Kac–Siegert formula for oscillatory random processes

A general scheme for calculating the characteristic functions of random variables represented by quadratic functionals of the trajectories of elementary Gaussian processes based on the Feynman—Kac method is described. This scheme is applied to the oscillatory random process $\langle{\tilde x}(t)$, $t \in {\Bbb R}\rangle$. The characteristic function $Q(-i\lambda,t)$ of the random variable $\mathsf{J}_t[{\tilde x}(s)]=\int_0^t (d {\tilde x}(s)/ds )^2 ds$ of its random trajectories ${\tilde x}(t)$ is calculated.