New conserved quantities derived from symmetry for stochastic dynamical systems

Approximate symmetries and conservation laws for Itô and Stratonovich dynamical systems

“…(1) and (43). Since the Fokker-Planck equation is a deterministic partial differential equation its approximate symmetries are found by using the method given in Section 2.1.…”

“…Conserved quantities of Stratonovich dynamical systems were considered in [1] and [2] without recourse to Hamiltonian formulation. Here we give an alternative theorem for approximate conservation laws.…”

“…(59) are written as ξ 0 1,t + x 2 ξ 0 1,1 − x 1 ξ 0 1,2 − ξ 0 2 − x 2 τ 0 ,t − x 2 2 τ 0 ,1 = 0, ξ 0 2,t + x 2 ξ 0 2,1 − x 1 ξ 0 2,2 + ξ 0 1 + x 1 τ 0 ,t + x 1 x 2 τ 0 ,1 = 0, τ 0 ,2 = 0, ξ 0 1,2 = 0, ξ 0 2,2 − 1 2 τ 0 ,t + x 2 τ 0 ,1 , = 0, and yield τ 0 = C 1 , ξ 0 1 = C 2 cos t + C 3 sin t, ξ 0 2 = C 3 cos t − C 2 sin t. Now the determining equations (40) become ξ 1 1,t + x 2 ξ 1 1,1 − x 1 ξ 1 1,2 − ξ 1 2 − x 2 τ 1 ,t − x 2 2 τ 1 ,1 = 0, ξ 1 2,t + x 2 ξ 1 2,1 − x 1 ξ 1 2,2 + ξ 1 1 + x 1 τ 1 ,t + x 1 x 2 τ 1 ,1 = ξ 0 1 f ,1 + ξ 0 2 f ,2 ,…”

“…There has been a growing interest in the literature to extend Lie's theorems to stochastic differential equations (see, for instance, [1][2][3][4][5][6]). The symmetry definition given in [5] maps Itô differential to another Itô differential.…”

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].…”