page_title: Wiener process
section_title: Related processes
hierarchical_section_title: Wiener process / Related processes
caption_attribution_description: Brownian Motion on a Sphere. The generator of ths process is ½ times the Laplace-Beltrami-Operator
context_page_description: In mathematics, the Wiener process is a real valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown. It is one of the best known Lévy processes and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.
  The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution.
context_section_description: The stochastic process defined by
  is called a Wiener process with drift μ and infinitesimal variance σ². These processes exhaust continuous Lévy processes.
  Two random processes on the time interval [0, 1] appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of [0,1]. With no further conditioning, the process takes both positive and negative values on [0, 1] and is called Brownian bridge. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion. In both cases a rigorous treatment involves a limiting procedure, since the formula P(A|B) = P(A ∩ B)/P(B)  does not apply when P(B) = 0.
  A geometric Brownian motion can be written
  It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.
  The stochastic process
  is distributed like the Ornstein–Uhlenbeck process with parameters , , and .
  The time of hitting a single point x > 0 by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers x) is a left-continuous modification of a Lévy process. The right-continuous modification of this process is given by times of first exit from closed intervals [0, x].
  The local time L = (Lˣₜ)ₓ ∈ R, ₜ ≥ 0 of a Brownian motion describes the time that the process spends at the point x. Formally
  where δ is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems.
caption: The generator of a Brownian motion is ½ times the Laplace–Beltrami operator. The image above is of the Brownian motion on a special manifold: the surface of a sphere.
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