A NEW MCKEAN-VLASOV STOCHASTIC INTERPRETATION OF THE PARABOLIC-PARABOLIC KELLER-SEGEL MODEL: THE TWO-DIMENSIONAL CASE

• 成果类型: Article
• 署名作者: Tomasevic, Milica
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1594
• 发表日期: 2021
• 页码: 432-459
• 关键词: transition-probability densities; global existence; SYSTEM

摘要

Recently, we proposed a new stochastic interpretation of the parabolic-parabolic Keller-Segel system without cut-off via a McKean-Vlasov stochastic process. The process was defined through an original type of interaction kernel which involved, in a singular way, all its past time marginal distributions. In the present paper, we study this McKean-Vlasov representation in the two-dimensional case. In this setting, there exists a possibility of a blow-up in finite time for the Keller-Segel system if some parameters of the model are large. Indeed, we prove the global in time well-posedness of the McKean-Vlasov process under some constraints involving a parameter of the model and the initial datum. Under these constraints, we also prove the global well-posedness for the Keller-Segel model in the plane.