arXiv subjects

Christopher B. C. Dean

Publications and source records attributed to Christopher B. C. Dean.

5 recordsLinked to original sources

arXiv2026

Functional central limit theorems for non-local branching Markov processes

The aim of this paper is to study the fluctuations of a general class of supercritical branching Markov processes with non-local branching mechanisms. We establish functional central limit theorems and show that the limiting behaviour falls into three regimes, determined by the size of the spectral gap associated with the first-moment semigroup of the branching process. The main novelty is to develop a unified functional fluctuation theory for spatial branching Markov processes with non-local reproduction, allowing a general finite-dimensional spectral structure for the first-moment semigroup, including non-simple leading eigenvalues and nilpotent Jordan-type components. In doing so, we extend the classical small, critical and large fluctuation trichotomy beyond the finite-type and local spatial settings, and obtain limiting processes that capture the covariance structure induced by non-local offspring displacement.

math.PR↗

arXiv2026

SDE-based Monte Carlo dose calculation for proton therapy validated against Geant4

Objective: To assess the accuracy and computational performance of a stochastic differential equation (SDE)--based model for proton beam dose calculation by benchmarking against Geant4 in simplified phantom geometries. Approach: Building on Crossley et al. (2025), we implemented the SDE model using standard approximations to interaction cross sections and mean excitation energies, enabling straightforward adaptation to new materials and configurations. The model was benchmarked against Geant4 in homogeneous, longitudinally heterogeneous and laterally heterogeneous phantoms to assess depth--dose behaviour, lateral transport and material heterogeneities. Main results: Across all phantoms and beam energies, the SDE model reproduced the main depth--dose characteristics predicted by Geant4, with proton range agreement within 0.2 mm for 100 MeV beams and 0.6 mm for 150 MeV beams. Voxel--wise comparisons yielded gamma pass rates exceeding 95% under 2%/0.5 mm criteria with a 1% dose threshold. Differences were localised to steep dose gradients or material interfaces, while overall lateral beam dispersion was well reproduced. The SDE model achieved speed-up factors of about 2.5--3 relative to single-threaded Geant4. Significance: The SDE approach reproduces key dosimetric features with good accuracy at lower computational cost and is amenable to parallel and GPU implementations, supporting fast proton therapy dose calculations.

physics.med-ph↗

arXiv2025

A moment approach to the law of large numbers for supercritical branching Markov processes

We offer a new proof of the classical law of large numbers for a general class of branching Markov processes based on the asymptotic behaviour of the moments developed in \cite{bmoments, gonzalez2022erratum}. Moreover, we show that the law of the limiting random variable, that is the almost sure limit of the classical additive martingale, is completely determined by its moments.

math.PR↗

arXiv2025

More on the asymptotic behaviour of moments of branching Markov processes

Consider a branching Markov process, $X = (X(t), t \ge 0)$, with non-local branching mechanism. Studying the asymptotic behaviour of the moments of X has recently received attention in the literature [6, 7] due to the importance of these results in understanding the underlying genealogical structure of $X$. In this article, we generalise the results of [7] to allow for a non-simple leading eigenvalue and to also study the higher order fluctuations of the moments of $X$. These results will be useful for proving central limit theorems and extending well-known LLN results.

math.PR↗

arXiv2022

Functional limit theorems for Pólya urns with growing initial compositions

In this paper, we prove functional limit theorems for Pólya urn processes whose number of draws and initial number of balls tend to infinity together. This is motivated by recent work of Borovkov [5], where they prove a functional limit theorem for this model when the urn has identity replacement rule. We generalize this result to arbitrary balanced replacement rules (the total number of balls added to the urn is deterministic). Three asymptotic regimes are possible depending on how one lets the number of initial balls scale with the number of draws of the urn. In each regime, we show a first order deterministic limit and Gaussian second order fluctuations, where the behaviour of these limit processes depend on the regime, the initial composition of the urn, and the urns replacement rule. To prove our main results, we embed the process in continuous-time and use martingale theory. Although these methods are classical since the works of Athreya & Karlin [1] and Janson [11], our setting with initial growing composition necessitates many new ideas. The main difference in proving limiting results for the continuous time embedding in our setting is that, when the initial composition is large compared to the number of draws, the branching process does not have time to reach equilibrium. Because of this, translating the results back to discrete-time is also much harder than in Janson [11]. Interestingly, our continuous-time results hold under weaker assumptions on the replacement structure than classical results for multi-type branching processes; in particular, we do not need any "irreducibility" assumption.

math.PR↗