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Starting from this question which is basically exactly what I was wanting to ask initially, and then its very-well written top answer, I want to make sure everything is clear to me.

Consider the Brownian motion $B_t$.

Do we agree :

  • that, for any given, specific value of $t$, $dB_t$ follows a normal distribution ?

  • the covariance of which is infinite (multiple of a Dirac) ?

Or does that not even make sense, thinking of these objects as "functions" of $t$ ?

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Parker Lewis
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  • Heuristically what you say is correct. However, when you are actually being rigorous, it makes no sense whatsoever. The main point illustrated in that answer is as follows: Brownian motion is differentiable nowhere, hence we can never hope to define its derivative in the classical sense. However, if you consider Brownian motion as a random distribution, then you can take its "generalized" derivative and the random distribution you get is called white noise. This distribution is not regular enough to be defined at individual point measures, but you can integrate it against smooth test functions – shalin Aug 11 '16 at 19:25
  • ...and still obtain a sensible value. So, in order to rigorously understand the definition of white noise, you have to be familiar with the basics of distribution theory and calculus with distributions. – shalin Aug 11 '16 at 19:28
  • And moreover, this random distribution is Gaussian (in the sense that $\phi(f)$ is normally distributed for any $f \in C^{\infty}_c$) and the covariance structure is given by the $L^2$ inner product, i.e., $E[\phi(f)\phi(g)] = \int fg$. – shalin Aug 11 '16 at 19:36
  • Thanks for the reply. At least it confirms what I was getting and/or suspecting. "you have to be familiar with the basics of distribution theory and calculus with distributions" : unfortunately never had a course on the subject, which as it is is what prevents me from a clear understanding of the whole thing. – Parker Lewis Aug 11 '16 at 20:25
  • It might be worth learning about if you like probability. White noise is only one example of a stochastic process which doesn't contain enough regularity to be evaluated at individual points. Another important example of a random distribution is the Gaussian Free Field, which comes up in areas like conformally invariant processes and field theory. – shalin Aug 12 '16 at 03:28

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