Stochastic canonical flows (Q2747959)

\textit{P. Talker} [Ann. Phys. 167, 390 (1986)] argued that there exists no quantum regression hypothesis. The paper is critique of the Langevin equations as they are applied in standard stochastic analysis to physical systems. NEWLINENEWLINENEWLINELet \(\Gamma\) be a phase space which has a Poisson structure. For \(f\in C^{\infty}(\Gamma)\) the Hamiltonian vector field generated by \(f\) is defined by \(X_f:=\{\cdot,f\}\). Let \(H,F_1,\dots,F_m\) be the corresponding generating Hamiltonian functions. For \(f\in C^{\infty}(\Gamma)\) one obtains on conversion back to Itô calculus NEWLINE\[NEWLINEdf_t=(X_H+\tfrac{1}{2}X_{F_{\alpha}}\circ X_{F_{\alpha}})(f)|_{X_t} dt + X_{F_{\alpha}}|_{X_t} dB^{\alpha}_t. NEWLINE\]NEWLINE This flow is called the canonical stochastic flow generated by \((H,F_{\alpha})\). The author considers several cases when \(F_{\alpha}\) is linearly coupling, quadratically coupling and nonlinear coupling. In the second case one obtains differential equations from the Langevin's one. In the last case it is given a necessary and sufficient condition for a canonical stochastic flow to be static. Time-reversal symmetry, anti-symmetry and stationary states of canonical stochastic flows are also considered.