Editing Random walk (section)

===Relation to Wiener process===

[[File:Brownian hierarchical.png|thumb|right|196px|Simulated steps approximating a Wiener process in two dimensions]]

A [[Wiener process]] is a stochastic process with similar behavior to [[Brownian motion]], the physical phenomenon of a minute particle diffusing in a fluid. (Sometimes the [[Wiener process]] is called "Brownian motion", although this is strictly speaking a confusion of a model with the phenomenon being modeled.)

A Wiener process is the [[scaling limit]] of random walk in dimension 1. This means that if there is a random walk with very small steps, there is an approximation to a Wiener process (and, less accurately, to Brownian motion). To be more precise, if the step size is ε, one needs to take a walk of length ''L''/ε<sup>2</sup> to approximate a Wiener length of ''L''. As the step size tends to 0 (and the number of steps increases proportionally), random walk converges to a Wiener process in an appropriate sense. Formally, if ''B'' is the space of all paths of length ''L'' with the maximum topology, and if ''M'' is the space of measure over ''B'' with the norm topology, then the convergence is in the space ''M''. Similarly, a Wiener process in several dimensions is the scaling limit of random walk in the same number of dimensions.

A random walk is a discrete fractal (a function with integer dimensions; 1, 2, ...), but a Wiener process trajectory is a true fractal, and there is a connection between the two. For example, take a random walk until it hits a circle of radius ''r'' times the step length. The average number of steps it performs is ''r''<sup>2</sup>.{{citation needed|date=April 2012}} This fact is the ''discrete version'' of the fact that a Wiener process walk is a fractal of [[Hausdorff dimension]]&nbsp;2.{{citation needed|date=April 2012}}

In two dimensions, the average number of points the same random walk has on the ''boundary'' of its trajectory is ''r''<sup>4/3</sup>. This corresponds to the fact that the boundary of the trajectory of a Wiener process is a fractal of dimension 4/3, a fact predicted by [[Benoît Mandelbrot|Mandelbrot]] using simulations but proved only in 2000
by [[Greg Lawler|Lawler]], [[Oded Schramm|Schramm]] and [[Wendelin Werner|Werner]].<ref>{{cite journal|year =  2000|doi = 10.1126/science.290.5498.1883|pmid = 17742050|title = MATHEMATICS: Taking the Measure of the Wildest Dance on Earth|journal = Science|volume = 290|issue = 5498|pages = 1883–4|last1 = MacKenzie|first1 = D.|s2cid = 12829171}} {{erratum|doi=10.1126/science.291.5504.597|checked=yes}}</ref>

A Wiener process enjoys many [[symmetry|symmetries]] a random walk does not. For example, a Wiener process walk is invariant to rotations, but the random walk is not, since the underlying grid is not (random walk is invariant to rotations by 90 degrees, but Wiener processes are invariant to rotations by, for example, 17 degrees too). This means that in many cases, problems on a random walk are easier to solve by translating them to a Wiener process, solving the problem there, and then translating back. On the other hand, some problems are easier to solve with random walks due to its discrete nature.

Random walk and [[Wiener process]] can be [[Coupling (probability)|''coupled'']], namely manifested on the same probability space in a dependent way that forces them to be quite close. The simplest such coupling is the [[Skorokhod's embedding theorem|Skorokhod embedding]], but there exist more precise couplings, such as [[Komlós–Major–Tusnády approximation]] theorem.

The convergence of a random walk toward the Wiener process is controlled by the [[central limit theorem]], and by [[Donsker's theorem]].  For a particle in a known fixed position at ''t''&nbsp;=&nbsp;0, the central limit theorem tells us that after a large number of [[statistical independence|independent]] steps in the random walk, the walker's position is distributed according to a [[normal distribution]] of total [[variance]]:

<math display="block">\sigma^2 = \frac{t}{\delta t}\,\varepsilon^2,</math>

where ''t'' is the time elapsed since the start of the random walk, <math>\varepsilon</math> is the size of a step of the random walk, and <math>\delta t</math> is the time elapsed between two successive steps.

This corresponds to the [[Green's function]] of the [[diffusion equation]] that controls the Wiener process, which suggests that, after a large number of steps, the random walk converges toward a Wiener process.

In 3D, the variance corresponding to the [[Green's function]] of the diffusion equation is:
<math display="block">\sigma^2 = 6\,D\,t.</math>

By equalizing this quantity with the variance associated to the position of the random walker, one obtains the equivalent diffusion coefficient to be considered for the asymptotic Wiener process toward which the random walk converges after a large number of steps:
<math display="block">D = \frac{\varepsilon^2}{6 \delta t}</math> (valid only in 3D).

The two expressions of the variance above correspond to the distribution associated to the vector <math>\vec R</math> that links the two ends of the random walk, in 3D. The variance associated to each component <math>R_x</math>, <math>R_y</math> or <math>R_z</math> is only one third of this value (still in 3D).

For 2D:<ref>[http://engineering.dartmouth.edu/~d30345d/courses/engs43/Chapter2.pdf Chapter 2 DIFFUSION]. dartmouth.edu.</ref>

<math display="block">D = \frac{\varepsilon^2}{4 \delta t}.</math>

For 1D:<ref>[http://nebula.physics.uakron.edu/dept/faculty/jutta/modeling/diff_eqn.pdf Diffusion equation for the random walk] {{Webarchive|url=https://web.archive.org/web/20150421024157/http://nebula.physics.uakron.edu/dept/faculty/jutta/modeling/diff_eqn.pdf |date=21 April 2015 }}. physics.uakron.edu.</ref>

<math display="block">D = \frac{\varepsilon^2}{2 \delta t}.</math>