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Jump Processes with Self-Interactions: Large Deviation Asymptotics

Amarjit Budhiraja, Francesco Coghi

TL;DR

The paper develops a Level 2.5 large deviation principle for self-interacting continuous-time Markov jump processes on a finite state space, where jump rates depend on the occupation measure. It introduces a dynamical, non-convex rate function that generalizes the Donsker–Varadhan functional via a weighted, discounted variational formulation tied to Poisson random measures and stochastic control. A novel time-reversal and rescaling framework is used to derive both Laplace upper and lower bounds, with careful construction of near-optimal controls, nondegeneracy, and continuity properties to establish the lower bound. The results elucidate a multiscale local equilibrium mechanism and yield contractions to the empirical current, offering a rigorous tool for analyzing non-Markovian memory effects in self-interacting systems. The framework is demonstrated through illustrative examples including autochemotaxis-inspired dynamics and congestion-modulated routing, highlighting potential applications in statistical physics and applied probability.

Abstract

We consider a pure jump process $\{X_t\}_{t\ge 0}$ with values in a finite state space $S= \{1, \ldots, d\}$ for which the jump rates at time instant $t$ depend on the occupation measure $L_t \doteq t^{-1} \int_0^t δ_{X_s}\,ds$. Such self-interacting chains arise in many contexts within statistical physics and applied probability. Under appropriate conditions, a large deviation principle is established for the pair $(L_t, R_t)$, as $t \to \infty$, where $R_t$ is the empirical flux process associated with the jump process. We show that the rate function takes a simple form that can be viewed as a dynamical generalization of the classical Donsker and Varadhan rate function for the analogous quantities in the setting of Markov processes, in particular, unlike the Markovian case, the rate function is not convex. Since the state process is non-Markovian, different techniques are needed than in the setting of Donsker and Varadhan and our proofs rely on variational representations for functionals of Poisson random measures and stochastic control methods.

Jump Processes with Self-Interactions: Large Deviation Asymptotics

TL;DR

The paper develops a Level 2.5 large deviation principle for self-interacting continuous-time Markov jump processes on a finite state space, where jump rates depend on the occupation measure. It introduces a dynamical, non-convex rate function that generalizes the Donsker–Varadhan functional via a weighted, discounted variational formulation tied to Poisson random measures and stochastic control. A novel time-reversal and rescaling framework is used to derive both Laplace upper and lower bounds, with careful construction of near-optimal controls, nondegeneracy, and continuity properties to establish the lower bound. The results elucidate a multiscale local equilibrium mechanism and yield contractions to the empirical current, offering a rigorous tool for analyzing non-Markovian memory effects in self-interacting systems. The framework is demonstrated through illustrative examples including autochemotaxis-inspired dynamics and congestion-modulated routing, highlighting potential applications in statistical physics and applied probability.

Abstract

We consider a pure jump process with values in a finite state space for which the jump rates at time instant depend on the occupation measure . Such self-interacting chains arise in many contexts within statistical physics and applied probability. Under appropriate conditions, a large deviation principle is established for the pair , as , where is the empirical flux process associated with the jump process. We show that the rate function takes a simple form that can be viewed as a dynamical generalization of the classical Donsker and Varadhan rate function for the analogous quantities in the setting of Markov processes, in particular, unlike the Markovian case, the rate function is not convex. Since the state process is non-Markovian, different techniques are needed than in the setting of Donsker and Varadhan and our proofs rely on variational representations for functionals of Poisson random measures and stochastic control methods.
Paper Structure (18 sections, 13 theorems, 225 equations)

This paper contains 18 sections, 13 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Model and Notation
  4. Main Result
  5. Laplace Asymptotics and Variational Representation
  6. Rescaling and time-reversed representation
  7. Dynamics of the time-reversed rescaled controlled empirical occupation measure
  8. Laplace upper bound
  9. Preparation for the Lower bound
  10. Alternative form for the rate function
  11. Selection of near minimizer
  12. Ensuring nondegeneracy.
  13. Finite time truncation and continuity of controls
  14. Lower Bound Proof
  15. Key Estimates
  16. ...and 3 more sections

Key Result

Theorem 3.4

The function $I$ is a rate function, namely it has compact sublevel sets. As $t \to \infty$, the collection $\{(L_t, R_t), t>0\}$ satisfies the LDP upper bound with speed $t$ and rate function $I$, namely for all closed sets $F \in \mathcal{P}(S) \times \mathbb{R}_+^a$, Suppose in addition that Assumption assu:lowbd is satisfied. Then, as $t \to \infty$, the collection $\{(L_t, R_t), t>0\}$ also

Theorems & Definitions (26)

  • Remark 3.3
  • Theorem 3.4
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 6.2
  • proof
  • Lemma 6.3
  • ...and 16 more