Parastochastics

The generalized commutation relations A_iB_j − λB_jA_i = Γ_ijI are introduced, where A_i and B_i are adjoint of each other, I is the identity, and Γ_ij is a real covariance. For λ = +1 (−1, or 0, respectively) the parastochastic function A_i + B_i is given an interpretation in terms of Gaussian random functions (a two‐valued stationary Markov process, or infinite symmetric random matrices of the type considered by Wigner in connection with energy levels of heavy nuclei, respectively). In the Fock‐space realization, A_i and B_i appear as destruction and creation operators for bosons (fermions or boltzmannons, i.e., distinguishable particles, respectively). A few purely algebraic theorems are proved, which are applied to linear stochastic equations (equations with random coefficients). Existence and uniqueness being presupposed, mean Green's functions are shown to satisfy closed master equations. A linear functional differential master equation is obtained for equations with Gaussian coefficients. It is shown that the often‐used first cumulant‐discard closure assumption, which leads to a very simple master equation, is exact for differential equations with a two‐valued stationary Markovian coefficient. A parastochastic reformulation of the theory of Kraichnan is given, and his nonlinear master equations are, for the first time, rigorously derived without any recourse to perturbation theory or to diagrams. Kraichnan's random‐coupling model is obtained by replacing scalar stochastic quantities by Wigner matrices or, equivalently, bosons by boltzmannons (i.e., changing λ from +1 to 0). Finally, the nonlinear Kraichnan equation, $$\mathrm{dy(t)/dt=-}{\displaystyle {\int }_0^t}\mathrm{y(t-t\prime )\Gamma (t-t\prime )y(t\prime )dt\prime }$$ , is reduced to a linear parastochastic equation in the Fock space; existence, uniqueness, boundedness, and asymptotic behavior are obtained.