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Optimal Fluctuations for Discrete-time Markov Jump Processes

Feng Zhao, Jinjie Zhu, Yang Li, Xianbin Liu, Dongping Jin

Abstract

In the last few decades, noise-induced large fluctuations and transition phenomena have garnered significant attention in a variety of scientific contexts. The concept of prehistory probability has been proposed within the framework of Langevin dynamics to illustrate the focusing effect of large fluctuation paths onto a deterministic trajectory known as the optimal path. The present paper is devoted to showing that such a focusing effect persists within the framework of discrete-time Markov jump processes. Our proof leverages large deviation theory and the concept of time reversal for Markov jump processes. A key finding is the relationship identified between the optimal path and the time reversal of a specific family of probability distributions. This theoretical framework elucidates how an essentially deterministic mechanism can emerge from rare stochastic events in discrete-time Markov jump systems.

Optimal Fluctuations for Discrete-time Markov Jump Processes

Abstract

In the last few decades, noise-induced large fluctuations and transition phenomena have garnered significant attention in a variety of scientific contexts. The concept of prehistory probability has been proposed within the framework of Langevin dynamics to illustrate the focusing effect of large fluctuation paths onto a deterministic trajectory known as the optimal path. The present paper is devoted to showing that such a focusing effect persists within the framework of discrete-time Markov jump processes. Our proof leverages large deviation theory and the concept of time reversal for Markov jump processes. A key finding is the relationship identified between the optimal path and the time reversal of a specific family of probability distributions. This theoretical framework elucidates how an essentially deterministic mechanism can emerge from rare stochastic events in discrete-time Markov jump systems.
Paper Structure (14 sections, 16 theorems, 142 equations, 12 figures, 4 tables)

This paper contains 14 sections, 16 theorems, 142 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Large Deviation Principle
  4. Optimal Fluctuations on Finite Time Intervals
  5. Time Reversal of Discrete-time Markov Jump Processes
  6. Time Reversal of a Given Family of Probability Densities
  7. Conditional Probability Density of the Reversed Process
  8. Non-stationary Prehistory Probability Density
  9. Prehistorical Description of Optimal Fluctuations on Finite Time Intervals
  10. Examples
  11. Conclusions
  12. Proof of Lemma \ref{['theo020203']}
  13. Proof of Proposition \ref{['theo040101']}
  14. An Algorithm for Calculating the Non-stationary Prehistory Probability Density

Key Result

Theorem 2.1

Suppose that: Then the family of processes $\{\boldsymbol{x}^{\varepsilon}(t),P^{\varepsilon}_{\boldsymbol{x}_{0}}\}$ satisfies a large deviation principle with rate function $I_{[0,T]}$ (also called the normalized action functional) and normalizing coefficient $\varepsilon^{-1}$, uniformly with respect to the in

Figures (12)

  • Figure 1: (a) The Hamiltonian vector field, together with the stable (blue) and unstable (green) manifolds of the fixed points $(-1,0)$ and $(1,0)$. The unique (magenta) solution of the constrained Hamiltonian problem for $x_0=-1$, $x_T=1$, $T=5$. (b) Plot of $T(\alpha_0)$ versus $\alpha_0$ for $x_0=-1$, $x_T=1$. The monotonicity indicates that each $T$ corresponds to a unique NOP.
  • Figure 2: NPPDs and their peak trajectories for (a) $\varepsilon=0.2$, (b) $\varepsilon=0.05$, (c) $\varepsilon=0.01$. (d) Convergence of these peak trajectories to the NOP. Parameters: $x_0=-1$, $x_T=1$, $T=5$.
  • Figure 3: NPPDs and their peak trajectories for (a) $\varepsilon=0.2$, (b) $\varepsilon=0.06$, (c) $\varepsilon=0.02$. (d) Convergence of these peak trajectories to the NOP. Parameters: $x_0=0$, $x_T=1$, $T=3$, $\kappa=1.5$.
  • Figure 4: NPPDs and their peak trajectories for (a) $\varepsilon=0.2$, (b) $\varepsilon=0.06$, (c) $\varepsilon=0.02$. (d) Convergence of these peak trajectories to the NOP. Parameters: $x_0=0$, $x_T=1$, $T=3$, $\kappa=3$.
  • Figure 5: (a) The approximated Hamiltonian vector field, together with the stable (blue) and unstable (green) manifolds of the fixed points $(-1,0)$ and $(1,0)$. The unique (magenta) solution of the constrained Hamiltonian problem for $x_0=-1$, $x_T=1$, $T=5$, $\kappa=1.5$, $k=10$. (b) Plot of $T(\alpha_0)$ versus $\alpha_0$ for $x_0=-1$, $x_T=1$. The monotonicity indicates that each $T$ corresponds to a unique NOP.
  • ...and 7 more figures

Theorems & Definitions (37)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.2
  • Definition 2.1
  • Proposition 2.3
  • Remark 2.4
  • Lemma 2.4
  • Lemma 2.5
  • ...and 27 more