What is Brownian Motion (Weiner process)?

[woundedtiger4]

Hi all,

When random walk takes the steps at random times, and in that case the position X_t is defined as the continuum of times t≥0, isn't this concept/phenomenon/rule is Brownian motion (Weiner process)? At http://en.wikipedia.org/wiki/Wiener_process in section "Characterizations of the Wiener process" denote it as "W_t", does it tell the position at time "t" just like we find the position in random walks? In the same section, I don't understand the second property, particularly "t→W_t" , what is this?

Thanks in advance

Note: The text says that the mathematical treatment of Brownian motion is called Weiner process, therefore I am thinking that both Brownian motion & Weiner are same, am I correct?

 

[mathman]

t -> W_t is part of a sentence which says the Weiner process is a continuous (almost everywhere) function of time (t).

Essentially Brownian motion and Weiner process refer to the same things. There may be some quibble, since physical Brownian motion also refers to processes with jumps.

 

[woundedtiger4]

t -> W_t is part of a sentence which says the Weiner process is a continuous (almost everywhere) function of time (t).

Essentially Brownian motion and Weiner process refer to the same things. There may be some quibble, since physical Brownian motion also refers to processes with jumps.

thanks for the reply, you are always very helpful.


At http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/McKnight.pdf the 3rd property says that Brownian motion is continuous everywhere and differentiable nowhere, if I am not wrong then BM (Brownian motion) is not differentiable nowhere because there are no jumps or discontinuity anywhere in BM or roughly speaking the graph of BM is too rough or it has too much spikes but at http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx the last theorem (at the bottom of the page) says that if f(x) is differentiable at x=a then f(x) is continuous at x=a, why does it contradict? Is it because BM is random variable whereas the continuity theorem is about normal variables?

thanks in advance

 

[mathman]

differentiability => continuity, but not the reverse. In fact it is possible to construct continuous functions which are nowhere differentiable - this is a general fact, not particularly for Brownian motion. See the following:

http://en.wikipedia.org/wiki/Weierstrass_function

 

[woundedtiger4]

differentiability => continuity, but not the reverse. In fact it is possible to construct continuous functions which are nowhere differentiable - this is a general fact, not particularly for Brownian motion. See the following:

http://en.wikipedia.org/wiki/Weierstrass_function

GOT IT :)
Thank you Sir.

 

[ImaLooser]

Hi all,

When random walk takes the steps at random times, and in that case the position X_t is defined as the continuum of times t≥0, isn't this concept/phenomenon/rule is Brownian motion (Weiner process)? At http://en.wikipedia.org/wiki/Wiener_process in section "Characterizations of the Wiener process" denote it as "W_t", does it tell the position at time "t" just like we find the position in random walks? In the same section, I don't understand the second property, particularly "t→W_t" , what is this?

Thanks in advance

Note: The text says that the mathematical treatment of Brownian motion is called Weiner process, therefore I am thinking that both Brownian motion & Weiner are same, am I correct?

I don't really recall, but my impression is that a Weiner process is the limit of Brownian motion with infinitely small particles.