TY - JOUR

T1 - Well-posedness and stationary solutions of McKean-Vlasov (S)PDEs

AU - Angeli, Letizia

AU - Barré, Julien

AU - Kolodziejczyk, Martin

AU - Ottobre, Michela

N1 - Funding Information:
L.A. and M.O. have been supported by the Leverhulme Trust grant RPG–2020–09 . J.B. acknowledges support by the project RETENU ANR-20-CE40-0005-01 of the French National Research Agency (ANR). M.O. gratefully acknowledges support from the EPSRC grant EP/W034220/1 .
Publisher Copyright:
© 2023 The Author(s)

PY - 2023/10/15

Y1 - 2023/10/15

N2 - This paper is composed of two parts. In the first part we consider McKean-Vlasov Partial Differential Equations (PDEs), obtained as thermodynamic limits of interacting particle systems (i.e. in the limit
N
→
∞
, where N is the number of particles). It is well-known that, even when the particle system has a unique invariant measure (stationary solution), the limiting PDE very often displays a phase transition: for certain choices of (coefficients and) parameter values, the PDE has a unique stationary solution, but as the value of the parameter varies multiple stationary states appear. In the first part of this paper, we add to this stream of literature and consider a specific instance of a McKean-Vlasov type equation, namely the Kuramoto model on the torus perturbed by a symmetric double-well potential, and show that this PDE undergoes the type of phase transition just described, as the diffusion coefficient is varied. In the second part of the paper, we consider a rather general class of McKean-Vlasov PDEs on the torus (which includes both the original Kuramoto model and the Kuramoto model in double well potential of part one) perturbed by (strong enough) infinite-dimensional additive noise. To the best of our knowledge, the resulting Stochastic PDE, which we refer to as the Stochastic McKean-Vlasov equation, has not been studied before, so we first study its well-posedness. We then show that the addition of noise to the PDE has the effect of restoring uniqueness of the stationary state in the sense that, irrespective of the choice of coefficients and parameter values in the McKean-Vlasov PDE, the Stochastic McKean-Vlasov PDE always admits at most one invariant measure.

AB - This paper is composed of two parts. In the first part we consider McKean-Vlasov Partial Differential Equations (PDEs), obtained as thermodynamic limits of interacting particle systems (i.e. in the limit
N
→
∞
, where N is the number of particles). It is well-known that, even when the particle system has a unique invariant measure (stationary solution), the limiting PDE very often displays a phase transition: for certain choices of (coefficients and) parameter values, the PDE has a unique stationary solution, but as the value of the parameter varies multiple stationary states appear. In the first part of this paper, we add to this stream of literature and consider a specific instance of a McKean-Vlasov type equation, namely the Kuramoto model on the torus perturbed by a symmetric double-well potential, and show that this PDE undergoes the type of phase transition just described, as the diffusion coefficient is varied. In the second part of the paper, we consider a rather general class of McKean-Vlasov PDEs on the torus (which includes both the original Kuramoto model and the Kuramoto model in double well potential of part one) perturbed by (strong enough) infinite-dimensional additive noise. To the best of our knowledge, the resulting Stochastic PDE, which we refer to as the Stochastic McKean-Vlasov equation, has not been studied before, so we first study its well-posedness. We then show that the addition of noise to the PDE has the effect of restoring uniqueness of the stationary state in the sense that, irrespective of the choice of coefficients and parameter values in the McKean-Vlasov PDE, the Stochastic McKean-Vlasov PDE always admits at most one invariant measure.

KW - Ergodic theory for SPDEs

KW - McKean Vlasov PDE

KW - Stationary solutions of PDEs

KW - Stochastic McKean Vlasov equation

KW - Stochastic partial differential equations

UR - http://www.scopus.com/inward/record.url?scp=85152682342&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2023.127301

DO - 10.1016/j.jmaa.2023.127301

M3 - Article

SN - 0022-247X

VL - 526

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 127301

ER -