Lévy's modulus of continuity
Encyclopedia
In mathematics
, Lévy's modulus of continuity theorem gives a result about the almost sure
behaviour of an estimate of the modulus of continuity
for the Wiener process
, which models Brownian motion
. It is due to the French
mathematician
Paul Pierre Lévy
.
be a standard Wiener process. Then, almost surely
,
In other words, the sample paths of Brownian motion have modulus of continuity
with probability one, and for sufficiently small
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Lévy's modulus of continuity theorem gives a result about the almost sure
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
behaviour of an estimate of the modulus of continuity
Modulus of continuity
In mathematical analysis, a modulus of continuity is a function\omega:[0,\infty]\to[0,\infty]used to measure quantitatively the uniform continuity of functions. So, a function f:I\to\R admits \omega as a modulus of continuity if and only if|f-f|\leq\omega,for all x and y in the domain of f...
for the Wiener process
Wiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
, which models Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
. It is due to the French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Paul Pierre Lévy
Paul Pierre Lévy
Paul Pierre Lévy was a Jewish French mathematician who was active especially in probability theory, introducing martingales and Lévy flights...
.
Statement of the result
Let be a standard Wiener process. Then, almost surelyAlmost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
,
In other words, the sample paths of Brownian motion have modulus of continuity
with probability one, and for sufficiently small
.See also
- Some properties of sample paths of the Wiener process