Abstract
The method for calculation of the characteristic functions \(Q(-i\lambda)\) and \(\lambda\in\mathbb{R}\) of random value defined by quadratic functionals \(\mathsf{J}[x]\) on the space \(\mathbb{L}_{2}[0,T]\) of trajectories \(x(t)\) of gaussian random processes is proposed. The functional representing the convolution of the function \(x(t)\) on the segment \([0,T]\) is under consideration. The general formula for the Fredholm determinant corresponding such characteristic functions is found. By the generalization of the reconstruction method, the calculation of the function \(Q(-i\lambda)\) connected with the convolution functional \(\mathsf{J}[x]\) is fulfilled. In a result, in the case when \(\{x(t)\) ; \(t\in[0,T]\}\) is the Ornstein–Uhlenbeck process, the formula of \(Q(-i\lambda)\) is obtained.
