Some remarks on integral parameters of Wiener process

TL;DR

This paper investigates the distribution of integral parameters of Wiener processes with generalized multipliers, showing a chi-squared distribution under certain conditions and providing a counterexample for non-trace-class multipliers.

Contribution

It establishes the distribution of integral parameters for Wiener processes with trace-class multipliers and constructs an example for non-trace-class cases, advancing understanding of Wiener process integrals.

Findings

Integral parameters have chi-squared distribution for trace-class multipliers.

Counterexample demonstrates non-trace-class multipliers do not follow this distribution.

Provides insights into the role of multipliers in Wiener process integrals.

Abstract

We consider integrals $\tau_{\rho}=\int_0^1\rho\xi^2\,dx$, where $\xi$ is Wiener process and $\rho$ is generalized function from some class of multipliers. In the case when multiplier $\rho$ belongs to the trace-class, it is shown that $\tau_{\rho}$ has $\chi^2$-distribution (or analogous). An example of multiplier $\rho$ not belonging to the trace-class is constructed.