Forming invariant stochastic differential systems with a given first integral

TL;DR

This paper introduces a method to construct stochastic differential systems that preserve a specified smooth manifold, using symbolic computation and basis selection to ensure invariance, with practical examples and simulations demonstrating its effectiveness.

Contribution

It presents a novel algorithm for forming invariant stochastic differential systems with a given manifold, addressing basis degeneration issues and enabling symbolic implementation.

Findings

The method successfully constructs invariant stochastic systems for specified manifolds.

Numerical simulations confirm the theoretical invariance properties.

The approach handles basis degeneration to ensure stable system formation.

Abstract

This article proposes a method for forming invariant stochastic differential systems, namely dynamic systems with trajectories belonging to a given smooth manifold. The It\^o or Stratonovich stochastic differential equations with the Wiener component describe dynamic systems, and the manifold is implicitly defined by a differentiable function. A convenient implementation of the algorithm for forming invariant stochastic differential systems within symbolic computation environments characterizes the proposed method. It is based on determining a basis associated with a tangent hyperplane to the manifold. The article discusses the problem of basis degeneration and examines variants that allow for the simple construction of a basis that does not degenerate. Examples of invariant stochastic differential systems are given, and numerical simulations are performed for them.