What probability density is used in Brownian motion?

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1. Dec 4, 2015

Earthland

1. The problem statement, all variables and given/known data

I have a free Brownian particle and its coordinate is given as a function of time:

And its first moment, or mean, is given as

But what kind of probability density was used to calculate this first moment?

2. Relevant equations

I know that the first moment is calculated by using probability density

<x> = ∫x*W(x)dx from -infinity to +infinity.

X itself is dependent on t, so is it W(x) or W(x,t) or W(t)? I'm little lost on this one.

3. The attempt at a solution

I did find some kind of density function in regards to Brownian motion

But I can't see how this would yield the correct result. And still, must I integrate over x or t?

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2. Dec 4, 2015

Orodruin

Staff Emeritus
Just take the expectation value of your first expression (expectation values are linear) keeping in mind that the only actually random variable in it is the driving force $\zeta(t')$.