The probabilitiy distribution density of random values of squared functional on Wiener process trajectories

TL;DR

This paper develops a method to derive the probability distribution density of quadratic functionals of Wiener process trajectories using the Kac formula for their characteristic functions.

Contribution

It introduces a novel approach to obtain the distribution density of quadratic functionals on Wiener trajectories based on the Kac formula.

Findings

Derived the distribution density for quadratic functionals of Wiener processes.

Provided a new method for analyzing the probabilistic behavior of quadratic functionals.

Enhanced understanding of the distribution properties of Wiener process functionals.

Abstract

On the basis of the Kac formula of the characteristic function of the random variable defined by values of the quadratic functional $J_T [w]$ on trajectories of the standard Wiener process $\{w\}$, the method of its distribution density is proposed.