On Mean Field Convergence and Stationary Regime

This paper is motivated by results on mean field interaction models and stochastic approximation algorithms which obtain convergence of a family of stochastic processes to a deterministic limit (Kurtz, 1970;Sandholm, 2006;Bordenave et al., 2007;Graham and Méléard, 1994;Benaïm and Le Boudec, 2008). Often, convergence is over finite time horizons, which asks the question of whether the convergence extends to the stationary regime. In this paper, we show that some form of convergence of the stationary regimes follows systematically from convergence at any fixed and finite time horizon, under very weak assumptions. Previous answers to this question exist with stronger assumptions than here; for example, the space is finite dimensional and the deterministic limit is a dissipative ODE (Benaïm, 1998), or the set of invariant distributions is tight (Fort and Pages, 1999). Such assumptions cannot always be made, consider for example the cases in (Chaintreau et al., 2009;Bordenave et al., 2007); our result requires much weaker assumptions and appears to be more general.

More precisely, we consider a family of stochastic processes Y N , indexed by N = 1, 2, … over some Polish space E; we assume that the processes have the property that, as N → ∞ and the initial conditions y N (0) → y(0), the marginals of the process Y N (t) converge in distribution to some deterministic y(t), where convergence is at every fixed time t (see Hypothesis 1 below). We show that, under the (mild) assumption that the processes are Feller, this is sufficient to obtain that any limit point of an invariant probability of Y N is an invariant probability of the deterministic process (and thus its support is included in its Birkhoff center). Note that we do not need to assume any semi-flow nor continuity property for the limiting deterministic process.

In the special case where the deterministic process has a unique limit point y * = lim t→∞ y(t) and where the sequence of invariant probabilities Π N is tight, it follows imme-diately that Π N converges to the Dirac mass at y * . This result is known in the context of stochastic approximation algorithms; our results here extend it to a more general setting.

Our result is also more general as it applies to other cases. Mean field interaction models were often used as practical approximations of complex interacting object systems, where the stationary distribution of the system Y N is approximated by the stationary regime of an ordinary differential equation (ODE); this was applied for example to TCP connections (Tinnakornsrisuphap and Makowski, 2003;Baccelli et al., 2006;Graham and Robert, 2009), HTTP flows (Baccelli et al., 2004), bandwidth sharing between streaming and file transfers (Kumar and Massoulie, 2007), mobile networks (Chaintreau et al., 2009), robot swarms (Martinoli and Easton, 2002), transportation systems (Afanassieva et al., 1997), reputation systems (Le Boudec et al., 2007), just to name a few. Previous results are obtained when the ODE has a unique limit point to which all trajectories converge. Not only does our result extends this finding to more general spaces, it also extends it to the cases where there is not a unique limit point. For example, in (Bordenave et al., 2007), the ODE a unique limit point under some restrictive assumptions on the model parameters; if these assumptions do not hold, the ODE may have limit cycles, as shown in (Cho et al., 2010), and nothing can be concluded from (Bordenave et al., 2007). Using our results, it follows that the limit points of any invariant probability has a support included in the limit cycles.

Let (E, d) be a Polish space and P(E) the set of probability measures on E, endowed with the topology of weak convergence. Let C b (E) be the set of bounded continuous functions from E to R.

We are given a collection of probability spaces (Ω N , F N , P N ) indexed by N = 1, 2, 3, … and for every N we have a process Y N defined on (Ω N , F N , P N ). Time is either discrete or continuous. In the discrete time case, Y N (t) is a collection of random variables indexed by N = 1, 2, 3… and t ∈ N. In the continuous time case, let

We denote by Y N (t) the random value of

We assume that, for every N, the process Y N is Feller, in the sense that for every t ≥ 0 and h

Further, let ϕ be a deterministic process, i.e. a mapping

where T = N or T = [0, ∞).

We assume that ϕ t is measurable for every fixed time t ≥ 0. Note that there is no assumption here that ϕ is continuous nor that it is a flow. Definition 2. A probability Π ∈ P(E) is invariant for ϕ if for every h ∈ C b (E) and every t ≥ 0:

We assume that, for every fixed t the processes Y N converge in distribution to the deterministic process ϕ as N → ∞ for every collection of converging initial conditions. More precisely: Hypothesis 1. For every y 0 in E, every sequence (y N 0 ) N =1,2,… such that y N 0 ∈ E N and lim N →∞ y N 0 = y 0 , and every t ≥ 0, the conditional law of Y N (t) gi

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