Pub Date : 2025-09-12DOI: 10.1016/j.spa.2025.104769
Eric A. Carlen , Gustavo Posta , Imre Péter Tóth
We prove a spectral gap inequality for the stochastic exchange model studied by Gaspard and Gilbert and by Grigo, Khanin and Szász in connection with understanding heat conduction in a deterministic billiards model. The bound on the spectral gap that we prove is uniform in the number of particles, as had been conjectured. We adapt techniques that were originally developed to prove spectral gap bounds for the Kac model with hard sphere collisions, which, like the stochastic exchange model, has degenerate jump rates.
{"title":"Spectral gap for the stochastic exchange model","authors":"Eric A. Carlen , Gustavo Posta , Imre Péter Tóth","doi":"10.1016/j.spa.2025.104769","DOIUrl":"10.1016/j.spa.2025.104769","url":null,"abstract":"<div><div>We prove a spectral gap inequality for the stochastic exchange model studied by Gaspard and Gilbert and by Grigo, Khanin and Szász in connection with understanding heat conduction in a deterministic billiards model. The bound on the spectral gap that we prove is uniform in the number of particles, as had been conjectured. We adapt techniques that were originally developed to prove spectral gap bounds for the Kac model with hard sphere collisions, which, like the stochastic exchange model, has degenerate jump rates.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104769"},"PeriodicalIF":1.2,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104279","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 : 2025-09-12DOI: 10.1016/j.spa.2025.104770
Chang Liu , Dejun Luo
We consider the globally modified stochastic (hyperviscous) Navier–Stokes equations with transport noise on 3D torus. We first establish the existence and pathwise uniqueness of the weak solutions, and then show their convergence to the solutions of the deterministic 3D globally modified (hyperviscous) Navier–Stokes equations in an appropriate scaling limit. Furthermore, we prove a large deviation principle for the stochastic globally modified hyperviscous system.
{"title":"Scaling limit and large deviation for 3D globally modified stochastic Navier–Stokes equations with transport noise","authors":"Chang Liu , Dejun Luo","doi":"10.1016/j.spa.2025.104770","DOIUrl":"10.1016/j.spa.2025.104770","url":null,"abstract":"<div><div>We consider the globally modified stochastic (hyperviscous) Navier–Stokes equations with transport noise on 3D torus. We first establish the existence and pathwise uniqueness of the weak solutions, and then show their convergence to the solutions of the deterministic 3D globally modified (hyperviscous) Navier–Stokes equations in an appropriate scaling limit. Furthermore, we prove a large deviation principle for the stochastic globally modified hyperviscous system.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"191 ","pages":"Article 104770"},"PeriodicalIF":1.2,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104565","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 : 2025-09-11DOI: 10.1016/j.spa.2025.104771
Jevgenijs Ivanovs , Guy Latouche , Peter Taylor
Dating from the work of Neuts in the 1980s, the field of matrix-analytic methods has been developed to analyse discrete or continuous-time Markov chains with a two-dimensional state space in which the increment of a level variable is governed by an auxiliary phase variable. More recently, matrix-analytic techniques have been applied to general Markov additive models with a finite phase space. The basic assumption underlying these developments is that the process is skip-free (in the case of QBDs or fluid queues) or that it is one-sided, that is it is jump-free in one direction.
From the Markov additive perspective, traditional matrix-analytic models can be viewed as special cases: for M/G/1 and GI/M/1-type Markov chains, increments in the level are constrained to be lattice random variables and for fluid queues, they have to be piecewise linear.
In this paper we discuss one-sided lattice and non-lattice Markov additive processes in parallel. Results that are standard in one tradition are interpreted in the other, and new perspectives emerge. In particular, using three fundamental matrices, we address hitting, two-sided exit, and creeping probabilities.
{"title":"One-sided Markov additive processes with lattice and non-lattice increments","authors":"Jevgenijs Ivanovs , Guy Latouche , Peter Taylor","doi":"10.1016/j.spa.2025.104771","DOIUrl":"10.1016/j.spa.2025.104771","url":null,"abstract":"<div><div>Dating from the work of Neuts in the 1980s, the field of matrix-analytic methods has been developed to analyse discrete or continuous-time Markov chains with a two-dimensional state space in which the increment of a <em>level</em> variable is governed by an auxiliary <em>phase</em> variable. More recently, matrix-analytic techniques have been applied to general Markov additive models with a finite phase space. The basic assumption underlying these developments is that the process is skip-free (in the case of QBDs or fluid queues) or that it is <em>one-sided</em>, that is it is jump-free in one direction.</div><div>From the Markov additive perspective, traditional matrix-analytic models can be viewed as special cases: for M/G/1 and GI/M/1-type Markov chains, increments in the level are constrained to be <em>lattice</em> random variables and for fluid queues, they have to be piecewise linear.</div><div>In this paper we discuss one-sided lattice and non-lattice Markov additive processes in parallel. Results that are standard in one tradition are interpreted in the other, and new perspectives emerge. In particular, using three fundamental matrices, we address hitting, two-sided exit, and creeping probabilities.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104771"},"PeriodicalIF":1.2,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104280","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 : 2025-09-08DOI: 10.1016/j.spa.2025.104766
Roger Tribe, Oleg Zaboronski
This paper considers the decay in particle intensities for a translation invariant two species system of diffusing and reacting particles on for . The intensities are shown to approximately solve modified rate equations, from which their polynomial decay can be deduced. The system illustrates that the underlying diffusion and reaction rates can influence the exact polynomial decay rates, despite the system evolving in a supercritical dimension.
{"title":"A+A→A, B+A→A","authors":"Roger Tribe, Oleg Zaboronski","doi":"10.1016/j.spa.2025.104766","DOIUrl":"10.1016/j.spa.2025.104766","url":null,"abstract":"<div><div>This paper considers the decay in particle intensities for a translation invariant two species system of diffusing and reacting particles on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. The intensities are shown to approximately solve modified rate equations, from which their polynomial decay can be deduced. The system illustrates that the underlying diffusion and reaction rates can influence the exact polynomial decay rates, despite the system evolving in a supercritical dimension.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104766"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104278","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 : 2025-09-03DOI: 10.1016/j.spa.2025.104767
Shui Feng, J.E. Paguyo
The hierarchical Dirichlet process is a discrete random measure used as a prior in Bayesian nonparametrics and motivated by the study of groups of clustered data. We study the asymptotic behavior of the power sum symmetric polynomials for the vector of weights of the hierarchical Dirichlet process as the concentration parameters tend to infinity. We establish central limit theorems and obtain explicit representations for the asymptotic variances, with the latter clearly showing the impact of the hierarchical structure. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl–Hirschman index in economics.
{"title":"Central limit theorems associated with the hierarchical Dirichlet process","authors":"Shui Feng, J.E. Paguyo","doi":"10.1016/j.spa.2025.104767","DOIUrl":"10.1016/j.spa.2025.104767","url":null,"abstract":"<div><div>The hierarchical Dirichlet process is a discrete random measure used as a prior in Bayesian nonparametrics and motivated by the study of groups of clustered data. We study the asymptotic behavior of the power sum symmetric polynomials for the vector of weights of the hierarchical Dirichlet process as the concentration parameters tend to infinity. We establish central limit theorems and obtain explicit representations for the asymptotic variances, with the latter clearly showing the impact of the hierarchical structure. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl–Hirschman index in economics.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104767"},"PeriodicalIF":1.2,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010502","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 : 2025-09-03DOI: 10.1016/j.spa.2025.104768
Anton Tiepner , Eric Ziebell
The coefficients of elastic and dissipative operators in a linear hyperbolic SPDE are jointly estimated using multiple spatially localised measurements. As the resolution level of the observations tends to zero, we establish the asymptotic normality of an augmented maximum likelihood estimator. The rate of convergence for the dissipative coefficients matches rates in related parabolic problems, whereas the rate for the elastic parameters also depends on the magnitude of the damping. The analysis of the observed Fisher information matrix relies upon the asymptotic behaviour of rescaled -functions generalising the operator cosine and sine families appearing in the undamped wave equation. In contrast to the energetically stable undamped wave equation, the -functions emerging within the covariance structure of the local measurements have additional smoothing properties similar to the heat kernel, and their asymptotic behaviour is analysed using functional calculus.
{"title":"Parameter estimation in hyperbolic linear SPDEs from multiple measurements","authors":"Anton Tiepner , Eric Ziebell","doi":"10.1016/j.spa.2025.104768","DOIUrl":"10.1016/j.spa.2025.104768","url":null,"abstract":"<div><div>The coefficients of elastic and dissipative operators in a linear hyperbolic SPDE are jointly estimated using multiple spatially localised measurements. As the resolution level of the observations tends to zero, we establish the asymptotic normality of an augmented maximum likelihood estimator. The rate of convergence for the dissipative coefficients matches rates in related parabolic problems, whereas the rate for the elastic parameters also depends on the magnitude of the damping. The analysis of the observed Fisher information matrix relies upon the asymptotic behaviour of rescaled <span><math><mrow><mi>M</mi><mo>,</mo><mi>N</mi></mrow></math></span>-functions generalising the operator cosine and sine families appearing in the undamped wave equation. In contrast to the energetically stable undamped wave equation, the <span><math><mrow><mi>M</mi><mo>,</mo><mi>N</mi></mrow></math></span>-functions emerging within the covariance structure of the local measurements have additional smoothing properties similar to the heat kernel, and their asymptotic behaviour is analysed using functional calculus.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104768"},"PeriodicalIF":1.2,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145026780","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}
This paper studies the convergence of the mirror descent algorithm for finite horizon stochastic control problems with measure-valued control processes. The control objective involves a convex regularisation function, denoted as , with regularisation strength determined by the weight . The setting covers regularised relaxed control problems. Under suitable conditions, we establish the relative smoothness and convexity of the control objective with respect to the Bregman divergence of , and prove linear convergence of the algorithm for and exponential convergence for . The results apply to common regularisers including relative entropy, -divergence, and entropic Wasserstein costs. This validates recent reinforcement learning heuristics that adding regularisation accelerates the convergence of gradient methods. The proof exploits careful regularity estimates of backward stochastic differential equations in the bounded mean oscillation norm.
{"title":"Mirror descent for stochastic control problems with measure-valued controls","authors":"Bekzhan Kerimkulov , David Šiška , Łukasz Szpruch , Yufei Zhang","doi":"10.1016/j.spa.2025.104765","DOIUrl":"10.1016/j.spa.2025.104765","url":null,"abstract":"<div><div>This paper studies the convergence of the mirror descent algorithm for finite horizon stochastic control problems with measure-valued control processes. The control objective involves a convex regularisation function, denoted as <span><math><mi>h</mi></math></span>, with regularisation strength determined by the weight <span><math><mrow><mi>τ</mi><mo>≥</mo><mn>0</mn></mrow></math></span>. The setting covers regularised relaxed control problems. Under suitable conditions, we establish the relative smoothness and convexity of the control objective with respect to the Bregman divergence of <span><math><mi>h</mi></math></span>, and prove linear convergence of the algorithm for <span><math><mrow><mi>τ</mi><mo>=</mo><mn>0</mn></mrow></math></span> and exponential convergence for <span><math><mrow><mi>τ</mi><mo>></mo><mn>0</mn></mrow></math></span>. The results apply to common regularisers including relative entropy, <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-divergence, and entropic Wasserstein costs. This validates recent reinforcement learning heuristics that adding regularisation accelerates the convergence of gradient methods. The proof exploits careful regularity estimates of backward stochastic differential equations in the bounded mean oscillation norm.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104765"},"PeriodicalIF":1.2,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144916858","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 : 2025-08-22DOI: 10.1016/j.spa.2025.104763
Yao Li , Molei Tao , Shirou Wang
This paper proposes a probabilistic approach to investigate the shape of landscapes of multi-dimensional potential functions. Under a suitable coupling scheme, two copies of the overdamped Langevin dynamics associated with the potential function are coupled, and the coupling times are collected. Assuming a set of intuitive yet technically challenging conditions on the coupling scheme, it is shown that the tail distributions of the coupling times exhibit qualitatively different dependencies on the noise magnitude for single-well versus multi-well potential functions. More specifically, for convex single-well potentials, the negative tail exponent of the coupling time distribution is uniformly bounded away from zero by the convexity parameter and is independent of the noise magnitude. In contrast, for multi-well potentials, the negative tail exponent decreases exponentially as the noise vanishes, with the decay rate governed by the essential barrier height, a quantity introduced in this paper to characterize the non-convex nature of the potential function. Numerical investigations are conducted for a variety of examples, including the Rosenbrock function, interacting particle systems, and loss functions arising in artificial neural networks. These examples not only illustrate the theoretical results in various contexts but also provide crucial numerical validation of the conjectured assumptions, which are essential to the theoretical analysis yet lie beyond the reach of standard technical tools.
{"title":"Essential barrier height and a probabilistic approach in characterizing potential landscape","authors":"Yao Li , Molei Tao , Shirou Wang","doi":"10.1016/j.spa.2025.104763","DOIUrl":"10.1016/j.spa.2025.104763","url":null,"abstract":"<div><div>This paper proposes a probabilistic approach to investigate the shape of landscapes of multi-dimensional potential functions. Under a suitable coupling scheme, two copies of the overdamped Langevin dynamics associated with the potential function are coupled, and the coupling times are collected. Assuming a set of intuitive yet technically challenging conditions on the coupling scheme, it is shown that the tail distributions of the coupling times exhibit qualitatively different dependencies on the noise magnitude for single-well versus multi-well potential functions. More specifically, for convex single-well potentials, the negative tail exponent of the coupling time distribution is uniformly bounded away from zero by the convexity parameter and is independent of the noise magnitude. In contrast, for multi-well potentials, the negative tail exponent decreases exponentially as the noise vanishes, with the decay rate governed by the <em>essential barrier height</em>, a quantity introduced in this paper to characterize the non-convex nature of the potential function. Numerical investigations are conducted for a variety of examples, including the Rosenbrock function, interacting particle systems, and loss functions arising in artificial neural networks. These examples not only illustrate the theoretical results in various contexts but also provide crucial numerical validation of the conjectured assumptions, which are essential to the theoretical analysis yet lie beyond the reach of standard technical tools.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104763"},"PeriodicalIF":1.2,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144907766","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 : 2025-08-21DOI: 10.1016/j.spa.2025.104764
Lu Yu , Arnak Dalalyan
We study the problem of sampling from a target probability density function in frameworks where parallel evaluations of the log-density gradient are feasible. Focusing on smooth and strongly log-concave densities, we revisit the parallelized randomized midpoint method and investigate its properties using recently developed techniques for analyzing its sequential version. Through these techniques, we derive upper bounds on the Wasserstein distance between sampling and target densities. These bounds quantify the substantial runtime improvements achieved through parallel processing.
{"title":"Parallelized midpoint randomization for Langevin Monte Carlo","authors":"Lu Yu , Arnak Dalalyan","doi":"10.1016/j.spa.2025.104764","DOIUrl":"10.1016/j.spa.2025.104764","url":null,"abstract":"<div><div>We study the problem of sampling from a target probability density function in frameworks where parallel evaluations of the log-density gradient are feasible. Focusing on smooth and strongly log-concave densities, we revisit the parallelized randomized midpoint method and investigate its properties using recently developed techniques for analyzing its sequential version. Through these techniques, we derive upper bounds on the Wasserstein distance between sampling and target densities. These bounds quantify the substantial runtime improvements achieved through parallel processing.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104764"},"PeriodicalIF":1.2,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896748","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 : 2025-08-07DOI: 10.1016/j.spa.2025.104757
Ajay Chandra , Léonard Ferdinand
We show how the flow approach of Duch (2021), with elementary differentials as coordinates as in Chandra and Ferdinand (2024), can be used to prove well-posedness for rough stochastic differential equations driven by fractional Brownian motion with Hurst index . A novelty appearing here is that we use coordinates for the flow that are indexed by trees rather than multi-indices.
{"title":"Rough differential equations in the flow approach","authors":"Ajay Chandra , Léonard Ferdinand","doi":"10.1016/j.spa.2025.104757","DOIUrl":"10.1016/j.spa.2025.104757","url":null,"abstract":"<div><div>We show how the flow approach of Duch (2021), with elementary differentials as coordinates as in Chandra and Ferdinand (2024), can be used to prove well-posedness for rough stochastic differential equations driven by fractional Brownian motion with Hurst index <span><math><mrow><mi>H</mi><mo>></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow></math></span>. A novelty appearing here is that we use coordinates for the flow that are indexed by trees rather than multi-indices.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"190 ","pages":"Article 104757"},"PeriodicalIF":1.2,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863748","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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