# Probability distribution of a stochastic variable

1. Jul 6, 2014

### Ravi Mohan

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 $V$ 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 $h$ is an arbitrary function and $\langle\rangle$ stands for expectation values evaluated by means of the probabil-
ity distribution of $V$.

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. Jul 6, 2014

### mathman

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

$$<\sqrt(A,A*)>$$, where A is the exponential integral.