A second order analysis of McKean-Vlasov semigroups

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A second order analysis of McKean-Vlasov semigroups

Marc Arnaudon, Pierre del Moral

To cite this version:

Marc Arnaudon, Pierre del Moral. A second order analysis of McKean-Vlasov semigroups. Annals of

Applied Probability, Institute of Mathematical Statistics (IMS), 2020, �10.1214/20-AAP1568�. �hal-

02151808v2�

A second order analysis of McKean-Vlasov semigroups

M. Arnaudon ¹ and P. Del Moral ²

1 Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence, France

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

Abstract

We propose a second order differential calculus to analyze the regularity and the stabil- ity properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolu- tion semigroup as well as the stochastic flow associated with this class of nonlinear diffusions.

Bismut-Elworthy-Li formulae for the gradient and the Hessian of the integro-differential opera- tors associated with these expansions are also presented.

The article also provides explicit Dyson-Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon.

We illustrate the impact of these results with a second order extension of the Alekseev-Gröbner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos prop- erties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities.

Keywords : Nonlinear diffusions, mean field particle systems, variational equations, logarith- mic norms, gradient flows, Taylor expansions, contraction inequalities, Wasserstein distance, Bismut-Elworthy-Li formulae.

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

1 Introduction

1.1 Description of the models

For any n ě 1 we let P n pR ^d q be the convex set of probability measures η, µ on R ^d with absolute n-th moment and equipped with the Wasserstein distance of order n denoted by W n pη, µq . Also let b _t px ₁ , x ₂ q be some Lipschitz function from R ^2d into R ^d and let W _t be an d-dimensional Brownian motion defined on some filtered probability space pΩ, pF t q tě0 , Pq . We also consider the Hilbert space H t pR ^d q :“ L 2 ppΩ, F t , Pq, R ^d q equipped with the L 2 inner product x . ^, . y _H

_{p R}

_q . Up to a probability space enlargement there is no loss of generality to assume that H t pR ^d q contains square integrable R ^d -valued variables independent of the Brownian motion.

For any µ P P ₂ pR ^d q and any time horizon s ě 0 we denote by X _s,t ^µ pxq the stochastic flow defined for any t P rs, 8r and any starting point x P R ^d by the McKean-Vlasov diffusion

dX _s,t ^µ pxq “ b _t `

X _s,t ^µ pxq, φ _s,t pµq ˘

dt ` dW _t with b _t px, µq :“

ż

µpdyq b _t px, yq (1.1)

In the above display, φ _s,t stands for the evolution semigroup on P ₂ pR ^d q defined by the formulae φ _s,t pµqpdyq “ µP _s,t ^µ pdyq :“

ż

µpdxq P _s,t ^µ px, dyq with P _s,t ^µ px, dyq :“ PpX _s,t ^µ pxq P dyq

We denote by L _t,φ

_pµq the generator of the stochastic flow X _s,t ^µ pxq . The existence of the stochastic flow X _s,t ^µ pxq is ensured by the Lipschitz property of the drift function see for instance [41, 47]. To analyze the smoothness of the semigroup φ s,t we need to strengthen this condition.

We shall assume that the function b t px 1 , x 2 q is differentiable at any order with uniformly bounded derivatives. In addition, the partial differential matrices w.r.t. the first and the second coordinate are uniformly bounded; that is for any i “ 1, 2 we have

}b ^ris } 2 :“ sup

tě0

sup

px

,x

qP R

}b ^ris _t px 1 , x 2 q} 2 ă 8 with b ^ris _t px 1 , x 2 q :“ ∇ _x

b t px 1 , x 2 q (1.2) In the above display, }A} ₂ :“ λ _max pAA ¹ q ^1{2 stands for the spectral norm of some matrix A, where A ¹ stands for the transpose of A, λ _max p . q and λ _min p . q the maximal and minimal eigenvalue. In the further development of the article, we shall also denote by A sym “ pA ` A ¹ q{2 the symmetric part of a matrix A. In the further development of the article we represent the gradient of a real valued function as a column vector, or equivalently as the transpose of the differential-Jacobian operator which is, as any cotangent vector, represented by a row vector. The gradient and the Hessian of a column vector valued function as tensors of type p1, 1q and p2, 1q , see for instance (3.1).

The mean field particle interpretation of the nonlinear diffusion (1.1) is described by a system of N-interacting diffusions ξ t “ pξ ⁱ _t q 1ďiďN defined by the stochastic differential equations

dξ _t ⁱ “ b t pξ _t ⁱ , mpξ t qq dt ` dW _t ⁱ with 1 ď i ď N and mpξ t q :“ 1 N

ÿ

1ďjďN

δ _ξ

t

(1.3)

In the above display, ξ ⁱ ₀ stands for N independent random variables ξ ₀ ⁱ with common distribution µ ₀ , and W _t ⁱ are N independent copie of the Brownian motion W _t .

McKean-Vlasov diffusions and their mean field type particle interpretations arise in a variety of application domains, including in porous media and granular flows [7, 8, 18, 67], fluid mechanics [58, 59, 61, 68], data assimilation [10, 26, 36], and more recently in mean field game theory [9, 14, 13, 15, 16, 17, 46, 43], and many others.

The origins of this subject certainly go back to the beginning of the 1950s with the article by Harris and Kahn [45] using mean field type splitting techniques for estimating particle transmission energies. We also refer to the pioneering article by Kac [50, 51] on particle interpretations of Boltzmann and Vlasov equations, and the seminal articles by McKean [58, 59] on mean field particle interpretations of nonlinear parabolic equations arising in fluid mechanics. Since this period, the analysis of this class of mean field type nonlinear diffusions and their discrete time versions have been developed in various directions. For a survey on these developments we refer to [15, 26, 65], and the references therein.

The McKean-Vlasov diffusions discussed in this article belong to the class of nonlinear Markov processes. One of the most important and difficult research questions concerns the regularity analysis and more particularly the stability and the long time behavior of these stochastic models.

In contrast with conventional Markov processes, one of the main difficulty of these Markov

processes comes from the fact that the evolution semigroup φ s,t pµq is nonlinear w.r.t. the initial

condition µ of the system. The additional complexity in the analysis of these models is that their

state space is the convex set of probability measures, thus conventional functional analysis and

differential calculus on Banach space cannot be directly applied.

The main contribution of this article is the development of a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. We also provide a refined analysis of the norm of these integro-differential operators with a series of exponential decays inequalities with respect to the time horizon.

The article is organized as follows:

The main contributions of this article are briefly discussed in section 1.2. The main theorems are stated in some detailed in section 2. Section 3 provides some pivotal results on tensor integral operators and on integro-differential operators associated with the second order Taylor expansions of the semigroup φ _s,t pµq . Section 4 is dedicated to the analysis of the tangent process associated with the nonlinear diffusion flow. We presents explicit Dyson-Phillips expansions as well as some spectral estimates. The last section, section 4 is mainly concerned with the proofs of the first and second order Taylor expansions. The proof of some technical results are collected in the appendix.

Detailed comparisons with existing literature on this subject are also provided in section 2.5.

1.2 Statement of some main results

One of the main contribution of the present article is the derivation of a second order Taylor expansion with remainder of the semigroup φ s,t on probability spaces. For any pair of measures µ 0 , µ 1 P P 2 pR ^d q , these expansions take basically the following form:

φ s,t pµ 1 q » φ s,t pµ 0 q ` pµ 1 ´ µ 0 qD µ

φ s,t ` 1

2 pµ 1 ´ µ 0 q ^b2 D _µ ²

φ s,t (1.4) In the above display, D ^k _µ

φ _s,t stands some first and second order operators, with k “ 1, 2. A more precise description of these expansions and the remainder terms is provided in section 2.2.

Section 2.3.1, also provides an almost sure second order Taylor expansions with remainder of the random state X _s,t ^µ pxq of the McKean diffusion w.r.t. the initial distribution µ. These almost sure expansions take basically the following form

X _s,t ^µ

pxq ´ X _s,t ^µ

pxq » ż

pµ 1 ´ µ 0 qpdyq D µ

X _s,t ^µ

px, yq ` 1 2

ż

pµ 1 ´ µ 0 q ^b2 pdzq D ² _µ

X _s,t ^µ

px, zq (1.5) for some random functions D ^k _µ

X _s,t ^µ

from R ^p1`kqd into R ^d , with k “ 1, 2. A more precise description of these almost sure expansions is provided in section 2.3.1 (see for instance (2.19) and theorem 2.6).

Given some random variable Y P H s pR ^d q with distribution µ P P ₂ pR ^d q , observe that the stochas- tic flow ψ s,t pY q :“ X _s,t ^µ pY q satisfies the H t pR ^d q-valued stochastic differential equation

dψ _s,t pY q :“ B _t pψ _s,t pY qq dt ` dW _t (1.6) In the above display, B _t stands for the drift function from H t pR ^d q into itself defined by the formula

B t pXq :“ E

` b t pX, Xq | X ˘

In the above display, X stands for an independent copy of X. The above Hilbert space valued representation of the McKean-Vlasov diffusion (1.1) readily implies that for any Y 1 , Y 0 P H s pR ^d q we have the exponential contraction inequality

}ψ _s,t pY ₁ q ´ ψ _s,t pY ₀ q} _H

_{p R}

_q ď e ^´λpt´sq }Y ₁ ´ Y ₀ } _H

_{p R}

_q

for some λ ą 0, as soon as the following condition is satisfied

xX 1 ´ X 0 , B t pX 1 q ´ B t pX 0 qy _H

_{p R}

_q ď ´2λ }X 1 ´ X 0 } ² _H

_pR

q (1.7) for any t ě 0 and any X 1 , X 0 P H t pR ^d q. In addition, in this framework the first order differential Bψ s,t pY q of the stochastic flow coincides with the conventional Fréchet derivative of functions from an Hilbert space into another. In addition, we shall see that the gradient of first order operator D µ φ s,t coincides with the dual of the tangent process associated with the Hilbert space-valued representation (1.6) of the McKean-Vlasov diffusion (1.1); that is, for any smooth function f we have that the dual tangent formula

Bψ _s,t pY q ^‹ ¨ ∇f pψ _s,t pY qq “ ∇D _µ φ _s,t pf qpY q (1.8) A more precise description of the Fréchet differential Bψ s,t pY q and the dual operator is provided in section 2.1 and section 4. A proof of the above formula is provided in theorem 4.8.

The Taylor expansions discussed above are valid under fairly general and easily verifiable condi- tions on the drift function. For instance, the regularity condition (1.2) is clearly satisfied for linear drift functions. As it is well known, dynamical systems and hence stochastic models involving drift functions with quadratic growth require additional regularity conditions to ensure non explosion of the solution in finite time.

Of course the expansions (1.4) and (1.5) will be of rather poor practical interest without a better understanding of the differential operators and the remainder terms. To get some useful approximations, we need to quantify with some precision the norm of these operators. A important part of the article is concerned with developing a series of quantitative estimates of the differential operators D ^k _µ

φ s,t and the remainder term; see for instance theorem 2.3 and theorem 2.4.

To avoid estimates that grow exponentially fast with respect to the time horizon, we need to estimate with some precision the operator norms of the differential operators in (1.4). To this end, we shall consider an additional regularity condition:

pHq : There exists some λ ₀ ą 0 and λ ₁ ą }b ^r2s } 2 such that for any px ₁ , x ₂ q P R ^2d and any time horizon t ě 0 we have

A _t px ₁ , x ₂ q sym ď ´λ ₀ I and b ^r1s _t px ₁ , x ₂ q sym ď ´λ ₁ I (1.9) In the above display, I stands for the identity matrix and A _t the matrix-valued function defined by

A t px 1 , x 2 q :“

«

b ^r1s _t px 1 , x 2 q b ^r2s _t px 2 , x 1 q b ^r2s _t px ₁ , x ₂ q b ^r1s _t px ₂ , x ₁ q

ff

and we set λ 1,2 :“ λ 1 ´ }b ^r2s } 2 (1.10)

Whenever (1.9) and (1.10) are met for some parameters λ 0 and λ 1 P R all the exponential estimates stated in the article remains valid but they grow exponentially fast with respect to the time horizon. More detailed comments on the above regularity conditions, including illustrations for linear drift and gradient flow models, as well as comparisons with related conditions used in the literature on this subject are also provided in section 2.4.

Under the above condition, we shall develop several exponential decays inequalities for the norm

of the differential operators D ^k _µ

φ s,t as well as for the remainder terms in the Taylor expansions. The

first order estimates are given in (2.6), the ones on the Bismut-Elworthy-Li gradient and Hessian

extension formulae are provided in (2.7) and (2.8). Second and third order estimates can also be

found in (2.12) and (2.15).

The second order differential calculus discussed above provides a natural theoretical basis to analyze the stability properties of the semigroup φ _s,t and the one of the mean field particle system discussed in (1.3).

For instance, a first order Taylor expansion of the form (1.4) already indicates that the sensi- tivity properties of the semigroup w.r.t. the initial condition µ are encapsulated in the first order differential operator D µ φ s,t . Roughly speaking, whenever pHq is satisfied, we show that there exists some parameter λ ą 0 such that

_ k“1,2 |||D _µ ^k

φ _s,t ||| » e ^´λpt´sq and therefore |||φ _s,t pµ ₁ q ´ φ _s,t pµ ₀ q||| » e ^´λpt´sq (1.11) for some operator norms ||| . ||| . For a more precise statement we refer to theorem 2.2 and the discussion following the theorem.

The second order expansion (1.4) also provides a natural basis to quantify the propagation of chaos properties of the mean field particle model (1.3). Combining these Taylor expansions with a backward semigroup analysis we derive a a variety of uniform mean error estimates w.r.t. the time horizon. This backward second order analysis can be seen a second order extension of the Alekseev- Gröbner lemma [1, 42] to nonlinear measure valued and stochastic semigroups. For a more precise statement we refer to theorem 2.7. As in (1.11), one of the main feature of the expansion (1.4) is that it allows to enter the stability properties of the limiting semigroup φ s,t into the analysis of the flow of empirical measures mpξ _t q .

Roughly speaking, this backward perturbation analysis can be interpreted as a second order variation-of-constants technique applied to nonlinear equations in distribution spaces. As in the Ito’s lemma, the second order term is essential to capture the quadratic variation of the processes, see for instance the recent articles [35, 48] in the context of conventional stochastic differential equation, as well as in [4, 31] in the context of interacting jump models.

The discrete time version of this backward perturbation semigroup methodology can also be found in chapter 7 in [25], a well as in the articles [27, 28, 30] and [34, 37] for general classes of mean field particle systems.

The central idea is to consider the telescoping sum on some time mesh t _n ď t _n`1 given by the interpolating formula

m _t

´ φ _t

_,t

pm _t

q “ ÿ

1ďkďn

“ φ _t

_,t

pm _t

q ´ φ _t

_,t

`

φ _t

_,t

pm _t

q ˘‰

with m _t

:“ mpξ _t

q Applying (1.4) and whenever pt _k ´ t _k´1 q » 0 we have the second order approximation

m t

´ φ t

,t

pm t

q » 1

? N ÿ

1ďkďn

∆M t

D m

φ t

,t

` 1 2N

ÿ

1ďkďn

p∆M t

q ^b2 D ² _m

tk´1

φ t

,t

with the local fluctuation random fields

∆M t

:“ ?

N pm t

´ m t

q and m t

:“ φ t

,t

` m t

˘ » m t

For discrete generation particle systems, ξ ⁱ _t

are defined by N conditionally independent variables given the system ξ _t

. For a more rigorous analysis we refer to section 2.3.2.

The above decomposition shows that the first order operator D _µ φ _s,t reflects the fluctuation

errors of the particle measures, while the second order term encapsulates their bias. In other words,

estimating the norm of second order operator D ² _µ φ _s,t allows to quantify the bias induced by the

interaction function, while the estimation of first order term is used to derive central limit theorems

as well as L p -mean error estimates.

As in (1.11), these estimates take basically the following form. For n ě 1 and any sufficiently regular function f we have

|||D _µ

φ _s,t ||| » e ^´λpt´sq ùñ |E r}m _t pf q ´ φ _0,t pm ₀ qpfq} ⁿ s ^1{n | ď c _n {

? N (1.12)

In addition, we have the uniform bias estimate w.r.t. the time horizon

|||D ² _µ

φ s,t ||| » e ^´λpt´sq ùñ |E rm t pf q ´ φ 0,t pm 0 qpf qs | ď c{N (1.13) In the above display, ||| . ||| stands for some operator norm, and pc, c _n q stands for some finite constants whose values doesn’t depend on the time horizon. We emphasize that the above results are direct consequence of a second order extension of the Alekseev-Gröbner type lemma for particle density profiles. For more precise statements we refer to theorem 2.7 and the discussion following the theorem.

1.3 Some basic notation

Let LinpB ₁ , B ₂ q be the set of bounded linear operators from a normed space B ₁ into a possibly different normed space B ₂ equipped with the operator norm ||| . ||| _B

_ÑB

. When B ₁ “ B ₂ we write Lin p B ₁ q instead of Lin p B ₁ , B ₁ q .

With a slight abuse of notation, we denote by I the identity pd ˆ dq-matrix, for any d ě 1, as well as the identity operator in Lin p B ₁ , B ₁ q . We also denote by } . } any (equivalent) norm on some finite dimensional vector space over R .

We also use the conventional notation B , B x

, B s , B t and so on for the partial derivatives w.r.t.

some real valued parameters , x i , s and t.

We let ∇f pxq “ rB _x

f pxqs _1ďiďd be the gradient column vector associated with some smooth function f pxq from R ^d into R . Given some smooth function hpxq from R ^d into R ^d we denote by

∇h “ “

∇h ¹ , . . . , ∇h ^d ‰

the gradient matrix associated with the column vector function h “ ph ⁱ q 1ďiďd . We also let p∇ b ∇q be the second order differential operator defined for any twice differentiable function gpx 1 , x 2 q on R ^2d by the Hessian-type formula

pp ∇ b ∇ qgq _i,j “ p ∇ _x

b ∇ _x

qpgq i,j “ p ∇ _x

b ∇ _x

qpgq j,i “ B _x

1

B _x

2

g (1.14)

We consider the space C ⁿ pR ^d q of n-differentiable functions and we denote by C _m ⁿ pR ^d q the subspace of functions f such that

sup

0ďkďn

} ∇ ^k f pxq} ď c w m pxq with the weight function w m pxq “ p1 ` }x}q ^m for some m ě 0.

We equip C _m ⁿ pR ^d q with the norm }f} _C

m

pR

q :“ ÿ

0ďkďn

}∇ ^k f {w m } 8 with }∇ ^k f {w m } 8 “ sup

xP R

}∇ ^k fpxq{w m pxq}

When there are no confusions, we drop to lower symbol } . } 8 and we write }f } instead of }f} 8 the supremum norm of some real valued function. We let epxq :“ x be the identify function on R ^d and for any µ P P _n pR ^d q and n ě 1 we set

}e} _µ,n :“

„ż

}x} ⁿ µpdxq

 _1{n

For any µ ₁ , µ ₂ P P _n pR ^d q , we also denote by ρ _n pµ ₁ , µ ₂ q some polynomial function of }e} _µ

_,n with i “ 1, 2. When µ ₁ “ µ ₂ we write ρ _n pµ ₁ q instead of ρ _n pµ ₁ , µ ₁ q .

Under our regularity conditions on the drift function, using elementary stochastic calculus for any n ě 2 and µ P P _n pR ^d q we check the following estimates

E

` }X _s,t ^µ pxq} ⁿ ˘ _1{n

ď c n ptq p}x} ` }e} µ,2 q which implies that φ s,t pµqp}e} ⁿ q ^1{n ď c n ptq }e} µ,n (1.15) In the above display and throughout the rest of the article, we write cptq, c ptq, c _n ptq, c _n, ptq, c _,n ptq and c m,n ptq with m, n ě 0 and P r0, 1s some collection of non decreasing and non negative functions of the time parameter t whose values may vary from line to line, but which only depend on the parameters m, n, , as well as on the drift function b _t . Importantly these contants do not depend on the probability measures µ. We also write c, c , c n , c n, , and c m,n when the constant do not depend on the time horizon.

2 Statement of the main theorems

2.1 First variational equation on Hilbert spaces

As expected, the Fréchet differential Bψ _s,t pY q of the stochastic flow ψ _s,t pY q associated with the stochastic differential equation (1.6) satisfies an Hilbert space-valued linear equation (cf. (4.1)).

The drift-matrix of this evolution equation is given by the Fréchet differential BB t pψ s,t pY qq of the drift function B _t evaluated along the solution of the flow. Mimicking the exponential notation of the solution of conventional homogeneous linear systems, the evolution semigroup (a.k.a. propagator) associated with the first variational equation is written as follows

Bψ s,t pY q “ e

ű

s

BB

pψ

pY qq du

P Lin pH s pR ^d q, H t pR ^d qq

The above exponential is understood as an operator valued Peano-Baker series [64]. A more detailed presentation of these models is provided in section 4.

The H t pR ^d q -log-norm of an operator T t P Lin pH t pR ^d q, H t pR ^d qq is defined by γ pT _t q :“ sup

}Z}

“1

xZ, pT _t ` T _t ^‹ q{2 ¨ Z y _H

_{p R}

_q

Our first main result is an extension of an inequality of Coppel [22] to tangent processes associ- ated with Hilbert-space valued stochastic flows.

Theorem 2.1. For any time horizon t ě s and any Y P H s pR ^d q we have the log-norm estimate

´ ż _t

s

γ p´BB u pψ s,u pY qqq du ď 1 t log |||e

ű

s

BB

pψ

pY qq du

||| _H

_pR

qÑH

pR

q ď ż _t

s

γ pBB u pψ s,u pY qqq du (2.1) In addition, we have

pHq ùñ BB t pXq sym ď ´λ 0 I ùñ 1 t log |||e

ű

s

BB

pψ

pY qq du

||| _H

_{p R}

_{qÑ H}

_{p R}

_q ď ´λ 0 (2.2) The proof of the above theorem in provided in section 4.1.

Let Y ₀ , Y ₁ P H s pR ^d q be a pair of random variables with distributions pµ ₀ , µ ₁ q P P ₂ pR ^d q ² . Also let µ be the probability distribution of the random variable

Y :“ p1 ´ q Y 0 ` Y 1 ùñ B ψ s,t pY q “ e

ű

s

BB

pψ

pY qq du ¨ pY 1 ´ Y 0 q (2.3)

This observation combined with the above theorem yields an alternative and more direct proof of an exponential Wasserstein contraction estimate obtained in [5]. Namely, using (2.2) we readily check the W 2 -exponential contraction inequality

BB t pXq sym ď ´λ 0 I ùñ W 2 pφ s,t pµ 1 q, φ s,t pµ 0 qq ď e ^´λ

^pt´sq W 2 pµ 0 , µ 1 q (2.4) For any function f P C ¹ pR ^d q with bounded derivative we also quote the first order expansion

rφ s,t pµ 1 q ´ φ s,t pµ 0 qs pf q “ ż 1

0

xBψ s,t pY q ^‹ ¨ ∇f pψ s,t pY qq, pY 1 ´ Y 0 qy _H

_{p R}

_q d

In the above display, x . ^, . y _H

_{p R}

_q stands for the conventional inner product on L 2 ppΩ, F t , Pq, R ^d q . The above assertion is a direct consequence of theorem 4.8.

2.2 Taylor expansions with remainder

The first expansion presented in this section is a first order linearization of the measure valued mapping φ _s,t in terms of a semigroup of linear integro-differential operators.

Theorem 2.2. For any m, n ě 1 and µ ₀ , µ ₁ P P _{m_2} pR ^d q , there exists a semigroup of linear operators D µ

,µ

φ s,t from C _m ⁿ pR ^d q into itself such that

φ s,t pµ 1 q “ φ s,t pµ 0 q ` pµ 1 ´ µ 0 qD µ

,µ

φ s,t (2.5) In addition, when pHq is satisfied we have the gradient estimate

} ∇D _µ

_,µ

φ _s,t pf q} ď c e ^´λpt´sq } ∇f } for some λ ą 0 (2.6) The proof of the above theorem with a more explicit description of the first order operators D µ

,µ

φ s,t are provided in section 4.3. In (2.6) we can choose λ “ λ 1,2 , with the parameter λ 1,2

introduced in (1.10). The semigroup property is a consequence of theorem 4.5 and the gradient estimates is a reformulation of the operator norm estimate discussed in (4.13).

We also provide Bismut-Elworthy-Li-type formulae that allow to extend the gradient and Hessian operators ∇ ^k D µ

,µ

φ s,t with k “ 1, 2 to measurable and bounded functions. When the condition pHq is satisfied we show the following exponential estimates

} ∇D _µ

_,µ

φ _s,t pfq} ď c `

1 _ 1{ ? t ´ s ˘

e ^´λpt´sq }f } for some λ ą 0 (2.7) In addition, we have the Hessian estimate

} ∇ ² D µ

,µ

φ s,t pfq} ď c p1 _ 1{pt ´ sqq e ^´λpt´sq }f } for some λ ą 0 (2.8) The proof of the first assertion can be found in remark 4.7 on page 31. The proof of the Hessian estimates is a consequence of the decomposition of ∇ ² D _µ

_,µ

φ _s,t discussed in (5.1) and the Hessian estimates (3.17) and (3.32).

It is worth mentioning that the semigroup property is equivalent to the chain rule formula D µ

,µ

φ s,t “ D µ

,µ

φ s,u ˝ D _φ

_pµ

_q,φ

_pµ

_q φ u,t (2.9) which is valid for any s ď u ď t. Without further work, theorem 2.2 also yields the exponential W 1 -contraction inequality

W 1 pφ s,t pµ 1 q, φ s,t pµ 0 qq ď c e ^´λpt´sq W 1 pµ 0 , µ 1 q (2.10)

with the same parameter λ a in (2.6). In the same vein, the estimate (2.7) yields the total variation estimate

}φ s,t pµ 1 q ´ φ s,t pµ 0 q} tv ď c `

1 _ 1{ ? t ´ s ˘

e ^´λpt´sq }µ 0 ´ µ 1 } tv

with the same parameter λ a in (2.7). In all the inequalities discussed above we can choose any parameter λ ą 0 such that λ ă λ _1,2 , with the parameter λ _1,2 introduced in (1.10). In the W 1 - contraction inequality (2.10) we can choose λ “ λ 1,2 . A more refined estimate is provided in section 2.4.

Next theorem provides a first order Taylor expansion with remainder.

Theorem 2.3. For any m, n ě 0 and µ 0 , µ 1 P P m`2 pR ^d q , there exists a linear operators D ² _µ

_,µ

φ s,t

from C _m <sup>n`2</sup> pR <sup>d</sup> q into C <sub>m`2</sub> ⁿ pR ^2d q such that

φ s,t pµ 1 q “ φ s,t pµ 0 q ` pµ 1 ´ µ 0 qD µ

φ s,t ` 1

2 pµ 1 ´ µ 0 q ^b2 D ² _µ

_,µ

φ s,t (2.11) with the first order operator D µ

φ s,t :“ D µ

,µ

φ s,t introduced in theorem 2.2. In addition, when pHq is satisfied we also have the estimate

}p∇ b ∇qD _µ ²

_,µ

φ _s,t pf q} ď c e ^´λpt´sq sup

i“1,2

}∇ ⁱ f } for some λ ą 0 (2.12) The proof of the above theorem in provided in section 5.2. A more precise description of the second order operator D _µ ²

_,µ

φ _s,t is provided in (5.9) and (5.13). Using (2.11) and arguing as in the proof of proposition 2.1 in [4], for any twice differentiable function f with bounded derivatives we check the backward evolution equation

B s φ _s,t pµqpf q “ ´µL _s,µ pD _µ φ _s,t pf qq (2.13) with the first order operator D µ φ s,t introduced in theorem 2.3. The above equation is a central tool to derive an extended version of the Alekseev-Gröbner lemma [1, 42] to measure valued semigroups and interacting diffusions (cf. theorem 2.7).

Next theorem provides a second order Taylor expansion with remainder.

Theorem 2.4. For any m, n ě 1 and µ ₀ , µ ₁ P P _m`4 pR ^d q , there exists a linear operators D ³ _µ

_,µ

φ _s,t from C _m <sup>n`3</sup> pR <sup>d</sup> q into C <sub>m`4</sub> ⁿ pR ^3d q such that

φ s,t pµ 1 q ´ φ s,t pµ 0 q

“ pµ 1 ´ µ 0 qD µ

φ s,t ` 1

2 pµ 1 ´ µ 0 q ^b2 D _µ ²

φ s,t ` pµ 1 ´ µ 0 q ^b3 D ³ _µ

_,µ

φ s,t

(2.14)

with the second order operator D _µ ²

φ _s,t :“ D _µ ²

_,µ

φ _s,t introduced in theorem 2.3. In addition, when pHq is satisfied we have the third order estimate

|pµ ₁ ´ µ ₀ q ^b3 D ³ _µ

_,µ

φ _s,t pfq|

ď c e ^´λpt´sq `

_ i“1,2,3 } ∇ ⁱ f } ˘

W 2 pµ 0 , µ 1 q ³ for some λ ą 0

(2.15)

The proof of the first part of the above theorem in provided in section 5.3. We can choose in

(2.15) any parameter λ ą 0 such that λ ă λ _1,2 , with the parameter λ _1,2 introduced in (1.10). The

proof of the third order estimate (2.15) is rather technical, thus it is provided in the appendix, on

page 44.

2.3 Illustrations

The first part of this section states with more details the almost sure expansions discussed in (1.5).

Up to some differential calculus technicalities, this result is a more or less direct consequence of the Taylor expansions with remainder presented in theorem 2.3 and theorem 2.4 combining with a backward formula presented in [5].

The second part of this section is concerned with a second order extension of the Alekseev- Gröbner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order stochastic perturbation analysis is also mainly based on the second order Taylor expansion with remainder presented in theorem 2.4 .

In the further development of this section without further mention we shall assume that condition pHq is satisfied.

2.3.1 Almost sure expansions

We recall the backward formula X _s,t ^µ

pxq ´ X _s,t ^µ

pxq “

ż _t

s

”

∇X _u,t ^φ

^pµ

^q ı

pX _s,u ^µ

pxqq ¹ rφ s,u pµ 1 q ´ φ s,u pµ 0 qs pb u pX _s,u ^µ

pxq, . qq du (2.16) The above formula combined with (2.4) and the tangent process estimates presented in section 3.3 yields the uniform almost sure estimates

}X _s,t ^µ

pxq ´ X _s,t ^µ

pxq} ď e ^´pλ

^{^λ}

^qpt´sq W 2 pµ 0 , µ 1 q (2.17) The above estimate is a consequence of (2.4) and conventional exponential estimates of the tangent process ∇X _s,t ^µ (cf. for instance (3.2)). A detailed proof of this claim and the backward formula (2.16) can be found in [5].

We extend the operators D _µ ^k φ s,t introduced in theorem 2.4 to tensor valued functions f “ pf i q iPrns

with i “ pi ₁ , . . . , i _n q P rns :“ t1, . . . , du ⁿ by considering the same type tensor function with entries D ^k _µ φ s,t pf q i :“ D _µ ^k φ s,t pf i q and we set d ^µ _s,t px, yq :“ D µ φ s,t pb t px, . qqpyq (2.18) for any px, yq P R ^2d . A brief review on tensor spaces is provided in section 3.1. We also consider the function

D _µ X _s,t ^µ px, yq :“

ż _t

s

”

∇X _u,t ^φ

^pµq ı

pX _s,u ^µ pxqq ¹ d ^µ _s,u pX _s,u ^µ pxq, yq du

Combining the first order formulae stated in theorem 2.3 with conventional Taylor expansions we check the following theorem.

Theorem 2.5. For any x P R ^d , µ 0 , µ 1 P P 2 pR ^d q and s ď t we have the almost sure expansion X _s,t ^µ

pxq ´ X _s,t ^µ

pxq “

ż

pµ ₁ ´ µ ₀ qpdyq D _µ

X _s,t ^µ

px, yq ` ∆ ^r2s,µ _s,t

^,µ

pxq (2.19) with the second order remainder function ∆ ^r2s,µ _s,t

^,µ

such that

}∆ ^r2s,µ _s,t

^,µ

} ď c e ^´λpt´sq W 2 pµ 0 , µ 1 q ² for some λ ą 0

The detailed proof of the above theorem is provided in the appendix, on page 48.

Second order expansions are expressed in terms of the functions defined for any px, yq P R ^2d and for any z P R ^2d by the formulae

d ^r1,1s,µ _s,t px, yq :“ D _µ φ _s,t pb ^r1s _t px, . q ¹ qpyq and d ^r2s,µ _s,t px, zq :“ D _µ ² φ _s,t pb _t px, . ^qqpzq

We associate with these objects the function D _µ ²

X _s,t ^µ

defined by D ² _µ X _s,t ^µ px, zq :“

ż _t

s

”

∇X _u,t ^φ

^pµq ı

pX _s,u ^µ pxqq ¹

”

d ^r2s,µ _s,u pX _s,u ^µ pxq, zq ` D ^r1,1s _µ X _s,u ^µ px, zq ı

du

` ż _t

s

”

∇ ² X _u,t ^φ

^pµq ı

pX _s,u ^µ pxqq ¹ D ^r2,1s _µ X _s,u ^µ px, zq du In the above display, D µ ^ri,1s X s,u ^µ stands for the functions given by

D ^r1,1s _µ X _s,u ^µ px, zq :“

”

d ^r1,1s,µ _s,u pX _s,u ^µ pxq, z ₂ q D _µ X _s,u ^µ px, z ₁ q ` d ^r1,1s,µ _s,u pX _s,u ^µ

pxq, z ₁ q D _µ X _s,u ^µ px, z ₂ q ı

D ^r2,1s _µ

X _s,u ^µ px, zq :“ “

D µ X _s,u ^µ px, z 1 q d ^µ _s,u pX _s,u ^µ pxq, z 2 q ` D µ X _s,u ^µ px, z 2 q d ^µ _s,u pX _s,u ^µ pxq, z 1 q ‰ We are now in position to state the main result of this section.

Theorem 2.6. For any x P R ^d , µ ₀ , µ ₁ P P ₂ pR ^d q and s ď t we have the almost sure expansion X _s,t ^µ

pxq ´ X _s,t ^µ

pxq

“ ż

pµ 1 ´ µ 0 qpdyq D µ

X _s,t ^µ

px, yq ` 1 2

ż

pµ 1 ´ µ 0 q ^b2 pdzq D ² _µ

X _s,t ^µ

px, zq ` ∆ ^r3s,µ _s,t

^,µ

pxq

(2.20)

with a third order remainder function ∆ ^r3s,µ _s,t

^,µ

such that

}∆ ^r3s,µ _s,t

^,µ

} ď c e ^´λpt´sq W 2 pµ ₀ , µ ₁ q ³ for some λ ą 0

The proof of the above theorem is provided in the appendix, on page 48. In the remainder term estimates presented in the above theorems, we can choose any parameter λ ą 0 such that λ ă λ 1,2 , with the parameter λ 1,2 introduced in (1.10).

2.3.2 Interacting diffusions

For any N ě 2, the N -mean field particle interpretation associated with a collection of generators L _t,η is defined by the Markov process ξ _t “ `

ξ _t ⁱ ˘

1ďiďN P pR ^d q ^N with generators Λ _t given for any sufficiently smooth function F and any x “ px ⁱ q 1ďiďN P pR ^d q ^N by

Λ _t pF qpxq “ ÿ

1ďiďN

L _t,mpxq pF _x

qpx ⁱ q (2.21)

with the function F _x

pyq :“ F `

x ¹ , . . . , x ^i´1 , y, x ^i`1 , . . . , x ^N ˘

and the measure mpxq “ 1 N

ÿ

1ďiďN

δ _x

We extend L _t,µ to symmetric functions F px ¹ , x ² q on R ^2d by setting

L ^p2q _t,µ pFqpx ¹ , x ² q :“ L t,µ pF px ¹ , . qqpx ² q ` L t,µ pF p . , x ² qqpx ¹ q In this notation, in our context we readily check that

F pxq “ mpxqpf q ùñ Λ _t p F qpxq “ mpxqL _t,mpxq pf q

F pxq “ mpxq ^b2 pF q ùñ Λ _t p F qpxq “ mpxq ^b2 L ^p2q _t,mpxq pF q ` 1

N mpxq rΓpF qs

(2.22)

for any symmetric function Fpx ¹ , x ² q “ Fpx ² , x ¹ q , with the function ΓpF q on R ^d defined for any y P R ^d by the formula

ΓpF qpyq :“ Tr ppr ∇ b ∇ s F q py, yqq “ ÿ

1ďiďd

´ B _x

1

B _x

2

F

¯ py, yq

ùñ Γ pf b gq pyq “ ÿ

1ďkďd

B y

f pyq B y

gpyq “ Tr `

∇fpyq ∇gpyq ¹ ˘

A proof of the above formula is provided in the appendix, on page 35. Applying Ito’s formula, for any smooth function g : t P r0, 8rÞÑ g t P C _b ² pR ^d q we prove that

m t :“ mpξ t q ùñ dm t pg t q “ rm t pB t g t q ` m t L t,m

pg t qs dt ` 1

? N dM t pgq In the above display, g ÞÑ M t pgq stands for a martingale random field with angle bracket

B t xM pf q, M pgqy _t :“ m _t pΓpf b gqq ùñ B _t xMpgqy _t “ ż

m _t pdxq }∇gpxq} ²

The above evolution equation is rather standard in mean field type interacting particle system theory, a detailed proof can be found in [29] (see for instance section 4.3). In the same vein, with some obvious abusive notation, using (2.22) we have

dm ^b2 _s pF q “ rm _s b dm _s ` dm <sub>s</sub> b m <sub>s</sub> ` pdm _s b dm _s qs pF q

“

„

m ^b2 _s L ^p2q _s,m

pF q ` 1

N m s rΓpF qs



ds ` martingale increment ùñ rdm _s b dm _s s pF q “ 1

N m _s rΓpF qs ds We fix a final time horizon t ě 0 and we denote by

s P r0, ts ÞÑ M _s pD _m . ^φ . ^{,t pfqq}

the martingale associated with the predictable function

s P r0, ts ÞÑ g s “ D m

φ s,t pf q

Combining the Itô formula with the tensor product formula (2.22) and with the backward formula (2.13) we obtain

d φ s,t pm s qpf q “ ´m s L s,m

pD m

φ s,t pf qq ds ` pdm s q pD m

φ s,t pf qq ` 1

2 pdm s b dm s qpD _m ²

φ s,t pf qq ds

This implies that

d φ s,t pm s qpfq “ 1

2 pdm s b dm s qpD _m ²

φ s,t pf qq ds ` 1

? N dM s pD m . ^φ . ^{,t pfqq}

This yields the following theorem.

Theorem 2.7. For any time horizon t ě 0, the interpolating semigroup s P r0, ts ÞÑ φ s,t pm s q satisfies for any f P C ² pR ^d q with sup _k“1,2 } ∇ ^k f } ď 1 the evolution equation

d φ s,t pm s qpf q “ 1 2N m s

“ Γ `

D _m ²

φ s,t pf q ˘‰

ds ` 1

? N dM s pD m . ^φ . ^{,t pf qq (2.23)}

The above theorem can be seen as a second order extension of the Alekseev-Gröbner lemma [1, 42]

to nonlinear measure valued and stochastic semigroups. This result also extends the perturbation theorem obtained in [4] (cf. theorem 3.6) in the context of interacting jumps processes to McKean- Vlasov diffusions. The discrete time version of the backward perturbation analysis described above can also be found in [27, 28, 30] in the context of Feynman-Kac particle models (see also [25, 26, 31]).

We end this section with some direct consequences of the above theorem. Firstly, using (2.6) and (2.12) we have the almost sure estimates

|B s xM . ^{,t pD m} . ^φ . ^{,t pf qqy s | ď c e ´2λpt´sq } ∇f} 2}

and }m _s “ Γ `

D ² _m

φ _s,t pf q ˘‰

} ď c e ^´λpt´sq sup

i“1,2

}∇ ⁱ f } for some λ ą 0

Without further work, the above inequality yields the uniform bias estimate stated in the r.h.s. of (1.13), for any twice differentiable function f with bounded derivatives. Using well known martingale concentration inequalities (cf. for instance lemma 3.2 in [60]), there exists some finite parameter c such that for any t ě 0 and any δ ě 1 the probability of the following event

|m t pf q ´ φ 0,t pm 0 qpfq ´ 1 2N

ż _t

0

m s

“ Γ `

D _m ²

φ s,t pf q ˘‰

ds| ď c c δ

N

is greater than 1 ´ e ^´δ . In addition, using the Burkholder-Davis-Gundy inequality, for any n ě 1 we obtain the time uniform estimates stated in the r.h.s. of (1.12). On the other hand, using (2.5) and (2.6) we have the almost sure exponential contraction inequality

W 1 pφ _0,t pm ₀ q, φ _0,t pµ ₀ qq ď c e ^´λt W 1 pm ₀ , µ ₀ q for some λ ą 0 This yields the bias estimates

|E rm _t pf q ´ φ _0,t pµ ₀ qpfqs | ď c ₁ N ` c ₂

N ^1{d e ^´λt

for any twice differentiable function f with bounded derivatives. The r.h.s. estimate comes from well known estimates of the average of the Wassertein distance for occupation measures, see for instance [38] and the more recent studies [40, 56]. The above inequality yields the following uniform bias estimate

sup

tě

log N

|E rm t pf q ´ φ 0,t pµ 0 qpf qs | ď c

N

2.4 Comments on the regularity conditions

We discuss in this section the regularity condition pHq introduced in (1.9). We illustrate these spectral conditions for linear-drift and gradient flow models. Comparisons with related conditions presented in other works are also provided.

Firstly, we mention that the condition stated in (1.9) has been introduced in the article [5] to derive several Wasserstein exponential contraction inequalities as well as uniform propagation of chaos estimates w.r.t. the time horizon.

Using the log-norm triangle inequality and recalling that the log-norm is dominated by the spectral norm we check that

λ max pA t px 1 , x 2 q sym q ď λ max pb ^r1s _t px 1 , x 2 q sym q ` 2 <sup>´1</sup> }b <sup>r2s</sup> <sub>t</sub> px 2 , x 1 q ` b ^r2s _t px 1 , x 2 q ¹ } 2

Choosing λ 0 and λ 1 as the supremum of the maximal eigenvalue functional of the matrices A _t px ₁ , x ₂ q sym and b ^r1s _t px ₁ , x ₂ q sym , the Cauchy interlacing theorem (see for instance [55] on page 294) yields λ 1 ě λ 0 ě λ 1,2 .

For linear drift functions

b _t px ₁ , x ₂ q “ B ₁ x ₁ ` B ₂ x ₂ (2.24) the matrix A _t px ₁ , x ₂ q sym reduces to the two-by-two block partitioned matrix

A t px 1 , x 2 q sym “

„ pB 1 q sym pB 2 q sym

pB 2 q sym pB 1 q sym



ùñ λ 0 ě λ 1 “ ´λ max ppB 1 q sym q and }b ^r2s } 2 “ }B 2 } 2

(2.25) In this situation the diffusion flow X _s,t ^µ pxq P R ^d is given by the formula

X _s,t ^µ pxq “ e ^pt´sqB

px ´ µpeqq ` e ^pt´sqrB

^`B

^s µpeq ` ż _t

s

e ^B

^pt´uq dW _u

In the one dimensional case we have

B ₁ ă 0 ă B ₂ ùñ B ₁ “ ´λ ₁ ď B ₁ ` B ₂ “ ´λ _1,2 “ ´λ ₀ Nonlinear Langevin diffusions are associated with the drift function

bpx 1 , x 2 q :“ ´ ∇U px 1 q ´ ∇V px 1 ´ x 2 q

ùñ b ^r1s px ₁ , x ₂ q “ ´ ∇ ² U px ₁ q ´ ∇ ² V px ₁ ´ x ₂ q and b ^r2s px ₁ , x ₂ q “ ∇ ² V px ₁ ´ x ₂ q

some confinement type potential function U (a.k.a. the exterior potential) and some interaction potential function V . In this context we have

´A _t px ₁ , x ₂ q sym “

„ ∇ ² U px ₁ q 0 0 ∇ ² U px 2 q



`

„ ∇ ² V px ₁ ´ x ₂ q ´p ∇ ² V px ₂ ´ x ₁ q ` ∇ ² V px ₁ ´ x ₂ qq{2

´p∇ ² V px 2 ´ x 1 q ` ∇ ² V px 1 ´ x 2 qq{2 ∇ ² V px 2 ´ x 1 q



When the potential function V is even and convex we have A t px 1 , x 2 q sym ď ´

„ ∇ ² U px ₁ q 0 0 ∇ ² U px 2 q



In the reverse angle, when the function V is odd we have the formula A t px 1 , x 2 q sym “ ´

„ ∇ ² U px ₁ q ` ∇ ² V px ₁ ´ x ₂ q 0

0 ∇ ² U px 2 q ` ∇ ² V px 2 ´ x 1 q



In both situations, condition pHq is satisfied when the strength of the confinement type potential dominates the one of the interaction potential; that is when we have that

∇ ² U px 1 q ` ∇ ² V px 2 q ě λ 1 ą } ∇ ² V } 2

The decay rate λ 0 in the W 2 -contraction inequality (2.4) is larger than the decay rate λ 1,2 in the W 1 -contraction inequality (2.10). In addition, the W 1 -exponential stability requires that λ 0

dominates the spectral norm of the matrix b ^r2s . Next we provide a more refined analysis based on the proof of the W 2 -contraction inequality presented in [5]. Using the interpolating paths pY , µ q introduced in (2.3) we set

X _s,t :“ X _s,t ^µ

pY q and X _s,t :“ X ^µ _s,t

pY q (2.26) In the above display pX ^µ _s,t

pxq, Y q stands for an independent copy of pX _s,t ^µ

pxq, Y q . Arguing as in [5]

we have

B t Ep}B X _s,t }q “ E

”

}B X _s,t } ^´1

´

xB X _s,t , b ^r1s pX _s,t , X _s,t qB X _s,t y ` xB X _s,t , b ^r2s pX _s,t , X _s,t qB X _s,t y

¯ı

We consider the symmetric and anti-symmetric matrices b ^r2s _t px 1 , x 2 q sym :“ 1

2

´

b ^r2s _t px 1 , x 2 q ` b ^r2s _t px 2 , x 1 q ¹

¯

b ^r2s _t px ₁ , x ₂ q asym :“ 1 2

´

b ^r2s _t px ₁ , x ₂ q ´ b ^r2s _t px ₂ , x ₁ q ¹

¯

and we set

pU _s,t , U _s,t q :“

¨

˝

B X _s,t b

}B X _s,t }

, B X _s,t b

}B X _s,t }

˛

‚ and pV _s,t , V _s,t q :“

˜ B X _s,t

}B X _s,t } , B X _s,t }B X _s,t }

¸

By symmetry arguments and using some elementary manipulations we check the formula 2 B t Ep}B X _s,t }q “ E

ˆBˆ U _s,t U _s,t

˙

, A _t pX _s,t , X _s,t q ˆ U _s,t

U _s,t

˙F

`

´b

}B X _s,t } ´ b

}B X _s,t }

¯ 2 A

V _s,t , b ^r2s _t pX _s,t , X _s,t q sym V _s,t E

` `

}B X _s,t } ´ }B X _s,t } ˘ A

V _s,t , b ^r2s _t pX _s,t , X _s,t q asym V _s,t E ¯

This shows that

B t Ep}B X _s,t }q ď ´ λ p _1,2 Ep}B X _s,t }q with the parameter p λ _1,2 given by

´ λ p 1,2 :“ sup

x

,x

”

λ max pA t px 1 , x 2 qq ` }b <sup>r2s</sup> <sub>t</sub> px 1 , x 2 q sym } 2 ` }b ^r2s _t px 1 , x 2 q asym } 2

ı

ď ´λ 1,2

We conclude that the W 1 -contraction inequality (2.10) is met with λ “ λ p _1,2 .

In a more recent article [69] the author presents some Wasserstein contraction inequalities of the same form as in (2.4) with λ 0 replaced by some parameter λ ^´ ₀ “ pκ 1 ´ κ 2 q , under the assumption

xx ₁ ´ y ₁ , b _t px ₁ , µ ₁ q ´ b _t py ₁ , µ ₂ qy ď ´κ ₁ }x ₁ ´ y ₁ } ² ` κ ₂ W 2 pµ ₁ , µ ₂ q ² for some κ ₁ ą κ ₂ Taking Dirac measures µ ₁ “ δ _x

and µ ₂ “ δ _y

we check that the above condition is equivalent to the fact that

xx ₁ ´ y ₁ , b _t px ₁ , x ₂ q ´ b _t py ₁ , y ₂ qy ď ´κ ₁ }x ₁ ´ y ₁ } ² ` κ ₂ }x ₂ ´ y ₂ } ² By symmetry arguments this implies that

xx 1 ´y 1 , b t px 1 , x 2 q´b t py 1 , y 2 qy`xx 2 ´y 2 , b t px 2 , x 1 q´b t py 2 , y 1 qy ď ´λ <sup>´</sup> <sub>0</sub> r}x 1 ´y 1 } <sup>2</sup> `}x 2 ´y 2 } ² s (2.27) For the linear drift model discussed in (2.25) the above condition reads

„ pB 1 q sym pB 2 q sym

pB ₂ q sym pB ₁ q sym



ď ´λ ^´ ₀ I which is implies that λ ₀ ě λ ^´ ₀ We also have p2.27q ùñ p1.7q with λ “ λ ^´ ₀ .

2.5 Comparisons with existing literature

The perturbation analysis developed in the article differs from the Otto differential calculus on pP 2 pR ^d q, W 2 q introduced in [61] and further developed by Ambrosio and his co-authors [2, 3] and Otto and Villani in [62]. These sophisticated gradient flow techniques in Wasserstein metric spaces are based on optimal transport theory.

The central idea is to interpret P 2 pR ^d q as an infinite dimensional Riemannian manifold. In this context, the Benamou-Brenier formulation of the Wasserstein distance provides a natural way to define geodesics, gradients and Hessians w.r.t. the Wasserstein distance. The details of these gradient flow techniques are beyond the scope of the semigroup perturbation analysis considered herein.

This methodology is mainly used to quantify the entropy dissipation of Langevin-type nonlinear diffusions. Thus, it cannot be used to derive any Taylor expansion of the form (1.4) nor to analyze the stability properties of more general classes of McKean-Vlasov diffusions.

Besides some interesting contact points, the methodology developed in the present article doesn’t rely on the more recent differential calculus on pP 2 pR ^d q, W 2 q developed by P.L. Lions and his co- authors in the seminal works on mean field game theory [14, 43]. In this context, the first order Lions differential of a smooth function from P ₂ pR ^d q into R is defined as the conventional derivative of lifted real valued function acting on the Hilbert space of square integrable random variables. In this interpretation, for a given test function, say f the gradient ∇D _µ φ _s,t pf qpY q of the first order differential in (1.4) can be seen as the Lions derivative pδu _s,t {δµqpY q of the lifted scalar function Y ÞÑ u s,t pY q :“ Epf pX _s,t ^µ pY qqq , for some random variable Y with distribution µ.

In the recent book [15], to distinguish these two notions, the authors called the random variable D _µ φ _s,t pf qpY q the linear functional derivative. For a more thorough discussion on the origins and the recent developments in mean field game theory, we refer to the book [15] as well as the more recent articles [13, 19, 23] and the references therein.

To the best of our knowledge, most of the literature on Lions’ derivatives is concerned with

existence theorems without a refined analysis of the exponential decays of these differentials w.r.t.

the time parameter. Last but not least, from the practical point of view all differential estimates we found in the literature are rather quite deceiving since after carefully checking, they grow expo- nentially fast with respect to the time horizon (cf. for instance [13, 19, 20, 23]).

Taylor expansions of the form (1.4) have already been discussed in the book [26] for discrete time nonlinear measure valued semigroups (cf. for instance chapters 3 and 10). We also refer to the more recent article [4] in the context of continuous time Feynman-Kac semigroups. In this context, we emphasize that the semigroup φ _s,t pµq is explicitly given by a normalization of a linear semigroup of positive operators. Thus, a fairly simple Taylor expansion yields the second order formula (1.4).

In contrast with Feynman-Kac models, McKean-Vlasov semigroups don’t have any explicit form nor an analytical description. As a result, none of above methodologies cannot be used to analyze nonlinear diffusions.

The second order perturbation analysis discussed in this article has been used with success in [27, 28, 30] to analyze the stability properties of Feynman-Kac type particle models, as well as the fluctuations and the exponential concentration of this class of interacting jump processes; see also [34, 37] for general classes of discrete generation mean field particle systems, a well as chapter 7 in [25] and [4, 31] for continuous time models.

These second order perturbation techniques have also been extended in the seminal book by V.N.

Kolokoltsov [52] to general classes of nonlinear Markov processes and kinetic equations. Chapter 8 in [52] is dedicated to the analysis of the first and the second order derivatives of nonlinear semigroups with respect to initial data. The use of the first and the second order derivatives in the analysis of central limit theorems and propagation of chaos properties respectively is developed in Chapters 9 and Chapter 10 in [52]. We underline that these results are obtained for diffusion processes as well as for jump-type processes and their combinations, see also [53, 54].

Nevertheless none of these studies apply to derive non asymptotic Taylor expansions (2.14) and (2.20) with exponential decay-type remainder estimates for McKean-Vlasov diffusions nor to estimate the stability properties of the associated semigroups. In addition, to the best of our knowledge the stochastic perturbation theorem 2.7 is the first result of this type for mean field type interacting diffusions.

Last but not least, the idea of considering the flow of empirical measures mpξ _t q of a mean field particle model as a stochastic perturbation of the limiting flow φ _0,t pµ ₀ q certainly goes back to the work by Dawson [24], itself based on the martingale approach developed by Papanicolaou, Stroock and Varadhan in [63], published in the end of the 1970’s. These two works are mainly centered on fluctuation type limit theorems. They don’t discuss any Taylor expansion on the limiting semigroup φ s,t nor any question related to the stability properties of the underlying processes.

3 Some preliminary results

The first part of this section provides a review of tensor product theory and Fréchet differential on Hilbert spaces. Section 3.1 is concerned with conventional tensor products and Fréchet derivatives.

Section 3.2 provides a short introduction to tensor integral operators.

In the second part of this section we review some basic tools of the theory of stochastic variational equations, including some differential properties of Markov semigroups. Section 3.3 is dedicated to variational equations. Section 3.5 discusses Bismut-Elworthy-Li extension formulae. We also provide some exponential inequalities for the gradient and the Hessian operators on bounded measurable functions.

The differential operator arising in the Taylor expansions (1.4) are defined in terms of tensor

integral operators that depend on the gradient of the drift function b t px 1 , x 2 q of the nonlinear diffu-