Some remarks on integral parameters of Wiener process

A. A. Vladimirov

Dalʹnevostočnyj matematičeskij žurnal, Tome 15 (2015) no. 2, pp. 156-165

Résumé

It is shown that if generalized function $\rho\in W_2^{-1}[0,1]$ is a multiplier of trace-class from space $W_2^1[0,1]$ to space $W_2^{-1}[0,1]$ then the distribution of stochastic variable $\int_0^1\rho\xi^2\,dt$ (where $\xi$ is a Wiener process) is determined by the spectrum of the boundary problem $$ -y''=\lambda\rho y,\qquad y(0)=y'(1)=0, $$ as in the case when $\rho$ is a measure. An example of generalized function $\rho\in W_2^{-1}[0,1]$ that is not a multiplier of trace-class from $W_2^1[0,1]$ to $W_2^{-1}[0,1]$ is also given.