[Paper Review] Non-autonomous interacting particle systems in continuum

Martin Friesen|arXiv (Cornell University)|Mar 23, 2016

Stochastic processes and statistical mechanics40 references3 citations

TL;DR

This paper constructs a conservative Feller evolution for non-autonomous, time-inhomogeneous interacting particle systems in continuum using a Foster-Lyapunov condition and weak continuity of transition kernels. The key contribution is the rigorous existence and uniqueness of a Feller process on the configuration space of finite particle systems in $\mathbb{R}^d$, applicable to general birth-and-death dynamics with spatial structure.

ABSTRACT

A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.

Motivation & Objective

• To establish a conservative Feller evolution for time-inhomogeneous, pure jump Markov processes on a locally compact Polish space.
• To extend semigroup methods to non-autonomous interacting particle systems in continuum, where particle positions are continuous in $\mathbb{R}^d$.
• To provide a rigorous mathematical framework for general birth-and-death processes with spatial interaction, including models like BDLP and Dieckmann-Law.
• To ensure the Feller property and strong Feller regularity under a Foster-Lyapunov type condition and weak continuity of transition kernels.
• To apply the abstract results to concrete models in mathematical ecology and population dynamics with spatial structure.

Proposed method

• Constructs a time-dependent generator $L(t)$ acting on bounded continuous functions via a transition kernel $Q(t,x,dy)$ on a locally compact Polish space $E$.
• Imposes a Foster-Lyapunov condition to control the tail behavior of the jump intensity and ensure stochastically boundedness.
• Uses weak continuity of the transition kernel $Q(t,x,dy)$ in space and time to ensure regularity of the evolution.
• Applies classical results from [20, 23, 17] to construct a conservative Feller evolution system $U(s,t)$ on $C_b(E)$, the space of bounded continuous functions.
• Establishes the Feller property via compact containment estimates and uniform control over time intervals.
• Applies the theory to configuration space $\Gamma_0$ of finite particle configurations in $\mathbb{R}^d$, treating particle jumps and birth-death events via a kernel $K(\xi,\eta,d\zeta)$.

Experimental results

Research questions

• RQ1 Can a conservative Feller evolution be constructed for non-autonomous, time-inhomogeneous pure jump processes on a locally compact Polish space?
• RQ2 Under what conditions does a time-dependent transition kernel $Q(t,x,dy)$ generate a Feller process on the configuration space $\Gamma_0$?
• RQ3 How can the Foster-Lyapunov criterion be adapted to non-autonomous systems to ensure tightness and regularity?
• RQ4 What are the sufficient conditions for the Feller property in the context of interacting particle systems in continuum with spatial dynamics?
• RQ5 How do the results apply to concrete models such as the BDLP and Dieckmann-Law models with spatially distributed particle interactions?

Key findings

• A conservative Feller evolution system $U(s,t)$ is constructed on $C_b(E)$ for a time-inhomogeneous pure jump process under a Foster-Lyapunov condition and weak continuity of the transition kernel.
• The Feller property is established via compact containment: for any $T>0$, compact $B\subset E$, and $\varepsilon>0$, there exists a compact $A\subset E$ such that $Q(t,\eta,A^c)<\varepsilon$ for all $t\in[0,T]$ and $\eta\in B$.
• The construction ensures existence and uniqueness of the associated Hunt process with state space $E$, extending classical semigroup methods to non-autonomous settings.
• The method applies to configuration space $\Gamma_0$ of finite particle systems in $\mathbb{R}^d$, enabling rigorous analysis of spatially structured birth-and-death processes.
• The Foster-Lyapunov condition is verified under sufficient decay of the jump rate kernel, such as $a^+(t,x) \leq C(t)/(\lambda(t)+|x|^2)^\alpha$ with $\alpha > d/2$.
• The results generalize previous work on lattice models and extend the applicability of Kolmogorov equations to non-autonomous, spatially continuous particle systems.