Characteristic Function of Quadratic Convolution Functional on the Space of Trajectories of Gaussian Processes

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.