Local convergence rates for Wasserstein gradient flows and McKean-Vlasov equations with multiple stationary solutions
Pierre Monmarché Julien Reygner


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Non-linear versions of log-Sobolev inequalities, that link a free energy to its dissipation along the corresponding Wasserstein gradient flow (i.e. corresponds to Polyak-Lojasiewicz inequalities in this context), are known to provide global exponential long-time convergence to the free energy minimizers, and have been shown to hold in various contexts. However they cannot hold when the free energy admits critical points which are not global minimizers, which is for instance the case of the granular media equation in a double-well potential with quadratic attractive interaction at low temperature. This work addresses such cases, extending the general arguments when a log-Sobolev inequality only holds locally and, as an example, establishing such local inequalities for the granular media equation with quadratic interaction either in the one-dimensional symmetric double-well case or in higher dimension in the low temperature regime. The method provides quantitative convergence rates for initial conditions in a Wasserstein ball around the stationary solutions. The same analysis is carried out for the kinetic counterpart of the gradient flow, i.e. the corresponding Vlasov-Fokker-Planck equation. The local exponential convergence to stationary solutions for the mean-field equations, both elliptic and kinetic, is shown to induce for the corresponding particle systems a fast (i.e. uniform in the number or particles) decay of the particle system free energy toward the level of the non-linear limit.




§ INTRODUCTION


We are interested in the long-time behavior of the granular media equation

    ∂_t ρ_t = ·σ^2∇ρ_t +  V + ρ_t ⋆ Wρ_t

or of more general McKean-Vlasov semilinear equations (see (<ref>) below). Here, ρ_t is a probability density over ^d, V∈𝒞^2(^d,), W∈𝒞^2(^d×^d,) are respectively a confining and interaction potential, W(x,y)=W(y,x) for all x,y∈^d, σ^2>0 stands for the temperature (or diffusivity) and ρ⋆ W(x) = ∫_^d_x W(x,y)ρ(y) y. We have typically in mind the double-well case where d=1,

    V(x) = x^4/4 - x^2/2 ,     W(x,y) = θ  (x-y)^2

for some θ∈. When θ⩽ 0 (repulsive interaction), or when θ> 0 and σ^2/θ is large enough (attractive interaction at high temperature or small interaction), there is a unique stationary solution to (<ref>), which is globally attractive with exponential rate. However for a fixed θ>0 there is a phase transition at some critical temperature σ_c^2>0 such that, for σ^2 <σ_c^2, (<ref>) with (<ref>) admits three distinct stationary solutions (one symmetric, unstable, and two non-symmetric, stable) <cit.>. We are interested in this second case, and more precisely on obtaining local convergence rates, namely to prove that solutions which start close to a stable stationary solution converge exponentially fast to it. Among other motivations, this is an important question both for the theoretical understanding of metastable interacting particle systems, where the stability property of the non-linear limit drives the short-time behavior and induces the metastable behaviour <cit.>, and for practical questions of optimization in the Wasserstein space, for instance in the mean-field modelling of artificial neural networks <cit.> (the loss function being convex only in toy models, like one-layer networks).



To illustrate the point, let us state a result obtained with our approach in the specific case of the symmetric double well (<ref>). In the next statement (proven in Section <ref>, see Remarks <ref> and <ref>) as in the rest of the work we write 𝒲_2 and ·_TV respectively the L2 Wasserstein distance and total variation norm and 𝒫_2(^d) the set of probability measures on ^d with finite second moment.


Consider the granular media (<ref>) in the case (<ref>) with θ>0.

    
  * If σ^2 < σ_c^2, let ρ_* be one of the two non-symmetric stationary solutions. There exist δ,λ,C>0 such that for any initial condition ρ_0∈𝒫_2() with 𝒲_2(ρ_0,ρ_*) ⩽δ, the corresponding solution to (<ref>) satisfies, for all t⩾ 0

    𝒲_2(ρ_t,ρ_*) + ρ_t - ρ_*_TV⩽ C e^-λ t .

 
  * If σ^2 = σ_c^2, denote by ρ_* the unique stationary solution. For any ρ_0∈𝒫_2(), there exists C_0>0 such that the corresponding solution to (<ref>) satisfies, for all t⩾ 0,  

    𝒲_2(ρ_t,ρ_*) + ρ_t - ρ_*_TV⩽C_0/t^1/3 .








Notice that, in the sub-critical case, there is no assumption on σ^2 other than σ^2 < σ_c^2, namely, the result holds arbitrarily close to the critical temperature (our results also apply in the simpler super-critical regime and provide, as soon as σ^2 > σ_c^2, a global quantitative exponential convergence toward the unique stationary solution, see Remark <ref>). This is in contrast to the results of <cit.>, also concerned with quantitative local convergence of (<ref>), which require the temperature to be sufficiently small (for a given θ). Moreover, the Wasserstein convergence in <cit.> only gives a convergence speed of order e^-λ√(t) for some λ>0, and additionally it requires θ > -inf V”, which our result does not (under this additional assumption we get that the rate λ is uniform over σ^2∈(0,σ_0^2] for any σ_0^2∈(0,σ_c^2), as in Proposition <ref>). The constants δ,λ,C in Proposition <ref> can be made explicit (see Remark <ref>), namely the result is quantitative.

Our method is the following. As recalled in the next section, it is now well-known that (<ref>) can be seen as the gradient flow in the Wasserstein space of some functional. To fix ideas, consider on ^d such a gradient flow

    ẋ_t = - f(x_t) .

Let x_* be a local minimizer of f. Assuming a Polyak-Lojasiewicz inequality, namely that there exists η>0 such that

    0 ⩽  f(x) - f(x_*) ⩽η | f(x)|^2
 
in a neighborhood 𝒜 of x_*, then by differentiating f(x_t)-f(x_*) we immediately get that, as long as x_t stays within 𝒜, 
  
    f(x_t)-f(x_*) ⩽ e^-t/η(f(x_0)-f(x_*))

  and
  
    ∂_t √(f(x_t) - f(x_*))⩽ -  1/2√(η)| f(x_t)| ,

  from which
  
    |x_t-x_0| =  |∫_0^t  f(x_s) s | ⩽ 2√(η f(x_0) - f(x_*)) .

Using that f is continuous so that the right hand side can be made arbitrarily small by taking x_0 sufficiently close to x_*, this shows that 𝒜 contains a ball centered at x_* such that, starting with an initial condition in this ball, the gradient flows remains in 𝒜 and thus the previous inequalities hold for all times t⩾ 0. Assuming furthermore that x_* is the unique critical point of f in 𝒜, by the LaSalle invariance principle, x_t converges to x_* and then letting t→∞ in (<ref>) and applying it with x_0 replaced by x_t gives

    |x_*-x_t|^2 ⩽ 4 η f(x_t) - f(x_*)⩽ 4 η e^-t/η f(x_0) - f(x_*) ,

which proves the exponential convergence to x_*. 

We show that this argument extends to the infinite dimensional settings of gradient flows over the Wasserstein space  (the lack of continuity of f with respect to the Wasserstein distance in this case being circumvented by the regularization properties of (<ref>) which, with the present notations, amounts to say that x_0 ↦ f(x_t) is continuous at x_* as soon as t>0). The key new ingredient is thus the (local) dissipation inequality (<ref>), which for elliptic McKean-Vlasov equations as (<ref>) corresponds to a (local) non-linear log-Sobolev inequality (LSI). Such inequalities  have been investigated in a number of works (see e.g. <cit.> and references within) but, to our knowledge, only in cases where they are global, corresponding in the finite-dimensional settings above to the case where   (<ref>) holds for all x∈^d (for instance when f is uniformly strongly convex). Our main contribution is thus to show that the method also works locally and, more importantly, to show that   it is indeed possible to establish such local dissipation inequalities in some cases where the global inequality fails. More specifically, the main results of this work are the following:


  * In the general framework of gradient flows with respect to 𝒲_2, under suitable regularity conditions, the local non-linear LSI that is the analogue of (<ref>) is shown to imply the exponential convergence (in 𝒲_2 and relative entropy) towards the local minimizer for all initial conditions in  a suitable 𝒲_2 ball (this is Theorem <ref>).

  * The same result is established in the kinetic case, i.e. for the Vlasov-Fokker-Planck equation (this is Theorem <ref>).

  * In the particular case of the granular media equation, when the interaction  is parametrized by some moments of the measure, a simple criterion for the local non-linear LSI is given in Proposition <ref>. It is then illustrated in two cases with quadratic interactions, the one-dimensional double-well case (Proposition <ref>) and the multi-well case in ^d (Proposition <ref>).

  * Under the same conditions as Theorem <ref> (or Theorem <ref> in the kinetic case), the free energy of the corresponding system of interacting particles is shown to decay fast below the level of the limit of the non-linear limit, as stated in Proposition <ref> (or Proposition <ref> in the kinetic case).


This work is organized as follows. The general framework is introduced in Section <ref>, where the main general result (Theorem <ref>) is stated and proven. Section <ref> addresses the question of establishing local non-Linear LSI, covering the granular media case (with a detailed study of the one-dimensional double-well potential). The Vlasov-Fokker-Planck equation is studied in Section <ref>, and Section <ref> is devoted to interacting particle systems.

To conclude this introduction, let us mention some perspectives of this work. The gradient descent structure underlying our study is quite flexible, as one may restrict the space over which the free energy is minimized (e.g. tensorized distributions, Gaussian distributions, distributions with some fixed marginals…), which is of interest for variational inference <cit.>, and one can modify the metric with respect to which the gradient is taken, which amounts to add some non-constant (possibly non-linear) diffusion matrix, allowing e.g. for slow or fast diffusion processes and congestion effects <cit.>. We expect most  of our analysis to extend to this kind of settings. Non-asymptotic bounds for discrete-time numerical schemes can also   be obtained as in <cit.>. Extension to time-varying temperature (for annealing or as a surrogate to stochastic gradient descent <cit.>) is straightforward, following e.g. <cit.> and references within.

Last, let us mention that for McKean-Vlasov equations which do not necessarily write as Wasserstein gradient flows, but under different assumptions than ours, a similar statement to Proposition <ref> was recently obtained in <cit.>. It provides exponential convergence when the initial condition is in a small 𝒲_1 neighborhood of a stationary solution. The method is completely different and relies on the differentiation, in the sense of Lions derivatives, of the drift of the underlying nonlinear SDE in the neighborhood of the considered invariant measure, which then provides a criterion for the stability of the invariant measure.



§ LOCAL CONVERGENCE RATES WITH LOG-SOBOLEV INEQUALITIES


The main result of this section is Theorem <ref>,  stated and proven in Section <ref>. Before that, the relevant  notions, conditions  and lemmas are gradually introduced.



 §.§ General settings, assumptions and notations


We consider an energy functional ℰ:𝒫_2(^d) → (-∞,∞] and, for a temperature σ^2>0, the free energy

    ℱ(ρ) = ℰ(ρ) + σ^2 ℋ(ρ)

for ρ∈𝒫_2(^d), where ℋ(ρ)  = ∫_^dρlnρ stands for the entropy (taken as +∞ if ρ is not absolutely continuous; otherwise we   also write ρ its density). 

[boundedness from below of the free energy]
  The free energy ℱ is bounded from below.


Given a functional 𝒢:𝒫_2(^d) → (-∞,∞], a measurable function δ𝒢/δμ:𝒫_2(^d)×^d → is called a linear functional derivative of 𝒢 if, for all  μ_1,μ_0 ∈𝒫_2(^d) with 𝒢(μ_0)+𝒢(μ_1)<∞,

    𝒢(μ_1) - 𝒢(μ_0) = ∫_0^1 ∫_^dδ𝒢(μ_t)/δμ(x) (μ_1-μ_0)( x) t ,

where μ_t = tμ_1 +(1-t) μ_0. 

[linear functional derivative of the energy]
  ℰ admits a linear functional derivative that we denote  E_ρ(x) = δℰ(ρ)/δμ(x). Moreover,
  
      
  (i) the function (x,ρ) ∈^d ×𝒫_2(^d) ↦ E_ρ(x) is continuous, where 𝒫_2(^d) is endowed with the 𝒲_2 distance;
      
  (ii) for any ρ∈𝒫_2(^d), E_ρ∈𝒞^2(^d);
      
  (iii) for all μ_0, μ_1 ∈𝒫_2(^d) with ℰ(μ_0) + ℰ(μ_1) < ∞, x ↦sup_t ∈ [0,1] |E_μ_t(x)| is in L^1(μ_0)∩ L^1(μ_1).
  

The last item of Assumption <ref> ensures that the integral in the right hand side of (<ref>) is well defined when 𝒢=ℰ, and moreover that

    lim_t → 0ℰ(μ_0 + t(μ_1-μ_0))-ℰ(μ_0)/t = ∫_^d E_μ_0(x)(μ_1-μ_0)( x).



 

We are interested in the McKean-Vlasov equation

    ∂_t ρ_t = σ^2Δρ_t +   ·ρ_t E_ρ_t ,

equivalently 

    ∂_t ρ_t =  ·ρ_t δℱ(ρ_t) /δμ .

In particular, the granular media equation  (<ref>) corresponds to the free energy

    ℱ(ρ) =  σ^2 ∫_^dρlnρ + ∫_^d Vρ + 1/2∫_^d×^d   W ρ^⊗ 2 ,
 
for which E_ρ(x) = V(x) + ρ⋆ W(x) and Assumptions <ref> and <ref> are satisfied as soon as, for instance, V is convex outside a compact set and W is lower bounded with at most a quadratic growth at infinity (other conditions can be considered, for instance in repulsive cases W may not be bounded below but Assumption <ref> still holds if V grows faster than -W at infinity)   Under suitable regularity  condition, a straightforward computation gives

    ∂_t ℱ( ρ_t)  = -  ∫_^d|δℱ(ρ_t)/δμ|^2 ρ_t   .

In particular, the free energy is decreasing along time, and the stationary solutions ρ_* are the critical points of ℱ, characterized by the fact that  δℱ/δμ(ρ_*) is constant, which, using that  δℋ/δμ(ρ) = lnρ,  is equivalent to the self-consistency equation

    ρ_* ∝exp - 1/σ^2 E_ρ_* .

In cases where ℱ satisfies a global non-linear log-Sobolev inequality, in the sense that there exists a constant >0 such that 

    ∀ρ∈𝒫_2(^d) ,    ℱ( ρ) - infℱ⩽∫_^d|δℱ(ρ)/δμ|^2 ρ ,
    G-NL-LSI

 we immediately get from (<ref>) that  the free energy decays exponentially fast toward its infimum. In particular such an inequality is clearly false when ℱ admits critical points which are not global minimizers, which is precisely the case we are interested in.
 
 [local equilibrium]
    For any ρ∈𝒫_2(^d), 
    
    Z_ρ = ∫_^dexp - 1/σ^2E_ρ < ∞.


For the granular media equation (<ref>) with free energy given by (<ref>), this assumption holds  if for instance W is lower bounded and ln|x| = o(V(x)) at infinity.  Under Assumption <ref>, denoting
 
    Γ(ρ) = Z_ρ^-1exp - 1/σ^2E_ρ ,

which we call the local equilibrium as it is the stationary solution of the linear equation

    ∂ρ̃_t = σ^2 Δρ̃_t + ·ρ̃_t   E_ρ ,

we can also interpret the free energy dissipation in (<ref>) as 

    ∫_^d|δℱ/δμ(ρ)|^2 ρ = σ^4 ℐρ|Γ(ρ) ,

where ℐ(ν|μ) stands for the Fisher information of ν with respect to μ,

    ℐ(ν|μ) = ∫_^d|lnν/μ|^2 ν .

 

 We write

    𝒦 = {ρ_* ∈𝒫_2(^d), ℱ(ρ_*)<∞, ρ_* = Γ(ρ_*)}

the set of critical points of ℱ.

 In this work we are interested in cases where the local equilibria satisfy a uniform (classical) log-Sobolev inequality, in the sense that there exist η >0 such that 

    ∀ν,ρ∈𝒫_2(^d) ,    ℋν|Γ(ρ)⩽ηℐν|Γ(ρ) ,
    U-LSI

where

    ℋ(ν|μ) = ∫_^dlnν/μν

stands for the relative entropy of ν with respect to μ. Indeed, contrary to (<ref>), there are many tools to establish (<ref>) and it can typically hold even if ℱ has several critical points. For instance, for the granular media equation (<ref>) in the double well case (<ref>) with attractive interaction (namely θ>0), we can decompose V=V_0+V_1 where V_0 is strongly convex and V_1 is bounded, so that,  for any ρ,

    V  + ρ⋆ W   = V_0 + V_1 + ρ⋆ W

is the sum of a strongly convex potential V_0+ρ⋆ W (with a lower bound on the curvature independent from ρ) and a bounded potential V_1 (independent from ρ), so that (<ref>) follows from classical Bakry-Emery and Holley-Stroock results <cit.>. One motivation of the present work is to understand the difference between the uniform classical log-Sobolev inequality (<ref>) and the  non-linear log-Sobolev inequality (<ref>).



 The well-posedness for (<ref>) for specific cases is a standard question that we do not address here, we refer the interested reader to <cit.> for general considerations on Wasserstein gradient flows.  
 
 
 
 [well-posedness and regularity of (<ref>)]
 For all ρ_0∈𝒫_2(^d), (<ref>) has a unique strong solution, continuous in time for 𝒲_2, which is the gradient flow of ℱ in the sense of <cit.>; for all t>0, ρ_t has a continuous positive density, ℱ(ρ_t) and ℋ(ρ_t|Γ(ρ_t)) are finite; for almost all times t>0, ℐ(ρ_t|Γ(ρ_t)) is finite and (<ref>) holds. 
 
 
The fact that the free energy, local relative entropy and Fisher information become finite instantaneously will be stated in more quantitative ways along the study. In particular, the results stated in Section <ref>, combined with the next statement   from <cit.>, show  that Assumption <ref> is met in particular in the granular media case (<ref>) in the settings of Proposition <ref>.



In the case (<ref>) with ∇ ^2 V ≥λ I_d for some λ∈, W(x,y) = w(x-y) with an even and convex function w : ^d → [0,∞) which has the doubling property w(x+y) ≤ c_w(1+w(x)+w(y)), the following holds. For any initial condition ρ_0 ∈𝒫_2(^d), there is a unique distributional solution to (<ref>), which is the gradient flow of ℱ and  possesses the following properties:

    
  * for any t > 0, ρ_t has a density with respect to the Lebesgue measure on ^d;
    
  * ρ_t →ρ_0 when t → 0 in 𝒫_2(^d);
    
  * ρ_·∈ L^1_loc((0,∞),W^1,1_loc(^d)); 
  * for any 0 < s < t < ∞,
    
    ℱ(ρ_s) = ℱ(ρ_t) + σ^4 ∫_s^t ℐ(ρ_r|Γ(ρ_r))  r < ∞.




With the present conditions on V and W the assumptions of <cit.> are satisfied, so that <cit.> applies, which gives all the points of Proposition <ref>.









From now on, Assumptions <ref>, <ref>, <ref> and <ref> are systematically enforced.



 §.§ Two known cases with global convergence



To further motivate our study, let us now highlight that it is known that (<ref>) is in fact sufficient to conclude (concerning the long-time behavior of the equation) in two cases: when ℰ is functional-convex, or when the interaction is sufficiently small (two cases which, of course, do not allow for multiple stationary solutions).



We start with functional-convexity (to be distinguished from displacement-convexity), which by definition means that

    ℰ(tμ_0 + (1-t) μ_1) ⩽ t ℰ(μ_0) +(1-t)ℰ(μ_1)     t∈[0,1] ,

for all  μ_0,μ_1 ∈𝒫_2(^d). As stated in e.g. <cit.>, assuming furthermore that ℱ admits a minimizer ρ_*∈𝒦 then the latter is unique (thanks to the strict convexity of the entropy), and for all ρ∈𝒫_2(^d), the following entropy sandwich inequalities hold:

    σ^2 ℋ(ρ|ρ_*) ⩽ℱ(ρ) - ℱ(ρ_*) ⩽σ^2ℋρ|Γ(ρ) ,

see also Lemma <ref> below.  In particular, the second inequality, together with  (<ref>), implies (<ref>) with = η/σ^2, hence the exponential decay of ℱ(ρ_t) to its global minimum which, by the first inequality of (<ref>), gives the exponential convergence of ℋ(ρ_t|ρ_*) to zero (which in turns implies the exponential convergence of ρ_t to ρ_* in total variation and, because ρ_* satisfies a classical LSI thanks to (<ref>) and thus a Talagrand inequality  <cit.>, in Wasserstein 2 distance). 

The convexity condition (<ref>) is known to hold in various settings, in particular for mean-field models of one-layer neuron networks <cit.>. In the granular media case (<ref>), assume that the interaction potential is of the form

    W(x,y) = W_0(x) + W_0(y) +2∑_k∈ r_k(x)r_k(y) -2 ∑_k∈ a_k(x)a_k(y)

for some   functions W_0,a_k,r_k (as in e.g. the quadratic case (<ref>); the letters a and r refers to attractive and repulsive by analogy with the quadratic case). Then we see that, for ℰ(ρ)= ∫ (V + 1/2ρ⋆ W)ρ and t∈[0,1],

    ℰ(tμ_0 + (1-t) μ_1) -  t ℰ(μ_0) - (1-t)ℰ(μ_1) 
    
    = t(1-t) ∑_k∈∫_^d a_k (μ_0-μ_1)^2  - ∫_^d r_k (μ_0-μ_1)^2 ,

so that (<ref>)  holds for all μ_0,μ_1,t  if and only if a_k=0 for all k∈, which corresponds to repulsive interaction (and for quadratic potentials (<ref>) to the case where θ⩽ 0, i.e. W is concave). See also <cit.> for further examples satisfying (<ref>).

In the general non-functional-convex cases, assuming that ρ_*∈𝒦 is a global minimizer of ℱ,   notice that the first inequality in (<ref>)  cannot hold if there is another global minimizer ρ'∈𝒦 (since we would have ℋ(ρ'|ρ_*)>0 while ℱ(ρ')=ℱ(ρ_*)) and the second inequality (which is the one we used to get the global non-linear LSI (<ref>) from the uniform classical LSI (<ref>)) cannot hold as soon as there is a critical point of ℱ which is not a global minimizer (since ℋ(ρ|Γ(ρ))=0 when ρ∈𝒦). 



Second, consider the non-convex but small Lipschitz interaction settings. We assume that  μ↦ E_μ is uniformly Lipschitz, namely there exists L>0 such that

    ∀ρ,ρ'∈𝒫_2(^d) ,     E_ρ -  E_ρ'_∞⩽ L 𝒲_2(ρ,ρ') .Lip
  
 We still assume   (<ref>), which by <cit.> implies  the uniform T_2 Talagrand inequality

    ∀ν,ρ∈𝒫_2(^d) ,    𝒲_2^2ν, Γ(ρ)⩽ 4 ηℋν|Γ(ρ) .

 Let ρ_* ∈𝒦.  A classical computation shows that

    ∂_t ℋ(ρ_t |ρ_*) = -σ^2 ℐ(ρ_t|ρ_*) + ∫_^d E_ρ_*-  E_ρ_t·lnρ_t/ρ_*ρ_t .

 Using the Cauchy-Schwarz, log-Sobolev  and Talagrand inequalities,

    ∂_t ℋ(ρ_t |ρ_*)    ⩽    -σ^2 ℐ(ρ_t|ρ_*) +  E_ρ_*-  E_ρ_t_∞√(ℐ(ρ_t|ρ_*))
       ⩽     -σ^2/2ℐ(ρ_t|ρ_*) + L^2/2σ^2𝒲_2^2(ρ_t,ρ_*)
       ⩽    -σ^2/2η + 2 L^2η/σ^2ℋ(ρ_t|ρ_*) ,

 In particular, as soon as Lη <2 σ^2, we get an exponential convergence in relative entropy (hence total variation and 𝒲_2) of ρ_t to ρ_*, and in particular uniqueness of the critical point ρ_*.
 






 §.§ Lower-bounded curvature



To revisit and generalize the two previous cases, let us consider the case where there exists a cost functional 𝒞:𝒫_2(^d)×𝒫_2(^d)→ (-∞,∞] with 𝒞(μ_0,μ_1)=𝒞(μ_1,μ_0) and such that 

    ∀μ_0,μ_1∈𝒫_2(^d),t∈[0,1],
    ℰ(tμ_0 + (1-t) μ_1) ⩽ t ℰ(μ_0) +(1-t)ℰ(μ_1) + t(1-t)𝒞(μ_0,μ_1)  .    𝒞-curv-ℰ

Functional-convexity corresponds to 𝒞=0. If 𝒞 = λ𝒲_2^2 for some λ∈, namely if

    ∀μ_0,μ_1∈𝒫_2(^d),t∈[0,1],
    ℰ(tμ_0 + (1-t) μ_1) ⩽ t ℰ(μ_0) +(1-t)ℰ(μ_1) + λ t(1-t)𝒲_2^2(μ_0,μ_1)  ,    𝒲_2^2-curv-ℰ

 it  can be interpreted in terms of Otto calculus by saying that the Hessian of ℰ is lower-bounded by -λ. In the case (<ref>) a bound (<ref>) follows from (<ref>), and if a_k is L_k-Lipschitz for all k∈ with λ=∑_k∈ L_k^2 <∞, (<ref>) holds (in fact, a stronger inequality holds, with 𝒲_2 replaced by 𝒲_1, which could be of interest in some cases; in this work we use 𝒲_2 everywhere for simplicity).


Consider (in dimension one for simplicity) the Gaussian kernel interaction potential

    W(x,y) = e^-(x-y)^2  = ∑_k∈(-1)^k e^-x^2x^k y^k/k! e^-y^2 ,

which appears for instance in the Adaptive Biasing Potential method <cit.> and more generally in regularized approximations of processes which are influenced by the local density of particles (e.g. <cit.>). It is of the form (<ref>) with a_k(x)=e^-x^2 x^k / √(k!) for odd k∈ and r_k(x)=e^-x^2 x^k / √(k!) for even k∈. An elementary analysis and the Stirling formula shows that a_k'_∞ = 𝒪(k/2^k/2), which means that (<ref>) holds. 
 


Considering the settings of <cit.>, let  be a Hilbert space with norm ·_, φ:^d→, R:→ and V_0:^d→. Consider 

    ℰ(μ) = ∫_^d V_0(x) μ( x) +  R∫_^dφ(x) μ( x) ,

where the integral of an -valued function is understood in Bochner's sense, and we take the convention that ℰ(μ)=∞ if ∫_^dφ(x)_μ( x)=∞. As discussed in <cit.>, this encompasses many cases of interest in optimization, machine learning and stastistics in high dimension. Assume that the curvature of R is lower-bounded in the sense that there exists θ>0 such that R+θ·_^2 is convex. This  implies  that (<ref>) holds with 

    𝒞(μ_0,μ_1)  = θ∫_^dφμ_0 - ∫_^dφμ_1^2_ .

This generalizes the quadratic finite-dimensional case (<ref>). Assuming furthemore that φ is Lipschitz continuous (namely φ(x)-φ(y)_⩽ℓ |x-y| for all x,y∈^d), we get (<ref>) with λ = θℓ^2. 

 
 
  The following generalizes the entropy sandwich inequality (<ref>) when 𝒞≠ 0.
 
 
 
 Under (<ref>), for all μ_0,μ_1∈𝒫_2(^d) and ρ_*∈𝒦,
 
    ℱ(μ_0)   ⩽ℱ(μ_1) +  σ^2 ℋμ_0|Γ(μ_0)+   𝒞(μ_0,μ_1) ,

 and
 
    ℱ(ρ_*) +  σ^2 ℋμ_0|ρ_*⩽ℱ(μ_0)  + 𝒞(μ_0,ρ_*)     .

 
  To prove each inequality, we assume that the right-hand side is finite, otherwise the result is trivial.
 
   Dividing by t and sending t to zero in (<ref>) yields (using (<ref>)),

    ∫_^d E_μ_1 (μ_0-μ_1) + ℰ(μ_1) ⩽ℰ(μ_0)  +     𝒞(μ_0,μ_1) ,

and thus, interverting the roles of μ_1 and μ_0,

    ∫_^d E_μ_1 (μ_0-μ_1) -     𝒞(μ_0,μ_1) ⩽ℰ(μ_0) -  ℰ(μ_1) ⩽∫_^d E_μ_0 (μ_0-μ_1) +     𝒞(μ_0,μ_1) .

Then

    ℱ(μ_0) -  ℱ(μ_1)    ⩽   ∫_^d E_μ_0 (μ_0-μ_1) +     𝒞(μ_0,μ_1) + σ^2 ℋ  (μ_0) - σ^2 ℋ(μ_1)
        =    σ^2 ℋμ_0|Γ(μ_0) - σ^2 ℋμ_1|Γ(μ_0) +   𝒞(μ_0,μ_1) 
       ⩽   σ^2 ℋμ_0|Γ(μ_0)+   𝒞(μ_0,μ_1) ,

and

    ℱ(μ_0) -  ℱ(μ_1)    ⩾   ∫_^d E_μ_1 (μ_0-μ_1) -     𝒞(μ_0,μ_1) + σ^2 ℋ  (μ_0) - σ^2 ℋ(μ_1)
        =    σ^2 ℋμ_0|Γ(μ_1) - σ^2 ℋμ_1|Γ(μ_1) -  𝒞(μ_0,μ_1)  .

The second term of the right hand side vanishes if μ_1∈𝒦, which concludes.
 
 
This result will prove useful in our cases of interest, namely when ℱ has several critical points. Nevertheless, for now, in the spirit of the previous section, let us discuss how (<ref>) may already give a global convergence in some cases under (<ref>). 
 

Indeed, assuming a Talagrand type inequality

    𝒞(μ_0,ρ_*) ⩽η' ℋ(μ_0|ρ_*)

for some η'>0 (which, if 𝒞⩽λ𝒲_2^2 for some λ>0, is implied by  (<ref>) with η'=4ηλ), using (<ref>) and then (<ref>),

    𝒞(μ_0,ρ_*)  ⩽η' ℋ(μ_0|ρ_*)    ⩽   η'/σ^2ℱ(μ_0) -  ℱ(ρ_*) +   𝒞(μ_0,ρ_*) 
       ⩽   η' ℋμ_0|Γ(μ_0) +  2η'/σ^2𝒞(μ_0,ρ_*)  .

Hence, under the condition 2η ' < σ^2 (i.e., again, high temperature or small interaction), we obtain a non-linear transport inequality

    𝒞(μ_0,ρ_*) ⩽η' /1-2η '/σ^2 ℋμ_0|Γ(μ_0) .

Together with the LSI for ρ_* and (<ref>), this proves (<ref>), since

    ℱ(μ_0) -  ℱ(ρ_*) ⩽σ^2 + η ' /1-2η '/σ^2 ℋμ_0|Γ(μ_0)⩽ησ^2 + η ' /1-2η' /σ^2 ℐμ_0|Γ(μ_0) .

Notice that, conversely, the non-linear transport inequality (<ref>) with 𝒞=𝒲_2^2 is shown to be a consequence of (<ref>) in <cit.> (see also Lemma <ref> below).



 §.§ Local non-linear log-Sobolev inequalities


Consider a non-empty set 𝒜⊂𝒫_2(^d) with  inf_𝒜ℱ<∞. Instead of the global condition (<ref>), the key ingredient in our approach is to obtain (local) non-linear LSI of the form

    ∀ρ∈𝒜 ,    ℱ( ρ) - inf_μ∈𝒜ℱ(μ)  ⩽σ^4  ℐρ|Γ(ρ)NL-LSI

for some >0. The fact that it is possible to get this in some subsets 𝒜 even in cases where ℱ has several critical points is the topic of Section <ref>. For now we assume that this holds in some   set 𝒜 and we discuss its consequences.

First, since the left hand side of  (<ref>) is non-negative and the  right hand side vanishes when ρ∈𝒦, we get that

    ρ_* ∈𝒜∩𝒦    ⇒    ℱ(ρ_*) = inf_μ∈𝒜ℱ(μ)  .

Second, the results which were known to hold under (<ref>) still hold as long as the solution of (<ref>) stays within 𝒜, as we state in the next lemma. For conciseness we write

    ℱ_𝒜(ρ) = ℱ( ρ) - inf_μ∈𝒜ℱ(μ) .

For ρ_0∈𝒫_2(^d), we write

    T_𝒜(ρ_0) = inf{t⩾ 0, ρ_t∉𝒜} ,

with (ρ_t)_t⩾ 0 the solution to (<ref>). The next lemma is the infinite-dimensional version of (<ref>) and (<ref>) and follows the same proof.


Assume that (<ref>) holds on 𝒜. Then, for all ρ_0∈𝒜 and t⩽ T_𝒜(ρ_0),

    ℱ_𝒜(ρ_t) ⩽ e^-t/ℱ_𝒜(ρ_0)

and

    𝒲_2^2(ρ_t,ρ_0) ⩽ 4 ℱ_𝒜(ρ_0) .



The first inequality is straightforward from (<ref>) and (<ref>). The proof of the second inequality is the same as the proof of <cit.> (although a factor 2 seems to disappear wrongly in the latter when using the Benamou-Brenier formula, which explains the difference with our result ; notice that our factor 4 is consistent with the classical linear case <cit.>, or the finite-dimensional settings (<ref>)). Since (<ref>) can be written as the continuity equation

    ∂_t ρ_t = ∇·ρ_t v_t     v_t := δℱ(ρ_t) /δμ ,

by the Benamou-Brenier formulation of the Wasserstein distance <cit.>, for any 0 ≤ s < t,

    𝒲_2^2(ρ_s,ρ_t) ≤ (t-s) ∫_s^t ∫_^d |v_r|^2 ρ_r  r = σ^4(t-s)∫_s^t ℐ(ρ_r|Γ(ρ_r)) r.

Thus,

    𝒲_2(ρ_s,ρ_t)/t-s≤σ^2 √(1/t-s∫_s^t ℐ(ρ_r|Γ(ρ_r)) r),

so by Lebesgue's differentiation theorem, the metric derivative of (ρ_t)_t ≥ 0 defined in <cit.> satisfies

    |ρ'_t| ≤σ^2 √(ℐ(ρ_t|Γ(ρ_t))),

t-almost everywhere. Since, by <cit.>, we have on the other hand

    𝒲_2(ρ_s,ρ_t) ≤∫_s^t |ρ'_r|  r,

we get

    𝒲_2(ρ_0,ρ_t)  ⩽σ^2 ∫_0^t √(ℐρ_s|Γ(ρ_s)) s .
 
Besides, for t⩽ T_𝒜(ρ_0), 

    ∂_t ℱ_𝒜(ρ_t)  =  - σ^4 ℐρ_t|Γ(ρ_t)
    ⩽  -  √(σ^ 4 ℐρ_t|Γ(ρ_t)ℱ_𝒜(ρ_t) /) ,

 from which

    √(ℱ_𝒜(ρ_t)) - √(ℱ_𝒜(ρ_0))⩽  -σ^2/2√()∫_0^t √(ℐρ_s|Γ(ρ_s)) s   ⩽  -1/2√()𝒲_2(ρ_t,ρ_0) .

  Rearranging the terms and using that √(ℱ_𝒜(ρ_t))⩾ 0 concludes.


As a consequence, when  (<ref>) holds in a set 𝒜⊂𝒫_2(^d) that contains a unique critical point ρ_*∈𝒦, in order to get the long-time convergence to ρ_* at exponential speed starting from an initial condition ρ_0∈𝒜,  it mainly remains to show that T_𝒜(ρ_0)=+∞. In specific cases, this can be shown by various means, for instance in dimension 1 by monotonicity arguments as in <cit.>, or in general using that the free energy decreases as in <cit.>, etc. In the following, we focus on a self-contained argument which establishes the exponential convergence towards ρ_* under the natural condition that 𝒲_2(ρ_*,ρ_0) is sufficiently small (in particular, without assuming that ℱ(ρ_0)<∞).


 

 §.§ Auxiliary regularization results

 
Let us recall two results. The first is proven in <cit.>.


Under (<ref>) and (<ref>), for all ρ∈𝒫_2(^d) and ρ_*∈𝒦, 

    ℋρ |Γ(ρ)⩽ 1+ η L + L^2 η^2/2ℋ(ρ|ρ_*) .
 




The second result is from <cit.>.

Assume that there exists κ_0,κ_1,κ_2>0 such that for all ν,μ∈𝒫_2(^d) and all x,y∈^d,








    | E_μ(0)|^2 + σ^2  ⩽κ_0 1+ ∫_^d |x|^2 μ( x)

and

    -2 E_μ(x) -  E_ν(y) · (x-y) ⩽κ_1 |x-y|^2 + κ_2 |x-y|𝒲_2(ν,μ) .

Then, given two solutions (ρ_t)_t⩾ 0 and (ρ_t')_t⩾ 0 of (<ref>), for all t>0,

    ℋρ_t|ρ_t'⩽ s_t𝒲_2^2 (ρ_0,ρ_0') ,

where

    s_t=1/σ^2 κ_1/1-e^-κ_1 t + 1/2 tκ_2^2 e^2(κ_1+κ_2)t .


Notice that, under the uniform Lipschitz condition (<ref>), since we can bound | E_μ(0)|⩽ | E_δ_0(0)| + L 𝒲_2(μ,δ_0) for any μ∈𝒫_2(^d), the assumptions of Proposition <ref> are fulfilled if additionally there exists κ_1>0 such that  the one-sided Lipschitz condition

    ∀μ∈𝒫_2(^d),x,y∈^d,     2 E_μ(x) -  E_μ(y) · (x-y) ⩾  - κ_1 |x-y|^2o-s-Lip

holds, and then we can take κ_2=2 L to have (<ref>).




Assume (<ref>), (<ref>), (<ref>) and (<ref>) for some λ, L,κ_1,η>0. Then, for all (ρ_t)_t⩾ 0 solution to (<ref>), all ρ_*∈𝒦 and all t>0,

    ℱ(ρ_t) - ℱ(ρ_*) ⩽ q_t 𝒲_2^2(ρ_0,ρ_*)

with 

    q_t =   1+ η L+ 4 ηλ/σ^2 + L^2 η^2/2κ_1/1-e^-κ_1 t + 1/2 tL^2 e^(2κ_1+4L)t .


Notice that q_t is of order 1/t as t→ 0. See <cit.> for a similar result.


Using (<ref>) in Lemma <ref> with 𝒞 = λ𝒲_2^2 and then Lemma <ref> and the Talagrand inequality (<ref>) for ρ_* = Γ(ρ_*) implied by (<ref>),

    ℱ(ρ_t) -  ℱ(ρ_*)    ⩽   σ^2 ℋρ_t|Γ(ρ_t)+   λ𝒲_2^2(ρ_t,ρ_*) 
       ⩽   σ^2   1+ η L+ 4 ηλ/σ^2 + L^2 η^2/2ℋ(ρ_t|ρ_*) .
 
Since t↦ρ_* solves (<ref>), conclusion follows from Proposition <ref>.





 §.§ Conclusion


For μ∈𝒫_2(^d) and r>0, write ℬ_𝒲_2(μ,r) = {ν∈𝒫_2(^d),𝒲_2(ν,μ)⩽ r}.


The conditions (<ref>), (<ref>), (<ref>) and (<ref>) hold for some λ, L,κ_1,η>0, and (<ref>) holds on some non-empty set  𝒜⊂𝒫_2(^d) for some >0. Furthermore  there exist ρ_* ∈𝒜∩𝒦 and δ>0 such that ℬ_𝒲_2(ρ_*,δ) ⊂𝒜.



Under Assumption <ref>, set 

    δ' =  δ/2( 2 √( q_1) + e^(κ_1/2+L)) ,

where q_1 is given by (<ref>) (with t=1). Then, for all ρ_0 ∈ℬ_𝒲_2(ρ_*,δ'), T_𝒜(ρ_0)=∞ and  in particular

    ℱ_𝒜(ρ_t)    ⩽   e^-t/ℱ_𝒜(ρ_0)

for all t⩾ 0. Furthermore, if we assume additionally that  ℬ_𝒲_2(ρ_*,δ)∩𝒦={ρ_*}, then:


  * The following non-linear Talagrand inequality holds:

    𝒲_2^2(ρ_0,ρ_*) ⩽ 4 ℱ_𝒜(ρ_0) .


  * Setting C= e^1/max(  4  q_1,e^κ_1+2L ), for all t⩾ 0,

    𝒲_2^2(ρ_t,ρ_*)     ⩽     C e^-t/𝒲_2^2(ρ_0,ρ_*)
    ℱ_𝒜(ρ_t)     ⩽   e^-t/minℱ_𝒜(ρ_0) , q_min(t,1) e^1/𝒲_2^2(ρ_0,ρ_*)
    σ^2 ℋ(ρ_t|ρ_*)    ⩽   ℱ_𝒜(ρ_t) + λ𝒲_2^2(ρ_t,ρ_*) .





 The last inequality on the relative entropy is just (<ref>) in Lemma <ref>, we simply recall it as it gives here the exponential convergence of ℋ(ρ_t|ρ_*) (hence of ρ_t-ρ_*_TV) to zero.
 
 
 Write 𝒜_∞(ρ_*) = {ρ_0∈𝒜: T_A(ρ_0)=∞ and 𝒲_2(ρ_t,ρ_*)→ 0  as t→∞}. Then it is clear by following the proof of Theorem <ref> that its conclusion (namely (<ref>), (<ref>) and (<ref>)) hold for any initial condition ρ_0∈𝒜_∞(ρ_*). Moreover, for any ρ_0 ∈𝒫_2(^d) such that 𝒲_2(ρ_t,ρ_*)→ 0 as t→∞, there exists a t_*>0 such that ρ_t_*∈ℬ_𝒲_2(ρ_*,δ) ⊂𝒜_∞(ρ_*), from which there exists C_0>0 such that for all t⩾ 1,
 
    𝒲_2^2(ρ_t,ρ_*) + ℋ(ρ_t|ρ_*) + ℱ(ρ_t) - ℱ(ρ_*) ⩽ C_0 e^-t/ .

 



Given a solution (ρ_t)_t⩾ 0 of (<ref>), we consider the time-inhomogeneous Markov diffusion process 

    X_t = - E_ρ_t (X_t)  t +√(2)σ B_t

where B is a Brownian motion  and X_0 is distributed according to ρ_0, so that X_t ∼ρ_t for all t⩾ 0. Existence of strong solutions for this SDE is justified by the fact (t,x)↦ E_ρ_t is continuous in time (by continuity of t↦ρ_t and (<ref>)) and 𝒞^1 in x, and explosion in finite time is prevented by (<ref>).  Considering a synchronous coupling (X,Y) (i.e. using the same Brownian motion for two processes) of such diffusions, the first one being associated with an arbitrary solution ρ, the second being associated to the stationary solution t↦ρ_*, we get that

    ∂_t |X_t-Y_t|^2 = -2 (X_t-Y_t) · E_ρ_t (X_t) -  E_ρ_* (Y_t)⩽ (κ_1+L) |X_t-Y_t|^2 + L 𝒲_2^2ρ_t,ρ_* ,

where we used (<ref>) and (<ref>). Taking the expectation, using the Gronwall Lemma, that 𝒲_2^2(ρ_t,ρ_*) ⩽𝔼|X_t-Y_t|^2 and taking the infimum over the coupling of the initial conditions, we obtain

    𝒲_2(ρ_t, ρ_*) ⩽ e^(κ_1/2+L)t𝒲_2(ρ_0, ρ_*) .

In view of the definition of δ', this implies that for ρ_0 ∈ℬ_𝒲_2(ρ_*,δ'), T_𝒜(ρ_0)⩾ 1 and ρ_1∈𝒜. On the other hand, using Corollary <ref> and the decay of the free energy  along time, for all t⩾ 1,

    ℱ(ρ_t) - ℱ(ρ_*) ⩽ q_1 (δ')^2 .

Hence, for all t ∈ [1,T_𝒜(ρ_0)],  using (<ref>), 

    𝒲_2(ρ_t,ρ_*) ⩽𝒲_2(ρ_t,ρ_1) + 𝒲_2(ρ_1,ρ_*) ⩽δ'   2 √( q_1) + e^(κ_1/2+L)  = δ/2 .

By contradiction, since t↦𝒲_2(ρ_t,ρ_*) is continuous, we get that T_𝒜(ρ_0)=∞.

Now, assume that ℬ_𝒲_2(ρ_*,δ)∩𝒦={ρ_*}. Since Equation (<ref>) admits ℱ as a strict Lyapunov function, the LaSalle
invariance principle established in <cit.> shows that the 𝒲_2 distance between ρ_t and 𝒦 vanishes as t→∞. Since the trajectory stays within ℬ_𝒲_2(ρ_*,δ/2) and thus at a distance at least δ/2 of 𝒦∖{ρ_*}, this implies the convergence towards ρ_*. We can then let t→∞ in (<ref>) to get (<ref>), that we use then to bound, for t⩾ 1,

    𝒲_2^2(ρ_t,ρ_*) ⩽ 4 ℱ_𝒜(ρ_t) ⩽ 4  e^-(t-1)/ℱ_𝒜(ρ_1)  ⩽ 4  e^-(t-1)/ q_1𝒲_2^2(ρ_t,ρ_*) .
 
For t⩽ 1, we simply use (<ref>). The bound on ℱ_𝒜(ρ_t) is straightforward from (<ref>) and Corollary <ref>.




§ LOCAL NON-LINEAR LSI IN A PARAMETRIC CASE


In this section we focus on the settings of Example <ref>, where we recall that

    ℰ(μ) = ∫_^d V_0(x) μ( x) +  R∫_^dφ(x) μ( x).

We assume that V_0∈𝒞^2(^d,), R∈𝒞^1(,) and φ∈𝒞^2(^d, ). In this case, denoting by _  and ⟨·,·⟩_ the gradient and scalar product over ,

    E_ρ(x) = V_0(x) + ⟨∇_ R(φ(ρ)), φ(x)⟩_,

with φ(ρ):=∫_^dφρ∈,
and Assumption <ref> is satisfied if, for any x ∈^d, φ(x)_≤ C(1+|x|^2).

The general idea that we develop in this section is that, since the non-linearity is somehow parametrized by φ(ρ), information for the Wasserstein gradient flow can be deduced from standard analysis of fixed point or optimization on Hilbert spaces. General considerations are gathered in Section <ref>, which are then applied to the granular media case (<ref>) in Section <ref>.


If R is convex then ℰ is functional convex. As discussed in Section <ref>, thanks to Lemma <ref>, in that case (<ref>) implies (<ref>) and then global convergence at constant rate towards a unique stationary solution.  Hence, this is not the case we are interested in.




 §.§ General results



  §.§.§ The associated fixed-point problem


In the case (<ref>), we get that Γ(ρ) = ρ_φ(ρ) where, for ψ∈,

    ρ_ψ(x) ∝exp -1/σ^2 V_0(x) + ⟨_ R(ψ), φ(x)⟩_ .

Let us assume that ℱ(ρ_ψ)<∞ for all ψ∈. This implies in particular that 
    f(ψ) = φρ_ψ = ∫_^dφ(x) ρ_ψ(x) x

is well-defined, and we see that φΓ(ρ)  =fφ(ρ). In particular, any ρ_*∈𝒦 satisfies ρ_* = ρ_φ(ρ_*). Thus, denoting by 𝒦' = {ψ∈,  f(ψ)=ψ} the set of fixed-points of f, 

    𝒦 = {ρ_ψ, ψ∈𝒦'} .

One may hope that the stability properties of ρ_* = ρ_ψ_*∈𝒦 as a stationary solution of the (continuous-time) Wasserstein gradient descent is related to the stability of ψ_* as a fixed-point of the (discrete-time) dynamical system ψ↦ f(ψ) on , which is a   classical question. In fact, we can find examples where ρ_ψ_* is stable although ψ_* is not (see the granular media equation with repulsive interaction in Figure <ref> below). However, on the contrary, the next result shows that (<ref>) holds in a neighborhood ρ_ψ_* (which is thus stable thanks to Theorem <ref>) if ψ_* is   a geometrically attracting fixed-point.


Let ψ_*∈𝒦', 𝒜' ⊂ and α∈[0,1) be such that for all ψ∈𝒜',

    f(ψ)-ψ_*_⩽αψ - ψ_*_ .

Then, for any ψ∈𝒜', 

    ψ-ψ_*_⩽1/1-αf(ψ)-ψ_ .

Assuming moreover that φ is ℓ-Lipschitz continuous, then, for all ρ∈𝒫_2(^d) such that φ(ρ)∈𝒜',

    φ(ρ)-ψ_*_⩽ℓ/1-α𝒲_2ρ,Γ(ρ) .

Finally, assuming furthermore (<ref>) and that R+θ·_^2 is convex for some θ>0, then, for all ρ∈𝒫_2(^d) such that φ(ρ)∈𝒜',

    ℱ(ρ) - ℱ(ρ_ψ_*) ⩽ησ^2   +  4ηθℓ^2/(1-α)^2ℐρ|Γ(ρ) ,

i.e. (<ref>) holds over 𝒜={ρ∈𝒫_2, φ(ρ)∈𝒜'}.



The first inequality (<ref>)  simply follows from the triangular inequality

    ψ-ψ_*_⩽ψ-f(ψ)_ + f(ψ)-ψ_*_⩽ψ-f(ψ)_ + αψ-ψ_*_ .

Applying this to ψ=φ(ρ) reads

    φ(ρ)-ψ_*_⩽1/1-αφ(ρ)-φΓ(ρ)_⩽ℓ/1-α𝒲_2ρ,Γ(ρ) .

Finally, applying Lemma <ref> (with 𝒞(μ_0,μ_1)= θφ(μ_0)-φ(μ_1)_^2 as in Example <ref>),

    ℱ(ρ) - ℱ(ρ_ψ_*)    ⩽   σ^2 ℋρ|Γ(ρ)  + θφ(ρ)-ψ_*_^2
       ⩽   σ^2 ℋρ|Γ(ρ)  +  θℓ^2/(1-α)^2𝒲_2^2ρ,Γ(ρ)
       ⩽   σ^2   +  4ηθℓ^2/(1-α)^2ℋρ|Γ(ρ)

thanks to Talagrand inequality (<ref>). Conclusion follows from (<ref>).



The constant  in  (<ref>) obtained 
when applying Proposition <ref>, and thus the quantitative convergence estimates obtained when applying Theorem <ref>, are explicit in terms of the constants in the assumptions. In particular there is no additionnal dependency in the dimension.


We have thus identified the following general conditions:



There exist θ,ℓ,η>0 such that φ is ℓ-Lipschitz continuous, R+θ·_^2 is convex and (<ref>) holds. Moreover, ℱ(ρ_ψ) < ∞ for any ψ∈.



As discussed in Example <ref>, these conditions imply (<ref>) with λ=θℓ^2. Moreover, since 

    E_ρ(x) =  V_0(x) + ⟨_ Rφ(ρ), φ(x)⟩_ ,

assuming that _ R is L'-Lipschitz continuous, we get that, for any x∈^d,

    | E_μ_0(x)  -  E_μ_1(x) |⩽ℓ L' φ(μ_0)-φ(μ_1)_⩽ℓ^2 L ' 𝒲_2(μ_0,μ_1) ,

which is (<ref>).



Under the general conditions of Assumption <ref>, given a specific problem, what remains to be done is  to establish the contraction (<ref>). A simple way to get it locally at some fixed point ψ_* is to check that |_ f(ψ_*)|, the operator norm of the Jacobian matrix of f, is strictly less than 1 at ψ_*. We end up with the following.



Under Assumption <ref>, let ψ_*∈𝒦'. Assume that f is differentiable at ψ_* with |_ f(ψ_*)|<1. Then there exist δ,>0 such that (<ref>) holds on 𝒜= ℬ_𝒲_2(ρ_ψ_*,δ).


Using that ψ_* is a fixed point of f and a Taylor expansion,

    f(ψ)- ψ_*_  = _ f(ψ_*) (ψ-ψ_*)_ + o(ψ-ψ_*_) ⩽αψ-ψ_*_

for some α<1 uniformly over some neighborhood 𝒩 of ψ_*. The function φ being Lipschitz continuous, ρ↦φ(ρ) is Lipschitz continuous for the 𝒲_2 distance, which means that φ(ρ) lies in 𝒩 for all ρ∈ℬ_𝒲_2(ρ_ψ_*,δ) for δ small enough. Conclusion follows from Proposition <ref>.
 



  §.§.§ Decomposition of the free energy


The difference between (<ref>) and (<ref>) obviously lies in the difference between ℋ(ρ|Γ(ρ)) and ℱ(ρ). In the case (<ref>), we can decompose

    ℱ(ρ) = σ^2  ℋρ|Γ(ρ) + gφ(ρ)

where

    g(ψ) =  Rψ - ⟨_ Rψ , ψ⟩_ - σ^2 ln∫_^d e^- 1/σ^2  V_0(x) +  ⟨_ Rψ , φ(x) ⟩_ x .

This second part of the free energy only depends on ρ through the parameter φ(ρ). 



If R is concave, then M(ψ_1,ψ_2):= R(ψ_2)-R(ψ_1) + ⟨_ R(ψ_2),ψ_1-ψ_2⟩_⩾ 0 (this is the so-called Bregman divergence associated with -R). Assuming furthermore that V_0(x) = V(x) - R(φ(x)) with V such that e^-V/σ^2 is integrable, and that M(ψ_1,ψ_2) → +∞ when ψ_2→∞, we get by dominated convergence that

    e^-1/σ^2 g(ψ) =    ∫_^d e^- 1/σ^2  V(x) + M φ(x), ψ x  ψ_→ +∞⟶  0 .

As a consequence, g is lower bounded and goes to infinity at infinity. Moreover, it is continuous, and thus in particular when  has finite dimension then g reaches its minimum. Notice that, in the quadratic case where R(ψ)=-θψ_^2, M(ψ_1,ψ_2)=θψ_1-ψ_2_^2.


Assuming that R∈𝒞^2( ,),

    _ g(ψ) =  _^2 R(ψ)   f(ψ) - ψ ,

where we used that

    ∫_^d_^2 R(ψ)  φ e^-1/σ^2[V_0(x) + ⟨_ Rψ,   φ(x)⟩_] x/∫_^d e^-1/σ^2[V_0(x) + ⟨_ Rψ,   φ(x)⟩_] x = _^2 R(ψ)   φρ_ψ =  _^2 R(ψ)   f(ψ) .

As a consequence, if ψ_* is a critical point of g such that _^2 R(ψ_*) is non-singular (which is in particular the case for all critical points of g if R is strictly convex or concave, which covers many cases of interest) then necessarily ψ_* ∈𝒦'. In that case, if moreover ψ_* is a local minimizer of g, then ρ_ψ_* is a minimizer of both parts of the free energy, ρ↦ℋ(ρ|Γ(ρ)) and ρ↦ gφ(ρ) (at least locally in the latter). It is thus a good candidate to be a stable stationary solution for the Wasserstein gradient descent for the free energy. This is indeed the case:


Assume (<ref>), (<ref>), (<ref>) and (<ref>) for some λ, L,κ_1,η>0   and φ is ℓ-Lipschitz continuous. 
Let ψ_*∈𝒦' be a  proper isolated local minimizer of g, in the sense that for all ε>0 small enough, ζ(ε):=inf{g(ψ)-g(ψ_*), ψ∈, ψ-ψ_*_=ε}>0 and 𝒦' ∩ℬ(ψ_*,ε) = {ψ_*}. Then there exists δ>0 such that for all initial conditions ρ_0 ∈ℬ_𝒲_2(ρ_ψ_*,δ), the flow (<ref>) converges in long-time to ρ_ψ_*.




Let ε>0 be small enough so that ζ(ε)> 0 and  𝒦' ∩ℬ(ψ_*,ε) = {ψ_*}.  Notice that, for ψ,ψ̃∈𝒦', ψ-ψ̃_ = φ(ρ_ψ)-φ(ρ_ψ̃)_≤ℓ𝒲_2(ρ_ψ,ρ_ψ̃), so that the second condition implies that 𝒦∩ℬ_𝒲_2(ρ_ψ_*,ε/ℓ) = {ρ_ψ_*}, and thus ρ_ψ_* is an isolated critical point of ℱ. For any ρ∈𝒫_2 with φ(ρ)-ψ_*_=ε, 

    ℱ(ρ) ⩾ gφ(ρ)⩾ g(ψ_*) + ζ(ε) = ℱ(ρ_ψ_*) + ζ(ε) .

As a consequence, given a solution (ρ_t)_t⩾ 0 of (<ref>), if there is a time t_0 such that φ(ρ_t_0) - ψ_*_ < ε and ℱ(ρ_t_0) - ℱ(ρ_ψ_*) < ζ(ε) then the monotonicity of the free energy along the flow together with the continuity of t↦φ(ρ_t)-ψ_*_ implies that φ(ρ_t)-ψ_*_<ε for all t⩾ t_0. Conclusion would thus follow from  LaSalle
invariance principle <cit.> using that ρ_ψ_* is isolated.

Hence, let us prove that the previous conditions are met for initial conditions that are sufficiently close to ρ_ψ_*. We follow arguments similar to the proof of Theorem <ref>. Indeed, for all t⩾ 0, using (<ref>)
,

    φ(ρ_t) - ψ_*_≤ℓ  e^(κ_1/2+L)t𝒲_2ρ_0 ,ρ_ψ_* .

By taking δ <  ε e^-κ_1/2-L /ℓ, we ensure that 
φ(ρ_t_0) - ψ_*_ <ε at time t_0=1 for all ρ_0 ∈ℬ_𝒲_2(ρ_ψ_*,δ). On the other hand, thanks to Corollary <ref>, by taking δ small enough, we can also ensure that   ℱ(ρ_t_0) - ℱ(ρ_ψ_*) < ζ(ε) with such initial conditions.

 

The argument that initial conditions that start with a free energy sufficiently close to ℱ(ρ_*) will stay within a ball centered at ρ_* has been used e.g. in <cit.>. Here we combine it with the small-time regularization result of Corollary <ref> to get a result only in terms of a 𝒲_2 ball.

 
 In fact, we can now reinterpret Proposition <ref> in light of Proposition <ref>. Indeed, differentiating again (<ref>) (assuming suitable regularity) at some ψ_*∈𝒦' yields
 
    _^2 g(ψ_*) = _^2 R(ψ_*) _ f(ψ_*) - I

 (where we used that  f(ψ_*)=ψ_*). For brevity, denote ^2R_*=_^2 R(ψ_*).  For any u∈,
 
    ⟨ u,_^2 g(ψ_*) u ⟩_ = ⟨ u , ^2 R_* _ f(ψ_*)u  ⟩_ - ⟨ u , ^2 R_* u⟩_ .

 Assuming that ^2 R_* is negative definite, we can consider the norm u_* = √(-⟨ u, ^2 R_*u⟩_) and the associated operator norm _ f(ψ_*)_*. Then, 
  
    ⟨ u, _^2 g(ψ_*) u ⟩_⩾ 1 - _ f(ψ_*)_*u_*^2  .

 We get that ψ_* is a non-degenerate local minimizer of g if _ f_* <1. On the other hand, if  _ f_* <1  and (^2 R_*)^-1 is bounded, then we can assume   that ·_* = ·_ (which amounts to the linear change of coordinates given by (^2 R_*)^-1/2) and apply Proposition <ref> or, alternatively, we can use the contraction (<ref>) expressed with norm ·_* to get  (<ref>) with the same norm and then with the initial ·_ by the equivalence of the norms  and then proceed with Proposition <ref>). At least, this suggests that in some cases, in order to apply Proposition <ref> at some point ψ_*, it may be more natural to work with the norm ·_* associated to ^2 R_* (when the latter is negative definite).
 
 
 Besides, we compute that
 
    _ f(ψ) = -1/σ^2∫_^dφ_^2 R(ψ) φ^T ρ_ψ - f(ψ) _^2 R(ψ) f(ψ) ^T ,
 
 from which, for u_1,u_2∈, writing w_i=^2 R_* u_i for i=1,2,
 
    ⟨ u_1, ^2 R_* _ f(ψ_*) u_2⟩_   =    - 1/σ^2⟨ w_1 ,  ∫_^dφφ^T ρ_ψ_* - f(ψ_*)  f(ψ_*) ^T w_2 ⟩_
       =    - 1/σ^2⟨ w_1 , covar_ρ_ψ_*φ(X)   w_2⟩_ .

  In particular, we see that _ f(ψ_*) is symmetric and nonnegative  for the 
scalar product associated to ·_*, from which

    _ f(ψ_*)_* = sup_u≠ 0⟨ u, ^2 R_* _ f(ψ_*) u⟩_/⟨ u, ^2 R_*   u⟩_ .

 
Furthermore, if we work with suitable coordinates in order to enforce that ^2 R_*=-I, we get that, for ψ_*∈𝒦',
 
    _^2 g(ψ_*) = I - 1/σ^2covar_ρ_ψ_*(φ(X))  .




  §.§.§ Degenerate minima


There is no convergence rate in Proposition <ref>, and we cannot expect one in general if we do not assume that ψ_* is a non-degenerate local minimizer of g. Assuming again that _^2 R(ψ_*) is non-singular, in view of (<ref>), if _^2 g(ψ_*) is singular then |_ f(ψ_*)| ⩾  1 (since _ f(ψ_*) u = u for   u∈ker_^2 g(ψ_*), and thus this does not depend on the norm we use). In other words, degenerate minima of g fall in the limit case where we cannot apply Proposition <ref> anymore. Then we can weaken the contraction condition (<ref>) to allow for high-order stable fixed points. The proof of the following is the same as the one of Proposition <ref> (hence omitted).


Let ψ_*∈𝒦', 𝒜' ⊂ and β>0,ν∈(0,1) be such that for all ψ∈𝒜',

    ψ-ψ_*_⩽βf(ψ)-ψ_^ν .

Assuming   furthermore (<ref>), that R+θ·_^2 is convex for some θ>0 and that  φ is ℓ-Lipschitz continuous, then, for all ρ∈𝒫_2(^d) such that φ(ρ)∈𝒜',

    ℱ(ρ) - ℱ(ρ_ψ_*) ⩽ησ^2   ℐρ|Γ(ρ) +  θβ^2 (4η^2ℓ^2 )^νℐ^νρ|Γ(ρ) .



Under the settings of Proposition <ref>, assume that (<ref>) holds in a neighborhood of ψ_*. Thanks to Proposition <ref>, there exists δ>0 such that for all initial conditions ρ_0 ∈ℬ_𝒲_2(ρ_ψ_*,δ), the gradient flow converges to ρ_* and the inequality (<ref>) holds for ρ_t for all time t⩾ 0. Combined with Corollary <ref>, this yields algebraic convergence rates of the form

    ℱ(ρ_t) - ℱ(ρ_ψ_*) ⩽C/𝒲_2^2r(ρ_0,ρ_ψ_*) + t  ^1/r ,

with r=1/ν-1>0 and some constant C>0, for all t⩾ 1. From this algebraic decay of the free energy, we can get convergence of ρ_t to ρ_ψ_* since (<ref>) in Lemma <ref> gives

    ℋρ_t|ρ_ψ_*⩽ℱ(ρ_t) - ℱ(ρ_ψ_*) + θφ(ρ_t)-ψ_*_^2 .

When following the proof of Proposition <ref> to get Proposition <ref>, we see that we  obtain as an intermediary inequality that 

    φ(ρ_t)-ψ_*_^2 ⩽β^2 (4ηℓ^2)^νℋ^νρ_t|Γ(ρ_t)
 
as long as φ(ρ_t) ∈𝒜' (hence for all times here since ρ_0 ∈ℬ_𝒲_2(ρ_ψ_*,δ)). Moreover, since ψ_* is a local minimizer of g, by taking δ sufficiently small we get that g(φ(ρ_t)) ⩾ g(ψ_*) for all t⩾ 0, in other words

    ℋ(ρ_t|Γ(ρ_t)) ⩽ℱ(ρ_t) - ℱ(ρ_ψ_*) .

Gathering all these bounds give 

    ℋρ_t|ρ_ψ_*⩽C_0/t^ν^2/(1-ν)

for some C_0>0 for all t⩾ 0.




 §.§ Application to granular media equation




In this section we focus on the granular media equation (<ref>) on ^d, which corresponds to  the free energy (<ref>) (in fact we will consider a slightly more general case in Section <ref>). For clarity, for now, let us focus on the quadratic interaction case where W(x,y)=θ|x-y|^2 for some θ> 0 (recall from (<ref>) that, for θ⩽0, ℰ is convex and thus (<ref>) follows from (<ref>) thanks to Lemma <ref>). This is a particular case of (<ref>) with ℋ=^d,

    V_0(x) = V(x) + θ |x|^2 ,    φ(x)=x ,     R(ψ) = -θ |ψ|^2 .

Assumption <ref> is satisfied  since R+θ|·|^2 is convex and φ is Lipschitz continuous (which, as discussed in Remark <ref>, also gives (<ref>) and (<ref>)). We assume that V_0 is strongly convex outside a compact set, so that (<ref>) holds (as explained in Section <ref>) and also (<ref>). Assumptions <ref>, <ref> and <ref> are readily checked, and Assumption <ref> follows from Propositions <ref> and <ref>.

 Using the notations introduced in Section <ref> (except that we use m, as in mean, to denote the parameter, instead of ψ as in the general case), for m∈^d,  

    ρ_m(x) = e^- 1/σ^2  V(x)+ θ|x|^2 -2θ x · m/∫_^d e^- 1/σ^2  V(y)+θ|y|^2 -2θ y · m y ,     f(m) = ∫_^d xρ_m(x) x ,

and

    g(m) =  θ |m|^2 - σ^2 ln∫_^d e^ - 1/σ^2 V(x) + θ|x|^2 - 2 θ x· m  x  =  -σ^2 ln∫_^d e^- 1/σ^2 V(x) + θ|x-m|^2  x   .


The graph of g and f are represented in Figures <ref>,  <ref> and <ref>, in dimension 1, either in the double-well case V(x)= x^4/4-x^2/2 (Figures <ref> and <ref>), or the convex case V(x)=x^2/2 (Figure <ref>), in each case with W(x,y)=θ|x-y|^2 (with θ=1 in Figures <ref> and <ref> and θ=-1 in Figure <ref>), at various temperatures.

 









  §.§.§ The one-dimensional double-well case


In this section, d=1, θ>0 and V(x)=x^4/4-x^2/2. In this one-dimensional attractive double-well case, the function f has been precisely studied in e.g. <cit.> (see also <cit.>), and the following holds (see Figure <ref>).


  * There exists σ_c^2>0 such that, for all σ^2 ⩾σ_c^2, the unique fixed point of f is m=0 and, for all σ^2 <σ_c^2, there are three such fixed points, 0,m_+,m_-, with m_+>0 and  m_-=-m_+.

  * When σ^2 < σ_c^2, f is an increasing function with  0< f'(m_+)=f'(m_-) <1 < f'(0), f'(m) → 0 as |m|→∞, f(m) > m for m∈(-∞,m_-)∪ (0,m_+) and f(m)<m for m∈ (m_-,0)∪(m_+,∞).

  * For σ^2 = σ_c^2, f is an increasing function with f'(m)<1 for all m≠ 0,  f'(m) → 0 as |m|→∞, f'(0)=1, f”(0)=0 and f^(3)(0) <0.

  * For σ^2 > σ_c^2, f is an increasing function with f'(m)∈(0,1) for all m∈ and  f'(m) → 0 as |m|→∞.

Moreover, since g'(m)= 2θ (m-f(m)),  the critical points of g are the  fixed points of f and, for σ^2 < σ_c^2, g is  decreasing  on (-∞,m_-] and [0,m_+] and  increasing on [m_-,0] and [m_+,∞). As a consequence of all this, below the critical temperature, we get a non-linear log-Sobolev inequality for any  non-centered ρ∈𝒫_2(^d).  We introduce the notation

    m(ρ) = ∫_ x ρ(x) x.



In the double-well case  with quadratic interaction (<ref>) (with θ>0, d=1) with σ^2 <σ_c^2, for any ε>0, there exists α_ε∈[0,1) and _ε>0 such that for any m⩾ε,

    |f(m)-m_+| ⩽α_ε |m-m_+|

and for any ρ∈𝒫_2() with |m(ρ)|⩾ε,

    ℱ(ρ) - infℱ⩽_εσ^4 ℐρ|Γ(ρ) .

In other words,  for any ε>0,  (<ref>) holds over 𝒜_ε = {ρ∈𝒫_2(), |m(ρ)|⩾ε}.



In the present setting, Assumption <ref> is satisfied and therefore Theorem <ref> applies so that we get an exponential convergence starting from a 𝒲_2-ball centered on ρ_m_+ or ρ_m_-. This gives the first item of  Proposition <ref>. 
 


Since  f(m) > m for m∈  (0,m_+) and f(m)<m for m∈ (m_+,∞), the function α given by

    α(m) = f(m)-m_+/m-m_+

for m≠ m_+ and α(m_+)=f'(m_+) is continuous on _+ with values in (0,1) on (0,∞). Moreover, since f'(m) → 0 as m→∞, so does α. As a consequence, for any ε>0, α_ε:= sup_m>εα(m) <1, which proves (<ref>).  Proposition <ref> then gives (<ref>) for ρ∈𝒫_2() with m(ρ)⩾ε. The case m(ρ)⩽ -ε is obtained by symmetry.




The same arguments work in the super-critical case where σ^2 >σ_c^2 because in that case f'(0)<1. In that case the contraction |f(m)|⩽α |m| holds globally for m∈ for some α<1. In other words, (<ref>) holds for all σ^2 >σ_c^2.



 To get explicit values for δ,λ,C in Proposition <ref>, it is sufficient to compute f'(m_+) and to bound f”_∞ (as in e.g. <cit.>) to get an explicit δ>0 such that  f'(m) ⩽ (1+f'(m_+))/2=:α <1 for all m with |m-m_+|⩽δ. From this, Theorem <ref> gives explicit estimates in terms of σ^2,θ,δ,α.



Denoting 𝒜_ε = {ρ∈𝒫_2(), |m(ρ)|⩾ε}, the previous result does not mean that ℱ(ρ_t) - infℱ⩽ e^-t/_εℱ(ρ_t) - infℱ for all t⩾ 0 as soon as ρ_0 ∈𝒜_ε, because the  non-linear LSI (<ref>) does not prevent  T_𝒜_ε(ρ_0) < ∞. In fact, we provide a counter-example in the next statement. 
 

In the setting of Proposition <ref>, the precise determination of the basins of attraction of ρ_m_+ and ρ_m_- for the McKean-Vlasov dynamics (<ref>) remains an open question. The first item of Proposition <ref> provides a sufficient condition, together with an exponential rate of convergence. On the other hand, in <cit.>, it is shown that if ℱ(ρ_0) is sufficiently small, then the sign of m(ρ_0) suffices to determine the limit of ρ_t. To complement these statements, we now show that without this condition on ℱ(ρ_0), the sign of m(ρ_t) may vary with t, so it is not enough to determine the limit of ρ_t. To our knowledge, this fact is known empirically but has never been explicitly evidenced.


Consider the granular media equation with d=1, V(x)=x^4/4-x^2/2 and W(x,y)=θ(x-y)^2 for some θ>0. Then, for any σ^2>0, there exist solutions of (<ref>) with m(ρ_0)>0 and m(ρ_t_0) < 0 for some t_0>0.





 

  For ε∈[0,1], we consider an initial condition ρ_0 = (1-ε) δ_-1 + εδ_2/ε-1 and write m_t = ∫_ xρ_t(x) x. In particular, m_0=1 independently from ε. We consider the time-inhomogeneous SDE

    X_t = -V'(X_t)  t - (X_t - m_t)  t  + √(2)σ B_t =  -X_t^3 + m_t t + √(2)σ B_t  .

Let ρ_t^+ and ρ_t^- be the law of the solution of (<ref>) with respective initial conditions δ_2/ε-1 and δ_-1. Then, m_t = (1-ε) ρ_t^- + ερ_t^+ for all t⩾ 0. Assume by contradiction that m_t ⩾ 0 for all t∈[0,1]. Consider the solution of

    Y_t =   -Y_t^3   t + √(2)σ B_t

(with the same Brownian motion as (<ref>)) with Y_0 = -1. Taking X_0 distributed according to m_0 in (<ref>) (in particular, Y_0 ⩽ X_0 and X_t∼ m_t for all t⩾ 0), we get by monotonicity that Y_t ⩽ X_t for all t∈[0,1], and thus

    m_t ⩾𝔼(Y_t) ,    ∫_ x^3 ρ_t( x) ⩾𝔼(Y_t^3) .

 Notice that the law of Y is independent from ε. Now,
 
    ∂_t m_t = - ∫_ x^3 ρ_t( x) + m_t ⩽ -𝔼(Y_t^3) + m_t ,

 which together with m_0=1 shows that M:=sup_ε∈[0,1]sup_t∈[0,1] m_t <∞. By comparing the solution of (<ref>) initialized with X_0=-1 (so that X_t∼ρ_t^-) to the solution of 
 
    Z_t =   (-Z_t^3+M)   t + √(2)σ B_t

 with Z_0 = -1 (whose law is again independent from ε), we get that X_t ⩽ Z_t for all t∈[0,1] and since t↦𝔼(Z_t) is continuous we get that there exists t_0∈(0,1], independent from ε∈[0,1], such that
 
    ∫_ x ρ_t_0^-( x) ⩽ -1/2 .

Now, writing h(t) = ∫_ x^2 ρ_t^+( x), we see that

    h'(t)  = - 2 ∫_ x^4 ρ_t^+( x)+ 2 m_t^2 + 2 σ^2 ⩽ -2 h^2(t) + 2(M+σ^2) .

As a consequence, if h^2 goes below 2(M+σ^2) at some time, it stays below afterwards, and while h^2 is above 2(M+σ^2) it holds

    h'(t) ⩽  -h^2(t) ,

hence h(t) ⩽ (t + 1/h(0))^-1⩽ 1/t. In any case, for all t∈[0,1],

    h(t) ⩽1/t + 2(M+σ^2) .

By Cauchy-Schwarz, 

    sup_ε∈[0,1]∫_ x ρ_t_0^+ ( x) ⩽sup_ε∈[0,1]√(h(t_0))⩽√(1/t_0 + 2(M+σ^2)) ,

and thus

    m_t_0⩽ -1-ε/2 + ε√(1/t_0 + 2(M+σ^2)) ,

which is negative for ε small enough, leading to a contradiction with the assumption that m_t⩾ 0 for all t∈[0,1].


To conclude the discussion in the one-dimensional double-well case, let us notice that the critical temperature σ^2= σ_c^2 provides an example of the degenerate situation addressed in Section <ref>.


In the double-well case  with quadratic interaction (<ref>) (with θ>0, d=1) with σ^2 =σ_c^2,  there exist β>0 and >0 such that for any m∈,

    |m| ⩽β  |m-f(m)| + |m-f(m)|^1/3 ,

and for any ρ∈𝒫_2()

    ℱ(ρ) - infℱ⩽ℐρ|Γ(ρ) + ℐρ|Γ(ρ)^1/3 .




By contrast to (<ref>), which may only  hold locally, in (<ref>) we add a linear term, because in this very simple case this inequality in fact holds globally for all m∈ (and the linear term becomes dominant at long range). This does not change the conclusion on the free energy, in the sense that (<ref>) is similar to (<ref>). 



As in the proof of Proposition <ref>, we see that, for any ε>0, α(m) = f(m)/m is continuous on _+∖[-ε,ε] with values in (0,1), from which α_ε:= sup_|m|⩾εα(m) <1. Then, for m ∉[-ε,ε], |m|⩽ |m-f(m)|+|f(m)| ⩽ |m-f(m)| + α_ε |m|, which proves that (<ref>) holds for such m with β = 1/(1-α_ε).

Denoting s = -f^(3)(0)>0, we get that f(m) = m - s m^3 + o(m^3) as m→ 0. Let ε>0 be such that |f(m)-m+s m^3| ⩽ s|m|^3 /2 for all m ∈[-ε,ε]. Then, for such m,

    |m|^3  ⩽ 2|f(m)-m|/s ,

which concludes the proof of (<ref>).

The proof of (<ref>) is then exactly the same as the proof of   (<ref>) from (<ref>).



The second part of Proposition <ref> is a consequence of  Proposition <ref>, following the arguments that led to (<ref>) (the only difference being that it is not necessary to check that the solution starts and remains in some neighborhood of the stationary solution since (<ref>) holds uniformly over 𝒫_2(^d)).





  §.§.§ Localization in the low-temperature regime


Consider now the granular  media  equation (<ref>) on ^d with W(x,y) = |x-y|^2 (i.e. we take θ=1 for simplicity, which can always be enforced up to rescaling V and σ^2). Let a∈^d be a non-degenerate local minimizer of V such that a is a global minimizer of x↦ h_0(x)=V(x)+|x-a|^2. It is proven in <cit.> that, under additional technical conditions, there exists a family (ρ_*,σ)_σ>0 such that 𝒲_2(ρ_*,σ,δ_a) → 0 as σ→ 0 and, for all σ>0, ρ_*,σ is a stationary solution of  (<ref>)  (at temperature σ). We want to apply Corollary <ref> in this context to get that ρ_*,σ is stable and get local convergence rates.

In fact we will work in a slightly more general setting. Consider the general case (<ref>). For simplicity, we restrict the study to the finite-dimensional case where = ^p for some p⩾ 1 and  R(ψ)=-|ψ|^2. Moreover, we write V(x)= V_0(x) + Rφ(x). In other words, in all this section,

    ℰ(μ)    =    ∫_^d V(x) μ( x) + Rφ(μ) - ∫_^d Rφ(x)μ( x) 
       =   ∫_^d V(x)μ( x) + 1/2∫_^d |φ(x) - φ(y)|^2 μ( x)μ( y) .

Denoting by g_σ the function defined in (<ref>) to emphasize the dependency in the temperature, we see that, for any ψ∈,

    g_σ(ψ)    =    -|ψ|^2 + 2 |ψ|^2  - σ^2 ln∫_^d e^- 1/σ^2  V(x) + |φ(x)|^2 -2 ψ·φ(x) x
        =      - σ^2 ln∫_^d e^- 1/σ^2  V(x) + |φ(x)-ψ|^2  x .

As σ→ 0,

    g_σ(ψ) → g_0(ψ) := inf_x∈^d{ V(x) + |φ x -ψ|^2} ,

and this convergence holds uniformly for ψ in compact sets of ^p.
We are interested in critical points ρ_*,σ which are localized at low temperature at some point a∈^d, in the sense that ρ_*,σ→δ_a as σ vanishes. This implies that φ(ρ_*,σ) →φ(a). On the other hand, for a fixed ψ, if x↦ V(x) + |φ(x)-ψ|^2 has a unique global minimum x_0, then ρ_ψ converges as σ vanishes to a Dirac mass at x_0. Hence, we expect ρ_*,σ to converge to a Dirac mass at the minimizer of x↦ V(x)+|φ(x)-φ(a)|^2, and thus we need this minimizer to be a. We retrieve the condition of <cit.>.


If we do not assume that R(ψ)=-|ψ|^2 but still write V(x) = V_0(x) +R(φ(x)), the previous computation gives 

    g_σ(ψ) =   - σ^2 ln∫_^d e^- 1/σ^2 h(x,ψ)  x ,

with a modulated energy 

    h(x,ψ) =   R(ψ) - R(φ(x)) +  R(ψ) ·φ(x)  - ψ  +   V(x) = Mψ,φ(x) + V(x)   ,

with the notation of Remark <ref>. As σ vanishes, g_σ(ψ) converges toward inf_x∈^d h(x,ψ).



In view of the previous discussion, we work under the following setting:


In the case (<ref>), =^p, R(ψ)=-|ψ|^2, V_0(x) = V(x) - Rφ(x), V∈𝒞^3(^d,), φ∈𝒞^3(^d,^p), the following conditions hold:

    
  * a∈^d is a non-degenerate local minimizer of V (i.e. ^2 V(a)>0) and it is the unique global minimizer of x↦ V(x) + |φ(x)-φ(a)|^2.
    
  * V goes to infinity at infinity and ∫_^d |x|^2 e^- β V(x) x < ∞ for some β>0. 






Under Assumptions <ref> and <ref>, assuming moreover that (<ref>) holds for some κ_1>0, there exist r_0,σ_0^2>0 and a family (ρ_*,σ)_σ∈(0,σ_0] with 𝒲_2(ρ_*,σ,δ_a) → 0 as σ→ 0 such that the following holds: for each σ∈(0,σ_0], ρ_*,σ is a stationary solution of the McKean-Vlasov equation (<ref>) at temperature σ^2 and there exist C,>0 such that for all initial condition ρ_0 ∈ℬ_𝒲_2(δ_a,r_0), along the flow (<ref>)  at temperature σ^2, for all t⩾ 1,

    𝒲_2^2(ρ_t,ρ_*,σ) + σ^2 ℋ(ρ_t|ρ_*,σ) + ℱ(ρ_t) - ℱ(ρ_*,σ) ⩽ C e^-t/𝒲_2^2(ρ_0,ρ_*,σ) .

    
    Moreover,  if we assume furthermore that x↦ V(x) + |φ(x)-ψ|^2 is strongly convex uniformly over ψ in a neighborhood of φ(a), then the previous statement holds with ,C which are independent from  σ∈(0,σ_0].



In particular, if V has several local minimizers a_1,…,a_n and the interaction is strong enough so that a_j is the unique global minimizer of V+|φ-φ(a_j)|^2 for several j, then, at a sufficiently low temperature, there are several stable stationary solutions to the granular media equation. By contrast, as discussed in Section <ref>, in the same setting, at sufficiently high temperature or sufficiently weak interaction (i.e. multiplying φ by a sufficiently small factor ε>0), uniqueness holds. This shows that phase transitions occur as the temperature varies. See <cit.> on this topic. 



First, to get the existence of ρ_*,σ, we reason in a first step as in  Proposition <ref>, namely we identify that g_σ should have a local minimizer close to φ(a) for σ small enough. Second, in the rest of the proof, we show that  Corollary <ref> applies. This  gives a local non-linear LSI and thus a convergence rate thanks to Theorem <ref> (since, as discussed in Remark <ref>, in the present case where R is quadratic, the conditions (<ref>) and (<ref>) of Assumption <ref> are implied by Assumption <ref>).



Step 1. Fix r∈(0,1] (which later on will be assumed sufficiently small, but always independently from σ). For ψ∈^p, set

    ζ(ψ,r):= inf{V(x)+|φ(x)-ψ|^2, |x-a|⩾ r} .

Let M⩾ 1 be such that V(x)⩾sup{V(y), y∈ℬ(a,1)} + 
(ℓ+1)^2 for all x∈^d with |x|⩾ M. Then, for ψ∈ℬ(φ(a),1) and x,y∈^d with |x|⩾ M and |y-a|⩽ 1,

    V(y) + |φ(y)-ψ|^2    ⩽ V(y) + (|φ(y)-φ(a)| + |φ(a)-ψ|)^2
       ⩽ V(y) + (ℓ |y-a| + |φ(a)-ψ|)^2 
       ⩽ V(y) + (ℓ+1)^2 
       ⩽  V(x)  
       ⩽ V(x) +  |φ(x)-ψ|^2 .
 
As a consequence, since r∈(0,1], for  ψ∈ℬ(φ(a),1),

    ζ(ψ,r)    =   inf{V(x)+|φ(x)-ψ|^2, |x-a|⩾ r, |x|⩽ M}
       ⩾   ζφ(a),r - 2 |ψ - φ(a)| sup_|x|⩽ M|φ(x)-φ(a)| ,

where, for the second line, we have used that for x∈ℬ(0,M),

    V(x)+|φ(x)-ψ|^2     = V(x)+|φ(x)-φ(a)|^2 - 2 φ(x)-φ(a)·ψ-φ(a)   + |ψ- φ(a)|^2 
       ⩾ V(x)+|φ(x)-φ(a)|^2- 2 |ψ - φ(a)| sup_|x|⩽ M|φ(x)-φ(a)| .

On the other hand, since a is the unique global minimizer of x↦ V(x)+|φ(x)-φ(a)|^2, ζ(φ(a), r) > V(a). Thus, we can find ε_0>0 (which depends on r) such that for all ε∈(0,ε_0] and  ψ∈ℬ(φ(a),ε),

    ζψ,r > V(a) + ε^2

(notice that the condition becomes weaker as ε decreases). As a consequence, for all  ψ∈ℬ(φ(a), ε ),

    g_0(ψ) = inf_x∈ℬ(a,r){ V(x) + |φ x -ψ|^2}

(since, for x∉ℬ(a,r), V(x)+|φ(x)-ψ|^2 ⩾ζ(ψ,r) > V(a) + |φ(x)-ψ|^2). 
Assuming that r is small enough so that a is the unique global minimizer of V over ℬ(a,r), we get that g_0(ψ)>V(a)=g_0φ(a) for all ψ∈ℬ(φ(a), ε )∖{φ(a)}, and in particular by continuity

    inf{g_0(ψ), |ψ - φ(a)|=ε} - V(a)>0 .

By uniform convergence on compact sets of g_σ toward g_0 as σ vanishes, there exists σ_0>0 (depending on ε) such that 

    κ := inf{g_σ(ψ), |ψ - φ(a)|=ε,σ∈(0,σ_0]} - V(a)>0 ,

while g_σφ(a)→ V(a) as σ→ 0, and in particular g_σφ(a)⩽ V(a)+κ/2 for σ small enough. This means that, then, the minimum of g_σ over ℬφ(a),ε is not attained at its boundary, which implies that g_σ admits a critical point within this ball. Since we can take ε arbitrarily small, we can take at any σ∈(0,σ_0] such a critical point ψ_*,σ in such a way that  ψ_*,σ→φ(a) as σ→ 0. Thanks to (<ref>), ρ_*,σ:= ρ_ψ_*,σ is a stationary solution to (<ref>). This concludes the first step of the proof. 

Before proceeding with Step 2 of the proof, let us discuss a few points. First,

    𝒲_2^2(ρ_*,σ,δ_a) = ∫_^d |x-a|^2  e^- 1/σ^2  V(x) + |φ(x)-ψ_*,σ|^2  x /∫_^d  e^- 1/σ^2  V(x) + |φ(x)-ψ_*,σ|^2  x .

Using that ψ_*,σ→φ(a) and that a is the global minimizer of V+|φ-φ(a)|^2, it is then not difficult to see that 𝒲_2^2(ρ_*,σ,δ_a) vanishes with σ (see also the end of the proof for details of the argument in a more complicated case).

It remains to show that Corollary <ref> applies. To emphasize the dependency on the temperature, we write f_σ the function (<ref>). We have to prove that | f_σ(ψ_*,σ)|<1 for σ small enough. From (<ref>) and (<ref>), 

    | f_σ(ψ_*,σ)| = 2/σ^2sup_u∈𝕊^p-1var_ρ_*,σ u·φ(X) .

Note that by the variational characterization of the variance, for u∈𝕊^p-1,
 
    var_ρ_*,σ u·φ(X)⩽∫_^d |u·φ(x)-φ(z) |^2 ρ_*,σ( x)

for any z∈^d. Having in mind the Laplace approximation of ρ_*,σ as σ vanishes, it is natural to take z as the minimizer of x↦ h_σ(x):= V(x) + |φ(x)-ψ_*,σ|^2. In the rest of the proof, first, we justify that, for σ small enough, h_σ admits a unique minimizer z_σ, which converges to a as σ vanishes (this is Step 2) and then (in Step 3), we apply the Laplace approximation (which still works although h_σ and z_σ depends on σ) to get that,

    lim_σ→ 02/σ^2sup_u∈𝕊^p-1∫_^d |u·φ(x)-φ(z_σ) |^2 ρ_*,σ( x) < 1 .

This implies that the same holds for σ small enough, which concludes the proof. The technical justification of the Laplace approximation is postponed to Step 4.



Step 2. We write φ=(∂_x_iφ_j)_(i,j)∈ 1,d× 1,p, where i (resp. j) is the index of the line (resp. the column).  Denoting h_0(x)=V(x)+|φ(x)-φ(a)|^2, we compute that 

    ^2 h_σ(x)    =   ^2 V(x) + 2 φ(x) φ(x)^T + ∑_j=1^p  ^2 φ_j(x) φ_j(x)-ψ_j,*,σ
       =    ^2 h_0(x) + ϵ(x,σ)

with 

    ϵ(x,σ)  = ∑_j=1^p  ^2 φ_j(x) φ_j(a)-ψ_j,*,σ ,

so that ϵ(x,σ) vanishes with σ uniformly over x∈ℬ(a,r). Besides,

    ^2 h_0(a) = ^2 V(a) + 2  φ(a) φ(a)^T

is positive definite, and thus we can assume that r is small enough so that ^2 h_0(x) is uniformly bounded below by a positive constant over x∈ℬ(a,r). Thanks to the uniform convergence of ^2 h_σ to ^2 h_0 over ℬ(a,r), there is σ_0',κ>0 such that for all σ∈(0,σ_0'] and x∈ℬ(a,r), ^2 h_σ(x) ⩾ 2κ. On the other hand, we have seen when establishing (<ref>) that there exists κ'>0 such that for σ small enough, 

    inf{h_σ(x), |x-a|⩾ r} - inf{h_σ(x), |x-a|⩽ r}⩾κ'

(here we use that for any ε>0, ψ_*,σ∈ℬ(φ(a),ε) for σ small enough). The infimum of h_σ is thus attained (for σ small enough) in ℬ(a,r), where it is strongly convex, and thus it admits a unique global minimizer z_σ. Since r can be taken arbitrarily small (which changes the threshold σ_0'), we get that z_σ→ a as σ vanishes. Moreover, for σ small enough,  for all x∈ℬ(a,r),

    h_σ(x) ⩾ h_σ(z_σ) + κ |x-z_σ|^2 .

  Since h_σ admits a unique global maximum which is non-degenerate, the Laplace approximation heuristics suggest that expectations with respect to ρ_*,σ∝exp(-1/σ^2 h_σ) are equivalent to expectations with respect to the Gaussian measure ρ̂_σ∝exp(-1/σ^2ĥ_σ) with

    ĥ_σ(x) = h_σ(z_σ) + 1/2(x-z_σ)^T ^2 h_σ(z_σ) (x-z_σ) .

As mentioned above, we postponed this technical justification to the end of the proof and, for now, take for granted that

    ∫_^dφ-φ(z_σ)φ-φ(z_σ) ^T  ρ_*,σσ→ 0∼∫_^dφ-φ(z_σ)φ-φ(z_σ) ^T  ρ̂_σ .


Step 3. From this approximation, we have now to study the limit in (<ref>). First, writing

    1/σ^2∫_^d (φ(x)-φ(z_σ))(φ(x)-φ(z_σ))^T ρ̂_σ( x)
       = 𝔼[(φ(z_σ + σ√(D_σ^-1) G)-φ(z_σ)/σ)(φ(z_σ + σ√(D_σ^-1) G)-φ(z_σ)/σ)^T],

with G a standard d-dimensional Gaussian variable and  D_σ := ∇^2 h_σ(z_σ), the almost sure convergence

    lim_σ→ 0φ(z_σ + σ√(D_σ^-1) G)-φ(z_σ)/σ = (∇φ(a))^T √(D_0^-1) G,

together with the bound

    |φ(z_σ + σ√(D_σ^-1) G)-φ(z_σ)/σ|^2 ≤ℓ^2 |√(D_σ^-1) G|^2 σ→ 0⟶ℓ^2 |√(D_0^-1) G|^2

give by dominated convergence that

    lim_σ→ 0𝔼[(φ(z_σ + σ√(D_σ^-1) G)-φ(z_σ)/σ)(φ(z_σ + σ√(D_σ^-1) G)-φ(z_σ)/σ)^T]
       = 𝔼[((∇φ(a))^T √(D_0^-1) G)((∇φ(a))^T √(D_0^-1) G)^T] 
       = (∇φ(a))^T D_0^-1∇φ(a)
       = A^T  2 A A^T + ^2 V(a) ^-1 A ,

where A:=φ(a). 
Thanks to (<ref>),

    2/σ^2∫_^dφ(x)-φ(z_σ)φ(x)-φ(z_σ) ^T  ρ_*,σ( x) σ→0⟶ 2 A^T  2 A A^T + ^2 V(a) ^-1 A .
 

It remains to show that this limit is strictly smaller than I_p (in the sense of quadratic forms), from which the left-hand side will be uniformly strictly less than I_p for σ small enough, which will conclude.

Let u∈𝕊^p-1 and h=(2AA^T + ^2 V(a))^-1 Au.  Then

    h^TAu = h^T  (2AA^T + ^2 V(a))  h ⩾ (2+c) |A^T h|^2

for some c>0, using that ^2 V(a) is definite positive. Then, 

    h^TAu   ⩽ |u| |A^T h | ⩽1/√(2+c)|u| √(h^TAu ) ,

and thus

    2 u^T A^T  2 A A^T + ^2 V(a) ^-1 A u
    =2 u^T A^T h   ⩽2/2+c .

As discussed above, this concludes the proof.

Notice that c is independent from σ, and thus, in Proposition <ref>, 
(<ref>) holds on 𝒜'=ℬφ(a),r with r>0 and α∈[0,1) independent from σ∈(0,σ_0”] for some σ_0”>0. If, moroever, x↦ V(x) + |φ(x)-ψ|^2 is k-strongly convex for some k>0 for all ψ∈𝒜', the Bakry-Emery criterion shows that Γ(ρ) satisfies an LSI with constant η = k σ^2 for all ρ such that φ(ρ) ∈𝒜', and thus for all ρ∈𝒜:= ℬ_𝒲2(δ,r_0) provided r_0 is small enough (independently from σ). Following the proof of Proposition <ref> (which in fact does not require (<ref>) to hold  globally on 𝒫(^d), but only on 𝒜), we obtain for ρ∈𝒜 the local non-linear LSI

    ℱ(ρ) - ℱ(ρ_*,σ) ⩽ k σ^4  1   +  4 k  θℓ^2/(1-α)^2ℐρ|Γ(ρ) ,

where θ,ℓ,α,k are independent from σ∈(0,σ_0”], i.e. (<ref>) with  independent from σ. Similarly, using that η=kσ^2, we get that q_1 defined in (<ref>) with t=1 is bounded independently from σ small enough, for initial conditions ρ_0 ∈𝒜. Applying Theorem <ref> shows that (<ref>) holds with C, independent from σ∈(0,σ_0”] in this case (again, the proof only uses an LSI for Γ(ρ) uniformly over 𝒜, not over 𝒫(^d)).



Step 4. We now turn to the justification of (<ref>). Write
χ_0(x)=1 and, omitting the dependency in σ, χ_1(x)= φ(x)-φ(z_σ)φ(x)-φ(z_σ) ^T (notice that |χ_1(x)|⩽ C|x|^2+C for some C independent from σ∈(0,σ_0]). For i∈{0,1} and k∈{1,2,3,4} we write

    I_i,k = ∫_A_kχ_i   exp -1/σ^2 h_σ -  h_σ(z_σ) ,    Î_i,k = ∫_A_kχ_i  exp -1/σ^2ĥ_σ -  h_σ(z_σ) ,

with

    [        A_1 = {|x-z_σ|⩽√(σ)} A_2 = {√(σ) < |x-z_σ|⩽ 2r };     A_3 = {2r < |x-z_σ|⩽ M}      A_4 = {M < |x-z_σ| } , ]

where r is small enough for  (<ref>) to hold for some κ'  and for  (<ref>) to hold for all x∈ℬ(a,3r) for some κ (with r,κ,κ'>0 independent from σ) for all σ small enough, and M>1 is large enough so that 

    ∀ x∉ℬ(a,M-1) ,    1/2 V(x) ⩾ V(a)+2

For σ small enough, |z_σ-a|<r, and thus ℬ(a,r) ⊂ℬ(z_σ,2r), which together with 
(<ref>) gives that 

    inf{h_σ(x), |x-z_σ |⩾ 2 r}⩾  h_σ(z_σ) + κ' .
 
As a consequence, for i∈{0,1}, using that φ is bounded uniformly over ℬ(a,M+1) (hence over ℬ(z_σ,M) for σ small enough)

    I_i,3  = σ→ 0𝒪 e^-κ'/σ^2 .

Similarly, for σ small enough, ℬ(z_σ,2r)⊂ℬ(a,3r) and thus, thanks to (<ref>)  for x∈ℬ(a,2r), using that φ is bounded uniformly over ℬ(a,3r), for i∈{0,1},

    I_i,2   = σ→ 0𝒪 e^-κ/σ .

Next, using that ℬ(a,M-1)⊂ℬ(z_σ,M) and V(a)+1 ⩾ h_σ(z_σ) for σ small enough, (<ref>) implies that

    ∀ x∉ℬ(z_σ,M) ,       h_σ(x) ⩾ V(x) ⩾1/2 V(x)+1+ h_σ(z_σ) .
 
This gives, for σ⩽ 1/(2β),

    I_i,4⩽ e^-1/σ^2∫_|x-z_σ|> M  |χ_i(x)| e^-V(x)/2σ^2 x  ⩽ e^-1/σ^2∫_^d  |χ_i(x)| e^-β V(x)  x  = σ→ 0𝒪 e^-1/σ^2 ,

since e^-β V admits a second moment (here for simplicity we have assumed without loss of generality that V⩾ 0).

Finally, using that ^(3) h_σ  converges ^(3) h_0 uniformly over compact sets, we get that there exists C>0 such that

    |h_σ(x) - ĥ_σ(x)| ⩽ C |x-z_σ|^3

for all x∈ℬ(a,1) (hence all x∈ℬ(z_σ,√(σ))) for σ small enough. From this, for i∈{0,1},

    I_i,1 = Î_i,1 1+ σ→ 0𝒪σ .
 
On the other hand, denoting c_σ = (2π)^d/2√(det[^2 h_σ(z_σ)^-1]) (which converges to some c_0>0 as σ→ 0),
 by usual computations for Gaussian distributions,
 
    I_0 := ∑_k=1^4 I_0,k  =    Î_0,1 1+ σ→ 0𝒪σ +  σ→ 0𝒪 e^-κ/σ = σ^d c_σ +  σ→ 0𝒪σ^d+1 ,

and then

    ∑_k=1^4 I_1,k/I_0    =   σ^d c_σ +  σ→ 0𝒪σ^d+1^-1Î_1,1 1+ σ→ 0𝒪σ +  σ→ 0𝒪 e^-κ/σ
        =    Î_1,1/σ^d c_σ 1+ σ→ 0𝒪σ +  σ→ 0𝒪 e^-κ/2σ
        =    ∑_k=1^4 Î_1,k/σ^d c_σ 1+ σ→ 0𝒪σ +  σ→ 0𝒪 e^-κ/2σ
        =    σ^2  φ(a)^T  ^2 h_0(a)^-1φ(a)  + σ→ 0oσ^2 ,

as we computed in (<ref>). This concludes the proof of (<ref>), hence of Proposition <ref>. 




In the case (<ref>) with W(x,y)=θ|x-y|^2, taking θ > - inf^2 V/2 (as in <cit.>), we get that  (<ref>) holds with ,C that are uniform over σ small enough in a neighborhood of δ_a. At first this seems to be a desirable property for optimization (and in fact this is the idea underlying consensus-based optimization). This should be mitigated by the following observations.

First, this condition implies that x↦ b_a(x)= V(x) + θ|x-a|^2 is strongly convex for all a and thus, any local minimizer  a of V being a critical point of b_a, it is then the unique global minimizer of b_a. Hence, the Wasserstein gradient descent can be trapped in any local well of V (and thus it is not very different from a basic gradient descent for V in ^d). By contrast, if θ is taken smaller,  the condition that a has to be a global minimizer of b_a for a localization to occur  can be used as a way to  discard shallow local wells and select better solutions.

Besides, the decay  (<ref>) holds for solutions initialized in ℬ_𝒲_2(δ_a,r_0) for some small r_0. Now, assume for instance that we start close to δ_a' where a' is a local minimizer of V but not a global minimizer of b_a', and a≠ a' is the global minimizer of V and also the global minimizer of b_x for all x∈^d. For σ small enough, we expect all solutions to converge to a stationary solution close to δ_a. In particular, eventually, any trajectory will lie in  ℬ_𝒲_2(δ_a,r_0) and thus the convergence rate  in (<ref>) is independent from σ, as in Remark <ref>. However the constant C highly depends on the initial condition, reflecting the time needed to reach  ℬ_𝒲_2(δ_a,r_0). Starting close to δ_a', the additional convexity due to the interaction will increase the energy barrier the process has to overcome to move from a' to a. More quantitatively, the Arrhenius law indicates that the typical time for the solution to put most of its mass around a will be of order e^D/σ^2 where D is larger than in the case θ=0. In other words, an attractive interaction worsen the metastability of the process. If the goal is not to improve the local convergence rate but to enhance the exploration of the space by lowering energy barriers (as in the Adaptive Potential algorithms and related methods <cit.>), repulsive interaction (i.e. θ<0) seems more indicated, as studied in <cit.>.






§ VLASOV-FOKKER-PLANCK EQUATION


In this section, we consider the kinetic Vlasov-Fokker-Planck equation, which reads 

    ∂_t ρ_t + v·_x ρ_t  = _v·σ^2∇_v ρ_t +  v+  E_ρ_t^xρ_t ,

where ρ_t(x,v) is the probability density of particles at position x∈^d with velocity v∈^d, ρ^x(x)=∫_^dρ(x,v) v is the marginal density of the position, E_ρ^x is the linear functional derivative of an energy ℰ as in Section <ref> (it only depends on the position x and the marginal density ρ^x). 

Entropic long-time convergence for (<ref>) have been established in some cases (with the energy corresponding to the granular media case (<ref>)) in <cit.> (using uniform LSI for the associated particle system), <cit.> (with global non-linear LSI) or <cit.> (with uniform conditional LSI for the associated particle system and a smallness assumption on the interaction). All these works establish global convergence toward a unique stationary solution and thus do not cover cases with several stationary solutions. 

The Vlasov-Fokker-Planck is not the gradient flow of some free energy functional. However, we can still get a local convergence under the same assumptions as in the elliptic case (<ref>). This is the main result of this section, stated in Theorem <ref> below.

The free energy associated to (<ref>) is

    ℱ_k(ρ ) = σ^2 ℋ(ρ) + ℰ(ρ^x) + ℰ_k(ρ)

with the kinetic energy 

    ℰ_k(ρ) = ∫_^2d|v|^2/2ρ (x,v) x v .

In this section, Assumptions <ref>, <ref> and <ref>  remain in force (which implies that ℱ_k is also lower bounded, see Remark <ref> below). The corresponding local equilibria are given by

    Γ_k(ρ) = Γ(ρ^x) ⊗𝒩(0, σ^2 I_d)

where Γ(ρ^x) is the corresponding overdamped local equilibrium (<ref>) and 𝒩(0,σ^2 I_d) stands for the centered  d-dimensional Gaussian distribution with variance σ^2I_d. In other words,

    Γ_k(ρ) ∝exp - 1/σ^2  E_ρ^x(x) + 1/2|v|^2 x  v  .

 Along the flow (<ref>), under suitable regularity conditions, 

    ∂_t  ℱ_k(ρ_t) = - σ^4 ∫_^2d| _v lnρ_t/Γ_k(ρ_t)|^2 ρ_t   .

Hence, t↦ℱ_k(ρ_t) is non-increasing, but the free energy dissipation can vanish even if ρ≠Γ_k(ρ). To avoid ambiguity we write 𝒦_k = {ρ_*∈𝒫_2(^2d), ℱ_k(ρ_*)<∞, Γ_k(ρ_*)=ρ_*}. There is a one-to-one correspondence between 𝒦_k (the set of critical points of ℱ_k in 𝒫_2(^2d)) and 𝒦 (the set of critical points of ℱ=σ^2 ℋ + ℰ in 𝒫_2(^d)), since ρ_*∈𝒦_k if and only if ρ_*^x ∈𝒦 and ρ_*=ρ_*^x ⊗𝒩(0,σ^2 I_d). 

As in the elliptic case, we do not address well-posedness issues and work under the following conditions.

 
 For all ρ_0∈𝒫_2(^2d), (<ref>) has a unique strong solution, continuous in time for 𝒲_2, such that, for all t>0, ρ_t has a continuous positive density, ℱ_k(ρ_t), ℋ(ρ_t|Γ_k(ρ_t)) and  ℐ(ρ_t|Γ_k(ρ_t)) are finite and (<ref>) holds, with t↦ℐ(ρ_t|Γ(ρ_t)) continuous over _+^*. 
 
 
 To recover an entropy dissipation with a  full Fisher information instead of (<ref>), we work with a modified free energy of the form 
 
    ℒ(ρ_t) = ℱ_k(ρ_t)  +  a σ^4  ∫_^2d|(_x+_v) lnρ_t/Γ_k(ρ_t)|^2 ρ_t ,

 for some a>0 to be fixed.
 
 
  Assume that 

    M := sup_ρ∈𝒫_2(^d)^2 E_ρ_∞ <∞ .x-Lip

Then, taking a=(3+4M^2)^-1 in the definition of ℒ, for all t⩾ 0,

    ∂_t ℒ(ρ_t) ⩽ - σ^4/4ℐρ_t|Γ_k(ρ_t) .

 
 
 Set h_t = ρ_t/Γ_k(ρ_t). 
As computed in the proof of <cit.> (notice that the friction parameter γ of <cit.> is 1 in the present case, which besides can always be enforced by a suitable time rescaling, up to rescaling E_ρ and σ^2), for any 2d×2d matrix A of the form

    A= [ a_11 I_d a_12 I_d;  a_21I_d a_22 I_d ] ,

writing M'= M √(a_11^2 + a_21^2), it holds, for all r>0,

    ∂_t ∫_^2d|A ln h_t |^2 ρ_t ⩽ 2 ∫_^2dln h_t  · B ln h_t ρ_t

with

    B = A^T A [     M'r I_d           0;        -I_d (M'r-1) I_d ] + M'/r[   0   0;   0 I_d ] .

Taking  a_ij=1/√(2) for all i,j∈{1,2} we get that M'=M and

    B     =     [      (Mr-1) I_d      (Mr-1) I_d;      (Mr-1) I_d Mr-1 + M/r  I_d ]
       ≼   [            - 1/4 I_d                    0;                    0 1/2 + 2M^2 + 1   I_d ]

taking r=1/(2M), where ≼ stands for the order over symmetric matrices and we used that xv ⩽1/4x^2 +   v^2 to bound the cross term. In other words, we have obtained that

    ∂_t ∫_^d|(_x+_v) ln h_t |^2 ρ_t ⩽ - 1/4∫_^d|_x ln h_t |^2 ρ_t
     + 3/2 + 2M^2  ∫_^d|_v ln h_t |^2 ρ_t .

Conclusion follows from (<ref>).
 
 As a corollary, assuming in the kinetic case the local non-linear LSI 
 
    ∀ρ∈𝒜 ,    ℱ_k(ρ) - ℱ_k(ρ_*) ⩽σ^4 ℐ(ρ|Γ_k(ρ))k-NL-LSI

for some 𝒜⊂𝒫_2(^2d),  >0 and ρ_*∈𝒦_k, which, using that |(_x+_v) f|^2 ⩽ 2| f|^2, in turn implies 
 
    ∀ρ∈𝒜 ,    ℒ(ρ) - ℒ(ρ_*) ⩽ +2aσ^4 ℐ(ρ|Γ_k(ρ))  ,

we get an exponential decay of ℒ(ρ_t)-ℒ(ρ_*) for times t⩽ T_𝒜(ρ_0). 


For ρ∈𝒫_2(^2d),

    ℱ_k(ρ)     =    σ^2 ℋρ|Γ_k(ρ) - σ^2 ln Z_ρ^x (2πσ^2)^d/2 + ℰ(ρ^x) - ∫_^dρ^x E_ρ^x
        =    σ^2 ℋρ|Γ_k(ρ) - d/2σ^2 ln 2πσ^2 + 𝒢ρ^x

with

    𝒢(ρ^x) = ℰ(ρ^x) - ∫_^dρ^x  E_ρ^x - σ^2 ln Z_ρ^x ,

which is also such that, in the elliptic case, ℱ(ρ^x) = σ^2 ℋ(ρ^x|Γ(ρ^x)) + 𝒢(ρ^x). In particular, by the extensivity property of the relative entropy, ℱ_k(ρ) ⩾ℱ(ρ^x) - dσ^2/2 ln (2πσ^2).

If Γ(ρ^x) satisfies a LSI uniformly over ρ, so does Γ_k(ρ) by tensorization of LSI, and then both (<ref>) and (<ref>) follow from a bound of the form 𝒢(ρ^x)-𝒢(ρ_*^x) ⩽' ℐ(ρ^x|Γ(ρ^x)) for some '>0, as has been established in various situations in  Section <ref>.



In order to follow the argument that led in the elliptic case to Theorem <ref>, we need to provide an analogue of  Corollary <ref> in the present hypoelliptic case. This is done by   using the Wasserstein-to-entropy short time regularization of <cit.> instead of Proposition <ref> and the entropy-to-Fisher short time regularization of <cit.>, as we now detail.


The following is from <cit.> (the proof is based on computations similar to Lemma <ref>).

Assuming (<ref>) and (<ref>), there exists ε∈(0,1] (which depends only on M,L and σ^2) such that for any solution of (<ref>) with ℱ_k(ρ_0) <∞, writing

    ℰ(t,ρ) := ℱ_k(ρ)  +  ε t   ∫_^2d|_v  lnρ/Γ_k(ρ)|^2 +  |(ε t _x+  _v) lnρ/Γ_k(ρ)|^2  ρ ,

then t ↦ℰ(t,ρ_t) is non-increasing over t∈[0,1].


The following is a particular case  of <cit.>.


Assuming (<ref>) and (<ref>), there exists K>0 (which depends only on M,L and σ^2) such that for any solution of (<ref>) with ρ_0∈𝒫_2(^2d) and any ρ_*∈𝒦_k and t⩾ 0,

    ℋρ_t+1 |ρ_*⩽ K 𝒲_2^2(ρ_t,ρ_*) .




We can now state the analogue to Theorem <ref>.


The conditions (<ref>), (<ref>), (<ref>) and (<ref>)  hold for some λ, L,M,η>0, and (<ref>) holds on some non-empty set  𝒜⊂𝒫_2(^2d) for some >0. Furthermore  there exist ρ_* ∈𝒜∩𝒦_k and δ>0 such that ℬ_𝒲_2(ρ_*,δ) ⊂𝒜 and ℬ_𝒲_2(ρ_*,δ) ∩𝒦_k = {ρ_*}. The function ℒ is given by (<ref>) with a=(3+4M^2)^-1.



Under Assumption <ref>, there exist δ',c,C>0 (which depend only on λ, M, L, δ, η,  and σ^2)  such that, for all ρ_0 ∈ℬ_𝒲_2(ρ_*,δ'), T_𝒜(ρ_0)=∞; for all t⩾ 0,

    𝒲_2^2( ρ_t,ρ_*)    ⩽    C e^-ct𝒲_2^2 (ρ_0,ρ_*) 
    ℱ_k(ρ_t)-ℱ_k(ρ_*)    ⩽    C e^-ctℱ_k(ρ_0)-ℱ_k(ρ_*)
    ℋ(ρ_t|ρ_∞)    ⩽    C e^-ctℋ(ρ_0|ρ_∞)

and for all t⩾ 1,

    max𝒲_2^2 (ρ_0,ρ_*)   , ℋ(ρ_0|ρ_∞)  , ℱ_k(ρ_0)-ℱ_k(ρ_*)
    ⩽    C e^-ctmin𝒲_2^2 (ρ_0,ρ_*)   , ℋ(ρ_0|ρ_∞)  , ℱ_k(ρ_0)-ℱ_k(ρ_*) .




First, without loss of generality we can assume that (<ref>) holds with η⩾σ^2, in which case the same inequality holds for all ν,ρ∈𝒫_2(^2d) when Γ is replaced by Γ_k, due to the LSI satisfied by 𝒩(0,σ^2 I_d) and the tensorization property of LSI. In particular, ρ_* satisfies a LSI and thus a Talagrand inequality with constant η. Moreover, notice that (<ref>) implies the same condition with ℰ replaced by ℰ + ℰ_k, since μ↦ℰ_k(μ) is linear. In particular, in view of these two points, we can apply Lemmas <ref> and <ref> in the kinetic case (namely with ℱ replaced by ℱ_k,  Γ   by Γ_k,  ρ_*∈𝒦_k and μ_0,μ_1,ρ∈𝒫_2(^2d)). Using all these points, for any ρ∈𝒫_2(^2d),

    ℱ_k(ρ) -  ℱ_k(ρ_*)    ⩽   σ^2 ℋρ|Γ_k(ρ)+   λ𝒲_2^2(ρ,ρ_*) 
       ⩽   σ^2   1+ Lη +  4λη/σ^2 + L^2η^2 /2ℋ(ρ|ρ_*) .
 
Similarly, we can apply Theorem <ref> to the elliptic equation on ^2d

    ∂_t ρ̃_t = ·ρ̃_t δℱ_k(ρ̃_t) /δμ ,

from which (<ref>) reads 

    𝒲_2^2(ρ,ρ_*) ⩽ 4ℱ_k(ρ) - ℱ_k(ρ_*)

for all ρ∈ℬ_𝒲_2(ρ_*,δ_1') for δ_1'⩽δ small enough. Using that t↦ℱ_k(ρ_t) is non-increasing and Proposition <ref>, gathering the previous bounds gives that 

    𝒲_2^2(ρ_t,ρ_*)  ⩽  4σ^2   1+ Lη +  4λη/σ^2 + L^2η^2 /2 K  𝒲_2^2(ρ_0,ρ_*) .

for all t⩾ 1 such that ρ_t∈ℬ_𝒲_2(ρ_*,δ_1').  Using a synchronous coupling as in the proof of Theorem <ref>, it is easily seen that

    𝒲_2(ρ_t,ρ_*) ⩽ e^Ct𝒲_2(ρ_0,ρ_*)

for some C depending only on M and L. As a consequence, we can take δ'_2 ∈(0, δ_1'] such that for all ρ_0 ∈ℬ_𝒲_2(ρ_*,δ_2'), it holds that ρ_t ∈ℬ_𝒲_2(ρ_*,δ_1') for all t∈[0,1] and 

    𝒲_2(ρ_t,ρ_*)  ⩽δ_1'/2

for all t⩾ 1 such that ρ_t∈ℬ_𝒲_2(ρ_*,δ_1'). Using that t↦ρ_t is continuous with respect to 𝒲_2, we get by contradiction that ρ_t∈ℬ_𝒲_2(ρ_*,δ_1') for all t⩾ 0 if ρ_0 ∈ℬ_𝒲_2(ρ_*,δ_2'), and in particular T_𝒜(ρ_0) = +∞.

In the remaining of the proof we consider an initial condition  ρ_0 ∈ℬ_𝒲_2(ρ_*,δ_2'). As we used in  (<ref>), (<ref>)  implies that ρ_* is a minimizer of ℱ_k over ℬ_𝒲_2(ρ_*,δ), so that ℱ_k(ρ_t) ⩾ℱ_k(ρ_*) for all t⩾ 0.  Applying Proposition <ref> (but with initial condition ρ_t instead of ρ_0), we get that, for all t⩾ 0,

    ε^3/2ℐρ_t+1|Γ(ρ_t+1)   ⩽   ℰ(1,ρ_t+1) - ℱ_k(ρ_t+1) 
       ⩽   ℰ(0,ρ_t) - ℱ_k(ρ_*)   =  ℱ_k(ρ_t) - ℱ_k(ρ_*) ,

and thus, using that t↦ℱ_k(ρ_t) is non-increasing, there exists R>0 such that

    ℒ(ρ_t+1 ) ⩽ R ℱ_k(ρ_t) - ℱ_k(ρ_*)

for all t⩾ 0. Since Lemma <ref> together with (<ref>) implies an exponential decay of ℒ(ρ_t ) at some rate c>0, we can bound, for t⩾ 1,

    ℱ_k(ρ_t) - ℱ_k(ρ_*)    ⩽   ℒ(ρ_t) 
       ⩽    e^-c(t-1)ℒ(ρ_1)   ⩽ R e^-c(t-1)ℱ_k(ρ_0) - ℱ_k(ρ_*) .

For t∈[0,1] we can simply use that t↦ℱ_k(ρ_t) is non-increasing, and this concludes the proof of (<ref>). The results for 𝒲_2 and the relative entropy then follow by combining (<ref>) with the various bounds relating the different quantities. More specifically, (<ref>) is obtained simply by (<ref>) for t∈[0,1] and, for t⩾ 1, by combining  (<ref>), (<ref>), (<ref>) and Proposition <ref> to bound

    𝒲_2^2 (ρ_t,ρ_*) ⩽ 4  C e^-c(t-1)ℱ_k(ρ_1) - ℱ_k(ρ_*)⩽  C' e^-ct𝒲_2^2 (ρ_0,ρ_*)

for some C'. Similarly, (<ref>) is obtained by using first (<ref>), then (<ref>) and (<ref>) and finally (<ref>) and (<ref>) to bound

    σ^2 ℋ(ρ_t |ρ_*)    ⩽   ℱ_k(ρ_t) - ℱ_k(ρ_*)  + λ𝒲_2^2 (ρ_t,ρ_*) 
       ⩽    C e^-ctℱ_k(ρ_0) - ℱ_k(ρ_*)  + λ𝒲_2^2 (ρ_0,ρ_*)  
       ⩽    C' e^-ctℋ(ρ_0 |ρ_*)

for some C'. The proof of (<ref>) uses the same ingredients.




In the one-dimensional double-well case, Proposition <ref> remains true if the granular media equation (<ref>) is replaced by the Vlasov-Fokker-Planck equation (<ref>). In the subcritical case, it directly follows from Theorem <ref>, thanks to Proposition <ref> and Remark <ref>. In the critical case, the arguments are as in Remark <ref>.



 


§ FAST FREE ENERGY DECAY FOR INTERACTING PARTICLES




 §.§ Settings and results


We consider the particle approximations of (<ref>) and (<ref>), namely the overdamped Langevin process (X_t)_t ≥ 0 solving

    ∀ i∈ 1,N ,     X_t^i = -  E_π(X_t)(X_t^i) t + √(2)σ B_t^i

and the kinetic Langevin process (Y_t,V_t)_t ≥ 0 solving

    ∀ i∈ 1,N ,    {[                                            Y_t^i                                                =                                         V_t^i  t;                                            V_t^i                                                = -  E_π(Y_t)(Y_t^i) t - V_t^i  t + √(2)σ B_t^i  , ].

where in both cases B^1,…,B^N are independent d-dimensional Brownian motions and

    π(x) = 1/N∑_i=1^N δ_x_i

stands for the empirical distribution of x=(x_1,…,x_N)∈^dN. Under (<ref>) and, respectively, (<ref>) for (<ref>) and (<ref>) for (<ref>), both equations admit a unique global strong solution.

Considering first the overdamped case, denote by ρ^N_t the law of X_t=(X_t^1,…,X_t^N) solving (<ref>).
Assuming that

    ∫_^dNexp(-N/σ^2ℰ(π(x)))  x < ∞ ,

we consider the Gibbs measure

    ρ^N_∞∝exp(-N/σ^2ℰ(π(x))) x .

Using that

    ∀ i ∈ 1, N,     ∇_x_i Nℰ(π(x)) = ∇ E_π(x)(x) ,

by the properties of the linear derivative, we see that (<ref>) is an overdamped Langevin diffusion on ^dN with potential Nℰ(π(x)) and temperature σ^2. Its invariant measure is ρ_∞^N, and t ↦ℋ(ρ^N_t|ρ^N_∞) is decreasing (by Jensen inequality) and goes to zero as t→∞ under mild assumptions. Defining the N-particle  free energy of ρ^N ∈𝒫_2(^dN) as 

    ℱ^N(ρ^N) = σ^2ℋ(ρ^N)+ N ∫_^dNℰ(π(x)) ρ^N( x) ,

we see that

    σ^2 ℋ(ρ^N|ρ^N_∞) = ℱ^N(ρ^N)-ℱ^N(ρ^N_∞).

In particular, ρ_∞^N is the global minimizer of ℱ^N and t ↦ℱ^N(ρ^N_t|ρ^N_∞) is non-increasing.


As studied in <cit.>, when the non-linear dynamics (<ref>) admits several stable stationary solutions, the log-Sobolev constant of ρ_∞^N (assuming that a LSI holds, for instance if x↦ℰ(π(x)) is uniformly convex outside a compact set) goes to infinity   as N→∞, and thus we cannot expect a fast convergence of ℱ^N(ρ_t^N) to its infimum (which typically occurs at a time-scale of order e^aN for some a>0). However, in short time, initialized with independent initial conditions, the law of the particle system stays close to the solution of the non-linear problem (<ref>) and thus the local convergence of the latter to some critical point ρ_*∈𝒦 drives the initial behavior of the particle system. This gives the following.



Under Assumption <ref>, assume furthermore (<ref>) and that there exists λ'⩾ 0 such that  for all μ_0,μ_1∈𝒫_2(^d) and t∈[0,1],

    ℰ(tμ_0 + (1-t) μ_1) ⩾ t ℰ(μ_0) +(1-t)ℰ(μ_1) - t(1-t) λ' 𝒲_2^2(μ_0,μ_1)  .

Let ρ_0 ∈𝒫_2(^d) be such that the solution of (<ref>) converges to ρ_* in long time. There exist C,β>0 such that, for all N∈ and t⩾ 1,  considering the particle system (<ref>) with initial distribution ρ_0^N = ρ_0^⊗ N,

    1/Nℱ^N(ρ_t^N) - ℱ(ρ_*) ⩽ C  N^-β + e^-t/ .



Notice that, in view of (<ref>), (<ref>)  holds for instance in the case (<ref>) if the functions r_k are L_k-Lipschitz with ∑_k∈ L_k^2 <∞.




When the solution of the non-linear McKean-Vlasov equations converges to some ρ_*, under suitable  conditions (in particular of regularity), a quantitative weak convergence at rate 1/N for the empirical distribution of the corresponding system of N interacting particles  is shown in 
<cit.> to hold over times of order N^p for any p. This requires the initial condition to start close to ρ_* in the total variation sense. However, thanks to Proposition <ref>, if the initial condition is close to ρ_* in 𝒲_2, then it becomes close in the total variation sense after a time 1.



The same occurs in the kinetic case. For ρ^N ∈𝒫_2(^2dN), define the N-particle  kinetic free energy as 

    ℱ^N_k(ρ^N) =  σ^2ℋ(ρ^N)  N ∫_^2dNℰ(π(x)) + ℰ_k(π(v))ρ^N( x v) .



Under Assumption <ref>, assume furthermore   that there exists λ'⩾ 0 such that  (<ref>)  holds for all μ_0,μ_1∈𝒫_2(^d) and t∈[0,1]. Let ρ_0 ∈𝒫_2(^2d) be such that the solution of (<ref>) converges to ρ_* in long time. There exist C,β,γ>0 such that, for all N∈ and t⩾ 1,  considering the kinetic particle system (<ref>)  with initial distribution ρ_0^N = ρ_0^⊗ N,

    1/Nℱ_k^N(ρ_t^N) - ℱ_k(ρ_*) ⩽ C  N^-β + e^-γ t .







 §.§ Proofs


We mostly focus on the overdamped case in this section, denoting by ρ_t^N the law of (<ref>). We start with the following variation of Proposition <ref>.


For any ρ_* ∈𝒦, there exists C>0 such that for all N∈, t⩾ 0 and any initial distribution ρ_0^N ∈𝒫_2(^dN), 

    ℋρ_t+1^N | ρ_*^⊗ N⩽ C 𝒲_2^2ρ_t^N,ρ_*^⊗ N + C  .


Similarly  to Proposition <ref>, this follows from a Girsanov transform, see <cit.> or <cit.>.


To relate ℋρ_t^N | ρ_*^⊗ N to ℱ^N(ρ_t^N), we can rely on the following.


Assume that there exists λ'⩾ 0 such that (<ref>) holds for all μ_0,μ_1∈𝒫_2(^d) and t∈[0,1]. Then, for all ρ^N ∈𝒫_2(^dN) and ρ_*∈𝒦, 

    ℱ^N(ρ^N) - N ℱ(ρ_*) ⩽σ^2 ℋρ^N|ρ_*^⊗ N + 2 λ'𝒲_2^2 ρ^N,ρ^⊗ N_* + 2λ' N α(N)

where

    α(N) = ∫_^dN𝒲_2^2(π(x), ρ_*)   ρ_*^⊗ N( x) .


Thanks to <cit.>, α(N) ⩽ C N^-2/d for some C (but possibly for some particular ρ_* it may vanish faster).


 
   Dividing by t and sending t to zero in (<ref>) yields

    ∫_^d E_μ_1 (μ_0-μ_1) + ℰ(μ_1) ⩾ℰ(μ_0)  -    λ𝒲_2^2(μ_0,μ_1) .

Then, using this with μ_0 = π(x) and μ_1=ρ_*,

    ℱ^N(ρ^N) - N ℱ(ρ_*) 
        =     N ∫_^dNℰ(π(x)) - ℰ(ρ_*)ρ^N( x) + σ^2ℋ(ρ^N) - N σ^2ℋ(ρ_*) 
       ⩽    N ∫_^dNλ' 𝒲_2^2(π(x),ρ_*) + ∫_^2dE_ρ_*(y) (π(x)-ρ_*)( y)   ρ^N( x) + σ^2ℋ(ρ^N) - Nσ^2 ℋ(ρ_*).

Using that E_ρ_* = -σ^2(lnρ_* + ln Z_ρ_*), 

    N ∫_^dN∫_^2dE_ρ_*(y) (π(x)-ρ_*)( y)     ρ^N( x)     =     N σ^2 ℋ(ρ_*) - σ^2∑_i=1^N ∫_^dNlnρ_*(x_i)   ρ^N( x) 
        =     N σ^2ℋ(ρ_*) - σ^2 ∫_^dNlnρ_*^⊗ N(x)   ρ^N( x)  .

Plugging this in the previous inequality reads 

    ℱ^N(ρ^N) - N ℱ(ρ_*) ⩽σ^2ℋρ_N|ρ_*^⊗ N +  N λ'   ∫_^dN𝒲_2^2(π(x),ρ_*)   ρ^N( x)  .

Let μ( x y) be an optimal 𝒲_2 coupling of ρ^N and ρ_*^⊗ N. We bound

    ∫_^dN𝒲_2^2(π(x),ρ_*)   ρ^N( x)    ⩽    2 ∫_^dN𝒲_2^2(π(x),π(y))   μ( x y) + 2 α(N)  
       ⩽   2/N∑_i=1^N ∫_^dN    |x_i-y_i|^2  μ( x y) + 2 α(N)  
       =     2 𝒲_2^2 ρ^N,ρ^⊗ N_*  + 2 α(N) ,

which completes the proof.






Thanks to (<ref>) and standard finite propagation of chaos estimates (obtained by synchronous coupling), we bound 

    𝒲_2^2ρ_t^N,ρ_*^⊗ N   ⩽     2 𝒲_2^2ρ_t^N,ρ_t^⊗ N + 2 𝒲_2^2ρ_t^⊗ N,ρ_*^⊗ N
       ⩽    C e^Ct  + 2 N C_0 e^-t/

for some C>0. Proposition <ref> and Lemma <ref> yield
  
    ℱ^N(ρ_t^N) - N ℱ(ρ_*)  ⩽ C'   e^Ct  + N  e^-t/ + N α(N)

for all t⩾ 1 for some C'>0. Conclusion follows by using that t ↦ℱ^N(ρ_t^N) is non-decreasing (distinguishing whether t is above or under ln N /(2C)) and applying <cit.> to bound α(N).



The proof is the same as the proof of  Proposition <ref>. Proposition <ref> still holds in the kinetic case (with ρ_*∈𝒦_k; this is <cit.>), and so does Lemma <ref> (replacing ℱ by ℱ_k, with the same proof, noticing that the kinetic energy is linear and thus satisfies (<ref>) with λ'=0). We use Theorem <ref> instead of Theorem <ref>.





 §.§ Acknowledgments


The research of P. Monmarché is supported by the projects SWIDIMS (ANR-20-CE40-0022) and CONVIVIALITY (ANR-23-CE40-0003) of the French National Research Agency. The research of J. Reygner is supported by the projects QUAMPROCS (ANR-19-CE40-0010) and CONVIVIALITY (ANR-23-CE40-0003) of the French National Research Agency.

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