Pub Date : 2026-02-01Epub Date: 2025-10-21DOI: 10.1016/j.spa.2025.104811
David Clancy Jr.
Enriquez et al. (2025) have established process-level fluctuations for the giant of the dynamic Erdős–Rényi random graph above criticality and show that the limit is a centered Gaussian process with continuous sample paths. A random walk proof was recently obtained by Corujo et al. (2024). We show that a similar result holds for rank-one inhomogeneous models whenever the empirical weight distribution converges to a limit and its second moment converges as well.
Enriquez et al.（2025）建立了临界以上动态Erdős-Rényi随机图的巨型过程级波动，并表明其极限是一个样本路径连续的中心高斯过程。Corujo et al.（2024）最近获得了一个随机行走证明。我们表明，当经验权重分布收敛到一个极限并且其第二矩也收敛时，对于秩一非齐次模型也有类似的结果。
{"title":"Fluctuations of the giant of Poisson random graphs","authors":"David Clancy Jr.","doi":"10.1016/j.spa.2025.104811","DOIUrl":"10.1016/j.spa.2025.104811","url":null,"abstract":"<div><div>Enriquez et al. (2025) have established process-level fluctuations for the giant of the dynamic Erdős–Rényi random graph above criticality and show that the limit is a centered Gaussian process with continuous sample paths. A random walk proof was recently obtained by Corujo et al. (2024). We show that a similar result holds for rank-one inhomogeneous models whenever the empirical weight distribution converges to a limit and its second moment converges as well.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104811"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-10-21DOI: 10.1016/j.spa.2025.104806
Oskar Eklund, Annika Lang, Moritz Schauer
The smoothing distribution is the conditional distribution of the diffusion process in the space of trajectories given noisy observations made continuously in time. It is generally difficult to sample from this distribution. We use the theory of enlargement of filtrations to show that the conditional process has an additional drift term derived from the backward filtering distribution that is moving or guiding the process towards the observations. This term is intractable, but its effect can be equally introduced by replacing it with a heuristic, where importance weights correct for the discrepancy. From this Markov Chain Monte Carlo and sequential Monte Carlo algorithms are derived to sample from the smoothing distribution. The choice of the guiding heuristic is discussed from an optimal control perspective and evaluated. The results are tested numerically on a stochastic differential equation for reaction–diffusion.
{"title":"Guided smoothing and control for diffusion processes","authors":"Oskar Eklund, Annika Lang, Moritz Schauer","doi":"10.1016/j.spa.2025.104806","DOIUrl":"10.1016/j.spa.2025.104806","url":null,"abstract":"<div><div>The smoothing distribution is the conditional distribution of the diffusion process in the space of trajectories given noisy observations made continuously in time. It is generally difficult to sample from this distribution. We use the theory of enlargement of filtrations to show that the conditional process has an additional drift term derived from the backward filtering distribution that is moving or guiding the process towards the observations. This term is intractable, but its effect can be equally introduced by replacing it with a heuristic, where importance weights correct for the discrepancy. From this Markov Chain Monte Carlo and sequential Monte Carlo algorithms are derived to sample from the smoothing distribution. The choice of the guiding heuristic is discussed from an optimal control perspective and evaluated. The results are tested numerically on a stochastic differential equation for reaction–diffusion.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104806"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-10-10DOI: 10.1016/j.spa.2025.104788
Thomas Cavallazzi
We prove Itô’s formula for the flow of measures associated with a jump process defined by a drift, an integral with respect to a Poisson random measure and with respect to the associated compensated Poisson random measure. We work in , the space of probability measures on having a finite moment of order . As an application, we exhibit the backward Kolmogorov partial differential equation stated on associated with a McKean–Vlasov stochastic differential equation driven by a Poisson random measure. It describes the dynamics of the semigroup acting on functions defined on associated with the McKean–Vlasov stochastic differential equation, under regularity assumptions on it. Finally, we use the semigroup and the backward Kolmogorov equation to prove new quantitative weak propagation of chaos results for a mean-field system of interacting Ornstein–Uhlenbeck processes driven by i.i.d. -stable processes with .
{"title":"Itô’s formula for the flow of measures of Poisson stochastic integrals and applications","authors":"Thomas Cavallazzi","doi":"10.1016/j.spa.2025.104788","DOIUrl":"10.1016/j.spa.2025.104788","url":null,"abstract":"<div><div>We prove Itô’s formula for the flow of measures associated with a jump process defined by a drift, an integral with respect to a Poisson random measure and with respect to the associated compensated Poisson random measure. We work in <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, the space of probability measures on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> having a finite moment of order <span><math><mrow><mi>β</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></mrow></math></span>. As an application, we exhibit the backward Kolmogorov partial differential equation stated on <span><math><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> associated with a McKean–Vlasov stochastic differential equation driven by a Poisson random measure. It describes the dynamics of the semigroup acting on functions defined on <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>β</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> associated with the McKean–Vlasov stochastic differential equation, under regularity assumptions on it. Finally, we use the semigroup and the backward Kolmogorov equation to prove new quantitative weak propagation of chaos results for a mean-field system of interacting Ornstein–Uhlenbeck processes driven by i.i.d. <span><math><mi>α</mi></math></span>-stable processes with <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104788"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145326388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-11-07DOI: 10.1016/j.spa.2025.104825
Qingwei Liu , Nicolas Privault
We derive normal approximation bounds for generalized -statistics of the form where and are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erdős–Rényi model.
{"title":"Gaussian fluctuations of generalized U-statistics and subgraph counting in the binomial random-connection model","authors":"Qingwei Liu , Nicolas Privault","doi":"10.1016/j.spa.2025.104825","DOIUrl":"10.1016/j.spa.2025.104825","url":null,"abstract":"<div><div>We derive normal approximation bounds for generalized <span><math><mi>U</mi></math></span>-statistics of the form <span><span><span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>≔</mo><munder><mrow><mo>∑</mo></mrow><mrow><mfrac><mrow><mn>1</mn><mo>≤</mo><mi>β</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>β</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mo>≤</mo><mi>n</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>≠</mo><mi>β</mi><mrow><mo>(</mo><mi>j</mi><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≠</mo><mi>j</mi><mo>≤</mo><mi>k</mi></mrow></mfrac></mrow></munder><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msub><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mi>β</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>β</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mi>β</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msub></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>}</mo></mrow></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erdős–Rényi model.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104825"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-10-18DOI: 10.1016/j.spa.2025.104803
Attila Lovas
Nonlinear time series models with exogenous regressors are essential in econometrics, queuing theory, and machine learning, though their statistical analysis remains incomplete. Key results, such as the law of large numbers and the functional central limit theorem, are known for weakly dependent variables. We demonstrate the transfer of mixing properties from the exogenous regressor to the response via coupling arguments. Additionally, we study Markov chains in random environments with drift and minorization conditions, even under non-stationary environments with favorable mixing properties, and apply this framework to single-server queuing models.
{"title":"Transition of α-mixing in random iterations with applications in queuing theory","authors":"Attila Lovas","doi":"10.1016/j.spa.2025.104803","DOIUrl":"10.1016/j.spa.2025.104803","url":null,"abstract":"<div><div>Nonlinear time series models with exogenous regressors are essential in econometrics, queuing theory, and machine learning, though their statistical analysis remains incomplete. Key results, such as the law of large numbers and the functional central limit theorem, are known for weakly dependent variables. We demonstrate the transfer of mixing properties from the exogenous regressor to the response via coupling arguments. Additionally, we study Markov chains in random environments with drift and minorization conditions, even under non-stationary environments with favorable mixing properties, and apply this framework to single-server queuing models.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104803"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-10-16DOI: 10.1016/j.spa.2025.104794
Diego Goldsztajn , Sem C. Borst , Johan S.H. van Leeuwaarden
Consider a network of single-server queues where tasks arrive independently at each server at rate . The servers are connected by a graph that is resampled at rate in a way that is symmetric with respect to the servers, and each task is dispatched to the shortest queue in the graph neighborhood where it appears. We aim to gain insight in the impact of the dynamic network structure on the load balancing dynamics in terms of the occupancy process which describes the empirical distribution of the number of tasks across the servers. This process evolves on the underlying dynamic graph, and its dynamics depend on the number of tasks at each individual server and the neighborhood structure of the graph. We establish that this dependency disappears in the limit as when and , and prove that the limit of the occupancy process is given by a system of differential equations that depends solely on and the limiting degree distribution of the graph. We further show that the stationary distribution of the occupancy process converges to an equilibrium of the differential equations, and derive properties of this equilibrium that reflect the impact of the degree distribution. Our focus is on truly sparse graphs where the maximum degree is uniformly bounded across , which is natural in load balancing systems.
{"title":"Fluid limits for interacting queues in sparse dynamic graphs","authors":"Diego Goldsztajn , Sem C. Borst , Johan S.H. van Leeuwaarden","doi":"10.1016/j.spa.2025.104794","DOIUrl":"10.1016/j.spa.2025.104794","url":null,"abstract":"<div><div>Consider a network of <span><math><mi>n</mi></math></span> single-server queues where tasks arrive independently at each server at rate <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The servers are connected by a graph that is resampled at rate <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in a way that is symmetric with respect to the servers, and each task is dispatched to the shortest queue in the graph neighborhood where it appears. We aim to gain insight in the impact of the dynamic network structure on the load balancing dynamics in terms of the occupancy process which describes the empirical distribution of the number of tasks across the servers. This process evolves on the underlying dynamic graph, and its dynamics depend on the number of tasks at each individual server and the neighborhood structure of the graph. We establish that this dependency disappears in the limit as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span> when <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>n</mi><mo>→</mo><mi>λ</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><mi>∞</mi></mrow></math></span>, and prove that the limit of the occupancy process is given by a system of differential equations that depends solely on <span><math><mi>λ</mi></math></span> and the limiting degree distribution of the graph. We further show that the stationary distribution of the occupancy process converges to an equilibrium of the differential equations, and derive properties of this equilibrium that reflect the impact of the degree distribution. Our focus is on truly sparse graphs where the maximum degree is uniformly bounded across <span><math><mi>n</mi></math></span>, which is natural in load balancing systems.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104794"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145326463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-10-21DOI: 10.1016/j.spa.2025.104809
Kou Fujimori , Koji Tsukuda
We propose a two-step estimation procedure for stochastic process models with high-dimensional parameters of interest under heteroskedasticity. In low-dimensional settings, when a consistent estimator for a nuisance parameter that characterizes the conditional variance is available, one can construct an asymptotically normal estimator for the parameter of interest under appropriate conditions. Motivated by this fact, we extend the idea to high-dimensional settings. We first establish variable selection via the Dantzig selector, and then combine this with consistent estimation of the nuisance parameter to develop a two-step procedure that yields an asymptotically normal estimator. Our framework accommodates infinite-dimensional nuisance parameters in the conditional variance term. Therefore, this study extends sparse estimation methods to a broader class of stochastic process models. Applications to ergodic time series models, including integer-valued autoregressive models and ergodic diffusion processes, are presented.
{"title":"Two-step estimations via the Dantzig selector for models of stochastic processes with high-dimensional parameters","authors":"Kou Fujimori , Koji Tsukuda","doi":"10.1016/j.spa.2025.104809","DOIUrl":"10.1016/j.spa.2025.104809","url":null,"abstract":"<div><div>We propose<!--> <!-->a<!--> <!-->two-step estimation procedure for stochastic process models with high-dimensional parameters of interest under heteroskedasticity. In low-dimensional settings, when a consistent estimator for a nuisance parameter that characterizes the conditional variance is available, one can construct an asymptotically normal estimator for the parameter of interest under appropriate conditions. Motivated by this fact, we extend the idea to high-dimensional settings. We first establish variable selection via the Dantzig selector, and then combine this with consistent estimation of the nuisance parameter to develop a two-step procedure that yields an asymptotically normal estimator. Our framework accommodates infinite-dimensional nuisance parameters in the conditional variance term. Therefore, this study extends sparse estimation methods to a broader class of stochastic process models. Applications to ergodic time series models, including integer-valued autoregressive models and ergodic diffusion processes, are presented.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104809"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-10-23DOI: 10.1016/j.spa.2025.104812
G. Alsmeyer , F. Cordero , H. Dopmeyer
We consider a population with two types of individuals, distinguished by the resources required for reproduction: type-0 (small) individuals need a fractional resource unit of size , while type-1 (large) individuals require 1 unit. The total available resource per generation is . To form a new generation, individuals are sampled one by one, and if enough resources remain, they reproduce, adding their offspring to the next generation. The probability of sampling an individual whose offspring is small is , where is the proportion of small individuals in the current generation. We call this discrete-time stochastic model a two-size Wright–Fisher model, where the function can represent mutation and/or frequency-dependent selection. We show that on the evolutionary time scale, i.e. accelerating time by a factor , the frequency process of type-0 individuals converges to the solution of a Wright–Fisher-type SDE. The drift term of that SDE accounts for the bias introduced by the function and the consumption strategy, the latter also inducing an additional multiplicative factor in the diffusion term. To prove this, the dynamics within each generation are viewed as a renewal process, with the population size corresponding to the first passage time above level . The proof relies on methods from renewal theory, in particular a uniform version of Blackwell’s renewal theorem for binary, non-arithmetic random variables, established via -coupling.
{"title":"A two-size Wright–Fisher model: asymptotic analysis via uniform renewal theory","authors":"G. Alsmeyer , F. Cordero , H. Dopmeyer","doi":"10.1016/j.spa.2025.104812","DOIUrl":"10.1016/j.spa.2025.104812","url":null,"abstract":"<div><div>We consider a population with two types of individuals, distinguished by the resources required for reproduction: type-0 (small) individuals need a fractional resource unit of size <span><math><mrow><mi>ϑ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, while type-1 (large) individuals require 1 unit. The total available resource per generation is <span><math><mi>R</mi></math></span>. To form a new generation, individuals are sampled one by one, and if enough resources remain, they reproduce, adding their offspring to the next generation. The probability of sampling an individual whose offspring is small is <span><math><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mi>R</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>x</mi></math></span> is the proportion of small individuals in the current generation. We call this discrete-time stochastic model a two-size Wright–Fisher model, where the function <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> can represent mutation and/or frequency-dependent selection. We show that on the evolutionary time scale, i.e. accelerating time by a factor <span><math><mi>R</mi></math></span>, the frequency process of type-0 individuals converges to the solution of a Wright–Fisher-type SDE. The drift term of that SDE accounts for the bias introduced by the function <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> and the consumption strategy, the latter also inducing an additional multiplicative factor in the diffusion term. To prove this, the dynamics within each generation are viewed as a renewal process, with the population size corresponding to the first passage time <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> above level <span><math><mi>R</mi></math></span>. The proof relies on methods from renewal theory, in particular a uniform version of Blackwell’s renewal theorem for binary, non-arithmetic random variables, established via <span><math><mi>ɛ</mi></math></span>-coupling.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104812"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-10-22DOI: 10.1016/j.spa.2025.104808
Benny Avelin , Tuomo Kuusi , Patrik Nummi , Eero Saksman , Jonas M. Tölle , Lauri Viitasaari
We study unique solvability for one-dimensional stochastic pressure equation with diffusion coefficient given by the Wick exponential of log-correlated Gaussian fields. We prove well-posedness for Dirichlet, Neumann and periodic boundary data and the initial value problem, covering the cases of both the Wick renormalization of the diffusion and of point-wise multiplication. We provide explicit representations for the solutions in both cases, characterized by the -transform and the Gaussian multiplicative chaos measure.
{"title":"1D stochastic pressure equation with log-correlated Gaussian coefficients","authors":"Benny Avelin , Tuomo Kuusi , Patrik Nummi , Eero Saksman , Jonas M. Tölle , Lauri Viitasaari","doi":"10.1016/j.spa.2025.104808","DOIUrl":"10.1016/j.spa.2025.104808","url":null,"abstract":"<div><div>We study unique solvability for one-dimensional stochastic pressure equation with diffusion coefficient given by the Wick exponential of log-correlated Gaussian fields. We prove well-posedness for Dirichlet, Neumann and periodic boundary data and the initial value problem, covering the cases of both the Wick renormalization of the diffusion and of point-wise multiplication. We provide explicit representations for the solutions in both cases, characterized by the <span><math><mi>S</mi></math></span>-transform and the Gaussian multiplicative chaos measure.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104808"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-01Epub Date: 2025-10-21DOI: 10.1016/j.spa.2025.104790
Jakob E. Björnberg , Cécile Mailler
We introduce a new model of random tree that grows like a random recursive tree, except at some exceptional “doubling events” when the tree is replaced by two copies of itself attached to a new root. We prove asymptotic results for the size of this tree at large times, its degree distribution, and its height profile. We also prove a lower bound for its height. Because of the doubling events that affect the tree globally, the proofs are all much more intricate than in the case of the random recursive tree in which the growing operation is always local.
{"title":"A random recursive tree model with doubling events","authors":"Jakob E. Björnberg , Cécile Mailler","doi":"10.1016/j.spa.2025.104790","DOIUrl":"10.1016/j.spa.2025.104790","url":null,"abstract":"<div><div>We introduce a new model of random tree that grows like a random recursive tree, except at some exceptional “doubling events” when the tree is replaced by two copies of itself attached to a new root. We prove asymptotic results for the size of this tree at large times, its degree distribution, and its height profile. We also prove a lower bound for its height. Because of the doubling events that affect the tree globally, the proofs are all much more intricate than in the case of the random recursive tree in which the growing operation is always local.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104790"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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