Author： Kowalski Emmanuel
Publisher： Mathematical Association of America
ISSN： 1930-0972
Source： American Mathematical Monthly, Vol.113, Iss.10, 2006-12, pp. : 865-886
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Abstract
Brownian motion is one of the most fascinating objects in probability and indeed in mathematics as a whole. It provides a way of speaking of a continuous function "picked at random" and to describe what kind of properties it can have. We present a new and very elementary way to look at Brownian motion by means of the sequence of its Bernstein approximating polynomials. This provides a very intuitive proof of the existence of Brownian motion, and a new very simple proof that a random continuous function is not differentiable at any (given) point.
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