A variational approach to nonlinear and interacting diffusions

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A variational approach to nonlinear and interacting diffusions

Marc Arnaudon, Pierre del Moral

To cite this version:

Marc Arnaudon, Pierre del Moral. A variational approach to nonlinear and interacting diffusions.

2019. �hal-01950673v3�

A variational approach to nonlinear and interacting diffusions

M. Arnaudon

¹

and P. Del Moral

^∗2

1

Institut de Mathématiques de Bordeaux (IMB), Bordeaux University, France

2

INRIA, Bordeaux Research Center & CMAP, Polytechnique Palaiseau, France

Abstract

The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology com- bines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic mod- els including non homogeneous diffusions, as well as stochastic processes evolving on differen- tiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t.

the time parameter are also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed.

Keywords : Nonlinear diffusions, mean field particle systems, variational equations, logarith- mic norms, gradient flows, contraction inequalities, Wasserstein distance, Riemannian manifolds.

Mathematics Subject Classification : 65C35, 82C80, 58J65, 47J20.

1 Introduction

1.1 Description of the models

We denote by } A }

²

: “ λ

_max

p AA

¹

q

^1{2

, resp. } A }

^F

“ Tr p AA

¹

q

^1{2

and ρ p A q “ λ

_max

pp A ` A

¹

q{ 2 q the spectral norm, the Frobenius norm, and the logarithmic norm of some matrix A, where A

¹

stands for the transpose of A, and λ

max

p . _q the maximal eigenvalue. With a slight abuse of notation, we denote by I the identity p d ˆ d q -matrix, for any d ě 1.

Let b

_t

be some time varying differentiable vector field with Jacobian matrix ∇b

_t

on R

^d

, for some parameter d ě 1. Consider the deterministic flow t P r s, 8rÞÑ X

_s,t

p x q starting at X

_s,s

p x q “ x associated with the evolution equation

B

t

X

_s,t

p x q “ b

_t

p X

_s,t

p x qq ùñ B

t

∇X

_s,t

p x q “ ∇X

_s,t

p x q ∇b

_t

p X

_s,t

p x qq with ∇X

_s,s

p x q “ I (1.1) The r.h.s. equation is often called the first order variational equation associated with the flow X

_s,t

p x q along the trajectory X

_s,t

p x q . This equation plays a central role in the sensitivity analysis

∗P. Del Moral was supported in part by funding from the Chaire Stress Test, BNP Paribas SFTS and CMAP, Polytechnique Palaiseau, France

of nonlinear dynamical systems w.r.t. their initial conditions. For instance, the spectral norm of

∇X

_s,t

p x q can be estimated in terms of the logarithmic norm using the inequalities

´ ż

t

s

ρ p´ ∇b

u

p X

s,u

p x qqq du ď log } ∇X

s,t

p x q}

²

ď ż

t

s

ρ p ∇b

u

p X

s,u

p x qqq du (1.2) A proof of this assertion can be found in [14], see also [27] for extensions to Lipschitz functions on Banach spaces. Whenever ρ p ∇b

_u

p x qq ď ´ λ for some λ ą 0, the r.h.s. estimate in (1.2) readily implies the exponential stability estimate

X

s,t

p x q ´ X

s,t

p y q “ ż

1

0

x ∇X

s,t

p ǫx ` p 1 ´ ǫ q y q , p x ´ y qy dǫ ùñ } X

s,t

p x q ´ X

s,t

p y q} ď e

^´λpt´sq

} x ´ y }

(1.3)

The linearization technique discussed above is often referred as the Lyapunov first or indirect method to analyze the stability of nonlinear dynamical systems. For a more thorough discussion on this subject we refer to the pioneering work by Lyapunov [24], as well as to chapter 4 in the more recent monograph by Khalil [23].

The main objective of this article is to extend these results to nonlinear diffusions and their mean field particle interpretations on Euclidian as well as on differentiable manifolds. The differential analysis of conventional diffusions w.r.t. initial conditions is also one of the stepping stones of Bismut and Malliavin calculus. This framework is mainly designed to study the existence and the properties of smooth probability densities in terms of the differential properties of the diffusion semigroup. For a more thorough discussion on this subject we refer to [13, 30], and references therein.

The relevant mathematical apparatus for the description and the variational analysis of stochas- tic processes on manifolds being technically more sophisticated than conventional differential cal- culus, this introduction only discusses nonlinear and interacting diffusions on Euclidian spaces.

The extended versions of these models on Riemannian manifolds are discussed in some details in section 3.2, as well as in section 4.3.

Let P

₂

p R

^d

q be the set of Borel probability measures on R

^d

with finite second absolute moment, equipped with the 2-Wasserstein distance given by

W

₂

p η, µ q “ inf E p} X ´ Y }

²

q

^1{2

In the above display, the infimum is taken over all pairs of random variables p X, Y q with respective distributions η and µ P P

₂

p R

^d

q ; and } X ´ Y } stands for the Euclidian distance between X and Y on the product space R

^d

.

Also let b

_t

and σ

_t

be differentiable functions from R

^2d

into R

^d

and R

^dˆr

, for some r ě 1; and let W

_t

be an r-dimensional Brownian motion. For any µ P P

₂

p R

^d

q and any time horizon s ě 0 we denote by X

_s,t^µ

p x q be the stochastic flow defined for any t P r s, 8r and any starting point x P R

^d

by the McKean-Vlasov diffusion

dX

_s,t^µ

p x q “ b

t

`

φ

s,t

p µ q , X

_s,t^µ

p x q ˘

dt ` σ

t

`

φ

s,t

p µ q , X

_s,t^µ

p x q ˘

dW

t

(1.4)

In the above display, φ

_s,t

stands for the evolution semigroup φ

_s,t

p µ qp dy q “ µP

_s,t^µ

p dy q : “

ż

µ p dx q P

_s,t^µ

p x, dy q with P

_s,t^µ

p x, dy q : “ P p X

_s,t^µ

p x q P dy q

We further assume that the mean field drift and diffusion functions are given by b

_t

p η, y q : “

ż

η p dx q b

_t

p x, y q and σ

_t

p η, y q : “ ż

η p dx q σ

_t

p x, y q

We shall assume that the nonlinear diffusion flow (1.4) is well defined. For instance, the existence of this flow is ensured as son as b

t

and σ

t

are Lipschitz, see for instance [18, 22].

The mean field particle system associated with (1.4) is defined by the stochastic flow ξ

_s,t

p z q “ p ξ

ⁱ_s,t

p z qq

¹ďiďN

of a system of N interacting diffusions

dξ

ⁱ_s,t

p z q “ b

_t

`

m p ξ

ⁱ_s,t

p z qq , ξ

_s,tⁱ

p z q ˘

dt ` σ

_t

`

m p ξ

_s,tⁱ

p z qq , ξ

_s,tⁱ

p z q ˘

dW

_tⁱ

(1.5)

with the empirical measures

m p ξ

_s,tⁱ

p z qq : “ 1 N

ÿ

1ďjďN

δ

_ξj s,tpzq

In the above displayed formulae, ξ

_s,s

p z q “ z “ p z

ⁱ

q

1ďN

P p R

^d

q

^N

stands for the initial configuration and W

_tⁱ

are N independent copies of W

_t

.

1.2 Statement of some main results and article organisation

To motivate this study, the variational calculus developed in the article is illustrated with the following example

r “ d σ p x, y q “ σ

0

I and b p x, y q “ ´ ∇U p y q ´ ∇V p y ´ x q (1.6) for some σ

0

ą 0, some confinement type potential function U (a.k.a. the exterior potential) and some interaction potential function V . This class of nonlinear diffusions and the corresponding particle interpretations were introduced by H. P. McKean in [28, 29]. The extended versions of these Langevin-type nonlinear diffusions on Riemannian manifolds are discussed in the end of section 3.2 as well as in section 4.3.

Nonlinear diffusions (1.4) with constant diffusion and gradient-type drifts (1.6) arise in fluid mechanics, and more particularly in the modeling of granular flows [6, 7, 35, 42]. In this context, φ

_s,t

represents the evolution semigroup of the velocity of a diffusive particule interacting with the distribution of the particles around its location and following some confinement exterior potential. In this interpretation, the mean field particle model (1.5) can be seen as a particle-type representation of the granular flow.

In the last two decades, the analysis of the long time behavior of this particular class of gradient type flow diffusions have been developed in various directions:

The first articles on the long time behavior of these models are the couple of articles by Tamura [33, 34]. The stability properties of one dimensional models has been started in [4, 5]

as well as in [6], see also [9, 11, 35].

Since this period, several sophisticated probabilistic techniques have been developed to analyze

the long time behavior of these Langevin-type nonlinear diffusions, including log-Sobolev functional

inequalities [25, 26], entropy dissipation [10, 15], as well as gradient flows in Wasserstein metric

spaces and optimal transportation inequalities [8, 10, 12, 31], combining the functional Γ

2

Bakry-

Emery method [3], with the Otto-Villani approach [32]. The long time self-stabilizing behavior of

this class of processes in multi-wells landscapes has also been developed by J. Tugaut in a series of

articles [36, 37, 39, 40, 41]. For a more thorough discussion on this subject we refer to the recent

article [17], and the references therein.

Unfortunately, most of the probabilistic techniques discussed above only apply to gradient flow type diffusions of the form (1.6). The variational calculus developed in the present article is not restricted to this class of gradient-type nonlinear models. Nevertheless, because of their importance in practice this introduction illustrates some of our main results in this context.

Firstly, and rather surprisingly, the variational methodology developed in the present article applies directly to gradient flow models of the form (1.6), simplifying considerably both of their stability analysis as well as the convergence analysis of their mean field particle interpretations.

This framework also allows to relax unnecessary technical conditions such as the symmetry of the interaction potential function, or the invariance of the center of mass, currently used in the literature on this subject (see for instance [33], as well as section 2 in [10], and section 1 in the more recent article [8]). It also allows to derive uniform as well as almost sure exponential stability inequalities at the level of the nonlinear diffusion flow. For instance, when V is an even convex function with bounded Hessian } ∇

²

V }

²

: “ sup

_x

} ∇

²

V p x q}

²

ă 8 , and when ∇

²

U ě λ I, for some λ ą 0 we have the almost sure estimates

} X

_s,t^η

p x q ´ X

_s,t^µ

p y q} ď } ∇

²

V }

²

p t ´ s q e

^´λpt´sq

W

₂

p η, µ q ` e

^´λpt´sq

} x ´ y } (1.7) The above estimate is also met for odd interaction potential, as soon as ∇

²

U p y q` ∇

²

V p y ´ x q ě λ I . In the above display, it is implicitly assumed that the stochastic flows are driven by the same Brownian motion.

These almost sure inequalities are direct consequence of the contraction inequality (2.6), the remark (2.15) and the almost sure estimates stated in corollary 3.2.

To the best of our knowledge, the almost sure exponential decays (1.7) are the first result of this type for this class of nonlinear gradient flow diffusions.

Consider a pair of random variables p Z

0

, Z

1

q with distributions p µ

0

, µ

1

q on R

^d

and set

Z

_ǫ

: “ p 1 ´ ǫ q Z

₀

` ǫ Z

₁

µ

_ǫ

: “ Law p Z

_ǫ

q and X

_s,t^ǫ

: “ X

_s,t^µǫ

p Z

_ǫ

q (1.8) Under the assumptions on the potential functions discussed above, for any differentiable function f on R

^d

with bounded gradient we have the first order differential formula

r φ

_s,t

p µ

1

q ´ φ

_s,t

p µ

0

qs p f q “ ż

1

0

B

ǫ

φ

_s,t

p µ

_ǫ

qp f q dǫ (1.9) with the linear differential operator

B

^ǫ

φ

s,t

p µ

ǫ

qp f q : “ E `@

B

^ǫ

X

_s,t^ǫ

, ∇f p X

_s,t^ǫ

q D˘

s.t. |B

^ǫ

φ

s,t

p µ

ǫ

qp f q| ď e

^´λpt´sq

} ∇f }

For a more precise statement we refer to theorem 2.2. Almost sure and uniform estimates of the first order differential maps ǫ ÞÑ B

ǫ

X

_s,t^ǫ

are also provided in theorem 2.3.

Section 4.1 also presents a differential calculus to estimate the gradient ∇ξ

_s,t

p z q of the stochastic flow ξ

s,t

p z q of the interacting particle model (1.5). Under the assumptions on the potential functions discussed above, we shall prove the following uniform spectral norm estimate

} ∇ ξ

s,t

p z q}

²

ď e

^´λpt´sq

The above result is a direct consequence of theorem 4.1. The above estimate ensures that the N -

particle model converges exponentially fast to its invariant measure with some exponential decay

that doesn’t depends on the number of particles. The latter property can also be checked using

more sophisticated Logarithmic Sobolev inequalities [25]. To the best of our knowledge, the almost

sure exponential decays stated above are the first result of this type for this class of interacting Langevin-type diffusions.

Section 4.2 also provides a natural differential calculus to derive quantitative and uniform prop- agation of chaos estimates for nonlinear diffusions of the form (1.5). Applying these results to interacting Langevin-type diffusions, without further work we recover the uniform estimates stated in theorem 1.2 in [25].

We emphasize that the differential calculus presented in this article allows to consider nonlinear diffusions evolving in differential manifolds. This should not come as a surprise since our framework allows to enter the variations of the diffusion matrices associated with these stochastic models which encapsulates the Riemannian structure of the manifold.

We illustrate these comments in the end of section 2.2 with a rather detailed discussion of an elementary nonlinear geometric-type diffusion. The manifold version of (1.9) is also provided in theorem 3.14.

We also underline that the variational calculus on differentiable manifolds developed in sec- tion 3.2 provides another view and additional results for the diffusions in R

^d

endowed, when possi- ble, with the Riemannian metric under which these diffusions are Brownian motion with drift. In this context, different types of synchronous coupling lead to gradient flow estimates where gradients of the diffusion functions are replaced by Ricci curvatures.

Quantitative propagation of chaos estimates of mean field particle systems on Riemannian man- ifolds are provided in section 4.3. Special attention is paid to derive uniform estimates w.r.t. the time horizon.

2 Nonlinear diffusion semigroups

2.1 Some gradient flow estimates

This section presents some basic properties of the first variational equation associated with the nonlinear diffusion (1.4). Let σ

_k,t

be the k-th column vector of σ

_t

, and let ∇

_u

b

_t

p x, y q and ∇

_u

σ

_k,t

p x, y q be the gradient of the functions b

_t

p x, y q and σ

_k,t

p x, y q w.r.t. the coordinate u P t x, y u . We also let X

_s,t^i,µ

p x q be the i-th coordinate of the column vector X

_s,t^µ

p x q . The Jacobian ∇X

_s,t^µ

p x q of the diffusion flow X

_s,t^µ

p x q is given by the gradient p d ˆ d q -matrix

∇X

_s,t^µ

p x q : “ ´

∇X

_s,t^1,µ

p x q , . . . , ∇X

_s,t^d,µ

p x q ¯

ùñ d ∇X

_s,t^µ

p x q “ ∇X

_s,t^µ

p x q

«

∇b

_t

`

φ

_s,t

p µ q , X

_s,t^µ

p x q ˘

dt ` ÿ

1ďkďr

∇σ

_t,k

`

φ

_s,t

p µ q , X

_s,t^µ

p x q ˘ dW

_t^k

ff

Consider the regularity condition stated below:

p H

_A

q : There exists some λ

_A

P R such that for any x, y P R

^d

and t ě 0 we have A

t

p x, y q : “ ∇

_y

b

t

p x, y q ` ∇

_y

b

t

p x, y q

¹

` ÿ

1ďkďr

∇

_y

σ

_k,t

p x, y q ∇

_y

σ

_k,t

p x, y q

¹

ď ´ 2λ

A

I (2.1)

This spectral condition produces several gradient estimates. For instance, we have the following uniform estimate

p H

_A

q ùñ E `

} ∇X

_s,t^µ

p x q}

²2

˘

1{2

ď E `

} ∇X

_s,t^µ

p x q}

²F

˘

1{2

ď ?

d e

^´λApt´sq

(2.2)

In addition, we have the almost sure estimate

p H

_A

q and ∇

_y

σ

_k,t

p x, y q “ 0 ùñ } ∇X

_s,t^µ

p x q}

²

ď e

^´λApt´sq

(2.3) The proofs of the above assertions are provided in the appendix, on page 25. For the nonlinear Langevin diffusion discussed in (1.6) we have

p H

_A

q ðñ ∇

²

U p y q ` ∇

²

V p y ´ x q ě λ

_A

I ùñ } ∇X

_s,t^µ

p x q}

²

ď e

^´λApt´sq

(2.4) Arguing as in (1.3) we readily check the following proposition.

Proposition 2.1. Assume p H

A

q is satisfied. In this situation, we have E `

} X

_t^µ

p x q ´ X

_t^µ

p y q}

²

˘

1{2

ď ?

d e

^´λApt´sq

} x ´ y } (2.5)

In addition, we have the almost sure estimate

∇

_y

σ

_k,t

“ 0 ùñ } X

_t^µ

p x q ´ X

_t^µ

p y q} ď e

^´λApt´sq

} x ´ y } (2.6) Whenever λ

A

ă 0 the above estimates ensure that the transition semigroup P

_s,t^µ

is exponentially stable, that is we have that

W

₂

`

η

0

P

_s,t^µ

, η

1

P

_s,t^µ

˘

ď c exp r´ λ

_A

p t ´ s qs W

₂

p η

0

, η

1

q (2.7) These contraction inequalities quantify the stability of the stochastic flow X

_s,t^µ

p x q w.r.t. the initial state x, but they don’t give any information of the stability properties of the nonlinear semigroup φ

_s,t

p µ q w.r.t. the initial measure µ.

2.2 A first order differential calculus

This section presents a natural first order differential calculus to analyze the stability properties of the nonlinear semigroup φ

_s,t

p µ q . Consider the matrices

B

_t

p z

1

, z

2

q : “

» –

∇

_y

b

_t

p z

2

, z

1

q ∇

_x

b

_t

p z

1

, z

2

q

∇

_x

b

t

p z

2

, z

1

q ∇

_y

b

t

p z

1

, z

2

q fi

fl D

_t

: “ ÿ

1ďkďr

» –

∇

_x

σ

t,k

∇

_x

σ

_t,k¹

∇

_x

σ

t,k

∇

_y

σ

_t,k¹

∇

_y

σ

t,k

∇

_x

σ

¹_t,k

∇

_y

σ

t,k

∇

_y

σ

_t,k¹

fi fl (2.8) In this notation, our second regularity condition takes the following form:

p H

_C

q : There exists some λ

_C

P R such that for any x, y P R

^d

and t ě 0 we have C

t

p x, y q : “ 1

2

“ B

t

p x, y q ` B

t

p x, y q

¹

‰

` D

t

p x, y q ď ´ λ

C

I (2.9)

Let Z

_ǫ

be the collection of random variables with distribution µ

_ǫ

defined in (1.8). We also consider a couple of independent stochastic flows

X

_s,t^ǫ

: “ X

_s,t^µǫ

p Z

_ǫ

q and Y

_s,t^ǫ

: “ Y

_s,t^µǫ

p Z

_ǫ

q (2.10)

driven by independent Brownian motions, say W

_t

“ p W

_t^k

q

1ďkďd

and W

_t

“ p W

^k_t

q

1ďkďd

, and starting

from a couple of independent random variables Z

_ǫ

and Z

_ǫ

with the same law.

In the further development of this section, we denote by E

_X

p . _q the expectation operator w.r.t.

the Brownian motion W

_t

“ p W

_t^k

q

1ďkďd

and the random variable Z

_ǫ

. In this notation, we have dY

_s,t^ǫ

“ E

_X

“

b

_t

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘‰

dt ` E

_X

“ σ

_t

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘‰

dW

_t

This implies that

d “ B

^ǫ

Y

_s,t^ǫ

‰

“ E

_X

”

∇

_x

b

t

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘

1

B

^ǫ

X

_s,t^ǫ

` ∇

_y

b

t

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘

1

B

^ǫ

Y

_s,t^ǫ

ı dt

` ÿ

1ďkďr

E

_X

”

∇

_x

σ

_t,k

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘

1

B

^ǫ

X

_s,t^ǫ

` ∇

_y

σ

_t,k

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘

1

B

^ǫ

Y

_s,t^ǫ

ı dW

^k_t

(2.11) with the initial condition

B

ǫ

Y

_s,s^ǫ

“ B

ǫ

Z

_ǫ

“ Z

1

´ Z

0

A simple calculation yields the following estimate B

^t

E ”› › B

^ǫ

Y

_s,t^ǫ

› ›

²

ı

ď E ˆ “

B

^ǫ

X

_s,t^ǫ

, B

^ǫ

Y

_s,t^ǫ

‰

1

C

_t

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘ „ B

^ǫ

X

_s,t^ǫ

B

^ǫ

Y

_s,t^ǫ

˙

(2.12) The inequality in the above display can be turned into an equality when D

_t

“ 0. Also note that

p H

_C

q ùñ E `

} Y

_s,t⁰

´ Y

_s,t¹

}

²

˘ ď

ż

1

0

E ´› › B

^ǫ

Y

_s,t^ǫ

› ›

²

¯

dǫ ď e

^´2λCpt´sq

E `

} Z

1

´ Z

0

}

²

˘

Let C

_b¹

p R

^d

q be the set of differentiable functions on R

^d

with bounded derivative. A direct consequence of the fundamental theorem of calculus yields the following theorem.

Theorem 2.2. For any s ď t and any f P C

_b¹

p R

^d

q and µ

0

, µ

1

P P

₂

p R

^d

q we have the first order differential formula (1.9). In addition, we have the exponential contraction inequality

p H

_C

q ùñ W

₂

p φ

_s,t

p µ

0

q , φ

_s,t

p µ

1

qq ď e

^´λCpt´sq

W

₂

p µ

0

, µ

1

q (2.13) When λ

C

ą 0, the above theorem provides an alternative and rather elementary proof of the exponential asymptotic stability of time varying McKean-Vlasov diffusions with non necessarily homogenous diffusion functions. To the best of our knowledge this stability property is the first result of this type for this general class of nonlinear diffusions.

For the Langevin-type diffusion discussed in (1.6) we have D

_t

“ 0 and the matrix C

_t

reduces to

´ C

t

p z

1

, z

2

q “

» –

∇

²

U p z

1

q 0 0 ∇

²

U p z

2

q

fi fl

`

»

— —

— –

∇

²

V p z

₁

´ z

₂

q ´

“ ∇

²

V p z

2

´ z

1

q ` ∇

²

V p z

1

´ z

2

q ‰ 2

´

“ ∇

²

V p z

2

´ z

1

q ` ∇

²

V p z

1

´ z

2

q ‰

2 ∇

²

V p z

2

´ z

1

q

fi ffi ffi ffi fl

When V is odd we have

p H

_C

q ðñ ∇

²

U p z

1

q ` ∇

²

V p z

1

´ z

2

q ě λ

_C

I ðñ p H

_A

q (2.14)

In the reverse angle, if V is even and convex then we have

p H

C

q ðñ ∇

²

U ě λ

C

I ùñ p H

A

q (2.15)

As expected, explicit formulae are available for linear and Gaussian models. For instance, when b

_t

p x, y q “ A

1

x ` A

2

y and σ

_t

p x, y q “ R

^1{2

with A

1

, A

2

P R

^dˆd

and R ě 0 the diffusion flow X

_s,t^µ

p x q P R

^d

is linear w.r.t. µ and given for any x P R

^d

by the formula

X

_s,t^µ

p x q “ e

^A2pt´sq

p x ´ µ p e qq ` e

^{rA1`A2spt´sq}

µ p e q ` ż

t

s

e

^A2pt´uq

R

^1{2

dW

_u

In the above display, e p x q “ x stands for the identity function on R

^d

. In this context, the process X

_s,t^ǫ

defined in (2.10) is also given by the formula

X

_s,t^ǫ

“ e

^A2pt´sq

p Z

_ǫ

´ µ

_ǫ

p e qq ` e

^{rA1`A2spt´sq}

µ

_ǫ

p e q ` ż

t

s

e

^A2pt´uq

R

^1{2

dW

_u

ùñ B

^ǫ

X

_s,t^ǫ

“ e

^A2pt´sq

pp Z

1

´ Z

0

q ´ E p Z

1

´ Z

0

qq ` e

^{rA1`A2spt´sq}

E p Z

1

´ Z

0

q This yields the rather crude estimate

E `

}B

^ǫ

X

_s,t^ǫ

}

²

˘ ď ”

} e

^{rA1`A2spt´sq}

}

²2

` } e

^A2pt´sq

}

²2

ı

E p} Z

1

´ Z

0

}

²

q

Up to some constant, this shows that the r.h.s. Wasserstein contraction estimate in (2.13) is met with ´ λ

_C

“ ρ p A

1

` A

2

q_ ρ p A

2

q . Applying Coppel’s inequality (cf. Proposition 3 in [14]) we can also choose ´ λ

C

“ r ς p A

1

` A

2

q _ ς p A

2

qs p 1 ´ δ q for any 0 ă δ ă 1, where ς p A q : “ max

i

t Re r λ

i

p A qsu ď ρ p A q stands for the spectral abscissa of a square matrix A.

It may happen the stochastic flow (1.4) remains in some domain S Ă R

^d

. The simplest model we have in head is the geometric diffusion on S “ r 0, 8r associated with the parameters

b

t

p x, y q “ r a

1

´ a

2

x s y and σ

t

p x, y q “ σ

0

y with a

1

P R and a

2

, σ

0

ą 0 In this situation, the diffusion flow X

_s,t^µ

p x q P S is nonlinear w.r.t. µ and given for any x P S by

X

_s,t^µ

p x q “ ψ

_t´s

p µ q E

_s,t

p W q x with E

_s,t

p W q : “ exp

„

σ

0

p W

t

´ W

s

q ´ σ

₀²

2 p t ´ s q



(2.16) with the function ψ

_t

defined by

ψ

_t

p µ q “ 1

e

^´a1t

` a

2

µ p e q θ

_a1

p t q with θ

_a1

p t q : “ a

^´1₁

p 1 ´ e

^´a1t

q

In the above display, we have used the convention θ

0

p t q “ t. In this context, the process X

_s,t^ǫ

defined in (2.10) is also given by the formula

X

_s,t^ǫ

“ ψ

_t´s

p µ

_ǫ

q E

_s,t

p W q Z

_ǫ

ùñ B

ǫ

X

_s,t^ǫ

“ ψ

_t´s

p µ

_ǫ

q E

_s,t

p W qq rp Z

1

´ Z

0

q ´ a

2

θ

_a1

p t q ψ

_t´s

p µ

_ǫ

q Z

_ǫ

E p Z

1

´ Z

0

qs

Assume that a

1

ă 0 is chosen so that | a

1

| ą σ

₀²

{ 2. In this situation, for any x, y P S we have A

t

p x, y q “ 2 r a

1

´ a

2

x s ` σ

²₀

ď 2a

1

` σ

₀²

ùñ p H

A

q with λ

A

“ | a

1

| ´ σ

₀²

{ 2 ă 0 as well as

ψ

t

p µ q “ | a

1

| e

^´|a1|t

| a

1

| ` a

2

µ p e q `

1 ´ e

^´|a1|t

˘ ď e

^´|a1|t

This yields the estimate

E “

rB

ǫ

X

_s,t^ǫ

s

²

‰ ď ”

1 ` | a

^´1₁

a

2

| e

^´|a1|pt´sq

´

E p Z

₀²

q

^1{2

_ E p Z

₁²

q

^1{2

¯ı

2

e

^{´p2|a1|´σ20qpt´sq}

E pp Z

1

´ Z

0

q

²

q Up to some constant, this shows that the r.h.s. Wasserstein contraction estimate in (2.13) is met with λ

_C

“ | a

1

| ´ σ

₀²

{ 2.

The analysis of nonlinear diffusions on more general differentiable manifolds is based on more sophisticated differential techniques. The extension of the variational calculus developed above to this class of stochastic processes on manifolds is provided in section 3.2.

We end this section with some illustrations of our results on time homogeneous models p b

_t

, σ

_t

q “ p b, σ q satisfying condition p H

C

q . We set φ

t

: “ φ

0,t

, and P

_t^µ

: “ P

_0,t^µ

. By theorem 2.2, there exists an unique invariant measure

π “ φ

t

p π q and W

₂

p φ

t

p µ q , π q ď e

^´λCt

W

₂

p µ, π q

For the nonlinear Langevin diffusion discussed in (1.6) condition p H

C

q is met when (2.14) or (2.15) are satisfied. In this context, X

_t^π

: “ X

_0,t^π

is a conventional Langevin diffusion given by the time homogeneous stochastic differential equation

dX

_t^π

“ ´ ∇ V

^π

p X

_t^π

q dt ` σ

0

dW

t

with 2

^´1

V

^π

p y q “ U p y q ` ż

π p dx q V p y ´ x q In this situation, the unique invariant measure of X

_t^π

is given by

̟ p π qp dx q : “ 1 v

_π

exp

„

´ 1 σ

0

V

^π

p x q



dx with v

π

: “ ż

exp

„

´ 1 σ

0

V

^π

p x q

 dx

In the above display, dx stands for the Lebesgue measure on R

^d

. In this case the measure π “ φ

t

p π q “ πP

_t^π

is the unique solution of the equation π “ ̟ p π q . We underline that the uniqueness of the invariant measure is not ensured for double-well confinement potential functions and too small noise. Further details on this subject including a description of the invariant measures for small noise can be found in the series of articles [19, 20, 21].

Whenever p H

_C

q is met, we also have the uniform moment estimates

φ

t

p µ qp} e }

²

q

^1{2

ď π p} e }

²

q

^1{2

` W

₂

p µ, π q (2.17) In the same vein, when when p H

_A

q and p H

_C

q are met we have

E “

} X

_t^µ

p x q}

²

‰

1{2

ď π p} e }

²

q

^1{2

` W

₂

p δ

_x

P

_t^µ

, π q ď c “

π p} e }

²

q _ µ p} e }

²

q ‰

1{2

r 1 ` } x }s for some finite constant c. The last assertion comes from the fact that

W

₂

p δ

x

P

_t^µ

, π q ď W

₂

p δ

x

P

_t^µ

, φ

t

p µ qq ` W

₂

p φ

t

p µ q , π q ď c e

^{´pλA^λCqt}

r W

₂

p δ

x

, µ q ` W

₂

p µ, π qs

2.3 Some almost sure estimates

We fix the parameters ǫ and some given time horizon s ě 0, and we set y

t

: “ B

^ǫ

Y

_s,t^ǫ

, for any t P r s, 8r , with the process Y

_s,t^ǫ

defined in (2.11). Also consider the processes

dz

_t

: “ z

_0,t

dt ` ÿ

1ďkďr

z

_k,t

dW

^k_t

and dZ

_t

: “ Z

_0,t

dt ` ÿ

1ďkďr

Z

_k,t

dW

^k_t

with the collection of processes

z

_0,t

: “ E

_X

”

∇

_x

b

_t

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘

1

B

^ǫ

X

_s,t^ǫ

ı

z

_k,t

: “ E

_X

”

∇

_x

σ

_t,k

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘

1

B

^ǫ

X

_s,t^ǫ

ı Z

_0,t

: “ E

_X

”

∇

_y

b

t

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘

1

ı

and Z

_k,t

: “ E

_X

”

∇

_y

σ

_t,k

`

X

_s,t^ǫ

, Y

_s,t^ǫ

˘

1

ı In this notation, the evolution equation (2.11) reduces to

dy

_t

: “ dz

_t

` dZ

_t

y

_t

Let t P r s, 8rÞÑ E

_t

be the solution of the matrix evolution equation

dE

_t

: “ dZ

_t

E

_t

with E

_s

“ I and set @ t P r u, 8r E

_u,t

: “ E

_t

E

_u^´1

ùñ @ t P r u, 8r d E

_u,t

: “ dZ

t

E

_u,t

In this notation, we readily check that y

_t

“ E

_s,t

y

_s

`

ż

t

s

E

_u,t

˜

dz

_u

´ ÿ

1ďkďr

Z

_k,u

z

_k,u

du

¸

Whenever condition p H

A

q is met, for any given u ě 0 and any t P r u, 8r we have d “

E

_u,t¹

E

_u,t

‰

“ E

_u,t¹

«

Z

_0,t

` Z

_0,t¹

` ÿ

1ďkďr

Z

_k,t¹

Z

_k,t

ff

E

_u,t

dt ` ÿ

1ďkďr

E

_u,t¹

`

Z

_k,t

` Z

_k,t¹

˘

E

_u,t

dW

^k_t

ď ´ 2λ

_A

E

_u,t¹

E

_u,t

dt ` ÿ

1ďkďr

E

_u,t¹

`

Z

_k,t

` Z

_k,t¹

˘

E

_u,t

dW

^k_t

This shows that

p H

A

q and ∇

_y

σ

_k,t

“ 0 ùñ E

_u,t¹

E

_u,t

ď e

^´2λApt´uq

I

In addition, when ∇

_x

b

t

is uniformly bounded, ∇

_x

σ

_k,t

“ 0 and p H

C

q is met, using (2.12) we have almost sure estimate

}B

^ǫ

Y

_s,t^ǫ

} ď e

^´λApt´sq

} Z

1

´ Z

0

} ` } ∇

_x

b

t

}

²

ż

t

s

e

^´λApt´uq

E `

}B

^ǫ

X

_s,u^ǫ

} ˘ du ď e

^´λApt´sq

} Z

1

´ Z

0

} ` } ∇

_x

b

_t

}

²

λ

_A

´ λ

_C

´

e

^´λCpt´sq

´ e

^´λApt´sq

¯ E `

} Z

1

´ Z

0

}

²

˘

1{2

with the uniform spectral norm

} ∇

_x

b

t

}

²

: “ sup

x,y

} ∇

_x

b

t

p x, y q}

²

We summarize the above discussion with the following theorem.

Theorem 2.3. Assume that ∇

_x

b

t

is uniformly bounded, ∇

_x

σ

k,t

“ 0 “ ∇

_y

σ

k,t

and conditions p H

A

q and p H

_C

q are met. In this situation, we have the almost sure estimate

}B

^ǫ

X

_s,t^ǫ

} ď e

^´λApt´sq

} Z

1

´ Z

0

} ` p t ´ s q e

^´λpt´sq

} ∇

_x

b

t

}

²

E `

} Z

1

´ Z

0

}

²

˘

¹{2

with the process X

_s,t^ǫ

defined in (1.8) and the parameter λ : “ λ

_A

^ λ

_C

.

3 Some extensions

3.1 A backward variational formula

The stochastic transition semigroup associated with the flow X

_s,t^µ

p x q is defined for any mesurable function f on R

^d

by the formula

P

^µ_s,t

p f qp x q : “ f p X

_s,t^µ

p x qq ùñ P

_s,t^µ

p f qp x q “ E `

P

^µ_s,t

p f qp x q ˘ For twice differentiable function f we have the gradient and the Hessian formulae

∇ P

^µ_s,t

p f qp x q “ ∇X

_s,t^µ

p x q P

^µ_s,t

p ∇f qp x q

∇

²

P

^µ_s,t

p f qp x q “ “

∇X

_s,t^µ

p x q b ∇X

_s,t^µ

p x q ‰

P

^µ_s,t

p ∇

²

f qp x q ` ∇

²

X

_s,t^µ

p x q P

^µ_s,t

p ∇f qp x q In the above display, ∇

²

X

_s,t^µ

p x q stand for the tensors functions

∇

²

X

_s,t^µ

p x q

pi,jq,k

“ B

^i,j

X

_s,t^µ,k

p x q

“ ∇X

_s,t^µ

p x q b ∇X

_s,t^µ

p x q ‰

pi,jq,pk,lq

“ ∇X

_s,t^µ

p x q

i,k

∇X

_s,t^µ

p x q

j,l

Also recall that the infinitesimal generator L

_t,φs,tpµq

of the stochastic flow (1.4) is given for any twice differentiable function f by the second order operator

L

_t,φs,tpµq

p f qp x q : “ x b

_t

p φ

_s,t

p µ q , x q , ∇f p x qy ` 1 2 Tr “

∇

²

f p x q σ

_t

p φ

_s,t

p µ q , x q σ

_t

p φ

_s,t

p µ q , x q

¹

‰ Next theorem is an extension of a theorem by Da Prato-Menaldi-Tubaro [16] to nonlinear diffusions.

Theorem 3.1. Assume that b

_t

p x, y q and σ

_t

p x, y q are Lipschitz functions w.r.t. the parameters p t, x, y q . In this situation, for any µ P P

₂

p R

^d

q we have

P

^µ_s,t

p f qp x q “ f p x q ` ż

t

s

L

_u,φs,upµq

´

P

^φ_u,t^s,upµq

p f q ¯ p x q du

` ż

t

s

∇ P

^φ_u,t^s,upµq

p f qp x q

¹

σ

u

p φ

s,u

p µ q , x q d W p

u

(3.1) where d W p

_u

stands for the backward integration notation, so that the r.h.s. term in the above formula is a square integrable backward martingale.

The proof of the above formula follows the elegant stochastic backward variational analysis developed in [16]. A sketched proof is provided in the appendix, on page 26.

We further assume that ∇

_x

σ

_k,t

p x, y q “ 0. In this situation, using the backward formula (3.1) we check the stochastic interpolation formula

B

u

´

X

_u,t^φs,upµq

˝ X

_s,u^η

¯

p y q

¹

“ r φ

_s,u

p η q ´ φ

_s,u

p µ qs p b

_u

p . , X

_s,u^η

p y qqq

¹

”

∇X

_u,t^φs,upµq

ı

p X

_s,u^η

p y qq Equivalently, we have

X

_s,t^η

p x q ´ X

_s,t^µ

p x q “ ż

t

s

” ∇X

_u,t^φs,upµq

ı

p X

_s,u^η

p x qq

¹

r φ

_s,u

p η q ´ φ

_s,u

p µ qs p b

_u

p . ^{, X}

_s,u^η

_p ^x _qqq ^du (3.2)

Combining (2.2) and (2.3) with (2.13) we obtain the following corollary.

Corollary 3.2. Assume the conditions of theorem 3.1 are satisfied and we have ∇

_x

σ

_k,t

“ 0 and } ∇

_x

b

_t

p x, y q}

²

ď c, for some constant c ă 8 . Also assume that p H

_A

q and p H

_C

q are met for some parameters λ

_A

and λ

_C

. In this situation we have the exponential decay estimates

E `

} X

_s,t^η

p x q ´ X

_s,t^µ

p x q}

²

˘

1{2

ď c ?

d p t ´ s q e

^´λpt´sq

W

₂

p η, µ q with λ : “ λ

_A

^ λ

_C

In addition, when ∇

_y

σ

_k,t

“ 0 we have the uniform and almost sure estimates

} X

_s,t^η

p x q ´ X

_s,t^µ

p x q} ď c p t ´ s q e

^´λpt´sq

W

₂

p η, µ q 3.2 Diffusions on smooth manifolds

This section is concerned with the extension of our results to nonlinear diffusions on Riemannian manifolds. Let us begin with the general necessary facts about nonlinear diffusions in manifolds.

Our presentation will be made as similar as possible to the one in Euclidean space. For this, we will need Itô differentials of manifold valued diffusions, parallel translation, covariant differential of tangent bundle valued semimartingales.

Let M be a smooth manifold of dimension d. Stratonovich calculus is similar on M and on R

^d

. So we are able to deal with Stratonovich SDE’s of the type

˝ dX

_s,t^µ

p x q “ b

^S_t

p φ

_s,t

p µ q , X

_s,t^µ

p x qq dt ` σ p X

_s,t^µ

p x qqq ˝ dW

_t

, (3.3) where for y P M

b

^S_t

p η, y q “ ż

M

η p dx q b

^S_t

p x, y q , b

^S_t

p x, y q P T

y

M,

W

_t

is a R

^m

-valued Brownian motion and σ p y q is a linear map R

^m

Ñ T

_y

M. For simplicity σ will not depend on time, but the time-dependent σ can also be treated, we refer to [1] for this extension, and also for the details of the constructions below.

The only situation we will be interested in is when for all y P M the map p σσ

^˚

qp y q : T

_y^˚

M Ñ T

_y

M

is a linear diffeomorphism. In this situation a scalar product can be defined in T

_y^˚

M and then in T

_y

M , leading to a Riemannian structure on M . The scalar product in T

_y^˚

M is

g

^˚

p y qp α, β q “ x σ

^˚

p y qp α q , σ

^˚

p y qp β qy

^Rm

, (3.4) and the scalar product in T

_y

M is

g p y qp u, v q “ g

^˚

p y q `

p σσ

^˚

q

^´1

p y qp u q , p σσ

^˚

q

^´1

p y qp v q ˘

. (3.5)

Associated to the metric g is the Levi-Civita connection ∇, which will be used to define parallel transport, Itô equations, Itô covariant differentials. Recall that the parallel transport along a continuous M-valued semimartingale X is the linear map {{

^t

: T

X0

M Ñ T

Xt

M which satisfies {{

⁰

“ Id and the Stratonovich SDE ∇

_˝dXt

{{

t

“ 0. It is the natural extension to parallel transport along smooth paths, and it is an isometry. Parallel translation allows to anti-develop X

_t

in T

_X0

M with the Stratonovich integral

A p X q

t

“ ż

t

0

{{

^´1s

˝ dX

_s

The process A p X q takes its values in the vector space, it has an Itô differential dA p X q

^t

, which allows to define the Itô differential of X

_t

d

^∇

X

t

: “ {{

^t

d A p X q

^t

. (3.6)

This Itô differential is formally a vector which can be expressed in local coordinates as d

^∇

X

_t

“

ˆ

dX

_tⁱ

` 1

2 Γ

ⁱ_j,k

p X

_t

q d ă X

^j

, X

^k

ą

^t

˙ B

B x

ⁱ

p X

_t

q , with the Christoffel symbols Γ

ⁱ_j,k

. The next object to consider is Itô covariant derivative DU

_t

of a T

_Xt

M -valued continuous semi- martingale U

_t

:

DU

t

: “ {{

^t

d `

{{

^´t¹

U

t

˘

, (3.7)

easily defined from the fact that {{

^´1t

U

_t

is vector valued. From the isometry property of parallel translation we easily get the formula for V

_t

another T

_Xt

M-valued semimartingale and x¨ , ¨y : “ g,

d x U

_t

, V

_t

y “ x DU

_t

, V

_t

y ` x U

_t

, DV

_t

y ` x DU

_t

, DV

_t

y . (3.8) Defining b

_t

p x, y q : “ b

^S_t

p x, y q ` 1

2 ÿ

m k“1

∇σ

_k

p σ

_k

p y qq (where for two vector fields A, B, ∇A p B p y qq denotes the covariant derivative of A in the direction B p y q ), it is well known that the Stratonovich SDE (3.3) is equivalent to the Itô SDEs

d

^∇

X

_s,t^µ

p x q “ b

_t

p φ

_s,t

p µ q , X

_s,t^µ

p x qq dt ` σ p X

_s,t^µ

p x qq dW

_t

. (3.9) A remarkable fact concerning this equation, is that whenever it exists, a solution to equation (3.9) is a diffusion with nonlinear generator L

_t,φs,t

p µ q , where

L

_t,η

“ 1 2 ∆ `

ż

M

η p dx q b

_t

p x, y q . (3.10)

So we can consider that the starting point of our study is SDE (3.9) in a Riemannian manifold p M, g q .

Let us adapt the regularity conditions p H

_A

q and p H

_C

q :

Define A

^g_t

p x, y q : “ ∇

_y

b

_t

p x, y q ` ∇

_y

b

_t

p x, y q

¹

, where ∇

_y

b

_t

p x, y q is the covariant derivative with respect to the variable y, it is a linear map from T

y

M into itself, and ∇

_y

b

t

p x, y q

¹

is its adjoint with respect to the Riemannian metric.

p H

_A^g

q : There exists some λ

^g_A

P R such that for any x, y P M and t ě 0 we have

A

^g_t

p x, y q ´ Ric p y q ď ´ 2λ

^g_A

g p y q (3.11) where Ric is the Ricci curvature tensor of M .

Let B

_t^g

be as in (2.8) with gradient replaced by covariant derivative.

Define C

_t^g

p x, y q : “

¹2

“ B

_t^g

p x, y q ` B

_t^g

p x, y q

¹

‰ .

p H

_C^g

q : There exists some λ

^g_C

P R such that for any x, y P M and t ě 0 we have C

_t^g

p x, y q ´ 1

2 Ric

_MˆM

p x, y q ď ´ λ

^g_C

g

_MˆM

p x, y q (3.12)

where g

MˆM

p x, y q , Ric

_MˆM

p x, y q are the product metric and Ricci curvature on M ˆ M .

Theorem 3.3. We have the exponential expansion or contraction inequalities p H

_A^g

q ùñ W

₂

`

η

0

P

_s,t^µ

, η

1

P

_s,t^µ

˘

ď c e

^´λgApt´sq

W

₂

p η

0

, η

1

q (3.13) for some finite constant c. In addition, we have

p H

_C^g

q ùñ W

₂

p φ

_s,t

p µ

0

q , φ

_s,t

p µ

1

qq ď e

^´λgCpt´sq

W

₂

p µ

0

, µ

1

q (3.14) Remark: The results of Theorem 3.3 still hold when σ “ σ

_t

and g “ g

_t

depend on time, one just has to replace in p H

_A^g

q Ric by Ric ´ g 9 and in p H

_C^g

q Ric

_MˆM

by Ric

_MˆM

´ g 9

_MˆM

.

Proof. The proof of the first estimate is similar to the proof of Theorem 4.1 in [1] (where time dependent metrics are considered), so we will omit it. The proof of the second one is a combination of this proof and to the one of Theorem 2.2 in the present article. Let us go into the details.

Let Z

₀

, Z

₁

two random variables with values in M , and such that p Z

₀

, Z

₁

q minimizes E r d

²

p Z

₀

, Z

₁

qs under the condition that Z

0

has law µ

0

and Z

1

has law µ

1

. For all ω, let ǫ ÞÑ Z

ǫ

p ω q be a geodesic between Z

0

p ω q and Z

1

p ω q .

As in the proof of Theorem 2.2, let Y

_s,s^µ0

p x q “ x and t P r s, 8rÞÑ Y

_s,t^µ0

p x q solve the equation dY

_s,t^µ0

p x q “ b

_t

p φ

_s,t

p µ

0

q , Y

_s,t^µ0

p x qq dt ` σ p Y

_s,t^µ0

p x qq d W ¯

_t

where W ¯

t

is a R

^m

valued Brownian motion independent of W

t

. Let p Z ¯

ǫ

q

ǫPr0,1s

be independent of p Z

_ǫ

q

ǫPr0,1s

with the same law, Y

_s,s^ǫ

“ Z ¯

_ǫ

and Y

_s,t^ǫ

the solution to the Itô SDE

dY

_s,t^ǫ

“ E

_X

“

b

t

p X

_s,t^ǫ

, Y

_s,t^ǫ

q ‰

dt ` {{

s,t^0,ǫ

` σ p Y

_s,t⁰

q d W ¯

t

˘

, (3.15)

where ǫ ÞÑ {{

s,t^0,ǫ

p ω q is the parallel transport along the ǫ ÞÑ Y

_s,t^ǫ

p ω q . Notice that Y

_s,t⁰

” Y

_s,t^µ0

p Z ¯

0

q . The equation (3.15) is not an SDE on the manifold M, it is an SDE on C

¹

M -valued paths.

Existence of solutions have been established in [1]. The processes t ÞÑ Y

_s,t^ǫ

are obtained one from the others by infinitesimal synchronious coupling, and it is the only construction where a.s. the paths ǫ ÞÑ Y

_s,t^ǫ

p ω q has finite variation. Moreover, the derivatives of theses paths satisfy

D B

^ǫ

Y

_s,t^ǫ

“ E

_X

“

∇

_x

b

t

p X

_s,t^ǫ

, Y

_s,t^ǫ

qB

^ǫ

X

_s,t^ǫ

‰

dt ` E

_X

“

∇

_y

b

t

p X

_s,t^ǫ

, Y

_s,t^ǫ

q ‰

B

^ǫ

Y

_s,t^ǫ

dt ´ 1

2 Ric

⁷

pB

^ǫ

Y

_s,t^ǫ

q dt (3.16) where Ric

⁷

p u q is the vector such that x Ric

⁷

p u q , v y “ Ric p u, v q . The advantage of this construction is that the above covariant derivative has finite variation, and this implies

d }B

^ǫ

Y

_s,t^ǫ

}

²

“ 2 xB

^ǫ

Y

_s,t^ǫ

, D B

^ǫ

Y

_s,t^ǫ

y . Then the proof is similar to the one of Theorem 2.2:

B

^t

E “

}B

^ǫ

Y

_s,t^ǫ

}

²

‰

“ E

„Bˆ B

ǫ

X

_s,t^ǫ

B

^ǫ

Y

_s,t^ǫ

˙

, B

_t

p X

_s,t^ǫ

, Y

_s,t^ǫ

q

ˆ B

ǫ

X

_s,t^ǫ

B

^ǫ

Y

_s,t^ǫ

˙F

´ E “

Ric pB

^ǫ

Y

_s,t^ǫ

, B

^ǫ

Y

_s,t^ǫ

q ‰

“ E

„Bˆ B

^ǫ

X

_s,t^ǫ

B

ǫ

Y

_s,t^ǫ

˙

, B

t

p X

_s,t^ǫ

, Y

_s,t^ǫ

q

ˆ B

^ǫ

X

_s,t^ǫ

B

ǫ

Y

_s,t^ǫ

˙F

´ 1 2 E

„

Ric

MˆM

ˆˆ B

^ǫ

X

_s,t^ǫ

B

ǫ

Y

_s,t^ǫ

˙ ,

ˆ B

^ǫ

X

_s,t^ǫ

B

ǫ

Y

_s,t^ǫ

˙˙

ď ´ λ

^g_C

E

«› › › ›

ˆ B

^ǫ

X

_s,t^ǫ

B

ǫ

Y

_s,t^ǫ

˙›› › ›

2

ff

“ ´ 2λ

^g_C

E “

}B

^ǫ

Y

_s,t^ǫ

}

²

‰

.

This implies that E “

}B

ǫ

Y

_s,t^ǫ

}

²

‰ ď E

” }B

ǫ

|

ǫ“0

Z

_ǫ

}

²

ı

e

^{´2λgCpt´sq}

“ e

^{´2λgCpt´sq}

W

²₂

p µ

0

, µ

1

q . On the other hand, we have

W

²₂

p φ

_s,t

p µ

0

q , φ

_s,t

p µ

1

qq ď E

«ˆż

1

0

}B

^ǫ

Y

_s,t^ǫ

} dǫ

˙

²

ff

ď ż

1

0

E “

}B

ǫ

Y

_s,t^ǫ

}

²

‰

dǫ ď e

^{´2λgCpt´sq}

W

²

2

p µ

0

, µ

1

q This ends the proof of the theorem.

An important example of nonlinear diffusions in manifolds is again given by Langevin diffusions, defined as in (3.9), with now

b

_t

p x, y q “ ´ ∇U p y q ´ ∇ p F ˝ ρ

_x

qp y q (3.17) where U is a potential function, ρ is the Riemannian distance associated to the metric g, ρ

_x

is the distance to x and F : R

_`

Ñ R is a C

²

function. A sufficient condition b

t

p x, y q defined by (3.17) to be well defined and smooth is that the derivative of F is nul at the origin and the support of F is included in r 0, ı p M qq , where ı p M q denotes the injectivity radius of M. But smoothness of b

_t

p x, y q is not a necessary condition for defining nonlinear diffusions.

We find that for u, v P T

_y

M ,

∇

_y

b p u, v q “ ´ ∇

²

U p u, v q ´ ∇

²

p F ˝ ρ

_x

qp u, v q . (3.18) In this context, condition p H

_A^g

q reduces to

∇

²

U p y q ` ∇

²

p F ˝ ρ

_x

qp y q ` 1

2 Ric p y q ě λ

^g_A

g p y q . (3.19) If for instance M is simply connected with nonpositive curvature (which implies that the distance function ρ is convex), and F is nondecreasing, a sufficient condition is

∇

²

U p y q ` 1

2 Ric p y q ě λ

^g_A

g p y q . (3.20) The computation of B

_t

reveals that it is symmetric, and that for p u, v q P T

_x

M ˆ T

_y

M ,

B

t

p x, y qpp u, v q , p u, v qq “ ´ ∇

²

U p x qp u, u q ´ ∇

²

U p y qp v, v q ´ ∇

²

p F ˝ ρ qp x, y qpp u, v q , p u, v qq , (3.21) In this context condition p H

_C^g

q reduces to

∇

²

U

^‘2

p x, y q ` ∇

²

p F ˝ ρ qp x, y q ` 1

2 Ric

_MˆM

p x, y q ě λ

^g_C

g

_MˆM

p x, y q (3.22) where U

^‘2

p x, y q “ U p x q ` U p y q . Here again, when M is simply connected with nonpositive curva- ture, F is convex and nondecreasing, the above condition is met as soon as

∇

²

U p y q ` 1

2 Ric p y q ě λ

^g_C

g p y q . (3.23)