The Wiener sausage of radius is the random process defined bywhere here, is the standard Brownian motion in for and denotes the open ball of radius centered at . Named after Norbert Wiener, the term is also intended to describe visually: Indeed, for a given Brownian motion , is essentially a sausage-like tube of radius having as its central line.
The mean square displacement (MSD) of a set of displacements is given byIt arises particularly in Brownian motion and random walk problems. For two-dimensional random walks with unit steps taken in random directions, the MSD is given by
Let be a Wiener process. Thenwhere for , and .Note that while Ito's lemma was proved by Kiyoshi Ito (also spelled Itô), Ito's theorem is due to Noboru Itô.
A real-valued stochastic process is a Brownian motion which starts at if the following properties are satisfied: 1. . 2. For all times , the increments , , ..., , are independent random variables. 3. For all , , the increments are normally distributed with expectation value zero and variance . 4. The function is continuous almost everywhere. The Brownian motion is said to be standard if . It is easily shown from the above criteria that a Brownian motion has a number of unique natural invariance properties including scaling invariance and invariance under time inversion. Moreover, any Brownian motion satisfies a law of large numbers so thatalmost everywhere. Moreover, despite looking ill-behaved at first glance, Brownian motions are Hölder continuous almost everywhere for all values . Contrarily, any Brownian motion is nowhere differentiable almost surely.The above definition is extended naturally to get higher-dimensional Brownian..
Let be the probability that a random walk on a -D lattice returns to the origin. In 1921, Pólya proved that(1)but(2)for . Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that(3)(OEIS A086230), where(4)(5)(6)(7)(8)(9)(OEIS A086231; Borwein and Bailey 2003, Ch. 2, Ex. 20) is the third of Watson's triple integrals modulo a multiplicative constant, is a complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function.Closed forms for are not known, but Montroll (1956) showed that for ,(10)where(11)(12)and is a modified Bessel function of the first kind.Numerical values of from Montroll (1956) and Flajolet (Finch 2003) are given in the following table.OEIS3A0862300.3405374A0862320.1932065A0862330.1351786A0862340.1047157A0862350.08584498A0862360.0729126..
Let , , be one-dimensional Brownian motion. Integration with respect to was defined by Itô (1951). A basic result of the theory is that stochastic integral equations of the form(1)can be interpreted as stochastic differential equations of the form(2)where differentials are handled with the use of Itô's formula(3)(4)Hudson and Parthasarathy (1984) obtained a Fock space representation of Brownian motion and Poisson processes. The boson Fock space over is the Hilbert space completion of the linear span of the exponential vectors under the inner product(5)where and and is the complex conjugate of .The annihilation, creation and conservation operators , and respectively, are defined on the exponential vectors of as follows,(6)(7)(8)The basic quantum stochastic differentials , , and are defined as follows,(9)(10)(11)Hudson and Parthasarathy (1984) defined stochastic integration with respect to the noise differentials..