Continuous Branching Processes with Settlement in Cancer Metastasis: Stochastic Modelling and the Feller Property

TL;DR

This paper develops a stochastic model of cancer metastasis using multi-type branching processes that incorporate particle movement, settlement, and absorption, and investigates the Feller property of the process.

Contribution

It introduces a new multi-type branching process model with spatial and phase dynamics, rigorously constructs it, and analyzes its Feller property and generator.

Findings

Established the Markov property via embedding into a multidimensional process.

Proved the Feller property for a simplified model.

Derived an explicit generator enabling Feynman-Kac formulas.

Abstract

Motivated by models of cancer metastasis, this paper introduces a type of (multi-type) branching process that records the positions of particles, representing tumor cells or clusters. Particles may be absorbed (removed from the state space), move, or settle. The process is rigorously constructed, and the Markov property is established via embedding into a multidimensional process that tracks the labels, positions, and phases (moving or resting) of living particles. The Feller property for the associated semigroup is investigated. It is proved for a simplified model that tracks the number of particles in each class, and an explicit generator is derived, enabling Feynman-Kac-type formulas in this framework.