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Local first integrals for stochastic differential equations

Kaiyin Huang, Wenlei Li, Shaoyun Shi, Zhiguo Xu

Abstract

Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] first provide a link between the existence of analytic first integrals and the resonant relations for analytic dynamical systems. In this paper, we show that by appropriately selecting the definition of the stochastic local first integrals, we are able to obtain the stochastic version of Poincaré non-integrability theorem. More specifically, we introduce two definitions of local first integrals for stochastic differential equations (SDEs) in the sense of probability one and expectation, respectively. We present the necessary conditions for the existence of functionally independent analytic or rational first integrals of SDEs via the resonances. We also show that for given integrable ordinary differential equations with some nondegeneracy conditions, there exists a linear stochastic perturbation such that the corresponding disturbed SDEs have no any analytic first integrals. Some examples are given to illustrate our results.

Local first integrals for stochastic differential equations

Abstract

Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] first provide a link between the existence of analytic first integrals and the resonant relations for analytic dynamical systems. In this paper, we show that by appropriately selecting the definition of the stochastic local first integrals, we are able to obtain the stochastic version of Poincaré non-integrability theorem. More specifically, we introduce two definitions of local first integrals for stochastic differential equations (SDEs) in the sense of probability one and expectation, respectively. We present the necessary conditions for the existence of functionally independent analytic or rational first integrals of SDEs via the resonances. We also show that for given integrable ordinary differential equations with some nondegeneracy conditions, there exists a linear stochastic perturbation such that the corresponding disturbed SDEs have no any analytic first integrals. Some examples are given to illustrate our results.
Paper Structure (4 sections, 11 theorems, 71 equations)

This paper contains 4 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Examples
  4. Discussion

Key Result

Theorem 1.1

Assume that the analytic differential system has a singularity at $x=0$, i.e., $f(0)=0$. If the eigenvalues of the Jacobian matrix $Df(0)$ at $x=0$ do not satisfy any $\mathbb{Z^+}$-resonant conditions, then system $(1.1)$ has no analytic first integrals in a neighborhood of the origin.

Theorems & Definitions (24)

  • Theorem 1.1: Poincaré non-integrability theorem
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • ...and 14 more