Semi Log-Concave Markov Diffusions

Introduction and main results.

In this paper we shall investigate some properties of time marginals (at time $T$ finite or infinite) of Markov diffusion processes satisfying some logarithmic semi-convexity like property. The properties we are interested in are functional inequalities (Poincaré, log-Sobolev) or transportation inequalities. We shall also give some consequences for the long time behavior of such processes.

Our main tools are on one hand coupling techniques and on the other hand stochastic calculus. We shall mainly use the so called “synchronous” coupling, i.e. using the same Brownian motion, but we also give some new results by using the “mirror” coupling (or coupling by reflection) introduced by Lindvall and Rogers in . The main stochastic tool is (a very simple form of) Girsanov theory and $h$-processes.

The use of coupling techniques for obtaining analytic estimates is far to be new. It is impossible (and dangerous) to give here, even an account of the existing literature (see however and references therein). The use of Girsanov theory for this goal is not new too. We shall recall later some references. The conjunction of both techniques is not usual.

We deliberately decided to present in details the simplest situation of a Brownian motion with a gradient drift, for which almost everything is well known, and then to extend our method to new situations. Some specialists would certainly find that these parts of the present paper are lengthy, but we think that the understanding of how the method works in this simple case is an useful guide for generalizations.

The meaning of logarithmic semi-convexity will generalize the “usual” one we recall now.

This property is called $K$-semi-convexity of $U$. It is clearly equivalent to the convexity of $U(x)-K|x|^{2}$. We denote $\Upsilon(dx)=e^{-U(x)}dx$ the Boltzmann measure associated to the potential $U$. If $e^{-U}$ is $dx$ integrable, we also introduce the normalized $\mu(dx)=\frac{1}{Z_{U}}\,e^{-U(x)}\,dx$ which is a probability measure. If $U$ is semi-convex, $\mu$ is said to be semi log-concave.

Consider first the diffusion process, given by the solution of the Ito stochastic differential system

and $\Gamma$ denotes the carré du champ, namely here

It is known that $\Upsilon$ is a symmetric (reversible) measure for the diffusion process, and is actually the unique invariant (stationary) measure for the process. If $\Upsilon$ is bounded, $\mu$ is ergodic.

In the latter case, $P_{t}$ is thus a symmetric semi-group on $L^{2}(\mu)$. The domain $D(L)$ of its generator contains the algebra $\mathcal{A}$ generated by the constant functions and $C_{c}^{\infty}$. In particular, if $f\in\mathcal{A}$, $\partial_{t}P_{t}f=LP_{t}f=P_{t}Lf$ in $L^{2}(\mu)$, so that since $\partial_{t}-L$ is hypo-elliptic $(t,x)\mapsto P_{t}f(x)\in C^{\infty}$.

$L$ is the basic example of generator satisfying the celebrated $C(K/2,+\infty)$ Bakry-Emery curvature condition (see ). Indeed if we define

(H.C.K) is equivalent to $\Gamma_{2}(f)\geq(K/2)\,\Gamma(f)$.

This curvature condition is known to imply (and is in fact equivalent to) a lot of nice inequalities for the semi-group, in particular for all $T>0$ and all $x$, a commutation between $\Gamma$ and the semi group $P_{t}$ holds, namely

which in turn implies powerful functional inequalities such as

Recall that $\nu$ satisfies a log-Sobolev inequality with constant $C_{LS}$ if

(1.3) is exactly what is (a little bit improperly) called a “local” log-Sobolev inequality in (theorem 5.4.7). For further informations and more, see the forthcoming book . The reader has to be careful with the constants, since we are writing the log-Sobolev inequality (as well as all other inequalities) in the usual form, while the “Bakry-Emery” version uses $\Gamma$, hence with an extra factor 2.

It is well known that a log-Sobolev inequality implies a Poincaré inequality

with $C_{P}=\frac{1}{2}\,C_{LS}$, as well as a $T_{2}$ transportation inequality

with $C_{W}=\frac{1}{2}\,C_{LS}$. Here $W_{2}$ denotes the Wasserstein distance between the probability measures $\eta$ and $\nu$, i.e.

where $\pi$ is a coupling of $\eta$ and $\nu$ (i.e. has respective marginals equal to $\eta$ and $\nu$) and

denotes the Kullback-Leibler information or relative entropy of $\eta$ w.r.t. $\nu$. The latter property is due to Otto-Villani . Another approach and related properties were developed by Bobkov, Gentil and Ledoux . For a nice survey on transportation inequalities we refer to . One can find in all these references another remarkable consequence of semi log-concavity, namely that a log-Sobolev inequality derives from a transportation inequality. This is a consequence of the following (H.W.I) inequality that holds for any nice $\mu$ density of probability $h$,

As a consequence, if (H.C.K) holds for some $K\leq 0$, a $T_{2}$ transportation inequality for $\mu$ implies a log-Sobolev inequality with constant $C_{LS}\leq(4/C_{W})\,(1+(K/C_{W}))^{-2}$ provided $1+(K/C_{W})>0$, in particular if $K=0$. Let us finally remark that the starting point of this approach is the $\Gamma_{2}$ commutation property (1.2) which fails however to give a direct proof of the $T_{2}$ inequality.

Our first goal is to show that functional and transportation inequalities can be derived, in the previous situation, by using synchronous coupling and simple tools of stochastic calculus. This is done in section 2. The methods are then extended to a more general framework which is as natural for studying properties of time marginals as the $\Gamma_{2}$ framework.

Indeed, consider a classical diffusion process, given by the solution of an Ito stochastic differential system

$B_{.}$ being a standard brownian motion. For simplicity we assume that $\sigma$ is a squared matrix. We extend (H.C.K) to this new situation

Notice that if $\sigma$ and $b$ are $C$-Lipschitz, i.e. each component is $C$ Lipschitz, (H.C.K) is satisfied for $K=-(C^{2}n^{2}+C)$, but if $\sigma$ is $C$-Lipschitz, (H.C.K) can be satisfied for a non-negative $K$ provided $b$ is sufficiently repealing. Contrary to the case of a constant diffusion coefficient, (H.C.K) is not related to the Bakry-Emery curvature condition which involves in this situation controls on derivatives of higher order of the coefficients.

For simplicity in the sequel we shall assume that $\sigma\in C_{b}^{2}$ hence is $C$-Lipschitz and that $b$ is $C^{2}$, but not necessarily bounded nor with bounded derivatives. With these assumptions, once again if we assume that (H.C.K) is in force, (1.7) admits a unique non explosive strong solution using $x\mapsto x^{2}$ as a Lyapunov function for non explosion. We shall show this and other properties of the process in subsection 3.1.

We still use the notations introduced before, but now

Our first results can be gathered in Theorem 1.8 below, after introducing some additional assumptions.

Hypothesis (R). One of the following assumptions is satisfied (in addition to the fact that $\sigma\in C_{b}^{2}$ and $b\in C^{2}$).

$\sigma\in C_{b}^{\infty}$, $b\in C^{\infty}$ and has at most polynomial growth and $\partial_{t}-L$ is hypo-elliptic,

$\sigma=a^{1/2}$, i.e. $\sigma$ is symmetric.

Actually, assumptions (R1)-(R4) ensure that for $f\in\mathcal{A}$,

which is what we really need. (R5) is a limiting situation for (R3) as we will see in (sub)subsection 3.1.2. Of course the time marginal distributions only depend on $L$ and not on $\sigma$, but the constant $K$ is related to $a^{1/2}$ in (R5) (note that the Hilbert Schmidt norm of $\sigma(x)-\sigma(y)$ can change when we change $\sigma$ without modifying $a$).

Assume that (R) and (H.C.K) are satisfied. Let $M=\sup_{|u|=1}\,\sup_{x}\,|\sigma(x)u|^{2}$.

If $\mu_{0}$ satisfies a Poincaré inequality with constant $C_{P}(0)$ then $\mu_{T}$ satisfies a Poincaré inequality with constant

When $K=0$ one has to replace $\frac{(1-e^{-KT})}{K}$ by $T$. This applies in particular to $P(T,x,.)$ with $C_{P}(0)=0$.

$P(T,x,.)$ satisfies a $T_{2}$ transportation inequality with constant $C_{T}=C_{T}(M,K)$, bounded in time if $K>0$, linear in time if $K=0$ and exploding exponentially in time if $K<0$.

If $\mu_{0}$ satisfies $T_{2}$ with constant $C_{W}(0)$, $\mu_{T}$ satisfies $T_{2}$ with a constant $C_{W}(T)=C_{W}(T,M,K,C_{W}(0))$, bounded in time if $K>0$, linear in time if $K=0$ and exploding exponentially in time if $K<0$.

When $\sigma=Id$, if $\mu_{0}$ satisfies a log-Sobolev inequality with constant $C_{LS}(0)$, $\mu_{T}$ satisfies a log-Sobolev inequality with constant

This applies in particular to $P(T,x,.)$ with $C_{LS}(0)=0$.

Some other consequences, as for example convergence to equilibrium (when it exists) are also discussed in particular in subsection 3.7.

Of course, (1) is a weaker version of the commutation relation (1.2) and (5) is nothing else than (1.3) when $2b=-\nabla U$. When a diffusion coefficient is present, (1) is however very different from the usual commutation property. For example we will show that it holds even in the negative infinite curvature case, but that it still enables us to provide interesting local functional inequalities. (3) as well as the general version of (H.C.K) we have introduced appeared (for this kind of application and to our knowledge) for the first time in the paper by Djellout, Guillin and Wu , Theorem 5.6 and condition 4.5 therein, for $K>0$. Our scheme of proof for the transportation inequality, based on Girsanov theory, is actually a simplified version of the one in , but instead of looking at the full law of the process on a time interval we shall use $h$-processes in order to look at time marginals. What we shall show is that the same scheme of proof also furnishes functional inequalities. This unified treatment of functional inequalities and transportation inequalities using an ad-hoc coupling is the novelty here. It easily extends to time dependent coefficients as shown in section 4.

In addition in this section we show how to directly obtain convergence to equilibrium and properties of the invariant measure for non linear diffusions of Mc Kean-Vlasov type, simplifying arguments in .

The use of stochastic calculus in deriving such inequalities is not new but only a small number of papers dealt with. One can trace back to the paper of Borell , who used Girsanov theory to study the propagation of log-concavity along the Schrödinger dynamics (not the Fokker-Planck one we are looking at here). In addition to for transportation inequalities, one can also mention where similar ideas are used to study hyper-boundedness. More recently, using similar arguments, Lehec has studied gaussian functional inequalities and Fontbona and Jourdain obtained a pathwise version of the $\Gamma_{2}$ theory.

Let us come back to (1.1). (H.C.K) (or the $\Gamma_{2}$ theory) for $K>0$ applies to potentials $U$ which are the sum of $(K/2)|x|^{2}$ and of a concave (hence sub-linear) potential $V$. In particular, for “super-convex” potentials like $|x|^{\beta}$ with $\beta>2$, or more generally for (smooth) potentials $U$ which are uniformly convex at “infinity”, (H.C.K) holds but with a negative $K$ due to the behavior of $U$ near the origin, so that, according to theorem 1.8, $\mu_{T}$ satisfies functional inequalities but with exploding constants in $T$.

It is however well known, since $U=V+W$ with $V$ $K$-uniformly convex and $W$ bounded, that $\mu(dx)=e^{-U(x)}dx$ satisfies a log-Sobolev (and a Poincaré) inequality with a constant $C_{LS}=(2/K)\,\exp(\textrm{Osc}W)$ where Osc denotes the oscillation of $W$. One can thus expect that $C_{LS}(T)$ is bounded in $T$. In section 5, we introduce the following extension of (H.C.K).

When $\alpha(a)=1$, we may take $\varepsilon=0$ and we recognize (H.C.K). In Proposition 5.4 we show that $U(x)=|x|^{2\beta}$ (with $\beta\geq 1$) satisfies (H.$\alpha$.$K_{\beta}$) for $\alpha(a)=a^{\beta-1}$ and an explicit $K_{\beta}>0$.

The main result of this section is then that, for suitable functions $\alpha$,

if (H.$\alpha$.K) holds (for $K>0$), then $\mu$ satisfies a log-Sobolev inequality.

See theorems 5.20 and 5.24. These theorems thus (partly) extend the Bakry-Emery criterion (1.3) to some non uniformly convex potentials. However, they are dealing with the invariant measure only and not with the law at time $T$ (only incomplete results are proved in this section for these distributions).

The next section 6 is devoted to the use of mirror coupling. In a recent work , Eberle has adapted the mirror coupling to get estimates of $W_{1}$ convergence for drifted brownian motions when the drift satisfies some “convexity at infinity” property. We recall Eberle’s method and obtain some new consequences of his result. In addition, up to an extra condition, we show that his result (and all the consequences we derived) can be extended to general elliptic diffusion processes. We will also use this mirror coupling to show that we may get a weak version of the commutation property in the log concave case with the “convexity at infinity” property at least in dimension one, which is the first result we know of in this direction. Still in dimension one, we will also consider using mirror coupling for non linear diffusions.

Section 7 is peculiar. Using the results we have described for the Ornstein-Uhlenbeck process we show how to recover known results on the stability of functional inequalities under convolution (provided one of the terms is gaussian).

Semi log-concave drifted brownian motion.

In this first warming up section we shall look at the usual situation given by (1.1)

For functional inequalities the key is a commutation property of the gradient and the semi group. This commutation property is almost immediate using an appropriate coupling as explained below :

In the situation of (1.1), assume (H.C.K). Then for all $f\in\mathcal{A}$,

Applying Ito formula yields (almost surely)

for some $z_{t}$ sandwiched by $X_{t}^{x}$ and $X_{t}^{y}$. It remains to use the continuity (and boundedness) of $\nabla f$ and the fact that $X_{t}^{y}$ goes almost surely to $X_{t}^{x}$ as $y\to x$ to conclude. ∎

As is seen from the proof, in fact, the sole convergence of the Wasserstein distance is not sufficient to get the commutation property exposed here. It will however be our starting point for the result when a diffusion coefficient is present. The synchronous coupling here enables us however to get an almost sure “deterministic” control of $X_{t}^{x}-X_{t}^{y}$ which is far more powerful. $\diamondsuit$

We recall previously that (H.C.K) is exactly the $\Gamma_{2}$ condition of Bakry-Emery in this context, which is in fact equivalent to (2.3). However the proof is very different from ours: it relies on a tricky calculus on $\psi(s)=e^{-Ks/2}\,P_{s}\sqrt{\Gamma(P_{t-s}f)}$ to show that $\psi^{\prime}(s)\geq 0$. $\diamondsuit$

If instead of $(x,y)$ the processes start with initial distribution $\pi_{0}$ the “optimal coupling” between $\mu_{0}$ and $\nu_{0}$ for the $W_{2}$ distance, the previous shows that $W_{2}^{2}(\mu_{T},\nu_{T})\leq e^{-KT}\,W_{2}^{2}(\mu_{0},\nu_{0})$. As discussed in the Appendix, this result can be used to show the existence and uniqueness of the invariant measure. $\diamondsuit$

2. hℎh-processes and functional inequalities.

We now introduce the standard notion of $h$-process. Let $T>0$ and $h$ be a non-negative function such that $\int\,h\,d\mu_{T}=1$. For simplicity, we assume for the moment that there exist $c$ and $C$ such that $C\geq h\geq c>0$. We thus may define on the path-space up to time $T$ a new probability measure

In this situation, it is well known (Girsanov transform theory) that one can find a progressively measurable process $u_{s}$ such that

Actually, if $h\in\mathcal{A}$, it is immediate to check (applying Ito formula) that

If $h$ is smooth we may apply Proposition 2.2 in order to get

where we have used Cauchy-Schwarz inequality for the second inequality and the Markov property for the third one. The previous inequality then extends to any $h$ in $C^{1}$ for which the right hand side makes sense, by density. We have thus obtained the following

In the situation of (1.1), assume (H.C.K). If $\mu_{0}$ satisfies a log-Sobolev inequality with constant $C_{LS}(0)$, $\mu_{T}$ satisfies a log-Sobolev inequality with constant

When $K=0$ one has to replace $\frac{(1-e^{-KT})}{K}$ by $T$. This applies in particular to $\mu_{T}=P(T,x,.)$ since $\delta_{x}$ satisfies a log-Sobolev inequality with constant equal to .

Apply the log-Sobolev inequality to $\mu_{0}$. It furnishes (since $\int P_{T}hd\mu_{0}=1$),

similarly as what we did in (2.9). Hence the result applying (2.9). ∎

As we recalled in the introduction a log-Sobolev inequality implies a $T_{2}$ transportation inequality. It is interesting to see that one can directly obtain such an inequality for semi log-concave measures, by using the previous construction. But before to do this, just remark that the above proof using $h=1+\varepsilon g$ with $\int gd\mu_{T}=0$ allows us to obtain a similar result replacing the log-Sobolev inequality by a Poincaré inequality i.e.

In the situation of (1.1), assume (H.C.K). If $\mu_{0}$ satisfies a Poincaré inequality with constant $C_{P}(0)$, $\mu_{T}$ satisfies a Poincaré inequality with constant

When $K=0$ one has to replace $\frac{(1-e^{-KT})}{K}$ by $T$. This applies in particular to $\mu_{T}=P(T,x,.)$ since $\delta_{x}$ satisfies a Poincaré inequality with constant equal to .

Once again, the proof presented here is very different from the one based on the $\Gamma_{2}$ calculus of Bakry-Emery which relies on the commutation property and the control of the derivative of $\psi(s)=P_{s}(P_{t-s}f\log(P_{t-s}f))$ to get a local logarithmic Sobolev inequality. Note that considering rather $\psi(s)=P_{s}((P_{t-s}f)^{2})$ leads to a local Poincaré inequality. $\diamondsuit$

3. Transportation inequalities.

The existence of $u_{s}$ and (2.7) are ensured as soon as $H(h\mu_{T}|\mu_{T})<+\infty$ (see ). For our goal we do not need the explicit expression of $u_{s}$.

We have then different alternatives depending on the sign of $K$.

1) First in the case where $K>0$, one has using that $2ab\leq Ka^{2}+b^{2}/K$

so that we recover an uniform transportation inequality when $\eta_{0}=0$, which is moreover optimal for the invariant measure, considering logarithmic Sobolev inequality and Poincaré inequality. If $\mu_{0}$ satisfies some transportation inequality then one obtains that $\mu_{T}$ satisfies a transportation inequality with constant the sum of the initial constant plus $\frac{2}{K}$.

2) The previous simple argument has however a serious drawback in the sense that in positive curvature, $\mu_{T}$ does not forget the “initial measure”. Let us see how to deal with this problem. Start once again from the first estimation, but using Itô’s formula between $t$ and $t+\varepsilon$ and (H.C.K)

so that we may differentiate in time to get for all positive $\lambda$

Note that this is once again optimal for the limiting measure, and captures the fact that it forgets the initial condition. When $K<0$, we then have

Note however the presence of the additional parameter $\lambda$.

3) Let us see how a direct approach may get rid of this additional parameter, which is particularly important in negative curvature. Define

Since $\frac{e^{Kt}}{K}\leq\frac{e^{KT}}{K}$ we obtain

If $\eta_{0}=0$, (2.14) can be improved in

Again if $\eta_{0}=0$, (2.16) can be improved in

The previous inequalities then extend to any non-negative $h$ (not necessarily bounded below nor above).

If we choose $\mu_{0}=\delta_{x}$, we have $\mu_{T}=P(T,x,.)$, $1=\int hd\mu_{T}=P_{T}h(x)$ and so $\nu_{0}=\delta_{x}$ and $\nu_{T}=\mu_{T}$. Hence

In the situation of (1.1), assume (H.C.K). Then $P(T,x,.)$ satisfies a $T_{2}$ transportation inequality

If we choose $\nu_{0}=\mu_{0}$, we may use the convexity of $t\mapsto t\log t$, i.e

In the situation of (1.1), assume (H.C.K). If $\mu_{0}$ satisfies $T_{2}$ with constant $C(0)$, then $\mu_{T}$ satisfies $T_{2}$ with a constant $C(T)$ given,

and when $K\leq 0$, $C(T)=C_{T}+\frac{\sqrt{2}}{2}+B_{T}\,C(0)$ with

All what precedes holds even if $\Upsilon$ is not bounded (i.e. if the process is not positive recurrent), in which case of course, $K<0$. $\diamondsuit$

If we choose $\mu_{0}=\mu$ (assuming that $\Upsilon$ is bounded), we have to choose $\nu_{0}=P_{T}h\mu$ hence $\nu_{T}=P_{2T}h\,\mu_{0}$. After noticing that we can slightly refine the previous bound replacing $H(h\mu_{T}|\mu_{T})$ by $H(h\mu_{T}|\mu_{T})-H(P_{T}h\mu_{0}|\mu_{0})$ according to (2.7), we obtain

The latter has to be compared with remark 4.9 in which shows that the inequality

If $K>0$ we may let $T$ go to $+\infty$ in Proposition 2.10 and recover that $\mu$ satisfies a log-Sobolev inequality with constant $2/K$, hence a $T_{2}$ transportation inequality with constant $1/K$ (in particular we are loosing a factor $4$ in Proposition 2.18). Similarly, when $T\to+\infty$, (2.22) shows that if $K>0$, $\mu$ satisfies a $T_{2}$ inequality , and since $\mu$ is log-concave, satisfies a log-Sobolev inequality. This scheme of proof does not require Proposition 2.2, but the (H.W.I) inequality. Unfortunately it does not furnish the optimal constant. $\diamondsuit$

4. Transportation-Fisher Inequalities.

Let us see look now at another type of Transportation Information inequality recently introduced in , which is weaker but quite close to logarithmic Sobolev inequality (in fact equivalent under bounded curvature). We are obliged to come back to the initial inequality in (2.3) which becomes in our new situation

Replacing the pair $(0,t)$ by $(t,t+\varepsilon)$ we thus have

It follows that $t\mapsto\eta_{t}$ is differentiable and satisfies,

(for the second inequality, recall that (H.C.) is satisfied so that, for short, $|\nabla P_{s}|\leq P_{s}|\nabla|$.) To explore (2.25) we shall use the usual trick $ab\leq\lambda a^{2}+\frac{1}{\lambda}\,b^{2}$ for $a,b,\lambda$ positive. Hence

This inequality is close to what is called a $W_{2}I$ inequality (see definition 10.4 or for examples and details on properties of WI inequality). Here we obtain a defective $W_{2}I$ inequality. However, as $T\to+\infty$, we recover the true $W_{2}I$ inequality for the invariant distribution, which together with the (H.W.I) inequality allows us to recover the log-Sobolev inequality. Nevertheless, we get

Assume (H.C.K), then $P(T,x,\cdot)$ satisfies a WI inequality of constant $\frac{2(1-e^{-AT})}{A\lambda}$. If we suppose moreover that $\mu_{0}$ satisfies a WI inequality with constant $D(0)$ then $\mu_{T}$ satisfies a WI inequality with constant $D(T)=e^{-AT}D(0)+\frac{2(1-e^{-AT})}{A\lambda}$.

As remarked, under (H.C.K), the inequalities verified by the law $\mu_{T}$ depend on the inequalities verified by the initial measure, in the range between Poincaré and logarithmic Sobolev inequality. Indeed, a logarithmic Sobolev inequality implies a WI inequality, but to get the the WI inequality for $P_{T}$ we need only a WI inequality for the initial measure. As seen by the example of the Gaussian measure, which satisfies (H.C.K), no stronger inequalities can be obtained. $\diamondsuit$

Instead of the $h$-process one should consider Schrödinger bridges allowing to choose both the initial and the final time marginals. Indeed if $h$ is bounded it is known that one can find non-negative functions such that the measure

and the drift $u_{s}$ is given by $u_{s}=\nabla\,\log P_{T-s}g$. As before it is immediately checked that

For all this we refer to p.162 and section 6. Even if $h$ is bounded from below, we do not know whether $g$ inherits this property. Nevertheless, at least formally we have the relation

Unfortunately, these inequalities do not seem to give new results. $\diamondsuit$

In almost all what we did we may replace the drift $-\frac{1}{2}\,\nabla U$ by a general (non-gradient) smooth drift $b$ satisfying

All the results of this section remain true in this more general situation, as far as we do not use reversibility. The only result where we used reversibility actually is (2.22). Indeed, if the initial law is $\nu_{0}=P_{T}h\mu$, $\nu_{T}=P_{T}^{*}P_{T}h\,\mu$ where $P_{T}^{*}$ denotes the $\mu$ adjoint semi-group.

The only delicate point is the smoothness of $P_{t}h$ and the fact that $\partial_{t}(P_{t}h)=LP_{t}h$ in the usual sense. This will be discussed in an even more general setting in the next section, where we shall look at more general cases with non constant diffusion coefficient.

General diffusion processes.

We shall now extend the results of the previous section to the general situation of (1.7),

First of all we have to discuss some properties of the process and the associated quantities. As we said in the introduction, we need some regularity for $P_{t}f$ at least if $f\in\mathcal{A}$. So there is a technical price to pay. We decided to pay this price at the level of the study of the process, rather than in deriving inequalities.

Since we assume that $\sigma\in C_{b}^{2}$, when (H.C.K) is fulfilled, $b$ satisfies

Thus, if $S_{k}$ denotes the exit time from the ball $B(x,k)$, and $t_{k}=t\wedge S_{k}$ it holds

where, since $\sigma$ is bounded, we have defined

It is then easily seen that one can perform similar calculations with $g(t,x)=\exp(e^{-Ct}|x|^{2})$ for a large enough $C$ in order to kill the integrated term, i.e

It is interesting to notice that one can similarly obtain some “deviation” bound from the starting point. Indeed

so that arguing as we did in order to get (2.14) and (2.16) we obtain the existence of constants $\alpha(T,D)$ and $\beta(T,D)$ such that for $0\leq t\leq T$,

1.2. Properties of the semi-group.

Let us mimic what we did to get Proposition 2.2 i.e. apply Ito formula to get

Notice that with our assumptions, the right hand side of (3.4) is a (true) martingale, so that

Moreover, if $\sigma\sigma^{*}$ is positive then (3.7) implies back (H.C.K.). If we suppose moreover for some $m\geq 2$

The contraction in $W_{2}$ distance inherited from (H.C.K.) has already been proved. The contraction in $W_{m}$ distance is done exactly in the same way using once again synchronous coupling. The necessary part comes from (or more precisely section 4. in the Arxiv version 1110.3606). Let us explain the ideas of the proof. In fact, one may compute the time derivative of the Wasserstein distance: note $M:N=\sum_{i,j}M_{ij}N_{ij}$ when $M$ and $N$ are two matrices, then denoting $\nu_{t}$ and $\mu_{t}$ two solutions starting respectively from $\,u_{0}$ and $\mu_{0}$

where if $\nu_{t}=\nabla\phi_{t}\#\mu_{t}$,

Then the contraction property implies that at time 0 for $\nu_{0}=\delta_{y}$ and $\mu_{0}=\delta_{x}$

A clever choice of $\phi$ then enables to prove the result. ∎

Let $f$ be $C$-Lipschitz continuous. It holds $|f(X_{t}^{x})-f(X_{t}^{y})|\leq C\,|X_{t}^{x}-X_{t}^{y}|$ so that, using (3.5), $P_{t}f$ is Lipschitz continuous with Lipschitz constant less than $C\,e^{-Kt/2}$.

As we said at the end of the previous section, when $K>0$ one deduces the existence and uniqueness of an invariant probability measure $\mu_{\infty}$, to which $\mu_{T}$ converges weakly.

The rest of this (sub)subsection is devoted to give a proof of the following: if $f\in\mathcal{A}$ (see the introduction), then $(t,x)\mapsto P_{t}f(x)$ is regular and satisfies (for $t>0$)

The reader who takes this result as granted can skip what follows.

First if $f\in\mathcal{A}$, $Lf$ is $C_{c}^{0}$ and we have $P_{t}f(x)-f(x)=\int_{0}^{t}\,P_{s}(Lf)(x)\,ds$. It follows that $\lim_{s\to 0}\frac{1}{s}\,(P_{t+s}f(x)-P_{t}f(x))=P_{t}(Lf)(x)$ for all $x$, since $v\mapsto P_{v}Lf(x)$ is continuous. So $\partial_{t}P_{t}f=P_{t}Lf$. The first delicate point is of course the commutation of $L$ and $P_{t}$. The second delicate point is the smoothness of $(t,x)\mapsto P_{t}f(x)$.

This commutation property is known if $\sigma$ and $b$ are in $C_{b}^{\infty}$ (see p.254-258, boundedness of derivatives is important) in which case $(t,x)\mapsto P_{t}f(x)$ is actually $C^{\infty}$. But assuming boundedness of $b$ and its derivatives will exclude the cases of positive $K$.

We shall first show that $P_{t}f$ is a mild solution, provided $b$ does not grow too fast.

It follows from lemma 3.2 that if $f\in E$, $P_{t}f$ is bounded by $c(t)\parallel f\parallel$, hence $P_{t}$ is a continuous semi-group on $E$ (whose range is included into $C_{b}$). To see that $f\in\mathcal{A}$ belongs to the domain of the generator of $P_{t}$, we have to show that the convergence of $\lim_{s\to 0}\frac{1}{s}\,(P_{s}f(x)-f(x))$ holds for the norm defined on $E$. But

according to (3.3). Since $|b(x)|\leq C\,(1+|x|^{k})$, convergence holds for the norm on $E$. The proof is completed. ∎

If the coefficients are $C^{\infty}$, and $\partial_{t}-L$ is hypo-elliptic (for instance if $L$ is uniformly elliptic) it follows that $x\mapsto P_{t}f\in C^{\infty}$ and that the last equalities hold in the usual sense.

If we do not want to assume too much regularity on the coefficients, we have first to assume that $L$ is uniformly elliptic and call upon P.D.E. theory. If what follows is certainly well known by specialists, we include the argument.

Let $f\in\mathcal{A}$. For $k$ large enough, $B_{k}=B(0,k)$ contains the support of $f$. Consider the parabolic equation

Since $f=0$ on $\partial B_{k}$ this makes sense. If $L$ is uniformly elliptic, it is known that there exists an unique solution $u_{k}$ in $C^{1,2}((0,T)\times B_{k})$ of (3.10), and that this solution is represented as

where $S^{x}_{k}$ denotes the exit time from $B_{k}$ of $X_{t}^{x}$. For all this see , in particular Theorem 5.2. p.147. It follows in particular that for all $k$, $\parallel u_{k}\parallel_{\infty}\leq\parallel f\parallel_{\infty}$ and that for all $(t,x)\in(0,T)\times B_{j}$, $u_{k}(t,x)\to P_{T-t}f(x)$ as $k\to+\infty$ since $S_{k}^{x}\to+\infty$. Now let $j$ be fixed, and look at $k>j$. The parabolic Schauder estimate tells us that there exists a constant $C_{j}$ depending on $j$, the ellipticity constant and the $C^{2}(B_{j+1})$ norms of $\sigma$ and $b$ such that

where $C^{k,\frac{1}{2}}$ is the set of $C^{k}$ functions with $\frac{1}{2}$-Hölder $k^{th}$ derivatives. Arzela-Ascoli theorem tells us that a subsequence of $u_{k}$ converges in $C^{2}((0,T)\times B_{j})$, and since the limit is $P_{T-t}f$, that the latter is $C^{2}$.

If $L$ is not uniformly elliptic, we may approximate it by $L_{\varepsilon}=L+\frac{1}{2}\,\varepsilon\Delta$ for $\varepsilon\to 0$. If $\sigma\in C_{b}^{2}$, the diffusion matrix (field) $a_{\varepsilon}=\sigma_{\varepsilon}\,\sigma^{*}_{\varepsilon}+\varepsilon\,Id$ is $C_{b}^{2}$ and uniformly elliptic). It is known that its square root (in the sense of symmetric matrices) $a^{1/2}_{\varepsilon}$ is bounded and $C_{b}^{2}$ too. In addition $a^{1/2}_{\varepsilon}\to a^{1/2}$ as $\varepsilon\to 0$ the convergence taking place in $C_{b}^{1}$. In particular, (H.C.K($\varepsilon$)) holds for the new diffusion process with a constant $K(\varepsilon)$ going to $K$ as $\varepsilon\to 0$, provided (H.C.K) is satisfied for $\sigma=a^{\frac{1}{2}}$.

It follows that the convergence of time marginals holds in $W_{2}$ Wasserstein distance, hence in the weak topology.

The conclusion of the previous discussion is the following: if $a^{1/2}=\sigma\in C_{b}^{2}$, we may assume that $L$ is uniformly elliptic as far as the bounds we get do not depend on the ellipticity constant, and then go to the limit. We shall use this trick in the sequel.

2. Commutation property with the gradient

Once again, we will see how synchronous coupling or a contraction in Wasserstein distance provides the commutation property.

where the last step is done by using Theorem 3.6. Provided we know that $\nabla P_{t}f$ exists, we have thus obtained a weaker form of Proposition 2.2,

Assume (R) and (H.C.K) or the weaker contraction property (3.7). Let $f\in\mathcal{A}$. If $\nabla P_{t}f$ exists (which is true except possibly for (R5)), it holds

Notice that, contrary to the Bakry-Emery bounded curvature case, the previous commutation property holds with the usual gradient and not with the natural one i.e. $\Gamma^{\frac{1}{2}}$.

If Proposition 2.2 allowed us to obtain logarithmic Sobolev inequalities, the weaker Proposition 3.11 will allow us to obtain a weaker inequality, namely a Poincaré inequality.

It is worth mentioning here the following alternate proof of the commutation property, starting from Wasserstein contraction, as derived in the recent paper following our suggestion, i.e. using Kantorovitch-Rubinstein duality we have for all bounded Lipschitz $\phi$ denoting the inf convolution operator $Q_{t}\phi(x)=\inf_{y}\{\phi(y)+\frac{|x-y|^{2}}{2t}\}$ and initial measure $\mu_{0}$ and $\nu_{0}$

Choose now $\mu_{0}=\delta_{x}$, $\nu_{0}=\delta_{y}$ to get for all $y$

which by homogeneity of the inf-convolution operator gives

This assertion is in fact stronger than the gradient commutation property which can be deduced by using the fact that the inf-convolution operator is the Hopf-Lax solution of the Hamilton-Jacobi equation. $\diamondsuit$

3. h-processes and functional inequalities.

We now introduce the corresponding $h$-process. Let $T>0$ and $h>0$ be such that

We thus may define on the path-space up to time $T$ a new probability measure

For simplicity, we assume in what follows that there exist $c$ and $C$ such that $C\geq h\geq c>0$. In this situation, using again Girsanov transform theory, we know that we can find a progressively measurable process $u_{s}$ such that

If $h\in\mathcal{A}$ we may apply Proposition 3.11 in order to get (recall that $h\geq c$)

where we have used the Markov property for the second inequality.

Now let $g\in C_{c}^{\infty}$ be such that $\int gd\mu_{T}=\int P_{T}g\,d\mu_{0}=0$ and choose $h=1+\eta g\in\mathcal{A}$ so that $\int P_{T}hd\mu_{0}=1$ and $h>c>0$ for $\eta$ small enough. Actually we will let $\eta$ go to so that in the limit $c=1$. Standard manipulations thus yield

We can eventually use first the density of $C_{c}^{\infty}$ and then the trick we formerly described in order to relax the uniform ellipticity assumption (recall that $M=\sup_{y}\,\sup_{|u|=1}|\langle u,a(y)u\rangle|$ hence only depends on $a$ too).

Arguing as for Proposition 2.10 we have obtained

Assume that (R) and (H.C.K) are satisfied. Let $M=\parallel|\sigma|^{2}\parallel_{\infty}$. If $\mu_{0}$ satisfies a Poincaré inequality with constant $C_{P}(0)$ then $\mu_{T}$ satisfies a Poincaré inequality with constant

This applies in particular to $P(T,x,.)$ with $C_{P}(0)=0$.

Contrary to the log-Sobolev inequality, the Poincaré inequality does not furnish a transportation inequality, so we shall try to adapt what we did in subsection 2.3.

4. Transportation inequalities.

We may thus conclude as in the previous section

Assume that (R) and (H.C.K) are satisfied. Let $M=\parallel|\sigma|^{2}\parallel_{\infty}$. The conclusions of Proposition 2.18 and Proposition 2.19 are still true, replacing $C_{T}$ by $MC_{T}$

Actually, when (R5) holds, we have proven this result for $h\in\mathcal{A}$ and $L+\frac{\varepsilon}{2}\,\Delta$. But as we have seen, $\mu_{T}(\varepsilon)\to\mu_{T}$ in $W_{2}$ distance, so that if $h$ is bounded the same holds for $z_{\varepsilon}\,h\mu_{T}(\varepsilon)$ to $h\mu_{T}$ ($z_{\varepsilon}$ being a normalization constant). Finally if a $T_{2}$ inequality holds for all $h\in\mathcal{A}$ it extends to all $h$ using density and the fact that $W_{2}(\nu,\mu)\leq\liminf W_{2}(\nu_{n},\mu)$ if $\nu_{n}$ weakly converges to $\nu$.

Of course a $T_{2}$ inequality implies a Poincaré inequality, but the constant in Proposition 3.17 is better (in addition we only require that $\mu_{0}$ satisfies a Poincaré inequality).

One of the renowned consequence of such inequalities is the concentration of measure phenomenon for $\mu_{T}$. In particular, under the assumptions of Proposition 3.20, $\mu_{T}$ satisfies a gaussian type concentration property. In particular $|X_{T}^{x}|^{2}$ has some exponential moment, fact we have already shown in lemma 3.2. But this integrability does not reflect all the strength of the $T_{2}$ inequality whose tensorization property is particularly useful for statistical purposes.

When $L$ is uniformly elliptic, this concentration property follows from gaussian estimates for the transition kernel. Here we obtain much more explicit constants (even if they are certainly far from optimality) which do not depend on the ellipticity constant. $\diamondsuit$

Assume that $L$ is uniformly elliptic, i.e.

According to proposition 5.4.1, this is equivalent to the $CD(K/2,\infty)$ condition provided $M=e$ hence when $\sigma$ is constant times the identity. In the non constant diffusion case, our condition (H.C.K) seems to be really different from the Bakry-Emery curvature condition. $\diamondsuit$

5. An hypoelliptic example : kinetic Fokker-Planck equation

We present in this section an application of the techniques developed here in an hypoelliptic example where the Bakry-Emery curvature is negative and where (H.C.K.) may not be satisfied also. Let $(x_{t},v_{t})$ be the solution of the following SDE

also called stochastic Hamiltonian system. The long time behavior study of such a system has been considered for a long time and have been tackled by different techniques, see for example: hypocoercivity by Villani or Lyapunov function technique by Bakry$\&$al . However, due to its hight degeneracy, the Bakry-Emery curvature is $-\infty$ so that we may not apply the $\Gamma_{2}$ technique. Remark also that the (H.C.K.) condition reads for all $(x,v)$ and $(y,w)$

Let us first remark that if $\nabla V$ is Lipshitz continuous, (H.C.K) is verified for some negative $K$ and using synchronous coupling, one may remark that we are in the same situation than in Section2 so that we get that for some negative $K$ the gradient commutation property holds

and thus the logarithmic Sobolev inequality holds for $P_{t}((x,v),\cdot)$. Let us remark once again that those properties are written with the usual gradient and not the Carré-du-Champ operator $\Gamma(f)=|\nabla_{v}f|^{2}$.

One may then wonder if it is possible to get the gradient commutation property with $K>0$. In fact, using synchronous coupling and Itô’s formula applied to the function $N((x,v),(y,w))=a|x-y|^{2}+b\langle x-y,v-w\rangle+|v-w|^{2}$, following , we get that if $V(x)=|x|^{2}+W(x)$ where $\nabla W$ is $\delta$-Lipschitz with $\delta$ sufficiently small there exists $a,b$ and $K>0$ such that $H$ is equivalent to the euclidean norm and

so that we get as in Section 2 the commutation property for some $K>0$ and $A>1$

and thus a Logarithmic Sobolev inequality holds uniformly in time.

It is not hard to extend the result of this simplified setting to the case where the Brownian motion in the velocity has a diffusion coefficient which is bounded and $L$-Lipschitz. We may then obtain a weaker gradient commutation property

and local Poincaré type inequality or Transportation information inequality like in Propositions 3.17 or 3.20, and if $L$ is sufficiently small uniform in time version of these inequalities (using functional $N$).

6. Interpolation of the gradient commutation property and local Beckner inequality

We have seen here that we cannot recover a logarithmic Sobolev inequality by our technique when (H.C.K.) is in force. Remember however that we have introduced the stronger (H.C.K.m) condition which implies a contraction in Wasserstein distance $W_{m}$. It is then not hard to deduce some interpolation of the gradient commutation property

Assume (R) and (H.C.K.m) or the weaker contraction property (3.8). Let $f\in{\mathcal{A}}$, if $\nabla P_{t}f$ exists, it holds

Remark once again that this property does hold even if the diffusion coefficient is degenerate, so that variations of the hypoelliptic example of the previous subsection with a diffusion coefficient in the velocity enters into this framework. This contraction property may thus lead to a reinforcement of the Poincaré inequality to a Beckner inequality.

Assume (R) and (H.C.K.m) or the weaker (3.24). Let $M=\||\sigma|^{2}\|_{\infty}$. Then for all nice $f$, we have the following Beckner inequality

The proof unfortunately does not rely on the $h$-process introduced previously but on the $\Gamma_{2}$ type proof. Denote $p=\frac{2m}{m+2}$. By (3.24) and Hölder inequality, for all nice non negative$f$

7. Convergence to equilibrium in positive curvature.

Still in the uniform elliptic case, assume that $K>0$. We already mentioned that in this case $\mu_{T}$ weakly converges to the unique invariant probability measure $\mu_{\infty}$ (which exists). In particular , for all smooth $g$ (say $C_{b}^{2}$), $\textrm{Var}_{\mu_{T}}(g)\to\textrm{Var}_{\mu_{\infty}}(g)$ as well as $\int|\nabla g|^{2}\,d\mu_{T}\to\int|\nabla g|^{2}\,d\mu_{\infty}$. We deduce that

As we said, this result is not captured by the $\Gamma_{2}$ theory.

But we can obtain general convergence results, even in the non uniformly elliptic case. Indeed recall that in full generality

Assume that $\sigma$ is bounded. Then if (H.C.K) holds for some $K>0$, defining $M$ as before, there exists an unique invariant probability measure $\mu_{\infty}$ and for all nice enough function $g$ ,

In addition if $\mu_{\infty}$ is symmetric (i.e. $\int fLgd\mu_{\infty}=\int gLfd\mu_{\infty}$), it holds

Remark once again that what is used here is the weak gradient commutation property which is a consequence of (H.C.K.). The last part of the theorem follows from a result in recalled in the Appendix, lemma A.1 (2). Of course, unless we explicitly know the invariant measure, it is not easy to see wether $\mu_{\infty}$ is symmetric or not.

We may further extends the previous argument to the entropic convergence to equilibrium. Let us suppose that there exists an unique invariant measure $\mu_{\infty}$.

Assume that $\sigma$ is bounded and that the gradient commutation property

holds for some positive $K$. Then for all nice positive function $f$ (defining $M$ as before)

The proof is as for the $L_{2}$ decay quite standard. Indeed,

One of the important point here is that we do not suppose any non-degeneracy on the diffusion coefficient, so that the result applies to the kinetic Fokker-Planck equation. It then provides an alternative to the approach by Villani , where he obtained such kind of convergence by completely different techniques with assumptions quite similar to the ones described in Section 3.5. One may then complete the approach by regularization of the Fisher Information in small time to obtain an entropic decay controlled by the initial entropy, see or . $\diamondsuit$

Let us point out that even in the symmetric case, such a control is not sufficient to recover a logarithmic Sobolev inequality as the analog of lemma A.1 is no more valid for the entropy. Remark however that we have shown in section 2 how to recover a logarithmic Sobolev inequality for $P_{t}$ using the strong commutation gradient property (3.30). If $K>0$, we may then let $t$ goes to infinity to recover a logarithmic Sobolev inequality for the invariant measure. It may be for example be used in the context of kinetic Fokker-Planck equation with non gradient coefficient, for which the invariant measure is unknown. $\diamondsuit$

Let us consider, as in the Poincaré case via (H.C.K.) condition, a particular class of test function $g$ such that $g\geq\varepsilon>0$, so that $P_{t}g\geq\varepsilon$. We then see adapting the preceding proof that a weak commutation of gradient property

obtained for example under (H.C.K.) condition implies that

Non homogeneous diffusion processes.

In the authors extended the $\Gamma_{2}$ theory to time dependent coefficients (non homogeneous diffusions). Considering the Ito system

If $f$ only depends on $x$, the proof of proposition 2.2 (resp. 3.11) is unchanged using the process starting from $(0,x)$ and $(0,y)$ and replacing $Kt$ by $K(t)$. To obtain the analogue of proposition 2.10 and proposition 3.17, it suffices to remark that $\sigma\nabla$ is equal to $\nabla_{x}$, and use what precedes for $h$ depending on $x$ only. ∎

For the transportation inequality we have to slightly modify the method in subsection 2.3. With the notations therein, (2.3) has become,

so that, as in the previous section we have to come back to

Using as usual $(ab)^{\frac{1}{2}}\leq\lambda a+\frac{1}{\lambda}b$ we obtain (see the details of the derivation in the previous section) that for all increasing function $\lambda(t)$

from which we deduce, provided we choose $K(0)=\lambda(0)=0$,

If $\mu_{0}$ satisfies a $T_{2}$ inequality with constant $C_{T}(0)$, then

The best choice of $\lambda$ is not clear. If $K^{\prime}(t)$ is not positive on the whole $[0,T]$, it seams that taking $\lambda(T)=\lambda T$ for some $\lambda>0$ is enough. If $K^{\prime}(t)>0$ for all $t$ (but not necessarily bounded from below by a positive constant), $\lambda(t)=\lambda K(t)$ seems to be natural.

Assume that $K(t)\to+\infty$ as $t\to+\infty$ and that $C=\int_{0}^{+\infty}e^{-K(s)}\,ds<+\infty$. Then, for all $t$, $P(t,x,.)$ (the distribution of the process starting from $x$ and $t_{0}=0$) satisfies a Poincaré inequality (and a log-Sobolev inequality when $\sigma=Id$) with a constant bounded by $MC$ (or $2C$). The family $(P(t,x,.))_{t>0}$ is then tight, but we do not know whether it is weakly convergent or not. Nevertheless any weak limit satisfies the same functional inequality. When $\sigma=Id$ we know that $|X_{t}^{x}-X_{t}|\leq e^{-K(t)}|x-X_{0}|$ for any initial random variable $X_{0}$. It follows that if a sequence $P(t_{k},x,.)$ is weakly convergent to some $\mu$, the sequence $\mu_{t_{k}}$ weakly converges to the same limit. In particular if we consider $\sigma=Id$, $b(t,x)=-\frac{1}{2}\,(\nabla U(x)+K^{\prime}(t)\,x)$, for some convex potential $U$, (H.C.K(t)) is satisfied, so that any weak limit satisfies a log-Sobolev inequality. If $d\mu=e^{-U}dx$ does not satisfy a log-Sobolev inequality, it cannot be a weak limit, even if $K^{\prime}(t)\to 0$. In this situation one should expect that the “perturbation” of $\nabla U$ being smaller and smaller when $t$ growths, the convergence to $\mu$ will still hold. This is not the case. $\diamondsuit$

2. Application to some non-linear diffusions.

We shall now discuss an example that does partly enter the framework of the beginning of this section.

Following consider the following non-linear stochastic differential equation

This is a non-linear diffusion of Mc Kean-Vlasov type modeling, for instance, granular media. We refer to the introduction of for details and motivations. One can approximate the solution of (4.6) by the first coordinate of a linear large particle system with mean field interactions. This is what is done in to study the long time behavior of $X_{t}$.

Let we see how to apply what we have just done. First, under some conditions on $V$ and $W$ (we later shall give some of them) existence and weak uniqueness of (4.6) are ensured, provided the initial law admits some big enough polynomial moment. This will imply for all $x$, the existence and uniqueness of $q_{t}^{x}$ solution of (4.7) with initial condition $\delta_{x}$. As usual for these non-linear equations, if we consider the linear time inhomogeneous S.D.E.

the pathwise unique solution (up to explosion) $Z_{.}^{x,x}$ is shown to satisfy (4.6) (i.e. $\mathcal{L}(Z^{x,x}_{t})=q_{t}^{x}$) so that it coincides with $X^{x}_{t}$. So, once $q_{t}^{x}$ and $q_{t}^{y}$ are built, we may build our synchronous coupling $(X_{t}^{x},X_{t}^{y})$ as before. Now introduce an independent copy $(\bar{X}_{t}^{x},\bar{X}_{t}^{y})$ of $(X_{t}^{x},X_{t}^{y})$.

If we assume in addition (as usual) that $W(-x)=W(x)$, and remember that $\bar{X}$ is a copy of $X$, it is still equal to

$V$, $W$ and their first two derivatives have at most polynomial growth of order $m$ and $W(-x)=W(x)$,

$V$ satisfies (H.C.$K_{V}$) and $W$ satisfies (H.C.$K_{W}$).

Let $a=\max(m(m+3),2m^{2})$. If $\mu_{0}$ and $\nu_{0}$ have a polynomial moment of order $a$, there exist an unique solution of (4.6) and an unique solution of (4.7) among the set of probability flows having having a polynomial moment of order $a$ with initial condition $\mu_{0}$ or $\nu_{0}$. Furthermore

If $V=0$ and $\int x\mu_{0}(dx)=\int x\nu_{0}(dx)$ then

$V=0$, $\int x\mu_{0}(dx)=\int x\nu_{0}(dx)$ and $K_{W}>0$.

The moment condition ensuring existence and uniqueness is described in section 2.

Of course we may replace the initial $\delta_{x}$ and $\delta_{y}$ by probability distributions $\mu_{0}$ and $\nu_{0}$ satisfying the required moment conditions. This immediately furnishes the first assertion about the upper bound for the Wasserstein distance.

If $V=0$ it is easily seen that $\int xq_{t}^{\mu_{0}}(x)dx=\int x\mu_{0}(dx)$ for all $t>0$, hence

provided the same holds at time . This furnishes the second assertion for the upper bound.

Finally the convergence under strict positivity of our new “curvature” condition ensures the existence of the limiting measure $\mu_{\infty}$. To see that $\mu_{\infty}=q^{\infty}(x)dx$ is actually invariant, one can for instance use the following trick: first consider the solution $q_{t}^{\infty}$ of (4.7) with initial condition $q^{\infty}$. Similar bounds for the Markov non homogeneous process $Z_{.}^{q^{\infty},y}$ (when we replace $q_{t}^{x}$ by $q_{t}^{\infty}$) are obtained applying the results of the beginning of this section. Hence the law of $Z_{T}^{q^{\infty},q^{\infty}}$ (which is exactly $\mu_{T}$ starting with $\mu_{\infty}$ as we explained before) converges to some limiting measure $\mu_{\infty}^{\mu_{\infty}}$ which in turn is equal to $\mu_{\infty}$ and is invariant for $Z_{.}^{q_{\infty},y}$. This achieves the proof. ∎

The proof of the above result is new and direct, while the result is mainly contained in using particle approximation. Notice that in the $\Gamma_{2}$ approach is developed for the non homogeneous Markov diffusion $Z_{.}$ and not for $X_{.}$. Also notice that some direct study of the decay to equilibrium in $W_{2}$ distance for granular media is done in . $\diamondsuit$

for which $\mu_{\infty}(dx)=q^{\infty}(x)dx$ the invariant probability measure. Using the results in section 2 we thus have