The random Wigner distribution of Gaussian stochastic processes with covariance in $S_0 (\R^{2d})$

BENGT PATRIK MARTIN WAHLBERG

Wolfgang Hörmann

Patrik Wahlberg

We define the Wigner distribution of a tempered generalized stochastic process that is complex-valued symmetric Gaussian. This gives a time-frequency generalized stochastic process defined on the phase space. We study its covariance and our main result is a formula for the Weyl symbol of the covariance operator, expressed in terms of the Weyl symbol of the covariance operator of the original generalized stochastic process.

R. Simon, E. C. G. Sudarshan, Narasimhaiengar Mukunda

Ioannis M. Besieris, F. D. Tappert

Within the framework of the first-order smoothing approximation and the long-time, Markovian approximation, kinetic equations are derived for the stochastic Wigner equation (the exact equation of evolution of the phase-space Wigner distribution function) and the stochastic Liouville equation (correspondence limit approximation) associated with the quantized motion of a particle described by a stochastic Schrödinger equation. In the limit of weak fluctuations and long times, the transport equation for the average probability density of the particle in momentum space which was reported recently by Papanicolaou is recovered. Also, on the basis of the Novikov functional formalism, it is established that several of the approximate kinetic equations derived in this paper are identical to the exact statistical equations in the special case that the potential field is a δ-correlated (in time), homogeneous, wide-sense stationary, Gaussian process.

Elena Cordero, Maurice A. de Gosson, Monika Dörfler +1 more

We study the covariance property of quadratic time-frequency distributions with respect to the action of the extended symplectic group. We show how covariance is related to and in fact in competition with the possibility of damping the interferences which arise due to the quadratic nature of the distributions. We also show that the well-known fully covariance property of the Wigner distribution in fact characterizes it (up to a constant factor) among the quadratic distributions $L^{2}(\mathbb{R}^{n})\rightarrow C_{0}({ \mathbb{R}^{2n}})$. A similar characterization for the closely related Weyl transform is given as well. The results are illustrated by several numerical experiments for the Wigner and Born--Jordan distributions of the sum of four Gaussian functions in the so-called diamond configuration.