Well-posedness and stationary solutions of McKean-Vlasov (S)PDEs

Angeli, L.; Barré, Julien; Kolodziejczyk, M.

doi:10.1016/j.jmaa.2023.127301

Cited by 4 publications

(1 citation statement)

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“…[7]. Depending on the choice of drift and on the strength of the noise coefficient, these equations may exhibit a phase transition [8][9][10][11]: typically, for large values of the noise the dynamics has only one invariant measure but for small values of the noise the SDE can have multiple invariant measures. If a McKean-Vlasov equation of this type was used to model the fast process (and the slow variable appeared in the noise coefficients of the fast process, a scenario which is at the moment work in progress from the second author and collaborators) then clearly the assumption that we make in this paper would not hold.…”

Section: Note 22mentioning

confidence: 99%

On the study of slow–fast dynamics, when the fast process has multiple invariant measures

Goddard,

Ottobre,

Painter

et al. 2023

Proc. R. Soc. A.

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Motivated by applications to mathematical biology, we study the averaging problem for slow–fast systems, in the case in which the fast dynamics is a stochastic process with multiple invariant measures . We consider both the case in which the fast process is decoupled from the slow process and the case in which the two components are fully coupled. We work in the setting in which the slow process evolves according to an ordinary differential equation (ODE) and the fast process is a continuous time Markov process with finite state space and show that, in this setting, the limiting (averaged) dynamics can be described as a random ODE (i.e. an ODE with random coefficients).

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