Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits

[2405.15142] Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2405.15142 (math) [Submitted on 24 May 2024] Title: Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits Authors: Patrícia Gonçalves , Kohei Hayashi , Makiko Sasada View a PDF of the paper titled Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits, by Patr\'icia Gon\c{c}alves and 2 other authors View PDF HTML (experimental) Abstract: We consider partial exclusion processes~(PEPs) on the one-dimensional square lattice, that is, a system of interacting particles where each particle random walks according to a jump rate satisfying an exclusion rule that allows up to a certain number of particles can exist on each site. Particularly, we assume that the jump rate is given as a product of two functions depending on occupation variables on the original and target sites. Our interest is to study the limiting behavior, especially to derive some macroscopic PDEs by means of (fluctuating) hydrodynamics, of fluctuation fields associated with PEPs, starting from an invariant measure. The so-called gradient condition, meaning that the symmetric part of the instantaneous current is written in a gradient form, and that the invariant measures are given as a product measure is technically crucial. Our first main result is to clarify the relationship between these two conditions, and we show that the gradient condition and the existence of product invariant measures are mutually equivalent, provided the jump rate is given in the above simple form, as it is imposed in most of the literature, and the dynamics is asymmetric. Moreover, when the width of the lattice tends to zero and the process is accelerated in diffusive time-scaling, we show that the family of fluctuation fields converges to the stationary energy solution of the stochastic Burgers equation (SBE), under the setting that the jump rate to the right neighboring site is a bit larger than the one to the left side, of which discrepancy is given as square root of the width of the underlying lattice. This fills the gap at the level of universality of SBE since it has been proved for the exclusion process (a special case of PEP) and for the zero-range process. Comments: 25 pages Subjects: Probability (math.PR) MSC classes: 60H15, 60K35, 82B44 Cite as: arXiv:2405.15142 [math.PR] (or arXiv:2405.15142v1 [math.PR] for this version) https://doi.org/10.48550/arXiv.2405.15142 Focus to learn more arXiv-issued DOI via DataCite Journal reference: ALEA, Lat. Am. J. Probab. Math. Stat. 22, 331-356 (2025) Related DOI : https://doi.org/10.30757/ALEA.v22-12 Focus to learn more DOI(s) linking to related resources Submission history From: Kohei Hayashi [ view email ] [v1] Fri, 24 May 2024 01:35:27 UTC (38 KB) Full-text links: Access Paper: View a PDF of the paper titled Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits, by Patr\'icia Gon\c{c}alves and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2024-05 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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