Quite recently, a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics, which is an extension of Nelson’s stochastic mechanics. In this article a ‘‘Noether’s theorem’’ is formulated within this calculus of variations. Conservation laws of momentum, angular momentum, and energy are proved, which are related with the same laws in quantum mechanics.
An energy conservative stochastic difference scheme is proposed for a one-dimensional stochastic Hamilton dynamical system governed by a stochastic differential equations in which the energy function, i.e. Hamiltonian, becomes a conserved quantity. The scheme is given by an stochastic extension of Greenspan's scheme which leaves Hamiltonians numerically invariant for deterministic Hamilton dynamical systems. The local error of accuracy of numerical solutions derived from the stochastic difference scheme is investigated. An illustrative numerical experiment for the scheme to a simple stochastic Hamilton system is also given.
“Symplectic” schemes for stochastic Hamiltonian dynamical systems are formulated through “composition methods (or operator splitting methods)” proposed by Misawa (2001). In the proposed methods, a symplectic map, which is given by the solution of a stochastic Hamiltonian system, is approximated by composition of the stochastic flows derived from simpler Hamiltonian vector fields. The global error orders of the numerical schemes derived from the stochastic composition methods are provided. To examine the superiority of the new schemes, some illustrative numerical simulations on the basis of the proposed schemes are carried out for a stochastic harmonic oscillator system.
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