[Paper Review] Long-time behaviour of generalised Zig-Zag process

Ninon Fétique|arXiv (Cornell University)|Oct 3, 2017

Stochastic processes and statistical mechanics6 references3 citations

TL;DR

This paper establishes exponential ergodicity and exponential moment bounds for a generalized Zig-Zag process—a piecewise-deterministic Markov process modeling bacterial chemotaxis with velocity in the unit ball. Using Lyapunov functions and semi-regenerative process theory, it proves exponential convergence to the invariant measure under mild conditions on the jump rate and velocity kernel, with explicit moment bounds in both one and higher dimensions.

ABSTRACT

We study the long-time behaviour of a class of piecewise-deterministic Markov processes which are an extension of some recent works. These $d$-dimensional processes, d>=1, can especially be used to model the motion of a bacterium in presence of a chemo-attractant, with parameters depending both on the position and the velocity of the bacterium. Using the method of Meyn and Tweedie, we show that under some good assumptions on the parameters of the model, such a process converges exponentially fast towards its invariant measure. We also establish the existence of exponential moments of the invariant measure using results on semi-regenerative processes. The one-dimensional case is studied separately since complementary results can be obtained in that particular case.

Motivation & Objective

• To analyze the long-time behavior of a generalized Zig-Zag process with velocity in the unit ball, extending models of bacterial motion in chemo-attractant fields.
• To establish exponential ergodicity for the process under mild regularity and growth conditions on the jump rate and velocity kernel.
• To derive explicit bounds on the exponential moments of the invariant measure using semi-regenerative process techniques.
• To compare the one-dimensional and higher-dimensional cases, highlighting structural differences in ergodicity proofs and assumptions.
• To demonstrate that the method applies to processes not covered by prior assumptions, such as non-monotonic jump rates, by constructing a counterexample satisfying the new conditions.

Proposed method

• Employs the Lyapunov function method of Meyn and Tweedie to establish geometric ergodicity of the PDMP.
• Models the process as a semi-regenerative process by considering hitting times of the origin in one dimension.
• Uses upper bounds on the exponential moments of the hitting time of the origin to verify conditions for moment bounds on the invariant measure.
• Applies Theorem 4.1 from semi-regenerative process theory to derive integrability of exponential moments under specific rate and kernel conditions.
• Constructs a Lyapunov function based on the sign of the position-velocity scalar product in one dimension, exploiting the process's tendency to return to the origin.
• Compares the one-dimensional case with higher dimensions by analyzing the role of $X_t \cdot V_t$ in determining return behavior and deriving distinct assumptions.

Experimental results

Research questions

• RQ1 Under what conditions does the generalized Zig-Zag process converge exponentially fast to its invariant measure?
• RQ2 How can exponential moments of the invariant measure be bounded for this class of PDMPs?
• RQ3 What are the structural differences in ergodicity proofs between the one-dimensional and higher-dimensional cases?
• RQ4 Can the ergodicity results be extended to processes with non-monotonic jump rates, such as those depending on position and velocity in a non-trivial way?
• RQ5 To what extent do the assumptions in this work cover models not addressed by prior studies, such as those with non-symmetric or non-product jump rates?

Key findings

• Under suitable assumptions on the jump rate $\lambda(x,v)$ and velocity kernel $Q(x,v,\cdot)$, the generalized Zig-Zag process converges exponentially fast to its invariant measure.
• Exponential moments of the invariant measure exist for all $\alpha < \gamma$ and $\beta > 0$, with $\int_{\mathbb{R} \times [-1,1]} e^{\alpha|x| + \beta|v|} \pi(dx,dv) < \infty$, as established via semi-regenerative process techniques.
• In one dimension, the process is exponentially ergodic even when the jump rate is not monotonic, as demonstrated by a constructed example satisfying Assumption $(\mathcal{A}_4)$ but not $(\mathcal{H}_4)$.
• The one-dimensional proof relies on the sign of $X_t \cdot V_t$ to determine return behavior, which does not generalize directly to higher dimensions.
• The method yields a quantitative bound on the exponential moment via the expression $\frac{1}{2}\int_{-1}^{1} G(\alpha,v) dv \leq J_* < 1$, which holds for $\alpha$ in a specific interval $I_*$.
• The paper shows that the one-dimensional analysis can be extended to higher dimensions, but the converse is not true due to the lack of a simple sign-based return criterion in higher dimensions.