PlanetMath: Brownian motion

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Brownian motion

(Definition)

Definition 1 One-dimensional Brownian motion is a stochastic process

, defined for $t\in [0,\infty)$ such that

almost surely
The sample paths $t \mapsto W(t)$ are almost surely continuous.
For any finite sequence of times $t_0 < t_1 < \cdots <t_n$ , the increments
$\displaystyle W(t_1)-W(t_0), W(t_2)-W(t_1), \dotsc, W(t_n)-W(t_{n-1})$
are independent.
For any times , is normally distributed with mean zero and variance .

Definition 2 A

-dimensional Brownian motion is a stochastic process $W(t) = (W_1(t), \dots, W_d(t))$ in $\mathbb{R}^d$ whose coordinate processes

are independent one-dimensional Brownian motions.

**Figure:** Sample paths of a standard Brownian motion
$\includegraphics{brownian.eps}$

Anyone with an account can edit this entry. Please help improve it!

"Brownian motion" is owned by skubeedooo. [ full author list (5) ]

(view preamble)

Other names:

Wiener process

Keywords:

Levy characterization

Log in to rate this entry.
(view current ratings)

Cross-references: coordinate, variance, mean, independent, finite sequence, continuous, sample paths, almost surely, stochastic process
There are 20 references to this entry.

This is version 10 of Brownian motion, born on 2005-04-27, modified 2007-06-29.
Object id is 6974, canonical name is BrownianMotion.
Accessed 7596 times total.

Classification:

AMS MSC:

60J65 (Probability theory and stochastic processes :: Markov processes :: Brownian motion)

Pending Errata and Addenda

None.

Discussion

forum policy

Stefan Odobleja invented cybernetics before Norbert Wiener by zorba on 2005-12-06 19:44:02

From these sites:
http://www.topedge.com/home/licinfor/stefan.html
http://www.ici.ro/romania/en/stiinta/odobleja.html

Our mentor - the father of cybernetics
" Stefan Odobleja could be considered not only a Socrate of our
times, but also a second Columb, because he had the same destiny:
to discover the America of science and at the end
this America got another name, not his."
(I.C.Dragan)
________________________________________

Stefan Odobleja, the creator of psychocybernetics and the father of generalised cybernetics was born on the 13'th of october 1902 in the house of some poor and illiterate peasants from Izvorul Anestilor - Mehedinti.
He went to the high school in Drobeta Turnu Severin and to the college in Bucharest. He became a military doctor.
His most important creation "The consonantist psychology" was presented in Psychological Abstracts (1941) but it didn't receive the deserved echo.
The cybernetic model, begining from obsevations, intuition and rationality, created by Odobleja in 1938-1939, but used (as we know) ten years later, in the american literature and then in the european one, was used and applied in many scientific fields.
Begining with 1972, when he read Norbert Wiener's autobiography, Stefan Odobleja devoted his time to prove that the origin of cybernetics is in psychology. He published a special creation named "The consonantist psychology and cybernetics".
He died on the 4'th of september 1978 in misery.
His work was better appreciated after his death - in 1982 began the establishment of Stefan Odobleja General Cybernetics Academy and the participants (from Romania and from foreign countries) decided to establish such an Academy in Switzerland too.
In all his work, Odobleja tried to answer the questions based on old enigmas.
His true value was discussed and recognised all over the world with appreciations like :
"You have a golden man, he deserves
a golden statue".

[ reply | up ]

Interact