Pub Date : 2024-10-22DOI: 10.1016/j.spa.2024.104511
Lucas Amorim , Nicolai Haydn , Sandro Vaienti
We obtain quenched hitting distributions to be compound Poissonian for a certain class of random dynamical systems. The theory is general and designed to accommodate non-uniformly expanding behavior and targets that do not overlap much with the region where uniformity breaks. Based on annealed and quenched polynomial decay of correlations, our quenched result adopts annealed Kac-type time-normalization and finds limits to be noise-independent. The technique involves a probabilistic block-approximation where the quenched hit-counting function up to annealed Kac-normalized time is split into equally sized blocks which are mimicked by an independency of random variables distributed just like each of them. The theory is made operational due to a result that allows certain hitting quantities to be recovered from return quantities. Our application is to a class of random piecewise expanding one-dimensional systems, casting new light on the well-known deterministic dichotomy between periodic and aperiodic points, their usual extremal index formula , and recovering the Polya–Aeppli case for general Bernoulli-driven systems, but distinct behavior otherwise.
{"title":"Compound Poisson distributions for random dynamical systems using probabilistic approximations","authors":"Lucas Amorim , Nicolai Haydn , Sandro Vaienti","doi":"10.1016/j.spa.2024.104511","DOIUrl":"10.1016/j.spa.2024.104511","url":null,"abstract":"<div><div>We obtain quenched hitting distributions to be compound Poissonian for a certain class of random dynamical systems. The theory is general and designed to accommodate non-uniformly expanding behavior and targets that do not overlap much with the region where uniformity breaks. Based on annealed and quenched polynomial decay of correlations, our quenched result adopts annealed Kac-type time-normalization and finds limits to be noise-independent. The technique involves a probabilistic block-approximation where the quenched hit-counting function up to annealed Kac-normalized time is split into equally sized blocks which are mimicked by an independency of random variables distributed just like each of them. The theory is made operational due to a result that allows certain hitting quantities to be recovered from return quantities. Our application is to a class of random piecewise expanding one-dimensional systems, casting new light on the well-known deterministic dichotomy between periodic and aperiodic points, their usual extremal index formula <span><math><mrow><mo>EI</mo><mo>=</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>J</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, and recovering the Polya–Aeppli case for general Bernoulli-driven systems, but distinct behavior otherwise.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104511"},"PeriodicalIF":1.1,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554289","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 : 2024-10-21DOI: 10.1016/j.spa.2024.104510
David Clancy Jr.
We study the susceptible–infected–recovered (SIR) epidemic on a random graph chosen uniformly over all graphs with certain critical, heavy-tailed degree distributions. We prove process level scaling limits for the number of individuals infected on day on the largest connected components of the graph. The scaling limits contain non-negative jumps corresponding to some vertices of large degree. These weak convergence techniques allow us to describe the height profile of the -stable continuum random graph (Goldschmidt et al., 2022; Conchon-Kerjan and Goldschmidt, 2023), extending results known in the Brownian case (Miermont and Sen, 2022). We also prove model-independent results that can be used on other critical random graph models.
{"title":"Epidemics on critical random graphs with heavy-tailed degree distribution","authors":"David Clancy Jr.","doi":"10.1016/j.spa.2024.104510","DOIUrl":"10.1016/j.spa.2024.104510","url":null,"abstract":"<div><div>We study the susceptible–infected–recovered (SIR) epidemic on a random graph chosen uniformly over all graphs with certain critical, heavy-tailed degree distributions. We prove process level scaling limits for the number of individuals infected on day <span><math><mi>h</mi></math></span> on the largest connected components of the graph. The scaling limits contain non-negative jumps corresponding to some vertices of large degree. These weak convergence techniques allow us to describe the height profile of the <span><math><mi>α</mi></math></span>-stable continuum random graph (Goldschmidt et al., 2022; Conchon-Kerjan and Goldschmidt, 2023), extending results known in the Brownian case (Miermont and Sen, 2022). We also prove model-independent results that can be used on other critical random graph models.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104510"},"PeriodicalIF":1.1,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560842","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}
We prove a fluid limit describing coarsening for zero-range processes on a finite number of sites, with asymptotically constant jump rates. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is described by a piecewise linear trajectory in the simplex indexed by the sites. The linear coefficients are determined by the trace process of the underlying random walk on the subset of non-empty sites, and the trajectory reaches an absorbing configuration in finite time. A boundary of the simplex is called absorbing for the fluid limit if a trajectory started at a configuration in the boundary remains in it for all times. We identify the set of absorbing configurations and characterize the absorbing boundaries.
{"title":"Coarsening in zero-range processes","authors":"Inés Armendáriz , Johel Beltrán , Daniela Cuesta , Milton Jara","doi":"10.1016/j.spa.2024.104507","DOIUrl":"10.1016/j.spa.2024.104507","url":null,"abstract":"<div><div>We prove a fluid limit describing coarsening for zero-range processes on a finite number of sites, with asymptotically constant jump rates. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is described by a piecewise linear trajectory in the simplex indexed by the sites. The linear coefficients are determined by the trace process of the underlying random walk on the subset of non-empty sites, and the trajectory reaches an absorbing configuration in finite time. A boundary of the simplex is called absorbing for the fluid limit if a trajectory started at a configuration in the boundary remains in it for all times. We identify the set of absorbing configurations and characterize the absorbing boundaries.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104507"},"PeriodicalIF":1.1,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554288","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 : 2024-10-19DOI: 10.1016/j.spa.2024.104508
Xing Huang
The couplings by change of measure are applied to establish log-Harnack inequality(equivalently the entropy-cost estimate) for conditional McKean–Vlasov SDEs and derive the quantitative conditional propagation of chaos in relative entropy for mean field interacting particle system with common noise. For the log-Harnack inequality, two different types of couplings will be constructed for non-degenerate conditional McKean–Vlasov SDEs with multiplicative noise. As to the quantitative conditional propagation of chaos in relative entropy, the initial distribution of interacting particle system is allowed to be singular with that of limit equation. The above results are also extended to conditional distribution dependent stochastic Hamiltonian system.
{"title":"Coupling by change of measure for conditional McKean–Vlasov SDEs and applications","authors":"Xing Huang","doi":"10.1016/j.spa.2024.104508","DOIUrl":"10.1016/j.spa.2024.104508","url":null,"abstract":"<div><div>The couplings by change of measure are applied to establish log-Harnack inequality(equivalently the entropy-cost estimate) for conditional McKean–Vlasov SDEs and derive the quantitative conditional propagation of chaos in relative entropy for mean field interacting particle system with common noise. For the log-Harnack inequality, two different types of couplings will be constructed for non-degenerate conditional McKean–Vlasov SDEs with multiplicative noise. As to the quantitative conditional propagation of chaos in relative entropy, the initial distribution of interacting particle system is allowed to be singular with that of limit equation. The above results are also extended to conditional distribution dependent stochastic Hamiltonian system.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104508"},"PeriodicalIF":1.1,"publicationDate":"2024-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532082","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 : 2024-10-19DOI: 10.1016/j.spa.2024.104512
Elisabetta Candellero , Tom Garcia-Sanchez
<div><div>First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph <span><math><mi>G</mi></math></span> place a red particle at a reference vertex <span><math><mi>o</mi></math></span> and colorless particles (seeds) at all other vertices. The red particle starts spreading a <em>red first passage percolation</em> of rate 1, while all seeds are dormant. As soon as a seed is reached by the process, it turns red and starts spreading red first passage percolation. All vertices are equipped with independent exponential clocks ringing at rate <span><math><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></math></span>, when a clock rings the corresponding <em>red vertex turns black</em>. For <span><math><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, let <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> denote the size of the longest red path and of the largest red cluster present at time <span><math><mi>t</mi></math></span>. If <span><math><mi>G</mi></math></span> is the semi-line, then for all <span><math><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></math></span> almost surely <span><math><mrow><msub><mrow><mo>lim sup</mo></mrow><mrow><mi>t</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>log</mo><mo>log</mo><mi>t</mi></mrow><mrow><mo>log</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mo>lim inf</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span>. In contrast, if <span><math><mi>G</mi></math></span> is an infinite Galton–Watson tree with offspring mean <span><math><mrow><mi>m</mi><mo>></mo><mn>1</mn></mrow></math></span> then, for all <span><math><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></math></span>, almost surely <span><math><mrow><msub><mrow><mo>lim inf</mo></mrow><mrow><mi>t</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>log</mo><mi>t</mi></mrow><mrow><mi>t</mi></mrow></mfrac><mo>≥</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mo>lim inf</mo></mrow><mrow><mi>t</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>log</mo><mo>log</mo><mi>t</mi></mrow><mrow><mi>t</mi></mrow></mfrac><mo>≥</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span>, while <span><math><mrow><msub><mrow><mo>lim sup</mo></mrow><mrow><mi>t</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>c</mi><mi>t</mi></mrow></msup></mrow></mfrac><mo>≤</mo><mn>1</mn></mrow></math></span>, for all <span><math><mrow><mi>c</mi><mo>></mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span>.
带恢复的第一通道渗流是一种旨在模拟流行病传播的过程。在一个图 G 上，在参考顶点 o 处放置一个红色粒子，在所有其他顶点放置无色粒子（种子）。红色粒子开始传播速率为 1 的红色第一通道渗流，而所有种子处于休眠状态。一旦进程到达某个种子，它就会变成红色，并开始传播红色第一通道渗流。所有顶点都配有独立的指数时钟，以 γ>0 的速率振铃，当时钟振铃时，相应的红色顶点变黑。对于 t≥0，让 Ht 和 Mt 表示 t 时刻存在的最长红色路径和最大红色集群的大小。如果 G 是半直线，那么对于所有 γ>0 几乎肯定 lim suptHtloglogtlogt=1 和 lim inftHt=0。相反，如果 G 是一棵后代均值为 m>1 的无限加尔顿-沃森树，那么对于所有 γ>0，几乎可以肯定 lim inftHtlogtt≥m-1 和 lim inftMtlogtt≥m-1，而对于所有 c>m-1，lim suptMtect≤1。此外，如果我们把注意力限制在有界度图上，那么对于任何 ɛ>0 都有一个临界值 γc>0，这样对于所有 γ>γc，几乎可以肯定 lim suptMtt≤ɛ。
{"title":"First passage percolation with recovery","authors":"Elisabetta Candellero , Tom Garcia-Sanchez","doi":"10.1016/j.spa.2024.104512","DOIUrl":"10.1016/j.spa.2024.104512","url":null,"abstract":"<div><div>First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph <span><math><mi>G</mi></math></span> place a red particle at a reference vertex <span><math><mi>o</mi></math></span> and colorless particles (seeds) at all other vertices. The red particle starts spreading a <em>red first passage percolation</em> of rate 1, while all seeds are dormant. As soon as a seed is reached by the process, it turns red and starts spreading red first passage percolation. All vertices are equipped with independent exponential clocks ringing at rate <span><math><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></math></span>, when a clock rings the corresponding <em>red vertex turns black</em>. For <span><math><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, let <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> denote the size of the longest red path and of the largest red cluster present at time <span><math><mi>t</mi></math></span>. If <span><math><mi>G</mi></math></span> is the semi-line, then for all <span><math><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></math></span> almost surely <span><math><mrow><msub><mrow><mo>lim sup</mo></mrow><mrow><mi>t</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>log</mo><mo>log</mo><mi>t</mi></mrow><mrow><mo>log</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mo>lim inf</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span>. In contrast, if <span><math><mi>G</mi></math></span> is an infinite Galton–Watson tree with offspring mean <span><math><mrow><mi>m</mi><mo>></mo><mn>1</mn></mrow></math></span> then, for all <span><math><mrow><mi>γ</mi><mo>></mo><mn>0</mn></mrow></math></span>, almost surely <span><math><mrow><msub><mrow><mo>lim inf</mo></mrow><mrow><mi>t</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>log</mo><mi>t</mi></mrow><mrow><mi>t</mi></mrow></mfrac><mo>≥</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mo>lim inf</mo></mrow><mrow><mi>t</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>log</mo><mo>log</mo><mi>t</mi></mrow><mrow><mi>t</mi></mrow></mfrac><mo>≥</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span>, while <span><math><mrow><msub><mrow><mo>lim sup</mo></mrow><mrow><mi>t</mi></mrow></msub><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>c</mi><mi>t</mi></mrow></msup></mrow></mfrac><mo>≤</mo><mn>1</mn></mrow></math></span>, for all <span><math><mrow><mi>c</mi><mo>></mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span>. ","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104512"},"PeriodicalIF":1.1,"publicationDate":"2024-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142572117","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 : 2024-10-19DOI: 10.1016/j.spa.2024.104509
Adrian Falkowski
We study the existence, uniqueness, and approximation of solutions of general stochastic differential equations (SDEs) with two time-dependent reflecting barriers driven by optional semimartingales. We do not assume that the probability space has to satisfy the usual conditions. We define and solve an appropriate version of the deterministic Skorokhod problem for regulated functions. Applications to currency option pricing in financial models are given.
{"title":"SDEs with two reflecting barriers driven by optional processes with regulated trajectories","authors":"Adrian Falkowski","doi":"10.1016/j.spa.2024.104509","DOIUrl":"10.1016/j.spa.2024.104509","url":null,"abstract":"<div><div>We study the existence, uniqueness, and approximation of solutions of general stochastic differential equations (SDEs) with two time-dependent reflecting barriers driven by optional semimartingales. We do not assume that the probability space has to satisfy the usual conditions. We define and solve an appropriate version of the deterministic Skorokhod problem for regulated functions. Applications to currency option pricing in financial models are given.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104509"},"PeriodicalIF":1.1,"publicationDate":"2024-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532083","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 : 2024-10-18DOI: 10.1016/j.spa.2024.104506
D.C. Antonopoulou , D. Farazakis , G. Karali
In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat Equation with a space–time white noise, posed in a moving boundary domain with velocity given by the Stefan condition. This condition determines the dynamics of the spread, and the solid phase defines the bid–ask spread area wherein the transactions vanish. We introduce a reflection measure and prove existence and uniqueness of maximal solutions up to stopping times in which the spread stays a.s. non-negative and bounded. For this, we define an approximation scheme, and use some of the estimates of Hambly et al. (2020) for the Green’s function and the associated to the reflection measure obstacle problem. Analogous results are obtained for the equation without reflection corresponding to a signed density.
{"title":"Existence of maximal solutions for the financial stochastic Stefan problem of a volatile asset with spread","authors":"D.C. Antonopoulou , D. Farazakis , G. Karali","doi":"10.1016/j.spa.2024.104506","DOIUrl":"10.1016/j.spa.2024.104506","url":null,"abstract":"<div><div>In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat Equation with a space–time white noise, posed in a moving boundary domain with velocity given by the Stefan condition. This condition determines the dynamics of the spread, and the solid phase <span><math><mrow><mo>[</mo><msup><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><msup><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow></math></span> defines the bid–ask spread area wherein the transactions vanish. We introduce a reflection measure and prove existence and uniqueness of maximal solutions up to stopping times in which the spread <span><math><mrow><msup><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><msup><mrow><mi>s</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> stays a.s. non-negative and bounded. For this, we define an approximation scheme, and use some of the estimates of Hambly et al. (2020) for the Green’s function and the associated to the reflection measure obstacle problem. Analogous results are obtained for the equation without reflection corresponding to a signed density.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104506"},"PeriodicalIF":1.1,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532080","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 : 2024-10-18DOI: 10.1016/j.spa.2024.104504
Linshan Liu , Mateusz B. Majka , Pierre Monmarché
We show -Wasserstein contraction for the transition kernel of a discretised diffusion process, under a contractivity at infinity condition on the drift and a sufficiently high diffusivity requirement. This extends recent results that, under similar assumptions on the drift but without the diffusivity restrictions, showed -Wasserstein contraction, or -Wasserstein bounds for that were, however, not true contractions. We explain how showing a true -Wasserstein contraction is crucial for obtaining a local Poincaré inequality for the transition kernel of the Euler scheme of a diffusion. Moreover, we discuss other consequences of our contraction results, such as concentration inequalities and convergence rates in KL-divergence and total variation. We also study corresponding -Wasserstein contraction for discretisations of interacting diffusions. As a particular application, this allows us to analyse the behaviour of particle systems that can be used to approximate a class of McKean-Vlasov SDEs that were recently studied in the mean-field optimisation literature.
{"title":"L2-Wasserstein contraction for Euler schemes of elliptic diffusions and interacting particle systems","authors":"Linshan Liu , Mateusz B. Majka , Pierre Monmarché","doi":"10.1016/j.spa.2024.104504","DOIUrl":"10.1016/j.spa.2024.104504","url":null,"abstract":"<div><div>We show <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-Wasserstein contraction for the transition kernel of a discretised diffusion process, under a contractivity at infinity condition on the drift and a sufficiently high diffusivity requirement. This extends recent results that, under similar assumptions on the drift but without the diffusivity restrictions, showed <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-Wasserstein contraction, or <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-Wasserstein bounds for <span><math><mrow><mi>p</mi><mo>></mo><mn>1</mn></mrow></math></span> that were, however, not true contractions. We explain how showing a true <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-Wasserstein contraction is crucial for obtaining a local Poincaré inequality for the transition kernel of the Euler scheme of a diffusion. Moreover, we discuss other consequences of our contraction results, such as concentration inequalities and convergence rates in KL-divergence and total variation. We also study corresponding <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-Wasserstein contraction for discretisations of interacting diffusions. As a particular application, this allows us to analyse the behaviour of particle systems that can be used to approximate a class of McKean-Vlasov SDEs that were recently studied in the mean-field optimisation literature.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104504"},"PeriodicalIF":1.1,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.spa.2024.104498
Arnaud Guillin , Pierre Le Bris , Pierre Monmarché
In this article, we focus on two toy models : the Curie–Weiss model and the system of particles in linear interactions in a double well confining potential. Both models, which have been extensively studied, describe a large system of particles with a mean-field limit that admits a phase transition. We are concerned with the numerical simulation of these particle systems. To deal with the quadratic complexity of the numerical scheme, corresponding to the computation of the interactions per time step, the Random Batch Method (RBM) has been suggested. It consists in randomly (and uniformly) dividing the particles into batches of size , and computing the interactions only within each batch, thus reducing the numerical complexity to per time step. The convergence of this numerical method has been proved in other works.
This work is motivated by the observation that the RBM, via the random constructions of batches, artificially adds noise to the particle system. The goal of this article is to study the effect of this added noise on the phase transition of the nonlinear limit, and more precisely we study the effective dynamics of the two models to show how a phase transition may still be observed with the RBM but at a lower critical temperature.
{"title":"Some remarks on the effect of the Random Batch Method on phase transition","authors":"Arnaud Guillin , Pierre Le Bris , Pierre Monmarché","doi":"10.1016/j.spa.2024.104498","DOIUrl":"10.1016/j.spa.2024.104498","url":null,"abstract":"<div><div>In this article, we focus on two toy models : the <em>Curie–Weiss</em> model and the system of <span><math><mi>N</mi></math></span> particles in linear interactions in a <em>double well confining potential</em>. Both models, which have been extensively studied, describe a large system of particles with a mean-field limit that admits a phase transition. We are concerned with the numerical simulation of these particle systems. To deal with the quadratic complexity of the numerical scheme, corresponding to the computation of the <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> interactions per time step, the <em>Random Batch Method</em> (RBM) has been suggested. It consists in randomly (and uniformly) dividing the particles into batches of size <span><math><mrow><mi>p</mi><mo>></mo><mn>1</mn></mrow></math></span>, and computing the interactions only within each batch, thus reducing the numerical complexity to <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>N</mi><mi>p</mi><mo>)</mo></mrow></mrow></math></span> per time step. The convergence of this numerical method has been proved in other works.</div><div>This work is motivated by the observation that the RBM, via the random constructions of batches, artificially adds noise to the particle system. The goal of this article is to study the effect of this added noise on the phase transition of the nonlinear limit, and more precisely we study the <em>effective dynamics</em> of the two models to show how a phase transition may still be observed with the RBM but at a lower critical temperature.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104498"},"PeriodicalIF":1.1,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532078","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 : 2024-10-15DOI: 10.1016/j.spa.2024.104502
Sheng Luo , Wenqiang Li , Xun Li , Qingmeng Wei
In this paper, we focus on the stochastic representation of a system of coupled Hamilton–Jacobi–Bellman–Isaacs (HJB–Isaacs (HJBI), for short) equations which is in fact a system of coupled Isaacs’ type integral-partial differential equation. For this, we introduce an associated zero-sum stochastic differential game, where the state process is described by a classical stochastic differential equation (SDE, for short) with jumps, and the cost functional of recursive type is defined by a new type of backward stochastic differential equation (BSDE, for short) with two Poisson random measures, whose wellposedness and a prior estimate as well as the comparison theorem are investigated for the first time. One of the Poisson random measures appearing in the SDE and the BSDE stems from the integral term of the HJBI equations; the other random measure in BSDE is introduced to link the coupling factor of the HJBI equations. We show through an extension of the dynamic programming principle that the lower value function of this game problem is the viscosity solution of the system of our coupled HJBI equations. The uniqueness of the viscosity solution is also obtained in a space of continuous functions satisfying certain growth condition. In addition, also the upper value function of the game is shown to be the solution of the associated system of coupled Isaacs’ type of integral-partial differential equations. As a byproduct, we obtain the existence of the value for the game problem under the well-known Isaacs’ condition.
{"title":"Stochastic representation for solutions of a system of coupled HJB-Isaacs equations with integral–differential operators","authors":"Sheng Luo , Wenqiang Li , Xun Li , Qingmeng Wei","doi":"10.1016/j.spa.2024.104502","DOIUrl":"10.1016/j.spa.2024.104502","url":null,"abstract":"<div><div>In this paper, we focus on the stochastic representation of a system of coupled Hamilton–Jacobi–Bellman–Isaacs (HJB–Isaacs (HJBI), for short) equations which is in fact a system of coupled Isaacs’ type integral-partial differential equation. For this, we introduce an associated zero-sum stochastic differential game, where the state process is described by a classical stochastic differential equation (SDE, for short) with jumps, and the cost functional of recursive type is defined by a new type of backward stochastic differential equation (BSDE, for short) with two Poisson random measures, whose wellposedness and a prior estimate as well as the comparison theorem are investigated for the first time. One of the Poisson random measures <span><math><mi>μ</mi></math></span> appearing in the SDE and the BSDE stems from the integral term of the HJBI equations; the other random measure in BSDE is introduced to link the coupling factor of the HJBI equations. We show through an extension of the dynamic programming principle that the lower value function of this game problem is the viscosity solution of the system of our coupled HJBI equations. The uniqueness of the viscosity solution is also obtained in a space of continuous functions satisfying certain growth condition. In addition, also the upper value function of the game is shown to be the solution of the associated system of coupled Isaacs’ type of integral-partial differential equations. As a byproduct, we obtain the existence of the value for the game problem under the well-known Isaacs’ condition.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"179 ","pages":"Article 104502"},"PeriodicalIF":1.1,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532073","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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