[Paper Review] On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors

Jean Bertoin|arXiv (Cornell University)|Apr 11, 2018

Stochastic processes and statistical mechanicsMathematics33 references20 citations

TL;DR

This paper establishes a necessary and sufficient condition for Malthusian behavior in growth-fragmentation semigroups using a Feynman-Kac probabilistic approach, and provides a simple criterion for exponential convergence speed. It extends prior results by proving the necessity of the Malthusian condition and offering new, verifiable conditions for exponential ergodicity, even in cases previously unaddressed in the literature.

ABSTRACT

This work develops further a probabilist approach to the asymptotic behavior of growth-fragmentation semigroups via the Feynman-Kac formula, which was introduced in a joint article with A.R. Watson [4]. Here, it is first shown that the sufficient condition for a Malthusian behavior which was established in [4], is also necessary. We then provide a simple criterion to ensure exponential speed of convergence, which enables us to treat cases than were not covered previously in the literature.

Motivation & Objective

• To establish the necessity of the Malthusian condition previously shown to be sufficient for asymptotic behavior in growth-fragmentation semigroups.
• To develop a probabilistic framework using the Feynman-Kac formula to analyze the long-time behavior of growth-fragmentation processes.
• To provide a new, simple criterion ensuring exponential speed of convergence to the asymptotic profile.
• To extend applicability to cases not covered by prior spectral theory methods, particularly those with non-standard growth and fragmentation rates.
• To unify and strengthen existing results on spectral gaps and ergodicity in growth-fragmentation dynamics.

Proposed method

• Utilizes the Feynman-Kac formula to express the growth-fragmentation semigroup in terms of expectations under a time-changed Lévy process.
• Applies the strong Markov property and moment generating functions to control exit times and exponential moments.
• Introduces a Lyapunov-type condition using power functions $ f(x) = x^r $ and $ g(x) = x^{-q} $ to ensure recurrence and prevent absorption at 0 or ∞.
• Derives conditions (25) and (26) on growth and fragmentation rates to ensure that the process does not explode or collapse.
• Combines these with the Feynman-Kac representation to prove exponential convergence under the criterion (17), (25), and (26).
• Applies the Krein-Rutman theorem indirectly via probabilistic coupling to establish existence of eigenelements and spectral gaps.

Experimental results

Research questions

• RQ1 Is the sufficient condition for Malthusian behavior in growth-fragmentation semigroups also necessary?
• RQ2 Can a probabilistic Feynman-Kac approach yield a simple, verifiable criterion for exponential convergence speed?
• RQ3 Under what conditions on the growth rate $ c(x) $ and fragmentation kernel $ k(x,y) $ does the semigroup converge exponentially fast?
• RQ4 How do Foster-type conditions on $ c(x) $ and $ k(x,y) $ ensure recurrence and prevent explosion or extinction?
• RQ5 In the self-similar fragmentation case, what explicit conditions on $ c(x)/xK(x) $ ensure exponential convergence?

Key findings

• The Malthusian behavior condition previously shown to be sufficient is also necessary, closing a gap in the theoretical understanding of growth-fragmentation semigroups.
• A new criterion for exponential convergence is established, which applies to cases not covered by earlier spectral theory methods.
• Exponential ergodicity holds if the process satisfies recurrence conditions (25) and (26) and the Malthusian condition (17), ensuring fast convergence to the asymptotic profile.
• In the self-similar case $ k(x,y) = x^{-1}K(x)p(y/x) $, the criterion reduces to bounds on $ c(x)/(xK(x)) $, as in (27) and (28), providing explicit verifiable conditions.
• The method improves upon Proposition 7.1 in [4] for homogeneous fragmentation rates ($ K(x) eq 1 $), offering stronger convergence results.
• The eigenfunction $ h $ is shown to be continuous and bounded when $ x o h(x)/x $, under the Feynman-Kac framework, confirming regularity of the principal eigenfunction.