Abstract
A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itô integral context is pursued. The basis of this construction relies on Lie bracket relations on both the instantaneous drift and diffusion of an Itˆo stochastic ordinary differential equation (SODE).
Highlights
A conserved quantity in the context of an Ito integral implies an entity which is constant on all sample paths for all time indices
We show that the symmetries of the FP equations are projectable by using the methodology of Mahomed
This implies that the Lie algebra generated by the stochastic ordinary differential equation (SODE) can have non-projectable symmetries which will not belong to the Lie algebra generated by the FP equation
Summary
A conserved quantity in the context of an Ito integral implies an entity which is constant on all sample paths for all time indices. The Ito integral construction of the conserved quantities was later pursued by Unal [3] In this contribution Unal [3] uses both the (Fokker-Planck) FP equation and its associated SODE to obtain the conserved quantity. The work of Unal [3] shows that in the SODE context, the temporal infinitesimal need not be a function of time only This implies that the Lie algebra generated by the SODE can have non-projectable symmetries which will not belong to the Lie algebra generated by the FP equation. We first revisit the conserved quantity results of Unal [3] and juxtapose it with the new findings of our deliberations This scrutiny will be followed by an attempt to construct a conserved quantity based upon the methodology of [2] for Stratonovich integral SODEs
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