We establish a limit theorem for a new model of 3-dimensional random walk in an inhomogeneous lattice with random orientations. This model can be seen as a 3-dimensional version of the Matheron and de Marsily model [1]. This new model leads us naturally to the study of iterated random walk in random scenery, which is a new process that can be described as a random walk in random scenery evolving in a second random scenery. We use the french acronym PAPAPA for this process unprecedented in literature, and answer a question about its stochastic behaviour asked about twenty years ago by Stéphane Le Borgne.
{"title":"Iterated random walks in random scenery (PAPAPA)","authors":"Nadine Guillotin-Plantard , Françoise Pène , Frédérique Watbled","doi":"10.1016/j.spa.2025.104843","DOIUrl":"10.1016/j.spa.2025.104843","url":null,"abstract":"<div><div>We establish a limit theorem for a new model of 3-dimensional random walk in an inhomogeneous lattice with random orientations. This model can be seen as a 3-dimensional version of the Matheron and de Marsily model [1]. This new model leads us naturally to the study of iterated random walk in random scenery, which is a new process that can be described as a random walk in random scenery evolving in a second random scenery. We use the french acronym PAPAPA for this process unprecedented in literature, and answer a question about its stochastic behaviour asked about twenty years ago by Stéphane Le Borgne.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"193 ","pages":"Article 104843"},"PeriodicalIF":1.2,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790418","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 : 2025-12-03DOI: 10.1016/j.spa.2025.104846
Liping Li, Jiangang Ying
The primary aim of this article is to investigate the domination relationship between two L2-semigroups using probabilistic methods. According to Ouhabaz’s domination criterion, the domination of semigroups can be transformed into relationships involving the corresponding Dirichlet forms. Our principal result establishes the equivalence between the domination of Dirichlet forms and the killing transformation of the associated Markov processes, which generalizes and completes the results in [1] and [2]. Based on this equivalence, we provide a representation of the dominated Dirichlet form using the bivariate Revuz measure associated with the killing transformation and further characterize the sandwiched Dirichlet form within the broader Dirichlet form framework. In particular, our findings apply to the characterization of operators sandwiched between the Dirichlet Laplacian and the Neumann Laplacian. For the local boundary case, we eliminate all technical conditions identified in the literature [3] and deliver a complete representation of all sandwiched operators governed by a Robin boundary condition determined by a specific quasi-admissible measure. Additionally, our results offer a comprehensive characterization of related operators in the non-local Robin boundary case, specifically resolving an open problem posed in the literature [4].
{"title":"On domination for (non-symmetric) dirichlet forms","authors":"Liping Li, Jiangang Ying","doi":"10.1016/j.spa.2025.104846","DOIUrl":"10.1016/j.spa.2025.104846","url":null,"abstract":"<div><div>The primary aim of this article is to investigate the domination relationship between two <em>L</em><sup>2</sup>-semigroups using probabilistic methods. According to Ouhabaz’s domination criterion, the domination of semigroups can be transformed into relationships involving the corresponding Dirichlet forms. Our principal result establishes the equivalence between the domination of Dirichlet forms and the killing transformation of the associated Markov processes, which generalizes and completes the results in [1] and [2]. Based on this equivalence, we provide a representation of the dominated Dirichlet form using the bivariate Revuz measure associated with the killing transformation and further characterize the sandwiched Dirichlet form within the broader Dirichlet form framework. In particular, our findings apply to the characterization of operators sandwiched between the Dirichlet Laplacian and the Neumann Laplacian. For the local boundary case, we eliminate all technical conditions identified in the literature [3] and deliver a complete representation of all sandwiched operators governed by a Robin boundary condition determined by a specific quasi-admissible measure. Additionally, our results offer a comprehensive characterization of related operators in the non-local Robin boundary case, specifically resolving an open problem posed in the literature [4].</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"193 ","pages":"Article 104846"},"PeriodicalIF":1.2,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790417","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 : 2025-12-02DOI: 10.1016/j.spa.2025.104836
Wenping Cao , Zirong Zeng , Deng Zhang
We consider the 3D stochastic Navier-Stokes equations where the viscosity exponent can be larger than the Lions exponent 5/4. Though it is well-known that the Leray-Hopf solutions are unique in this high viscous regime, we prove that the uniqueness would fail in two scaling-supercritical regimes with respect to the Ladyžhenskaya-Prodi-Serrin criteria. The constructed solutions can be non-Leray-Hopf and very close to the Leray-Hopf solutions. Furthermore, we prove the vanishing noise limit result, which relates together the stochastic solutions and the deterministic convex integration solutions constructed by Buckmaster-Vicol [1] and the recent work [2].
{"title":"Non-Leray-Hopf solutions to 3D stochastic hyper-viscous Navier-Stokes equations: Beyond the lions exponent","authors":"Wenping Cao , Zirong Zeng , Deng Zhang","doi":"10.1016/j.spa.2025.104836","DOIUrl":"10.1016/j.spa.2025.104836","url":null,"abstract":"<div><div>We consider the 3D stochastic Navier-Stokes equations where the viscosity exponent can be larger than the Lions exponent 5/4. Though it is well-known that the Leray-Hopf solutions are unique in this high viscous regime, we prove that the uniqueness would fail in two scaling-supercritical regimes with respect to the Ladyžhenskaya-Prodi-Serrin criteria. The constructed solutions can be non-Leray-Hopf and very close to the Leray-Hopf solutions. Furthermore, we prove the vanishing noise limit result, which relates together the stochastic solutions and the deterministic convex integration solutions constructed by Buckmaster-Vicol [1] and the recent work [2].</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"193 ","pages":"Article 104836"},"PeriodicalIF":1.2,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737187","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 : 2025-11-29DOI: 10.1016/j.spa.2025.104835
Bruno Costacèque, Laurent Decreusefond
The functional characterization of a measure, an essential but delicate aspect of Stein’s method, is shown to be accessible for stable probability distributions on convex cones. This notion encompasses the usual stable distributions e.g. Gaussian, Pareto, etc. but also the max-stable distributions: Weibull, Gumbel and Fréchet. We use the definition of max-stability to define a Markov process whose invariant measure is the stable measure of interest. In this paper, we focus on the Gumbel distribution and show how this construction can be applied to estimate the rate of convergence in the classical coupon collector’s problem.
{"title":"Convergence rate for the coupon collector’s problem with Stein’s method","authors":"Bruno Costacèque, Laurent Decreusefond","doi":"10.1016/j.spa.2025.104835","DOIUrl":"10.1016/j.spa.2025.104835","url":null,"abstract":"<div><div>The functional characterization of a measure, an essential but delicate aspect of Stein’s method, is shown to be accessible for stable probability distributions on convex cones. This notion encompasses the usual stable distributions <em>e.g.</em> Gaussian, Pareto, <em>etc.</em> but also the max-stable distributions: Weibull, Gumbel and Fréchet. We use the definition of max-stability to define a Markov process whose invariant measure is the stable measure of interest. In this paper, we focus on the Gumbel distribution and show how this construction can be applied to estimate the rate of convergence in the classical coupon collector’s problem.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"193 ","pages":"Article 104835"},"PeriodicalIF":1.2,"publicationDate":"2025-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737188","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 : 2025-11-15DOI: 10.1016/j.spa.2025.104833
Juan Li , Zhanxin Li , Chuanzhi Xing
For general mean-field backward stochastic differential equations (BSDEs) it is well-known that we usually do not have the comparison theorem if the coefficients depend on the law of -component of the solution process . A natural question is whether general mean-field BSDEs whose coefficients depend on the law of have the comparison theorem for some cases. In this paper we establish the comparison theorems for one-dimensional mean-field BSDEs whose coefficients also depend on the joint law of the solution process . With the help of Malliavin calculus and a BMO martingale argument, we obtain two comparison theorems for different cases and a strong comparison result. In particular, in this framework, we compare not only the first component of the solution for such mean-field BSDEs, but also the second component . After a discussion of mean-field BSDEs whose terminal condition and the driving coefficient are Malliavin differentiable, the results are extended in a second phase to the case without assumption of Malliavin differentiability.
{"title":"Comparison theorems for mean-field BSDEs whose generators depend on the law of the solution (Y,Z)","authors":"Juan Li , Zhanxin Li , Chuanzhi Xing","doi":"10.1016/j.spa.2025.104833","DOIUrl":"10.1016/j.spa.2025.104833","url":null,"abstract":"<div><div>For general mean-field backward stochastic differential equations (BSDEs) it is well-known that we usually do not have the comparison theorem if the coefficients depend on the law of <span><math><mi>Z</mi></math></span>-component of the solution process <span><math><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></mrow></math></span>. A natural question is whether general mean-field BSDEs whose coefficients depend on the law of <span><math><mi>Z</mi></math></span> have the comparison theorem for some cases. In this paper we establish the comparison theorems for one-dimensional mean-field BSDEs whose coefficients also depend on the joint law of the solution process <span><math><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></mrow></math></span>. With the help of Malliavin calculus and a BMO martingale argument, we obtain two comparison theorems for different cases and a strong comparison result. In particular, in this framework, we compare not only the first component <span><math><mi>Y</mi></math></span> of the solution <span><math><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></mrow></math></span> for such mean-field BSDEs, but also the second component <span><math><mi>Z</mi></math></span>. After a discussion of mean-field BSDEs whose terminal condition and the driving coefficient are Malliavin differentiable, the results are extended in a second phase to the case without assumption of Malliavin differentiability.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"193 ","pages":"Article 104833"},"PeriodicalIF":1.2,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571942","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 : 2025-11-15DOI: 10.1016/j.spa.2025.104834
Haojie Hou , Yan-Xia Ren , Renming Song
<div><div>Consider a one dimensional critical branching Lévy process <span><math><mrow><mo>(</mo><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow></math></span>. Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction of some <span><math><mi>α</mi></math></span>-stable distribution 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>, and that the underlying Lévy process <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> is non-lattice and has finite <span><math><mrow><mn>2</mn><mo>+</mo><msup><mrow><mi>δ</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> moment for some <span><math><mrow><msup><mrow><mi>δ</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>></mo><mn>0</mn></mrow></math></span>. We first prove that <span><span><span><math><mrow><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mfenced><mrow><mn>1</mn><mo>−</mo><msub><mrow><mi>E</mi></mrow><mrow><msqrt><mrow><mi>t</mi></mrow></msqrt><mi>y</mi></mrow></msub><mfenced><mrow><mo>exp</mo><mfenced><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>∫</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>∫</mo><mi>g</mi><mfenced><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><msqrt><mrow><mi>t</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><msub><mrow><mi>Z</mi></mrow><mrow><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>)</mo></mrow></mrow></mfenced></mrow></mfenced></mrow></mfenced></mrow></math></span></span></span>converges as <span><math><mrow><mi>t</mi><mo>→</mo><mi>∞</mi></mrow></math></span> for any non-negative bounded Lipschitz function <span><math><mi>g</mi></math></span> and any non-negative directly Riemann integrable function <span><math><mi>h</mi></math></span> of compact support. Then for any <span><math><mrow><mi>y</mi><mo>∈</mo><mi>R</mi></mrow></math></span> and bounded Borel set <span><math><mi>A</mi></math></span> of positive Lebesgue measure with its boundary having zero Lebesgue measure,
{"title":"Local properties for 1-dimensional critical branching Lévy process","authors":"Haojie Hou , Yan-Xia Ren , Renming Song","doi":"10.1016/j.spa.2025.104834","DOIUrl":"10.1016/j.spa.2025.104834","url":null,"abstract":"<div><div>Consider a one dimensional critical branching Lévy process <span><math><mrow><mo>(</mo><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></mrow></math></span>. Assume that the offspring distribution either has finite second moment or belongs to the domain of attraction of some <span><math><mi>α</mi></math></span>-stable distribution 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>, and that the underlying Lévy process <span><math><msub><mrow><mrow><mo>(</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> is non-lattice and has finite <span><math><mrow><mn>2</mn><mo>+</mo><msup><mrow><mi>δ</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> moment for some <span><math><mrow><msup><mrow><mi>δ</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>></mo><mn>0</mn></mrow></math></span>. We first prove that <span><span><span><math><mrow><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mfenced><mrow><mn>1</mn><mo>−</mo><msub><mrow><mi>E</mi></mrow><mrow><msqrt><mrow><mi>t</mi></mrow></msqrt><mi>y</mi></mrow></msub><mfenced><mrow><mo>exp</mo><mfenced><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>∫</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>∫</mo><mi>g</mi><mfenced><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><msqrt><mrow><mi>t</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><msub><mrow><mi>Z</mi></mrow><mrow><mi>t</mi></mrow></msub><mrow><mo>(</mo><mi>d</mi><mi>x</mi><mo>)</mo></mrow></mrow></mfenced></mrow></mfenced></mrow></mfenced></mrow></math></span></span></span>converges as <span><math><mrow><mi>t</mi><mo>→</mo><mi>∞</mi></mrow></math></span> for any non-negative bounded Lipschitz function <span><math><mi>g</mi></math></span> and any non-negative directly Riemann integrable function <span><math><mi>h</mi></math></span> of compact support. Then for any <span><math><mrow><mi>y</mi><mo>∈</mo><mi>R</mi></mrow></math></span> and bounded Borel set <span><math><mi>A</mi></math></span> of positive Lebesgue measure with its boundary having zero Lebesgue measure, ","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104834"},"PeriodicalIF":1.2,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145578989","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 : 2025-11-12DOI: 10.1016/j.spa.2025.104831
Stein Andreas Bethuelsen , Malin Palö Forsström
Forsström et al. (2025) recently introduced a large class of -valued processes that they named Poisson representable. In addition to deriving several interesting properties for these processes, their main focus was determining which processes are contained in this class.
In this paper, we derive new characteristics for Poisson representable processes in terms of certain mixing properties. Using these, we argue that neither the upper invariant measure of the supercritical contact process on nor the plus state of the Ising model on within the phase transition regime is Poisson representable. Moreover, we show that on any non-extremal translation invariant state of the Ising model cannot be Poisson representable. Together, these results provide answers to questions raised in Forsström et al. (2025).
{"title":"Mixing for Poisson representable processes and consequences for the Ising model and the contact process","authors":"Stein Andreas Bethuelsen , Malin Palö Forsström","doi":"10.1016/j.spa.2025.104831","DOIUrl":"10.1016/j.spa.2025.104831","url":null,"abstract":"<div><div>Forsström et al. (2025) recently introduced a large class of <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math></span>-valued processes that they named Poisson representable. In addition to deriving several interesting properties for these processes, their main focus was determining which processes are contained in this class.</div><div>In this paper, we derive new characteristics for Poisson representable processes in terms of certain mixing properties. Using these, we argue that neither the upper invariant measure of the supercritical contact process on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> nor the plus state of the Ising model on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> within the phase transition regime is Poisson representable. Moreover, we show that on <span><math><mrow><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo></mrow></math></span> <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn><mo>,</mo></mrow></math></span> any non-extremal translation invariant state of the Ising model cannot be Poisson representable. Together, these results provide answers to questions raised in Forsström et al. (2025).</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104831"},"PeriodicalIF":1.2,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528794","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 : 2025-11-12DOI: 10.1016/j.spa.2025.104832
Anton Tiepner , Lukas Trottner
<div><div>We study a stochastic heat equation with piecewise constant diffusivity <span><math><mi>ϑ</mi></math></span> having a jump at a hypersurface <span><math><mi>Γ</mi></math></span> that splits the underlying space <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>d</mi></mrow></msup></math></span>, <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn><mo>,</mo></mrow></math></span> into two disjoint sets <span><math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>∪</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>.</mo></mrow></math></span> Based on multiple spatially localized measurement observations on a regular <span><math><mi>δ</mi></math></span>-grid of <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>d</mi></mrow></msup></math></span>, we propose a joint M-estimator for the diffusivity values and the set <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> that is inspired by statistical image reconstruction methods. We study convergence of the domain estimator <span><math><msub><mrow><mover><mrow><mi>Λ</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mo>+</mo></mrow></msub></math></span> in the vanishing resolution level regime <span><math><mrow><mi>δ</mi><mo>→</mo><mn>0</mn></mrow></math></span> and with respect to the expected symmetric difference pseudometric. As a first main finding we give a characterization of the convergence rate for <span><math><msub><mrow><mover><mrow><mi>Λ</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mo>+</mo></mrow></msub></math></span> in terms of the complexity of <span><math><mi>Γ</mi></math></span> measured by the number of intersecting hypercubes from the regular <span><math><mi>δ</mi></math></span>-grid. Furthermore, for the special case of domains <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> that are built from hypercubes from the <span><math><mi>δ</mi></math></span>-grid, we demonstrate that perfect identification with probability tending to one is possible with a slight modification of the estimation approach. Implications of our general results are discussed under two specific structural assumptions on <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>. For a <span><math><mi>β</mi></math></span>-Hölder smooth boundary fragment <span><math><mi>Γ</mi></math></span>, the set <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> is estimated with rate <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mi>β</mi></mrow></msup></math></span>. If we assume <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> to be convex, we obtain a <span><math><mi>δ</mi></math></span>-rate. While our approach only aims at optimal domain estimation rates, we also demonstrate consistency of ou
{"title":"Multivariate change estimation for a stochastic heat equation from local measurements","authors":"Anton Tiepner , Lukas Trottner","doi":"10.1016/j.spa.2025.104832","DOIUrl":"10.1016/j.spa.2025.104832","url":null,"abstract":"<div><div>We study a stochastic heat equation with piecewise constant diffusivity <span><math><mi>ϑ</mi></math></span> having a jump at a hypersurface <span><math><mi>Γ</mi></math></span> that splits the underlying space <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>d</mi></mrow></msup></math></span>, <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn><mo>,</mo></mrow></math></span> into two disjoint sets <span><math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>∪</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>.</mo></mrow></math></span> Based on multiple spatially localized measurement observations on a regular <span><math><mi>δ</mi></math></span>-grid of <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>d</mi></mrow></msup></math></span>, we propose a joint M-estimator for the diffusivity values and the set <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> that is inspired by statistical image reconstruction methods. We study convergence of the domain estimator <span><math><msub><mrow><mover><mrow><mi>Λ</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mo>+</mo></mrow></msub></math></span> in the vanishing resolution level regime <span><math><mrow><mi>δ</mi><mo>→</mo><mn>0</mn></mrow></math></span> and with respect to the expected symmetric difference pseudometric. As a first main finding we give a characterization of the convergence rate for <span><math><msub><mrow><mover><mrow><mi>Λ</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mo>+</mo></mrow></msub></math></span> in terms of the complexity of <span><math><mi>Γ</mi></math></span> measured by the number of intersecting hypercubes from the regular <span><math><mi>δ</mi></math></span>-grid. Furthermore, for the special case of domains <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> that are built from hypercubes from the <span><math><mi>δ</mi></math></span>-grid, we demonstrate that perfect identification with probability tending to one is possible with a slight modification of the estimation approach. Implications of our general results are discussed under two specific structural assumptions on <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>. For a <span><math><mi>β</mi></math></span>-Hölder smooth boundary fragment <span><math><mi>Γ</mi></math></span>, the set <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> is estimated with rate <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mi>β</mi></mrow></msup></math></span>. If we assume <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span> to be convex, we obtain a <span><math><mi>δ</mi></math></span>-rate. While our approach only aims at optimal domain estimation rates, we also demonstrate consistency of ou","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104832"},"PeriodicalIF":1.2,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528795","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 : 2025-11-10DOI: 10.1016/j.spa.2025.104826
Franziska Bielert
We prove a rough Itô formula for path-dependent functionals of -Hölder continuous paths for . Our approach combines the sewing lemma and a Taylor approximation in terms of path-dependent derivatives.
{"title":"Rough functional Itô formula","authors":"Franziska Bielert","doi":"10.1016/j.spa.2025.104826","DOIUrl":"10.1016/j.spa.2025.104826","url":null,"abstract":"<div><div>We prove a rough Itô formula for path-dependent functionals of <span><math><mi>α</mi></math></span>-Hölder continuous paths for <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. Our approach combines the sewing lemma and a Taylor approximation in terms of path-dependent derivatives.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104826"},"PeriodicalIF":1.2,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528792","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 : 2025-11-08DOI: 10.1016/j.spa.2025.104827
Wenxuan Chen , Linjie Zhao
We study a weakly asymmetric exclusion process with long jumps and with infinitely many extended reservoirs. We prove that the stationary fluctuations of the process are governed by the generalized Ornstein–Uhlenbeck process or the stochastic Burgers equation with Dirichlet boundary conditions depending on the strength of the asymmetry of the dynamics.
{"title":"Stationary fluctuations for the WASEP with long jumps and infinitely extended reservoirs","authors":"Wenxuan Chen , Linjie Zhao","doi":"10.1016/j.spa.2025.104827","DOIUrl":"10.1016/j.spa.2025.104827","url":null,"abstract":"<div><div>We study a weakly asymmetric exclusion process with long jumps and with infinitely many extended reservoirs. We prove that the stationary fluctuations of the process are governed by the generalized Ornstein–Uhlenbeck process or the stochastic Burgers equation with Dirichlet boundary conditions depending on the strength of the asymmetry of the dynamics.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"192 ","pages":"Article 104827"},"PeriodicalIF":1.2,"publicationDate":"2025-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528824","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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