 78th-largest sovereign state in the world. It is also the 22nd-most populous country, with an estimated 66.0 million inhabitants in 2017.

Notation

Let X be an Rⁿ-valued Itō diffusion satisfying an Itō stochastic differential equation of the form

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 unstable 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.

\mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} B_{t}.

Let P^x denote the law of X given the initial condition X₀ = x, and let E^x denote expectation with respect to P^x. Let L_X be the infinitesimal generator of X, i.e.

Probability measure Measure of total value one, generalizing probability distributions

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure is that a probability measure must assign value 1 to the entire probability space.

In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the same experiment it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up is 3.5 as the number of rolls approaches infinity. In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment.

In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a stochastic process is a partial differential operator that encodes a great deal of information about the process. The generator is used in evolution equations such as the Kolmogorov backward equation ; its L² Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation.

L_{X}=\sum _{i}b_{i}{\frac {\partial }{\partial x_{i}}}+{\frac {1}{2}}\sum _{i,j}{\big (}\sigma \sigma ^{\top }{\big )}_{i,j}{\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}.

Let D ⊆ Rⁿ be an open, bounded domain; let τ_D be the first exit time of X from D:

In mathematics, and more specifically in topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points ; however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets is open, the intersection of a finite number of open sets is open, and the space itself is open. These conditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, every set can be open, or no set can be open but the space itself and the empty set.

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded. The word bounded makes no sense in a general topological space without a corresponding metric.

\tau _{D}:=\inf\{t\geq 0|X_{t}\not \in D\}.

The Green measure

Intuitively, the Green measure of a Borel set H (with respect to a point x and domain D) is the expected length of time that X, having started at x, stays in H before it leaves the domain D. That is, the Green measure of X with respect to D at x, denoted G(x, ·), is defined for Borel sets H ⊆ Rⁿ by

G(x,H)=\mathbf {E} ^{x}\left[\int _{0}^{\tau _{D}}\chi _{H}(X_{s})\,\mathrm {d} s\right],

or for bounded, continuous functions f : D → R by

\int _{D}f(y)\,G(x,\mathrm {d} y)=\mathbf {E} ^{x}\left[\int _{0}^{\tau _{D}}f(X_{s})\,\mathrm {d} s\right]

The name "Green measure" comes from the fact that if X is Brownian motion, then

Brownian motion or pedesis is the random motion of particles suspended in a fluid resulting from their collision with the fast-moving molecules in the fluid.

G(x,H)=\int _{H}G(x,y)\,\mathrm {d} y,

where G(x, y) is Green's function for the operator L_X (which, in the case of Brownian motion, is ½Δ, where Δ is the Laplace operator) on the domain D.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols $\nabla\cdot\nabla$ , $\nabla 2$ , or $Δ$ . The Laplacian $Δ f (p)$ of a function $f$ at a point $p$ , is the rate at which the average value of $f$ over spheres centered at $p$ deviates from $f (p)$ as the radius of the sphere grows. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

The Green formula

Suppose that E^x[τ_D] < +∞ for all x ∈ D, and let f : Rⁿ → R be of smoothness class C² with compact support. Then

f(x)=\mathbf {E} ^{x}{\big [}f{\big (}X_{\tau _{D}}{\big )}{\big ]}-\int _{D}L_{X}f(y)\,G(x,\mathrm {d} y).

In particular, for C² functions f with support compactly embedded in D,

f(x)=-\int _{D}L_{X}f(y)\,G(x,\mathrm {d} y).

The proof of Green's formula is an easy application of Dynkin's formula and the definition of the Green measure:

\mathbf {E} ^{x}{\big [}f{\big (}X_{\tau _{D}}{\big )}{\big ]}

=f(x)+\mathbf {E} ^{x}\left[\int _{0}^{\tau _{D}}L_{X}f(X_{s})\,\mathrm {d} s\right]

=f(x)+\int _{D}L_{X}f(y)\,G(x,\mathrm {d} y).

Related Research Articles

In mathematics convolution is a mathematical operation on two functions to produce a third function that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either $f (x)$ or $g (x)$ is reflected about the y-axis; thus it is a cross-correlation of $f (x)$ and $g (- x)$ , or $f (- x)$ and $g (x)$ . For continuous functions, the cross-correlation operator is the adjoint of the convolution operator.

In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of

In mathematics, the Dirac delta function is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space(M, g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ g_p(X|_p, Y|_p) is a smooth function. The family g_p of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

Heat equation partial differential equation for distribution of heat in a given region over time

The heat equation is a parabolic partial differential equation that describes the distribution of heat in a given region over time.

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green, though its first proof is due to Bernhard Riemann and is the two-dimensional special case of the more general Kelvin–Stokes theorem.

An instanton is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime.

In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable.

The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. When Mark Kac and Richard Feynman were both Cornell faculty, Kac attended a lecture of Feynman's and remarked that the two of them were working on the same thing from different directions. The Feynman-Kac formula resulted, which proves rigorously the real case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still unproven.

In mathematics, the Hamilton–Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

In probability and statistics, given two stochastic processes $and, the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation; for the expectation operator, if the processes have the mean functions and, then the cross-covariance is given by$

In mathematics, the Cameron–Martin theorem or Cameron–Martin formula is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space.

In mathematics, the Clark–Ocone theorem is a theorem of stochastic analysis. It expresses the value of some function F defined on the classical Wiener space of continuous paths starting at the origin as the sum of its mean value and an Itō integral with respect to that path. It is named after the contributions of mathematicians J.M.C. Clark (1970), Daniel Ocone (1984) and U.G. Haussmann (1978).

In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion at a stopping time. It may be seen as a stochastic generalization of the (second) fundamental theorem of calculus. It is named after the Russian mathematician Eugene Dynkin.

In mathematics — specifically, in stochastic analysis — an Itô diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are named after the Japanese mathematician Kiyosi Itô.

In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic differential equation.

In the theory of stochastic processes, the filtering problem is a mathematical model for a number of state estimation problems in signal processing and related fields. The general idea is to establish a "best estimate" for the true value of some system from an incomplete, potentially noisy set of observations on that system. The problem of optimal non-linear filtering was solved by Ruslan L. Stratonovich, see also Harold J. Kushner's work and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter known as Zakai equation. The solution, however, is infinite-dimensional in the general case. Certain approximations and special cases are well understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the Extended Kalman Filter or the Assumed Density Filters, or more methodologically oriented such as for example the Projection Filters, some sub-families of which are shown to coincide with the Assumed Density Filters.

In mathematics, the Skorokhod integral, often denoted δ, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod. Part of its importance is that it unifies several concepts:

References

Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. MR 2001996 (See Section 9)

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.