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Transition pathways for a class of high dimensional stochastic dynamical systems with Lévy noise

Jianyu Hu, Jianyu Chen

Abstract

This work is devoted to deriving the Onsager-Machlup action functional for a class of stochastic differential equations with (non-Gaussian) Lévy process as well as Brownian motion in high dimensions. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. The Poincaré lemma is essential to handle such path representation problem in high dimensions. We provide a sufficient condition on the vector field such that this path representation holds in high dimensions. Moreover, this Onsager-Machlup action functional may be considered as the integral of a Lagrangian. Finally, by a variational principle, we investigate the most probable transition pathways analytically and numerically.

Transition pathways for a class of high dimensional stochastic dynamical systems with Lévy noise

Abstract

This work is devoted to deriving the Onsager-Machlup action functional for a class of stochastic differential equations with (non-Gaussian) Lévy process as well as Brownian motion in high dimensions. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. The Poincaré lemma is essential to handle such path representation problem in high dimensions. We provide a sufficient condition on the vector field such that this path representation holds in high dimensions. Moreover, this Onsager-Machlup action functional may be considered as the integral of a Lagrangian. Finally, by a variational principle, we investigate the most probable transition pathways analytically and numerically.

Paper Structure

This paper contains 8 sections, 2 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Onsager-Machlup action functional
  3. The framework
  4. Derivation of Onsager-Machlup action functional
  5. Euler-Lagrange equation and the most probable transition pathway
  6. An application to the Maier-Stein system
  7. Conclusion and discussion
  8. Proof of Theorem 2.5

Key Result

Lemma 2.2

A Lévy process with Lévy triplet $(A,\nu, b)$ has bounded variation if and only if $A=0$ and the jump measure satisfies $\int_{|x|<1}x\nu(dx)<\infty$.

Figures (3)

  • Figure 1: The patterns of the Maier-Stein system. The vector field is described by (a), the potential is shown as (b).
  • Figure 2: The most probable transition pathway from SN2 to SN1 for system (4.1). The small green circle in the figure represents the transition pathway. Initial data z(0)=(1,0) and finial data z(0)=(-1,0). The Lévy noise $L_1\sim S_{0.5}(1,0.5,0)$ and $L_2\sim S_{0.7}(1,0,0)$.
  • Figure 3: The patterns of sample paths and the most probable transition pathway.

Theorems & Definitions (13)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • proof
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • proof
  • ...and 3 more