Conserved quantities and symmetry for stochastic dynamical systems

On the definition of an admitted Lie group for stochastic differential equations

“…, r) are Brownian motions. The first approach [3][4][5][6] deals with fiber-preserving transformations…”

Energy conservative stochastic difference scheme for stochastic Hamilton dynamical systems

“…Let I = I (x, p) be a smooth function on R2 x R 1 . In terms of the definition of conserved quantities in stochastic dynamical systems proposed by the author [12], the function becomes a conserved quantity for stochastic Hamilton dynamical system (1), if it satisfies…”

“…Indeed, in stochastic dynamical systems mentioned above, the notion has been formulated in the various ways (e.g. [9,12,14]). Moreover, in the field of numerical analysis for deterministic dynamical systems described by ordinary differential equations, it is also recognized that the studies on the numerical preservation of the conserved quantities for dynamical systems are very important in performing reliable numerical calculations to the systems [5,7].…”

“…Hamiltonian) becomes a conserved quantity in the sense of the author's definition of conservations in stochastic systems [12]. In the new stochastic scheme for the system, the energy function is conserved numerically, and hence there exist energy invariant surface in the scheme for the Hamilton system.…”

On Lie Group Classification of a Scalar Stochastic Differential Equation

“…First, restricted cases of point transformations were considered [1,10,16,17]. Then, the theory for general point transformations was developed [8,9,[25][26][27].…”