On the McKean‐Vlasov Limit for Interacting Diffusions

Gärtner, Jürgen

doi:10.1002/mana.19881370116

Cited by 219 publications

(249 citation statements)

References 19 publications

(3 reference statements)

Supporting

Mentioning

245

Contrasting

Unclassified

Order By: Relevance

“…[22,26,34,35] for similar situations without disorder). Note also that this convergence is also valid for more general models (see e.g.…”

Section: The Microscopic Modelmentioning

confidence: 99%

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Luçon¹

2014

Journal of Functional Analysis

View full text Add to dashboard Cite

We investigate the long-time asymptotics of the fluctuation SPDE in the Kuramoto synchronization model. We establish the linear behavior for large time and weak disorder of the quenched limit fluctuations of the empirical measure of the particles around its McKean-Vlasov limit. This is carried out through a spectral analysis of the underlying unbounded evolution operator, using arguments of perturbation of self-adjoint operators and analytic semigroups. We state in particular a Jordan decomposition of the evolution operator which is the key point in order to show that the fluctuations of the disordered Kuramoto model are not self-averaging.

show abstract

“…[22,26,34,35] for similar situations without disorder). Note also that this convergence is also valid for more general models (see e.g.…”

Section: The Microscopic Modelmentioning

confidence: 99%

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Luçon¹

2014

Journal of Functional Analysis

View full text Add to dashboard Cite

show abstract

“…Moreover, for every t > 0 the measure ν t is absolutely continuous with respect to the Lebesgue measure with (strictly) positive density q t (·) and the function (t, θ) → q t (θ), from (0, ∞) × S to (0, ∞), is smooth and solves (1.9). Proposition 1.3 is a particular (and particularly easy) case of far more general results (see for example [8,15]). The derivation goes along proving tightness of {ν N,· } N ∈N and then proving uniqueness for the limiting equation (1.8).…”

Section: 1mentioning

confidence: 92%

“…and therefore (2.5) is equivalent to 8) where the intermediate step follows from the identity yI 2 (y) + 2I 1 (y) − yI 0 (y) = 0 [23]. But this is equivalent to I 1 (y)/I 0 (y) > I 2 (y)/I 1 (y) for y > 0, a fact that is proven in [11, (3.8)].…”

Section: Synchronization Stabilitymentioning

confidence: 99%

Dynamical Aspects of Mean Field Plane Rotators and the Kuramoto Model

2009

View full text Add to dashboard Cite

Abstract. The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N → ∞ limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N → ∞ limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value Kc. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K > Kc.

show abstract

“…We next introduce an alternative and more traditional model with spatially uncorrelated Brownian noise, following Oelschläger [29] and Gärtner [12]. To this end, choose a sequence of i.i.d.…”

Section: (S) L ε ϕ(· X ε (S) S) Dsmentioning

confidence: 99%

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

Kotelenez

Kurtz

2008

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita.

show abstract

On the McKean‐Vlasov Limit for Interacting Diffusions

Cited by 219 publications

References 19 publications

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Dynamical Aspects of Mean Field Plane Rotators and the Kuramoto Model

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

Contact Info

Product

Resources

About