Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2405.10193

The Lamperti transformation in the infinite-dimensional setting, self-similar populations, and coalescents

Arno Siri-Jégousse·Alejandro Hernández Wences
View original source ↗Read source document ↗Export metadata ↓

Abstract

We propose a change in focus from the prevalent paradigm based on the branching property as a tool to analyze the structure of population models, to one based on the self-similarity property, which we also introduce for the first time in the setting of measure-valued processes. By extending the well-known Lamperti transformation into the infinite dimensional setting, we were able to embed and extend known results in population genetics within the self-similarity framework: we describe the frequency process of a larger class of measure-valued SS populations in terms of general Lambda Fleming-Viot processes. Our results demonstrate the potential power of the self-similar perspective for the study of populations whose total size varies stochastically over time, and in which the reproduction dynamics of the individuals are not independent from one another but are modulated by the total size of the population, allowing for more complex and realistic models. We also uncover a new duality relation between measure-valued processes and Lambda-coalescents which extends the well-known duality relation between Lambda Fleming-Viot processes and Lambda coalescents.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Arno Siri-Jégousse, Alejandro Hernández Wences. 2026-04-13. The Lamperti transformation in the infinite-dimensional setting, self-similar populations, and coalescents. https://arxiv.org/abs/2405.10193

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bounded weak solutions to cross-diffusion semiconductor model with electron-hole scattering

Semiconductor model is a system of parabolic partial differential equations with cross-diffusion phenomenon. Previous results showed that a weak solution exists and is not bounded in general. So semiconductor model was categorized as a cross-diffusion system without bounded weak solutions. In this work, we show that once the initial value is bounded, there exists a weak solution that is also bounded. The entropy method is a major tool in global existence analysis of cross-diffusion systems. We notice that traditional entropies in volume-filling cases may not provide required positive semi-definiteness result for the existence proof. In this situation, a transformation of variables technique has been applied. The product between Hessian matrix of the entropy and replacement diffusion matrix is positive semi-definite, then we apply the entropy method to show semiconductor model has a bounded weak solution.

math.PR↗
arXiv2026

Global existence and uniqueness analysis of cross-diffusion multispecies chemotaxis system with volume-filling

The system of multispecies chemotaxis equations is a cross-diffusion system with volume-filling. In this work, we show that a weak solution of the two species chemotaxis system exists. The entropy method is a major tool in existence analysis of cross-diffusion systems. Previous investigations indicate that traditional entropies in volume-filling cases may not be able to provide required gradient estimates. In this situation, we upgrade existing matrix computation methods to derive gradient estimates. Due to the cross-diffusion phenomenon, the uniqueness of the weak solution to a cross-diffusion system is very difficult to prove in general. In this work, we apply the distance functional to show that when parameters of the chemotaxis system are identical, the weak solution is unique.

math.PR↗
arXiv2026

Self-normalized scaled quadratic variation

The concept of a scaled quadratic variation was originally introduced by E. Gladyshev in 1961 for processes with Gaussian increments. Using certain deterministic scaling, arrived at from the covariance of the process, Gladyshev showed that the sum of scaled square increments along the dyadic partition sequence converges almost surely to a finite limit. In this paper, we propose a pathwise counterpart in which the deterministic normalization is replaced by a self-normalizing factor built from the $p$-th variation of the path along a given sequence of partitions. The resulting quantity requires no probabilistic assumption and no knowledge of a covariance structure, and its scale is both path-dependent and sensitive to the partition sequence. Under a mild regularity condition on the limiting $p$-th variation, we show that the self-normalized and the classical deterministic normalizations are comparable, and for fractional Brownian motion the two agree up to a multiplicative constant. We establish a switching behaviour in the index, and prove that for $p \ge 2$ the self-normalized scaled quadratic variation obeys a smooth-transformation formula under $C^2$ maps; at $p=2$ this recovers the known transformation rule for quadratic variation. Since only squared increments are scaled, the construction polarizes, yielding a matrix-valued scaled quadratic variation for every $p \geq 1$ for $\mathbb R^d$ valued paths. We conclude with examples beyond the Gaussian setting.

math.PR↗