Lie-point symmetries and stochastic differential equations

Random Lie-point symmetries

“…This example was considered in [4], where Gaeta and Quintero compare its symmetries with the associated Fokker-Planck equation. Kozlov in [8] and [9] also mention the symmetries of the Brownian motion equation.…”

“…On the other hand, the connection between symmetry methods and stochastic differential equations is more recent. This connection started with Misawa ( [11], 1994) and Albeverio and Fei ( [1], 1995), and it was followed by many other authors until today ( [4], [5], [7], [8], [9], [10], [15], [17], [18]). …”

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].…”

“…In the latter case the transformation of the Brownian motion needs to be more deeply specified. In [11] there were introduced W -symmetries, which are fiber-preserving symmetries acting also on Wiener processes. It can be of interest to extend symmetry framework to very general transformations such as random diffeomorphisms of SDEs [3].…”

“…Symmetries of SDEs can be used to find symmetries of Fokker-Planck (FP) equation [10,26]. In the case of fiber-preserving symmetries a symmetry of SDEs can be extended to a symmetry of the associated FP equation.…”