A systematic approach to a class of problems in the theory of noise and other random phenomena--III: Examples

The method of Part I is applied to the problem of finding the characteristic function for the probability distribution of
\int_0^t \sum_{jk} x_j (\tau) K_{jl}(\tau)x_l(\tau) d\tau
, where
x_j(\tau)
denotes the
j
th component of a stationary n-dimensional Markoffian Gaussian process. The problem is reduced to the problem of solving
2n
first-order linear differential equations with initial conditions only. For the case of constant
K
, the explicit solution is given in terms of the eigenvalues and the first
2n - 1
powers of a constant
2n \times 2n
matrix. For the case of a symmetric correlation matrix which commutes with
K
, the problem is reduced to the one-dimensional case treated in Part II. For the case
K_{ij}(t) = \delta_{il} \delta_{jl} e^{-t}
, where the functional represents the output of a receiver consisting of a lumped circuit amplifier, a quadratic detector, and a single-stage amplifier, the solution has been obtained in a form which is more explicit than that provided by the earlier methods.