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Probability distribution of a stochastic variable

  1. Jul 6, 2014 #1
    I am studying an article which involves stochastic variables http://www.rmki.kfki.hu/~diosi/prints/1985pla112p288.pdf.

    The author defines a probability distribution of a stochastic potential [itex]V[/itex] by a generator functional
    G[h] = \left<exp\left(i\int V(\vec{r},t)h(\vec{r},t)d\vec{r}dt\right)\right>,
    where [itex]h[/itex] is an arbitrary function and [itex]\langle\rangle[/itex] stands for expectation values evaluated by means of the probabil-
    ity distribution of [itex]V[/itex].

    He, then equates it to (equation 1 in the article)
    G[h] = exp\left(-\frac{1}{2}\iint h(\vec{r},t)h(\vec{r}^{\prime},t)f(\vec{r}-\vec{r}^{\prime})d\vec{r}d\vec{r}^{\prime}dt\right).

    How do we mathematically work out the steps? Any relevant reference or hint will be of great help. Thanks.
  2. jcsd
  3. Jul 6, 2014 #2


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    Science Advisor

    I don't understand the author's justification. However it looks like something of the form:

    [tex]<\sqrt(A,A*)>[/tex], where A is the exponential integral.
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