Probability Theory and Related Fields最新文献

From ABC to KPZ.

IF 1.5 1区数学

Probability Theory and Related Fields Pub Date : 2025-01-01 Epub Date: 2024-10-22 DOI: 10.1007/s00440-024-01314-z

G Cannizzaro, P Gonçalves, R Misturini, A Occelli

引用次数: 0

The dynamical Ising-Kac model in 3D converges to Φ 3 4.

IF 1.5 1区数学

Probability Theory and Related Fields Pub Date : 2025-01-01 Epub Date: 2024-10-15 DOI: 10.1007/s00440-024-01316-x

P Grazieschi, K Matetski, H Weber

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">The dynamical Ising-Kac model in 3D converges to <ns0:math><ns0:msubsup><ns0:mi>Φ</ns0:mi> <ns0:mn>3</ns0:mn> <ns0:mn>4</ns0:mn></ns0:msubsup></ns0:math>.","authors":"P Grazieschi, K Matetski, H Weber","doi":"10.1007/s00440-024-01316-x","DOIUrl":"https://doi.org/10.1007/s00440-024-01316-x","url":null,"abstract":"We consider the Glauber dynamics of a ferromagnetic Ising-Kac model on a three-dimensional periodic lattice of size <math> <msup><mrow><mo>(</mo> <mn>2</mn> <mi>N</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>3</mn></msup> </math> , in which the flipping rate of each spin depends on an average field in a large neighborhood of radius <math> <mrow><msup><mi>γ</mi> <mrow><mo>-</mo> <mn>1</mn></mrow> </msup> <mo><</mo> <mspace></mspace> <mspace></mspace> <mo><</mo> <mi>N</mi></mrow> </math> . We study the random fluctuations of a suitably rescaled coarse-grained spin field as <math><mrow><mi>N</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> and <math><mrow><mi>γ</mi> <mo>→</mo> <mn>0</mn></mrow> </math> ; we show that near the mean-field value of the critical temperature, the process converges in distribution to the solution of the dynamical <math><msubsup><mi>Φ</mi> <mn>3</mn> <mn>4</mn></msubsup> </math> model on a torus. Our result settles a conjecture from Giacomin et al. (1999). The dynamical <math><msubsup><mi>Φ</mi> <mn>3</mn> <mn>4</mn></msubsup> </math> model is given by a non-linear stochastic partial differential equation (SPDE) which is driven by an additive space-time white noise and which requires renormalisation of the non-linearity. A rigorous notion of solution for this SPDE and its renormalisation is provided by the framework of regularity structures (Hairer in Invent Math 198(2):269-504, 2014. 10.1007/s00222-014-0505-4). As in the two-dimensional case (Mourrat and Weber in Commun Pure Appl Math 70(4):717-812, 2017), the renormalisation corresponds to a small shift of the inverse temperature of the discrete system away from its mean-field value.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"191 1-2","pages":"671-778"},"PeriodicalIF":1.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11850488/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143516413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Rearranged Stochastic Heat Equation.

IF 1.5 1区数学

Probability Theory and Related Fields Pub Date : 2025-01-01 Epub Date: 2024-10-24 DOI: 10.1007/s00440-024-01335-8

François Delarue, William R P Hammersley

{"title":"Rearranged Stochastic Heat Equation.","authors":"François Delarue, William R P Hammersley","doi":"10.1007/s00440-024-01335-8","DOIUrl":"10.1007/s00440-024-01335-8","url":null,"abstract":"The purpose of this work is to provide an explicit construction of a strong Feller semigroup on the space of probability measures over the real line that additionally maps bounded measurable functions into Lipschitz continuous functions, with a Lipschitz constant that blows up in an integrable manner in small time. Our construction relies on a rearranged version of the stochastic heat equation on the circle driven by a coloured noise. Formally, this stochastic equation writes as a reflected equation in infinite dimension. Under the action of the rearrangement, the solution is forced to live in a space of quantile functions that is isometric to the space of probability measures on the real line. We prove the equation to be solvable by means of an Euler scheme in which we alternate flat dynamics in the space of random variables on the circle with a rearrangement operation that projects back the random variables onto the subset of quantile functions. A first challenge is to prove that this scheme is tight. A second one is to provide a consistent theory for the limiting reflected equation and in particular to interpret in a relevant manner the reflection term. The last step in our work is to establish the aforementioned Lipschitz property of the semigroup by adapting earlier ideas from the Bismut-Elworthy-Li formula.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"191 1-2","pages":"41-102"},"PeriodicalIF":1.5,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11850558/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143516439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations 随机穿孔域中非线性德里赫特问题的均质化（关于穿孔大小的最小假设条件

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-09-17 DOI: 10.1007/s00440-024-01320-1

Lucia Scardia, Konstantinos Zemas, Caterina Ida Zeppieri

{"title":"Homogenisation of nonlinear Dirichlet problems in randomly perforated domains under minimal assumptions on the size of perforations","authors":"Lucia Scardia, Konstantinos Zemas, Caterina Ida Zeppieri","doi":"10.1007/s00440-024-01320-1","DOIUrl":"https://doi.org/10.1007/s00440-024-01320-1","url":null,"abstract":"In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of (mathbb {R}^n). Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite ((n-q))-moment, where (1<q<n) is the growth-exponent of the associated energy functionals. This assumption on the one hand ensures that the expectation of the nonlinear q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.\u0000","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"49 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142257708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements 关于维纳香肠空位集和布朗交错的唯一性问题

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-09-14 DOI: 10.1007/s00440-024-01315-y

Yingxin Mu, Artem Sapozhnikov

引用次数: 0

Weighted sums and Berry-Esseen type estimates in free probability theory 自由概率论中的加权和与贝里-埃森类型估计

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-08-21 DOI: 10.1007/s00440-024-01294-0

Leonie Neufeld

引用次数: 0

Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules 具有对称各向同性阈值规则的二维引导渗流的尖锐蜕变

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-08-21 DOI: 10.1007/s00440-024-01310-3

Hugo Duminil-Copin, Ivailo Hartarsky

引用次数: 0

Subexponential lower bounds for f-ergodic Markov processes f-ergodic Markov 过程的亚指数下界

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-08-21 DOI: 10.1007/s00440-024-01306-z

Miha Brešar, Aleksandar Mijatović

{"title":"Subexponential lower bounds for f-ergodic Markov processes","authors":"Miha Brešar, Aleksandar Mijatović","doi":"10.1007/s00440-024-01306-z","DOIUrl":"https://doi.org/10.1007/s00440-024-01306-z","url":null,"abstract":"We provide a criterion for establishing lower bounds on the rate of convergence in f-variation of a continuous-time ergodic Markov process to its invariant measure. The criterion consists of novel super- and submartingale conditions for certain functionals of the Markov process. It provides a general approach for proving lower bounds on the tails of the invariant measure and the rate of convergence in f-variation of a Markov process, analogous to the widely used Lyapunov drift conditions for upper bounds. Our key technical innovation produces lower bounds on the tails of the heights and durations of the excursions from bounded sets of a continuous-time Markov process using path-wise arguments. We apply our theory to elliptic diffusions and Lévy-driven stochastic differential equations with known polynomial/stretched exponential upper bounds on their rates of convergence. Our lower bounds match asymptotically the known upper bounds for these classes of models, thus establishing their rate of convergence to stationarity. The generality of the approach suggests that, analogous to the Lyapunov drift conditions for upper bounds, our methods can be expected to find applications in many other settings.","PeriodicalId":20527,"journal":{"name":"Probability Theory and Related Fields","volume":"47 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions 李雅普诺夫条件下非 Lipschitz 系数的 Kolmogorov 方程的正则性保持

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-08-20 DOI: 10.1007/s00440-024-01313-0

Martin Chak

引用次数: 0

The replica-symmetric free energy for Ising spin glasses with orthogonally invariant couplings 具有正交不变耦合的伊辛自旋玻璃的复制对称自由能

IF 2 1区数学

Probability Theory and Related Fields Pub Date : 2024-08-13 DOI: 10.1007/s00440-024-01309-w

Zhou Fan, Yihong Wu

引用次数: 0