Frank G Jacobitz
email: jacobitz@sandiego.edu

Kai Schneider
email: kai.schneider@univ-amu.fr

SCALE-DEPENDENT GEOMETRIC STATISTICS OF THE LAGRANGIAN AND EULERIAN ACCELERATIONS IN HOMOGENEOUS TURBULENT SHEAR FLOW

The alignment properties of different vector-valued flow quantities, including the Lagrangian, Eulerian, and convective accelerations in homogeneous turbulent shear flow, are quantified through the introduction of scale-dependent geometrical statistics. The vector fields are decomposed into an orthogonal wavelet series and the angles of the scale-wise contributions of different vector-valued flow quantities can be determined. This approach allows us to revisit the random Taylor hypothesis by examining the cancellation properties of Eulerian and convective accelerations at different flow scales. The results for homogeneous turbulent shear flow, computed by direct numerical simulation, show that Taylor's hypothesis holds at small scales of the flow as reflected by the anti-alignment of the Eulerian acceleration and the convective term. Such antialignment, however, is not observed at the largest scales of the turbulent motion, indicating that Taylor's hypothesis does not generally hold for homogeneous turbulent shear flow.

INTRODUCTION

Taylor's frozen-flow hypothesis [START_REF] Taylor | The spectrum of turbulence[END_REF] supposes that small-scale eddies in turbulent flows move downstream at a constant speed with little distortion. Temporal and spatial fluctuations in turbulent flows can thus be related, and, for example, temporal spectra in experimental measurements can be converted into spatial spectra. Hot-wire measurements in turbulent flows often implicitly assume its validity. Theoretically, it is also of utmost importance to understand convection in turbulent flow and the related temporal to spatial intermittency. Moreover, both play an important role in two-point closure turbulence models for space-time correlations in Eulerian and Lagrangian reference frames [START_REF] He | Space-time correlations and dynamic coupling in turbulent flows[END_REF]. However, Taylor's hypothesis has many limitations, such as the requirements of weak shear rates and low turbulence intensities. Discussions on its validity have a long history and are still subject to some controversy in the recent literature (e.g., [START_REF] Moin | Revisiting Taylor's hypothesis[END_REF][START_REF] Del Alamo | Estimation of turbulent convection velocities and corrections to Taylor's approximation[END_REF].

Motivated by the random Taylor hypothesis or sweeping decorrelation hypothesis, stating that 'small eddies in turbulent flow being swept past a stationary Eulerian observer,' (cf. [START_REF] Tennekes | Eulerian and Lagrangian time microscales in isotropic turbulence[END_REF] acceleration fluctuations and their different contributions have been studied in [START_REF] Pinsky | Accelerations in isotropic and homogeneous turbulence and Taylor's hypothesis[END_REF] and [START_REF] Tsinober | Random Taylor hypothesis and the behavior of local and convective accelerations in isotropic turbulence[END_REF] for isotropic turbulence. Their work is based on the prediction of [START_REF] Tennekes | Eulerian and Lagrangian time microscales in isotropic turbulence[END_REF], which states that the Lagrangian acceleration must be small, justified by considering Eulerian and Lagrangian time scales.

Homogeneous turbulent shear flow has been investigated as a prototypical example of turbulence due to the importance of shear production in the geophysical environment and in many engineering applications. In homogeneous flows, the statistical properties of turbulence do not change in the spatial directions (x, y, and z), but they evolve in time t. While this simplification has been used extensively as the basis of numerical simulations, homogeneous turbulent shear flow was first studied experimentally. [START_REF] Rose | Results of an attempt to generate a homogeneous turbulent shear flow[END_REF] and [START_REF] Champagne | Experiments on nearly homogeneous turbulent shear flow[END_REF] are credited with the first experimental studies of homogeneous turbulent shear flow. [START_REF] Rohr | An investigation of the growth of turbulence in a uniform-mean-shear flow[END_REF] established that the eventual evolution of homogeneous turbulent shear flow follows an exponential growth law. Using direct numerical simulations, [START_REF] Jacobitz | Direct numerical simulations of the turbulence evolution in a uniformly sheared and stably stratified flow[END_REF] confirmed the exponential evolution and also investigated the impact of buoyancy forces on the flow. This prototypical flow has been studied extensively including the effects of buoyancy and rotation as well as using a variety of statistical methods (e.g., [START_REF] Salhi | Nonlinear dynamics and anisotropic structure of rotating sheared turbulence[END_REF].

Motivated by our recent work in [START_REF] Jacobitz | Lagrangian and Eulerian accelerations in turbulent stratified shear flows[END_REF], in which we studied the Lagrangian and Eulerian acceleration properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification, we generalize this approach here to investigate scale dependent geometrical statistics. A wavelet-based scale-dependent decomposition of the Lagrangian and Eulerian accelerations is performed and the alignment properties are analyzed at different scales of the turbulent motion (e.g., [START_REF] Farge | Wavelet transforms and their applications to mhd and plasma turbulence: a review[END_REF]. The aim of this work is to address the applicability of Taylor's hypothesis at small scales in homogeneous turbulent shear flow.

The remainder of the manuscript first presents the numerical approach based on direct numerical simulations.

NUMERICAL APPROACH

The present study is based on a direct numerical simulation of homogeneous turbulent shear flow using the incompressible form of the Navier-Stokes equations for the fluctuating components velocity and pressure with an imposed constant vertical shear rate S = ∂U/∂ y. The equations of motion are transformed into a frame of reference moving with the mean velocity [START_REF] Rogallo | Numerical experiments in homogeneous turbulence[END_REF]. This transformation enables the application of periodic boundary conditions for the fluctuating components of velocity and pressure. A spectral collocation method is used for the spatial discretization and the solution is advanced in time with a fourth-order Runge-Kutta scheme. The simulations are performed using 512 3 grid points. The initial conditions are taken from a separate simulation of isotropic turbulence, which was allowed to develop for approximately one eddy turnover time. The initial value of the Taylor-microscale Reynolds number is Re λ = 89 and SK/ε = 2 for the shear number. The corresponding values at St = 10 are Re λ = 157 and SK/ε = 5.22. The results are analyzed at St = 10.

SCALE-DEPENDENT STATISTICS

The Lagrangian and Eulerian accelerations a a a L = a a a E + a a a C , where a a a C denotes the convective contribution, are defined as

a a a L = ∂ u u u ∂t + u u u • ∇u u u and a a a E = ∂ u u u ∂t , (1)
respectively. Both accelerations are computed as a volume average at a fixed time, which is an appropriate choice for homogeneous flows. The effect of shear is considered as an external force. The pressure-gradient term is given by a a a P = ∇(p/ρ 0 ), where ρ 0 is the ambient density.

For defining scale-dependent alignment statistics of the above quantities, we use a three-dimensional orthogonal vector-valued wavelet decomposition. To this end we consider a generic vector field a a a = (a 1 , a 2 , a 3 ) at a fixed time instant and decompose each component a α (x x x) into an orthogonal wavelet series,

a α (x x x) = ∑ λ a α λ ψ λ (x x x), (2)
where the wavelet coefficients are given by the scalar product a α = a α , ψ λ (e.g., [START_REF] Farge | Wavelet transforms and their applications to mhd and plasma turbulence: a review[END_REF]. The wavelets ψ λ with the multi-index λ = ( j, i i i, d) are well localized in scale L 0 2 -j (where L 0 corresponds to the size of the computational domain), around position L 0 i i i/2 j , and orientated in one of the seven directions d = 1, ..., 7, respectively. Reconstructing the three components a α at scale L 0 2 -j by summing only over the position i i i and direction d indices in eq. 2 yields the vector field a a a j at scale index j. By construction we have a a a = ∑ j a a a j , where the a a a j are mutually orthogonal. The scale-dependent statistical moments of the flow fields, including scale-dependent flatness, and scale-dependent pdfs, can thus be computed from a a a j using classical statistical estimators. For instance, the q-th order moment of a a a j (x x x) can be defined by, M q [a a a j ] = (a a a j ) q ,

(3) and by construction the mean value vanishes with a a a j = 0. The moments are thus central moments. These scaledependent moments are directly related to the q-th order structure functions [START_REF] Schneider | Spatial intermittency in two-dimensional turbulence: a wavelet approach[END_REF], where the increment size is ∝ 2 -j . The scale-dependent flatness, which measures the intermittency of a a a j at scale 2 -j , is defined by

Fl[a a a j ] = M 4 [a a a j ] (M 2 [a a a j ]) 2 . ( 4
)
For a Gaussian distribution the flatness equals three at all scales.

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12) Osaka, Japan, July 19-22, 2022 To understand the magnitude of the Eulerian and Lagrangian accelerations, we statistically assess, following [START_REF] Tsinober | Random Taylor hypothesis and the behavior of local and convective accelerations in isotropic turbulence[END_REF], the alignment properties of a a a E , a a a C , and its sum, corresponding to the Lagrangian acceleration a a a L = a a a E + a a a C . When the vectors of the Eulerian acceleration a a a E and the convective terms a a a C are anti-parallel, then the magnitude of the Lagrangian acceleration a a a L , is small compared to those of the Eulerian and convective contributions, since

a a a L , a a a L = a a a E + a a a C , a a a E + a a a C = a a a E , a a a E + a a a C , a a a C +2 cos(a a a E , a a a C ) ||a a a E || ||a a a C ||. (5)
Applying the wavelet decomposition to the different accelerations in eq. ( 2), we can study, in addition to the total, also the scale-dependent alignment properties of the Eulerian and convective accelerations, i.e., cos(a a a j E , a a a j C ) and check its implications for the Lagrangian acceleration.

We can directly conclude that, if a a a j E and a a a j C are antialigned and the cosine term is negative, the norm of a a a j L becomes minimal. This means that for perfect anti-alignment Taylor's hypothesis would hold as temporal variations thus correspond to the spatial variations. Quantifying the departure of alignment yields a measure for how well Taylor's approximation holds at different scales of motion, as detailed below.

RESULTS AND DISCUSSION Flow Evolution

Fig. 1 (left) shows the evolution of the turbulent kinetic energy K = 1/2 u u u, u u u normalized by its initial value K 0 . The turbulent kinetic energy initially decays due to the isotropic initial conditions, starts to grow at about St = 2, and eventually grows exponentially starting at about St = 4. The evolution equation for K can be written in the following nondimensional form:

γ = 1 SK dK dt = P SK - ε SK (6)
Here, γ is the growth rate of the turbulent kinetic energy, P/(SK) its normalized production rate, and ε/(SK) the normalized dissipation rate. The evolution of γ, P/(SK), and -ε/(SK) is shown in fig. 1

St

In the following, the flow is analyzed in the exponential growth regime with St ≥ 4.

Scale-Dependent Statistics

Fig. 2 show the joint probability distribution functions (pdfs) of the Eulerian acceleration a a a E and the convective acceleration a a a C as a function of scale j at nondimensional time St = 10. While the joint pdf of the total fields (top, left) shows some negative correlation, the joint pdfs clearly become more and more negatively correlated as the scale j increases and the scale of the motion decreases.

Table 1 shows the Pearson correlation coefficient r between the Eulerian acceleration a a a E and the convective acceleration a a a C as a function of scale j and nondimensional time St. For all times, a negative value of r = -0.8 is found for the total fields. For a given time St, the value of r decreases with increasing scale index j or decreasing scale of the turbulent motion. For the smaller scales of the turbulent motion with scale index j ≤ 4, the results obtained for r at different times St remain approximately the same. For the larger scales of the turbulent motion with scale indices 1 ≤ j ≤ 3, the results for r decrease with time St and a stronger anti-alignment is observed for later times. Only the results for j = 0 show a different trend, but only few wavelet modes contribute to this value for r. Table 1 also shows the Taylor-microscale Reynolds number corresponding to the different times in the flows evolution with later times corresponding to higher Reynolds numbers due to the exponential evolution of the flow.

Tables 2 and3 erations 'increase with scale for the turbulent flow, but the flatness of the Lagrangian acceleration is one order of magnitude larger than the flatness of the Eulerian acceleration, which shows the extreme intermittency of the former.' In a comparison to random fields with phase randomization, it was found that 'the flatness remains almost constant, around 5 for the Lagrangian acceleration and around 6 for the Eulerian acceleration, which confirms that the latter yields a Laplace distribution whose flatness is 6. This proves that the random fields are non intermittent, as no scale dependence can be observed, even if their pdfs of acceleration are strongly non-Gaussian.' Fig. 3 shows the pdf of the cosine of the angle between different vector-valued quantities for the total and scaledependent fields for j = 2, 4, 6, and 8 (from large to small scales). The following observations can be made. The Eulerian and convective accelerations have the tendency of being strongly anti-aligned, reflected in a peak at -1 in the pdf of cos(a a a E , a a a c ) (top, left). The peak is more strongly pronounced at small scales, and weak or no alignment is found at large scales. The reason for this observation is that the advection term is essentially a small scale quantity. This finding supports that Taylor's hypothesis holds at small scales in homogeneous turbulent shear flow. In contrast, for the pdf of cos(a a a L , a a a C ) (top, right), we observe a slight probability for alignment, which becomes weaker at small scales. Consequently, the Langrangian and Eulerian accelerations are somewhat aligned (center, left), corresponding to a peak at +1 in the pdf of cos(a a a L , a a a E ). This behavior becomes stronger for large scales. The reason is again that the convective term is a small scale quantity and thus supports Taylor's hypothesis.

In the pdf for cos(a a a L , a a a p ) we find likewise strong anti-alignment, which is most pronounced at intermediate scales and weakens at the largest and smallest scales (center, right). The alignment of the Lagrangian acceleration and the pressure-gradient confirms the results in [START_REF] Jacobitz | Lagrangian and Eulerian accelerations in turbulent stratified shear flows[END_REF] that the pressure-gradient is the dominant term contributing to the Lagrangian acceleration. The pdf of cos(a a a E , a a a P ) shows some anti-alignment, which is stronger at large scales, and no alignment is observed at small scales (bottom, left). A similar weak anti-alignment behavior is found for the pdf of cos(a C , a p ) (bottom, right).

CONCLUSIONS

We performed direct numerical simulation in order to revisit Taylor's hypothesis in homogeneous turbulent shear flow. We found strong anti-correlation of the Eulerian acceleration with the convective acceleration based on the joint PDF and the Pearson correlation coefficient. The scale dependence of the results was studied using an orthogonal wavelet decomposition. We observed that the anti-correlation of the Eulerian acceleration and the convective acceleration is strongest for the smallest scales of the turbulent motion. We also considered the PDF of the cosine of the angle between the Eulerian acceleration and the convective acceleration. The PDF shows again strong anti-alignment of the terms, indicating a cancellation between the two accelerations, which is again strongest at small scales. However, this result does not hold for the largest scales of the motion. Hence, the results confirm that Taylor's hypothesis holds at small scales and not at large scales of the turbulent motion, which is in agreement with [START_REF] Lin | On Taylor's hypothesis and the acceleration terms in the Navier-Stokes equation[END_REF], who wrote that 'there is no general extension of Taylor's hypothesis to the case of shear flow.'

Figure 2 .

 2 Figure 2. Joint probability distribution functions (pdfs) of the the Eulerian acceleration a a a E and the convective acceleration a a a C at nondimensional time St = 10 for the total fields (top, left) and for the decomposed fields at scales j = 2 (top, right), j = 3 (center, left), j = 4 (center, right), j = 6 (bottom, left), and j = 8 (bottom, right).

  = 6) -0.81143 -0.80741 -0.80288 r( j = 7) -0.86320 -0.86073 -0.85560 r( j = 8) -0.89679 -0.90279 -0.90486 obtain K = K * e γSt

Figure 3 .

 3 Fig.2show the joint probability distribution functions (pdfs) of the Eulerian acceleration a a a E and the convective acceleration a a a C as a function of scale j at nondimensional time St = 10. While the joint pdf of the total fields (top, left) shows some negative correlation, the joint pdfs clearly become more and more negatively correlated as the scale j increases and the scale of the motion decreases.Table1shows the Pearson correlation coefficient r between the Eulerian acceleration a a a E and the convective acceleration a a a C as a function of scale j and nondimensional time St. For all times, a negative value of r = -0.8 is found for the total fields. For a given time St, the value of r decreases with increasing scale index j or decreasing scale of the turbulent motion. For the smaller scales of the turbulent motion with scale index j ≤ 4, the results obtained for r at different times St remain approximately the same. For the larger scales of the turbulent motion with scale indices 1 ≤ j ≤ 3, the results for r decrease with time St and a stronger anti-alignment is observed for later times. Only the results for j = 0 show a different trend, but only few wavelet modes contribute to this value for r. Table1also shows the Taylor-microscale Reynolds number corresponding to the different times in the flows evolution with later times corresponding to higher Reynolds numbers due to the exponential evolution of the flow.Tables2 and 3show the flatness values for the Lagrangian and Eulerian accelerations, respectively. Both the values for the total accelerations and for their scale dependence are shown. The results are presented at three nondimensional times St = 4, St = 7, and St = 10, corresponding to three different Reynolds numbers Re λ = 103.43, Re λ = 136.40, and Re λ = 156.90. Except for the flatness values at the largest scale of motion ( j = 0), the flatness values increase with decreasing scale of motion (or increasing j). For a given scale, the flatness values of both accelerations increase with increasing nondimensional time St and increasing Reynolds number Re λ . While the flatness values of the total Lagragian acceleration is always larger than that of the corresponding total Eulerian acceleration, this result does not hold at all scales for

  Evolution of the normalized turbulent kinetic energy K/K 0 with normalized time St (left) as well as the growth rate of the turbulent kinetic energy γ, the normalized production rate P/(SK), and the normalized dissipation rate ε/(SK) (right).

				12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12) Osaka, Japan, July 19-22, 2022
		1.4							0.6
		1.2							0.4
		1							0.2
	0 K/K	0.6 0.8							0
		0.4							-0.2
		0 0.2	0	2	4	6	8	10	-0.6 -0.4	0	2	4	6	8 γ P/(SK) -ε/(SK)	10
					St							St
	Figure 1. and the scale-dependent alignment of Eulerian and convec-
	tive accelerations are considered. Results illustrate our find-
	ings, considering the direct numerical simulation data of tur-
	bulent homogeneous shear flow, and conclusions complete the
	manuscript.
															Then,
									Lagrangian and Eulerian accelerations are introduced, their
									corresponding orthogonal wavelet decomposition is presented,

Table 1 .

 1 Pearson correlation coefficient between the Eulerian acceleration a a a E and the convective acceleration a a a C as a function of scale j and nondimensional time St or Taylormicroscale Reynolds number Re λ .

	12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12) Osaka, Japan, July 19-22, 2022

(right)

and the three terms are approximately constant for St ≥ 4. Once the normalized production and dissipation rates have reaches constant values, resulting in a constant growth rate γ, eq. 6 can be integrated to

Table 2 .

 2 Scale-dependent flatness of the Lagrangian acceleration Fl a L as a function of scale j and nondimensional time St or Taylor-microscale Reynolds number Re λ .

	St	St = 4 St = 7 St = 10
	Re λ	103.43 136.40 156.90
	total	13.47	20.77	27.81
	Fl( j = 0)	3.77	6.35	6.72
	Fl( j = 1)	3.26	3.63	3.75
	Fl( j = 2)	4.30	4.55	4.63
	Fl( j = 3)	4.79	5.38	5.93
	Fl( j = 4)	5.91	6.29	7.25
	Fl( j = 5)	8.17	10.35	12.44
	Fl( j = 6) 16.55	25.42	30.54
	Fl( j = 7) 45.81 107.90 111.72
	Fl( j = 8) 82.07 298.47 327.36

Table 3 .

 3 Scale-dependent flatness of the Eulerian acceleration Fl a E as a function of scale j and nondimensional time St or Taylor-microscale Reynolds number Re λ .

	St	St = 4 St = 7 St = 10
	Re λ	103.43 136.40 156.90
	total	10.10	11.96	14.41
	Fl( j = 0)	3.71	5.51	5.22
	Fl( j = 1)	3.54	3.47	4.09
	Fl( j = 2)	4.35	4.36	4.72
	Fl( j = 3)	4.70	5.43	5.53
	Fl( j = 4)	5.86	6.01	6.55
	Fl( j = 5)	7.30	7.70	8.22
	Fl( j = 6)	9.69	11.35	12.45
	Fl( j = 7) 16.63	22.57	25.92
	Fl( j = 8) 32.88	47.88	56.43

ACKNOWLEDGEMENTS

KS acknowledges support from Agence Nationale de la Recherche, project CM2E (ANR-20-CE46-0010-01), and the French Research Federation for Fusion Studies within the framework of the European Fusion Development Agreement (EFDA). FGJ acknowledges support from the University Professor award and the Saber3 cluster computer at the University of San Diego.


Keigo Matsuda
email: k.matsuda@jamstec.go.jp

Katsunori Yoshimatsu
email: yoshimatsu@nagoya-u.jp

Kai Schneider
email: kai.schneider@univ-amu.fr

LARGE-SCALE CLUSTERING OF INERTIAL PARTICLES IN HOMOGENEOUS ISOTROPIC TURBULENCE

The nonlinear dynamics of inertial particles in high Reynolds number turbulence, and in particular clustering and void formation, are important fundamental processes, for instance, for droplets in atmospheric clouds and rain formation. Here we analyze particle data from three-dimensional direct numerical simulations of particle-laden homogeneous isotropic turbulence at high Reynolds number, up to Re λ = 678 and with up to 3.2 × 10 9 particles, computed at resolution 4096 3 . We show that the particle density spectrum of the analyzed data has a bump at large scales for sufficiently high Reynolds number, in addition to the near-dissipation scale bump observed so far. The large-scale bump is attributed to large-scale clustering of inertial particles. We find that this behavior is generic and independent of the forcing scheme used to maintain a statistically stationary flow.

INTRODUCTION

Inertial particles suspended in high Reynolds number turbulence are ubiquitous in geophysical or industrial flows, such as cloud droplets in atmospheric turbulence, dust particles in protoplanetary disks, and fuel droplets in spray combustion. Inertial particles show nonuniform distribution, i.e., clustering, in turbulence due to the centrifugal effect because particles are swept out from turbulent vortices and concentrate in low vorticity regions [START_REF] Maxey | The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields[END_REF][START_REF] Squires | Measurements of particle dispersion obtained from direct numerical simulations of isotropic turbulence[END_REF]. In atmospheric flows, turbulent clustering of cloud droplets can increase collision and coalescence frequency in raindrop formation process [START_REF] Onishi | Collision statistics of inertial particles in two-dimensional homogeneous isotropic turbulence with an inverse cascade[END_REF][START_REF] Ireland | The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. simulations with gravitational effects[END_REF]. The clustering can also affect the cloud radar reflectivity factor [START_REF] Matsuda | Influence of microscale turbulent droplet on radar cloud observations[END_REF][START_REF] Matsuda | Turbulent enhancement of radar reflectivity factor for polydisperse cloud droplets[END_REF]. Thus, for improving cloud microphysics models, it is important to understand the clustering behavior in high Reynolds number flow.

Even though the Reynolds-number dependence of inertial particle clustering has been studied extensively, the clustering behavior in high Reynolds number turbulence in atmospheric clouds (Re λ ∼ 10 3 -10 5 , where Re λ is the Taylor microscale Reynolds number) is not fully understood. Large-scale flows could affect the particle clustering at such high Reynolds number. [START_REF] Matsuda | Influence of microscale turbulent droplet on radar cloud observations[END_REF] computed the particle number density spectrum, using direct numerical simulation (DNS) of particle-laden homogeneous isotropic turbulence. They observed a bump near the dissipation scale. Moreover they showed that the number density spectrum is intensified at large scales, i.e. at small wavenumbers, as the Reynolds number increases. This observation can arise a question whether large-scale clustering does occur or not. In this work, we aim to clarify the Reynolds-number dependence of the number density spectrum and, particularly, the large-scale clustering, using three-dimensional DNS of particle-laden homogeneous isotropic turbulence at high Reynolds number.

In the following, we first briefly describe the governing equations and the DNS approach. Numerical results showing spectra of turbulent kinetic energy and particle number density are discussed. Finally, some conclusions are given, and future work is motivated.

DIRECT NUMERICAL SIMULATION

A statistically homogeneous velocity field u u u(x x x,t) of an incompressible fluid was obtained by the Navier-Stokes equation:

∂ u u u ∂t + (u u u • ∇)u u u = - 1 ρ ∇p + ν∇ 2 u u u + f f f , (1)
∇ • u u u = 0, (2)
where f f f (x x x,t) is an external solenoidal forcing, p(x x x,t) is the pressure, ν is the constant kinematic viscosity, and ρ is the constant fluid density. The particles were treated as point particles and tracked by the Lagrangian method. We assumed that the particle size is sufficiently smaller than the Kolmogorov scale η and the particle density ρ p is sufficiently larger than the fluid density ρ. Then, Lagrangian motion of inertial heavy particles was 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)

Osaka, Japan (Online), July [19][20][21][22]2022 tracked by integrating the following equations:

dx x x p dt = v v v, (3)
dv v v dt = - v v v -u u u(x x x p ) τ p , (4)
where x x x p and v v v are the position and velocity of a Lagrangian particle, respectively, and τ p is the relaxation time of particle motion.

Equations ( 1) and ( 2) were solved in a cubic computational domain with periodic boundary conditions. The domain has side length of 2π and is discretized by Cartesian staggered grids. The fourth-order central-difference schemes were used for the advection and viscous terms [START_REF] Morinishi | Fully conservative higher order finite difference schemes for incompressible flow[END_REF] and the second-order Runge-Kutta scheme was used for time integration. The velocity and pressure were coupled by the HSMAC method [START_REF] Hirt | Calculating three-dimensional flow around structures and over rough terrain[END_REF].

To clarify the influence of the external forcing for turbulence on the large-scale clustering, we applied two types of forcing schemes: large-scale linear forcing (LF) and largescale random forcing (RF). The large-scale linear forcing is applied to the large-scale flow where the wavenumber k is smaller than 2.5. The forcing f f f is proportional to the largescale velocity u u u k<2.5 [START_REF] Onishi | Large-scale forcing with less communication in finite-difference simulations of steady isotropic turbulence[END_REF], where u u u k<2.5 is the low-pass filtered velocity field, retaining the energy of Fourier modes only for k < 2.5. The large-scale random forcing is in accordance with the velocity forcing proposed by [START_REF] Yoshida | Inertial-subrange structures of isotropic incompressible magnetohydrodynamic turbulence in the lagrangian renormalized approximation[END_REF]. In this case, the forcing f f f is similarly applied for k < 2.5 but changes randomly, having a correlation time of T f . We set T f = 1 to be close to the large-eddy turnover time.

The important parameters in this study are the Taylormicroscale Reynolds number Re λ ≡ u λ /ν and the Stokes number St ≡ τ p /τ η , where u is the root-mean-square velocity fluctuation, λ is the Taylor microscale, and τ η is the Kolmogorov time. The DNS with the linear forcing was performed for Re λ = 204, 328, and 531, using 512 3 , 1024 3 , and 2048 3 grid points, respectively. The DNS with the random forcing was carried out for Re λ = 155, 251, 402, and 678, using 512 3 , 1024 3 , 2048 3 , and 4096 3 grid points, respectively. In all simulations, the Stokes number was set to 1.0. The number of particles is in maximum 3.2 × 10 9 . The initial particle distribution is random and homogeneous. The flow velocity and particle distribution data were sampled after the time integration of 10T f , and 10 snapshots of three-dimensional data were saved at interval of T f . The statistical results are obtained by averaging over 10 snapshots.

RESULTS AND DISCUSSIONS

Figure 1 shows time-averaged turbulent kinetic energy spectra for different Reynolds numbers, for two different forcing mechanisms, either large-scale linear forcing (LF) or random forcing (RF). In the large Reynolds number case, Re λ = 678, a k -5/3 power law behavior can be observed for about one decade in the normalized wavenumber kη.

We also calculate the particle number density spectrum. It should be noted that the number density spectrum is typically contaminated by Poisson noise. In this work, the Poisson noise is removed from the spectrum by applying an analytical Fourier transform directly to Lagrangian particle data [START_REF] Matsuda | Influence of microscale turbulent droplet on radar cloud observations[END_REF][START_REF] Matsuda | Turbulent enhancement of radar reflectivity factor for polydisperse cloud droplets[END_REF]. Due to high computational cost of the direct evaluation of Fourier transform, the spectrum values are calculated for 19 selected wavenumbers at k = 1,2,3,4,6,8,12,16,24,32,48,64,96,128,192, 256, 384, 512, and 768. Figure 2 shows the time-averaged number density spectra for St = 1.0. The number density spectra for the linear forcing illustrate the influence of Re λ at small wavenumbers (kη 0.03), as reported by [START_REF] Matsuda | Influence of microscale turbulent droplet on radar cloud observations[END_REF]. The spectra for the random forcing also show the similar Re λ dependence; i.e., for both forcing schemes, the particle density spectra exhibit two well pronounced bumps when Re λ is larger than 300. The bump at larger scale is attributed to large-scale clustering of inertial particles. Therefore, the largescale clustering is generic and independent of the two forcing 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)

/)Reλ = 2⟨4λt = ⟩ /)Reλ = 328λt = ⟩ /)Reλ = 53⟩λt = ⟩ 5)Reλ = ⟩55λt = ⟩ 5)Reλ = 25⟩λt = ⟩ 5)Reλ = 4⟨2λt = ⟩ 5)Reλ = 678λt = ⟩
Osaka, Japan (Online), July [19][20][21][22]2022 schemes used to maintain statistically stationary flows. Recently, in [START_REF] Matsuda | Scaledependent statistics of inertial particle distribution in high Reynolds number turbulence[END_REF], we performed wavelet analyses of inertial particle distribution, and a signature of void-pronounced structure was observed at kη 0.02 for Re λ > 300. Thus, we conjecture that the large-scale clustering could be related to such large-scale voids. For illustration, a snapshot of the spatial distribution of particles in a slice of thickness 4η is presented in figure 3 for Re λ = 678 using the RF scheme. The inhomogeneous particle distribution shows large pronounced void regions and accumulated clusters. To clarify the mechanism of the large-scale clustering, assessing the convergence and divergence of particles is crucial. For instance, [START_REF] Oujia | Divergence and convergence of inertial particles in high Reynolds number turbulence[END_REF] introduced a finite time measure to quantify the convergence and divergence of the particle velocity using Voronoi tessellation. They showed that divergence is the most prominent in cluster regions and less pronounced in void regions. It was also found that for larger volumes, negative divergence values are observed which represent cluster formation (i.e. particle convergence). For small volumes, positive divergence values are found which correspond to cluster destruction/void formation (i.e. particle divergence). Applying this technique to high Reynolds number particle-laden flows will yield deeper insight into the dynamics of large-scale clustering.

CONCLUDING REMARKS

Reynolds-number dependence of inertial heavy particle clustering has been investigated by using three-dimensional DNS of particle-laden homogeneous isotropic turbulence. The DNS was performed for several Reynolds numbers up to Re λ = 678 and two cases of large-scale forcing schemes (the linear and random forcing schemes) to maintain statistically stationary turbulence. The number of particles was in maximum 3.2 × 10 9 . The Stokes number was fixed to 1.0. The analyzed data show that for Re λ > 300 the particle density spectra exhibit two well pronounced bumps. The bump at larger scale is attributed to large-scale clustering of inertial particles. This behavior is generic and independent of the forcing scheme. Further investigation of the large-scale clustering mechanism will be performed, the influence of the Stokes number will be assessed and elaborated analyses of the spectral bumps will be published in forthcoming work.

Figure 1 .

 1 Figure 1. Turbulent kinetic energy spectra for different Reynolds numbers using the large-scale linear forcing (LF) and random forcing (RF) schemes as a function of kη. The dotted line corresponds to k -5/3 .

Figure 2 .

 2 Figure 2. Particle number density spectra for St = 1 at different Reynolds numbers using the large-scale linear forcing (LF) and random forcing (RF) schemes as a function of kη.

ACKNOWLEDGEMENTS

K.M. acknowledges financial support from Grant-in-Aid for Scientific Research (C) No. JP20K04298 from the Japan Society for the Promotion of Science (JSPS). K.S. acknowledges funding from the Agence Nationale de la Recherche (ANR), grant ANR-20-CE46-0010-01 and the French Research Federation for Fusion Studies within the framework of the European Fusion Development Agreement (EFDA). He also acknowledges kind hospitality during his visit at JAM-STEC and financial support. The present DNS was performed on the Earth Simulator supercomputer system of JAMSTEC.


Jordan Serres
email: jordan.serres@math.univ-toulouse.fr

Behavior of the Poincaré constant along the Polchinski renormalization flow

We control the behavior of the Poincaré constant along the Polchinski renormalization flow using a dynamic version of Γ-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by B. Klartag and E. Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general ϕ 4 -measures and discuss the interpretation in terms of transport maps.

Introduction

A probability measure µ on R d is said to satisfy a Poincaré inequality when there exists a positive finite constant C such that for all functions f in the Sobolev space H 1 (µ),

Var µ (f ) ≤ C |∇f | 2 dµ.
We denote by C P (µ) the smallest constant for which the above inequality holds. Poincaré inequalities have many applications (see for instance [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]), and are in particular related to the Kannan-Lovasz-Simonovits conjecture which states that all isotropic log-concave distributions have dimension-independent Poincaré constants (see [START_REF] Yin | The Kannan-Lovász-Simonovits conjecture[END_REF] for a survey). A classical method to obtain bounds on the Poincaré constant C P (µ) of a distribution µ is the perturbative approach: if C P (µ) is known, we want to compare it with the Poincaré constant C P (δµ) of a small perturbation δµ. Some results exist when δµ is a bounded perturbation of µ, i.e. when δµ has a bounded density with respect to µ (see [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]Proposition 4.2.7]). Another result is that if γ denotes the standard Normal distribution on R d (then C P (γ) = 1) and * denotes the convolution, then for all probability distributions µ we have C P (µ) ≥ C P (µ * γ) -C P (γ). This is a consequence of [START_REF] Ball | Entropy jumps in the presence of a spectral gap[END_REF]Proposition 1]. The reverse inequality [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] C P (µ) ≤ C P (µ * γ) + C P (γ) is more informative and was proved under a log-concavity assumption on the measure µ (see [START_REF] Cattiaux | On the Poincaré constant of log-concave measures[END_REF]Theorem 9.4.3]). In [START_REF] Klartag | Spectral monotonicity under Gaussian convolution[END_REF], B. Klartag and E. Putterman interpret the convolution with the Gaussian distribution as the action of the heat flow on µ. This interpretation together with the preservation of the log-concavity by the heat flow, allowed them to strengthen [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] and to prove by Γ-calculus that the Poincaré constant is non-decreasing along the heat flow acting on a log-concave distribution.

Theorem. [START_REF] Klartag | Spectral monotonicity under Gaussian convolution[END_REF]Theorem 1.1] If µ is log-concave, then for all 0 ≤ s ≤ t,

(2)

C P (µ * γ s ) ≤ C P (µ * γ t ),
where γ s denotes the Normal distribution with covariance matrix sI d .

They also proved (2) by constructing contractive transport maps, see Section 3 for a discussion of this method. Let us mention that, following this, the analysis of the evolution of a log-concave distribution along the heat flow allowed B. Klartag and J. Lehec to obtain the best result currently known towards the KLS conjecture (see [START_REF] Klartag | Bourgain's slicing problem and KLS isoperimetry up to polylog[END_REF]). In this paper, we study the behavior of the Poincaré constant along the Polchinski flow. The origins of this flow go back to the work of J. Polchinski [START_REF] Polchinski | Renormalization and effective Lagrangians[END_REF] on the Renormalization Group method. It was introduced in its current formulation with the multiscale Bakry-Emery criterion by R. Bauerschmidt and T. Bodineau in [START_REF] Bauerschmidt | Log-Sobolev inequality for the continuum sine-Gordon model[END_REF] in order to derive logarithmic-Sobolev inequality for the continuum sine-Gordon model. Subsequently, this flow has also been used to derive logarithmic-Sobolev inequalities for other models from Quantum Fields Theory, such as the ϕ 4 models [START_REF] Bauerschmidt | Log-Sobolev inequality for the ϕ 4 2 and ϕ 4 3 measures[END_REF] or the Ising models [START_REF] Bauerschmidt | Log-Sobolev inequality for near critical Ising models[END_REF]. Let ν 0 = e -V 0 (x) γ C∞ be a probability distribution with density e -V 0 with respect to the Gaussian distribution γ C∞ with covariance (positive definite) matrix C ∞ . A family (C t ) t≥0 of semidefinite matrices on R d increasing continuously as quadratic forms from C 0 = 0 to C ∞ is called a covariance decomposition. Given a covariance decomposition, we can define the renormalized potential

V t := -log γ Ct * e -V 0 .
The Polchinski flow is then defined as

ν t := e V∞(0)-Vt(x) γ C∞-Ct .
By definition, γ 0 := δ 0 is the Dirac delta function, hence ν ∞ = δ 0 and the Polchinski flow interpolates between ν 0 and δ 0 . We prove the following theorem.

Theorem. If the Polchinski flow (ν t ) t of a probability distribution ν 0 satisfies the multiscale Bakry-Emery criterion, then the Poincaré constant along the flow satisfies

∀0 < s ≤ t, C P (ν s ) ≤ e (αt-αs)-2(λt-λs) C P (ν t ),
where λ t are the multiscale curvatures, and α t are corrective terms due to the fact that the Polchinski flow admits the Dirac delta function as ergodic distribution (see Definition 1).

This paper is organized as follows. In Section 2, we introduce the general Γ-calculus method to study the behavior of the Poincaré constant along a flow of probability distributions driven by a diffusion semigroup. We also treat the case of higher order eigenvalues. In Section 3, we discuss the transport map method, and in particular why it cannot currently be applied to the Polchinski flow. In Section 4, we rigorously define the Polchinski flow, the multiscale Bakry-Emery criterion, the renormalized Poincaré constant along the flow, and then present a proof of the main theorem of this paper. Finally, in Section 5, we illustrate our theorem on the example of general ϕ 4 measures.

A general Γ-calculus approach

Let L be the generator of a Markov diffusion semigroup P t with carré du champ operator Γ and reversible measure µ on a manifold M . Let µ 0 be a probability distribution on M with density ρ with respect to µ. If we draw randomly X 0 with distribution µ 0 and let the semigroup evolve, then we obtain a Markov process X t , whose laws µ t interpolate between µ 0 and µ ∞ = µ. In this section, we describe a general method for obtaining bounds on the Poincaré constant along the flow µ t of the form

(3) ∀0 ≤ s ≤ t, C P (µ s ) ≤ e t s k s ds C P (µ t ),
where k t are curvature terms that will be described later. This monotonicity-type result for the Poincaré constant can be captured from the point of view of Γ-calculus. When the operator L has a discrete spectrum, this method also allows to treat the case of higher order eigenvalues.

The measure µ t has density P t (ρ) with respect to µ, and allows us to define the dual semigroup (4) Q t (φ) := P t (φρ) P t (ρ) .

This dual semigroup evolves in such a way that there exist a flow of diffusion operators (L t ) t satisfying

∂ t Q t = L t Q t .
We can see from a straightforward computation that (5) L t u = L u + 2 Γ (log P t (ρ), u) .

We denote by Γ t (f ) := 1 2 (L t (f 2 ) -2f L t f ) and Γ 2,t := 1 2 (L t Γ t (f ) -2Γ t (f, L t f )) its iterated first and second Γ-operators. It is immediate to see that Γ t = Γ does not depend on time. We also define the operators Λ t as the diffusion generators with reversible distribution µ t , and denote by

Γ Λ t (f ) := 1 2 (Λ(f 2 ) -2f Λ t f ) and Γ Λ 2,t = 1 2 Λ t Γ Λ t (f ) -2Γ Λ t (f, Λ t f
) its iterated first and second Γ-operators. An easy computation shows that [START_REF] Bauerschmidt | Log-Sobolev inequality for near critical Ising models[END_REF] Λ t u = 2L u + 2 Γ (log P t (ρ), u) , and moreover, Γ Λ t = 2 Γ does not depend on time. Although in this case none of the operators Γ t and Γ Λ t are time dependent, we still retain the notation in order to better understand the method we will use for the Polchinski flow where the operators will indeed be time dependent.

The key strategy to obtain (3) is then to show the quasi-decay of the Raylegh quotient

R φ (t) := E µt Γ Λ t (Q t φ) E µt [(Q t φ) 2 ]
, when φ : M → R is a smooth and compactly supported function. For this purpose, using that Γ Λ t = 2 Γ t , we can compute that

∂ t E µt (Q t φ) 2 = -2 E µt [Γ t (Q t φ)] = -E µt Γ Λ t (Q t φ) and ∂ t E µt Γ Λ t (Q t φ) = 4 E νt [Γ 2,t (Q t φ)] ,
and therefore

R φ (t) = E µt dΓ Λ t dt (Q t φ) -4 E µt [Γ 2,t (Q t φ)] E µt [(Q t φ) 2 ] + E µt Γ Λ t (Q t φ) 2 E µt [(Q t φ) 2 ] 2 = E µt Γ Λ t (Q t φ) 2 -E µt Γ Λ 2,t (Q t φ) E µt [(Q t φ) 2 ] E µt [(Q t φ) 2 ] 2 + E µt dΓ Λ t dt (Q t φ) + E µt Γ Λ 2,t -4 Γ 2,t (Q t φ) E µt [(Q t φ) 2 ]
.

We can easily see that the term

E µt Γ Λ t (Q t φ) 2 -E µt Γ Λ 2,t (Q t φ) E µt [(Q t φ) 2
] is non positive. Indeed, we have by integrating by parts and using the Cauchy-Schwarz inequality

E µt Γ Λ t (Q t φ) = -E µt [Q t φ(Λ t Q t φ)] ≤ E µt [(Q t φ) 2 ] E µt [(Λ t Q t φ) 2 ] = E µt [(Q t φ) 2 ] E µt Γ Λ 2,t (Q t φ) .
It appears therefore that under the dynamical curvature assumption

(7) dΓ Λ t dt + Γ Λ 2,t -4 Γ 2,t ≤ k t Γ t ,
the Gronwall lemma gives us that the Raylegh quotient satisfies R φ (t) ≤ e t 0 k s ds R φ (0) and this is enough to show (3). When the operator L has discrete spectrum, let us denote by λ k (µ), k ≥ 1 its k-th non-zero eigenvalue. By the min-max theorem, there exists a (k + 1)-dimensional subspace E ⊂ H 1 (µ) such that ∀φ ∈ E \ {0}, R φ (0) ≤ λ k (µ). If furthermore, for all t > 0, the dual operator Q t given by ( 4) is a bijection from L 1 (µ) onto L 1 (µ), then the space

E t := {Q t (φ) | φ ∈ E} is a (k + 1)-dimensional vector space in H 1 (µ t ) such that for all ψ = Q t φ ∈ E t , E µt Γ Λ t (ψ) E µt [ψ 2 ] = R φ (t) ≤ e t 0 k s ds R φ (0) ≤ e t 0 k s ds λ k (µ).
Again by the min-max theorem, we can conclude that

λ k (µ t ) ≤ e t 0 k s ds λ k (µ). When the operator L is of the form ∆ -∇Φ • ∇, then L t u = ∆u + ∇ (2 log P t ρ -Φ) • ∇u, Λ t u = 2 ∆u + 2 ∇ (log P t ρ -Φ) • ∇u, Γ t (u) = |∇u| 2 , Γ Λ t (u) = 2 |∇u| 2 , and Γ Λ 2,t = 4 Γ 2,t + ∇ 2 log P t (ρ). Consequently, the condition (7) becomes the classical Bakry-Emery condition ∇ 2 log P t (ρ) ≤ k I d .
In [START_REF] Klartag | Spectral monotonicity under Gaussian convolution[END_REF], B. Klartag and E. Putterman take the operator L as (half of) the Laplacian on R d . Therefore the reversible distribution µ is the Lebesgue measure on R d and the semigroup P t is the heat semigroup. They start from a log-concave distribution µ 0 with Lebesgue density ρ, and by stopping the flow at time t = 1, they obtain an interpolation between µ 0 and µ 1 = P 1 (ρ)dx = µ 0 * γ where γ denotes the normalized Gaussian distribution on R d and * denotes the convolution. The dynamical curvature assumption ∇ 2 log P t (ρ) ≤ 0 is then satisfied because the heat flow preserves the log-concavity. Hence k t can be chosen to be zero and (3) becomes an exact monotonicity result. Moreover, the dual operator Q t is indeed a bijection, since the quantity P t (φρ) = φρ * γ t allows to recover the Fourier transform of φρ. Therefore the monotonicity result is satisfied by all eigenvalues of any order.

If we take the Ornstein-Ulhenbeck generator L = ∆ -Aϕ • ∇, then the reversible distribution is the Normal distribution γ A -1 with covariance matrix A -1 , and the semigroup is given by P

t (f )(ϕ) = E γ C 2t f e tA ϕ + ζ where C t = A -1 -A -1 e -tA .
Consider then a probability distribution µ 0 which is absolutely continuous with respect to γ A -1 and with density e -V 0 . Then (µ t ) t is the Langevin flow which interpolates between µ 0 and the Gaussian γ A -1 , and the density of µ t with respect to γ A -1 is given by P t 2 e -V 0 (ϕ) = e -Vt(e t 2 A ϕ ), where V t is the renormalized potential which we will define in Section 4.1 (see [START_REF] Shenfeld | Exact renormalization groups and transportation of measures[END_REF]Lemma 2.1]). Therefore

∇ 2 log P t 2 e -V 0 = e t 2 A ∇ 2 V t e t 2 A
, and the dynamical curvature assumption ∇ 2 log P t (ρ) ≤ k t I d is no other than the multiscale Bakry-Emery criterion [START_REF] Klartag | Spectral monotonicity under Gaussian convolution[END_REF]. We have then showed that if µ 0 = e -V 0 dγ A -1 satisfies the multiscale Bakry-Emery criterion [START_REF] Klartag | Spectral monotonicity under Gaussian convolution[END_REF], then the Poincaré constant along the Langevin flow satisfies (3). Moreover, the dual operator

Q t (φ) = Pt(e -V 0 φ)
Pt(e -V 0 ) is again a bijection on L 1 (γ A -1 ). Therefore the quasi monotonicity result is satisfied by all eigenvalues of any order. These results can be also proved from [START_REF] Shenfeld | Exact renormalization groups and transportation of measures[END_REF]Theorem 2.2], where Y. Shenfeld shows that under the multiscale Bakry-Emery criterion [START_REF] Klartag | Spectral monotonicity under Gaussian convolution[END_REF], the Langevin transport map constructed as we will see in Section 3 is Lipschitz.

In the case of the Polchinski flow (ν t ) t (see Section 4.1), the previous framework does not apply. Indeed, this framework would apply to the flow (µ t ) t where dµ t (x) = e V∞(x)-Vt(x) dx, however the distributions ν t have in addition the Gaussian term e -1 2 <x, (C∞-Ct) -1 x> . Therefore, in order to study the behavior of the Poincaré constant along the flow, we have to introduce in addition to L t and Λ t , the diffusion generators L t defined in [START_REF] Kim | A generalization of Caffarelli's contraction theorem via (reverse) heat flow[END_REF]. As a result, we can derive a monotonicity-type formula such as [START_REF] Ball | Entropy jumps in the presence of a spectral gap[END_REF], but only at the cost of an additional term e (αt-αs) depending on the Gaussian factor e -1 2 <x, (C∞-Ct) -1 x> (see Theorem 2).

A transport map interpretation

When a probability measure ν is the pushforward by a Lipschitz function S of a probability distribution µ which satisfies some functional inequalities (such as the Poincaré inequality or the logarithmic-Sobolev inequality), then these functional inequalities are transfered to ν via the map S (see [START_REF] Cordero-Erausquin | Some applications of mass transport to Gaussian-type inequalities[END_REF][START_REF] Milman | Spectral estimates, contractions and hypercontractivity[END_REF]). Consequently, a powerful way to derive functional inequalities for a probability distribution is to construct a transport map from a classic probability distribution satisfying those functional inequalities. These are generally log-concave distributions (see [START_REF] Kim | A generalization of Caffarelli's contraction theorem via (reverse) heat flow[END_REF]), the Normal distribution (see e.g [START_REF] Mikulincer | On the Lipschitz properties of transportation along heat flows[END_REF][START_REF] Neeman | Lipschitz changes of variables via heat flow[END_REF]), or the Wiener measure (see [START_REF] Mikulincer | The Brownian transport map[END_REF]). Put in a general and informal way, the conditions on the measure ν to be realized as a transport of one of the previous measures, are a compromise between the smallness of its support and the magnitude of its curvature (as the convexity of its potential). The Bakry-Emery multiscale criterion [START_REF] Klartag | Spectral monotonicity under Gaussian convolution[END_REF] has recently been used as a sufficient condition for a measure ν to be the pushforward by a Lipschitz function of a Normal distribution, and this has been applied in particular to the case of the sine-Gordon model (see [START_REF] Shenfeld | Exact renormalization groups and transportation of measures[END_REF]). Let us briefly describe a classic method to construct such a Lipschitz transport map.

Let L be a generator of a Markov diffusion semigroup P t with reversible measure µ on a manifold M . If we draw randomly X 0 with distribution µ 0 and let the semigroup evolve, then we obtain a Markov process X t , whose laws µ t interpolate between µ 0 and µ ∞ = µ. If we find the vector field (ξ t ) t driving the flow µ t , i.e. such that the continuity equation

∂ t µ t + ∇ • (ξ t µ t ) = 0
holds (where ∇• denotes the divergence operator), then we can compute the integral curves (T t : M → M ) t of the velocity field by solving

∂ t T t (x) = ξ(t, T t (x)).
If all goes well (see [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF]Chapter 4] for rigorous details), then for all t ≥ 0, µ t is the pushforward of µ 0 by T t and has density P t (ρ)dµ, where ρ denotes the density of µ 0 with respect to µ. Moreover, for all t ≥ 0, T t are bijective with inverse functions S t , and therefore S := lim t→+∞ S t is a transport map from µ ∞ = µ to µ 0 . It was shown that this transport map is different than the Brenier optimal transport map (see [START_REF] Tanana | Comparison of transport map generated by heat flow interpolation and the optimal transport Brenier map[END_REF]). The curvature type assumptions on µ 0 are then used to show that S is Lipschitz via a control of the driving vector field ξ t .

If the transport maps S t : µ → µ t are e t 0 k s ds -Lipschitz for some function k : R + → R, then the transport map S : µ → µ 0 is e ∞ 0 k s ds -Lipschitz (see [START_REF] Mikulincer | On the Lipschitz properties of transportation along heat flows[END_REF]Lemma 1]). In particular, by taking the composition T t • S s for s < t, we obtain a e t s k s ds -Lipschitz transport map from µ s to µ t . In term of Poincaré constants, it implies the following control along the flow:

(8) C P (µ s ) ≤ e t s k s ds C P (µ t ).
We recognize the same result as (3) in Section 2. This transport map approach also allows to derive such results for higher order eigenvalues.

In the case of the Polchinski flow (ν t ) t defined in Section 4.1, the distributions ν t interpolate between ν 0 given in [START_REF] Cordero-Erausquin | Some applications of mass transport to Gaussian-type inequalities[END_REF] and ν ∞ = δ 0 the Dirac delta function. Therefore, this dynamic cannot give a transport map, since ν 0 cannot be realized as the pushforward of the Dirac delta function. However, a natural question is to find a rescaling of the Polchinski semigroup such that the ergodic distribution ν ∞ is no longer the Dirac delta function, so a transport map between ν 0 and ν ∞ could be constructed. The theorem 2, linked to the monotonicity result (8) valid when the transport maps exist and are Lipschitz, strongly suggests that such a rescaling is possible. Moreover, this rescaling must remove the term e (αt-αs) in [START_REF] Neeman | Lipschitz changes of variables via heat flow[END_REF], and thus the largest eigenvalue α t of the matrix (C t )

1 2 (∇ 2 V t + (C ∞ -C t ) -1 ) (C t ) 1 2
should characterize the order of magnitude of the rescaling. However, the problem of finding exactly such a rescaling is still open. [START_REF] Bauerschmidt | Log-Sobolev inequality for the continuum sine-Gordon model[END_REF] The Poincaré constant along the Polchinski flow

The Polchinski renormalization semigroup

In this section, we introduce the Polchinski renormalization semigroup. Let (C t ) t≥0 be a family of semidefinite matrices on R d . We assume that the C t matrices increase continuously as quadratic forms from C 0 = 0 to a matrix C ∞ . We also assume the family to be twice differentiable with respect to the parameter t, and moreover that its first derivative t → C t is bounded.

We denote by γ Ct the (possibly degenerate) Gaussian measure with covariance C t and mean zero. Let ( 9)

ν 0 = e -1 2 <x,C -1 ∞ x>-V 0 (x) dx
be a probability distribution on R d , where V 0 : R d → R is a smooth function. The part e -<x,C -1 ∞ x> represents the log-concave part of the measure, while e -V 0 represents the non logconcave part of ν 0 . We define

V t := -log γ Ct * e -V 0 P s,t f := e Vt γ Ct-Cs * f e -Vs ν t := e V∞(0)-Vt(x) γ C∞-Ct = e -1 2 <x, (C∞-Ct) -1 x>-Vt(x)+V∞(0) dx
where * denotes the convolution. By definition, γ 0 := δ 0 , hence ν ∞ = δ 0 . As in [4, Section 2.1], we assume that E νt g (P 0,t f ) is continuous in t, so that

E νt g (P 0,t f ) → t→∞ g (E ν 0 (f )) .
The renormalized potential V t satisfies the Polchinski equation (see [4, Proposition 2.1])

∂ t V t = 1 2 1 2 ∆ C t + L t V t ,
where

L t := 1 2 ∆ C t -< ∇V t , ∇ > C t ,
and the index C t in the Laplacian or the dot product denotes that these operations are computed with respect to C t on R d , i.e.

< U, V > C

t := i,j (C t ) i,j U i V j and ∆ C t f := i,j (C t ) i,j ∂ 2 f ∂x i,j .
The diffusion operator Λ t := 1 2 ∆ C t +L t admits e -Vt(x) dx as reversible distribution. Let us define the diffusion operator [START_REF] Kim | A generalization of Caffarelli's contraction theorem via (reverse) heat flow[END_REF] L

t := ∆ C t -< ∇V t + (C ∞ -C t ) -1 x, ∇ > C t ,
which admits ν t as reversible measure. The two operators Λ t and L t have the same carré du champ operator

|∇ • | 2 C t
, but different Γ 2 -operators. We will denote by Γ 2,t the Γ 2 -operator associated with L t and by Γ Lt 2 the one associated with L t . These operators will be used in the proof of Lemma 3.

The multiscale Bakry-Emery criterion

The multiscale Bakri-Emery criterion has been introduced in [START_REF] Bauerschmidt | Log-Sobolev inequality for the continuum sine-Gordon model[END_REF] in order to derive logarithmic Sobolev inequalities for certain non-log-concave probability distributions from Quantum Field Theory on which the classical Bakry-Emery criterion is unusable. One of these examples (the ϕ 4 model) is treated in Section 5. There are several different multiscale Bakry-Emery conditions, depending on the choice of the covariance decomposition (C t ) t in the Polchinski renormalization. We will consider the following one, which is the most general. Assumption 1. (Multiscale Bakry-Emery criterion) For all t ≥ 0, there exists λ t ∈ R (possibly negative) such that

(11) ∀x ∈ R d , C t ∇ 2 V t C t ≥ 1 2 C t + λ t C t ,
in the sense of quadratic forms.

The criterion (11) is satisfied for example by the ϕ 4 models under the Pauli-Villar decomposition (see Section 5). When V 0 = 0, then V t = 0, and one takes

C t = C ∞ -C ∞ e -tC -1 ∞ . Then C t = -C -1
∞ C t is negative semi definite, and then Assumption 1 is satisfied with

λ t ≤ λ 2
where λ is the smallest eigenvalue of C ∞ . When ∇ 2 V 0 ≥ 0, then ∇ 2 V t ≥ 0 for all t ≥ 0 (see [START_REF] Bauerschmidt | Log-Sobolev inequality for the continuum sine-Gordon model[END_REF]Example 1.3]). Hence by taking again

C t = C ∞ -C ∞ e -tC -1 ∞ as in the case V 0 = 0, Assumption 1 is satisfied again with λ t ≤ λ 2 .
The second and most classic multiscale Bakry-Emery criterion requires that [START_REF] Klartag | Spectral monotonicity under Gaussian convolution[END_REF] (C t )

1 2 HessV t (C t ) 1 2 ≥ µ t I d
and is more adapted to the heat kernel decomposition C t = e -tC -1 ∞ , where in that case it implies [START_REF] Klartag | Bourgain's slicing problem and KLS isoperimetry up to polylog[END_REF]. This criterion is satisfied for example by the sine-Gordon model.

The Poincaré inequality under renormalization

Under the multiscale Bakry-Emery criterion, the probability distribution ν 0 satisfies a Poincaré inequality with sharp constant [START_REF] Yin | The Kannan-Lovász-Simonovits conjecture[END_REF] C

P (ν 0 ) ≤ |C 0 | ∞ 0 e -2λt dt,
where |C 0 | denotes the spectral radius of the matrix C 0 . Let us sketch the proof of this fact. First, we decompose the variance under ν 0 along the renormalization flow ν t :

(14) Var ν 0 (F ) = - ∞ 0 ∂ ∂t E νt (P 0t F ) 2 dt = ∞ 0 E νt |∇P 0t F | 2 C t dt.
Next, it is possible to show that the multiscale Bakry-Emery criterion implies, as in the classical case, the following intertwining relation between the Polchinski semigroup and the gradient operator:

(15)

|∇P 0t F | 2 C t ≤ |C 0 |e -2λt P 0t |∇F | 2 .
Then ( 14) and ( 15) together give [START_REF] Yin | The Kannan-Lovász-Simonovits conjecture[END_REF]. The proof of ( 15) is based on classic semigroup calculations. Let t > 0 and let us define ψ(s) = e -2(λt-λs) P s,t |∇P 0,s F | 2 C s for all s ∈ [0, t]. One can compute that (see [START_REF] Bauerschmidt | Log-Sobolev inequality for the continuum sine-Gordon model[END_REF]Lemma 2.8])

ψ (s) = e -2(λt-λs) P s,t 2λ s |∇P 0,s F | 2 C s -(L s -∂ s ) |∇P 0,s F | 2 C s ≤ 0.
Therefore ψ(t) ≤ ψ(0), which is exactly [START_REF] Mikulincer | On the Lipschitz properties of transportation along heat flows[END_REF]. We are interested in the behavior of the Poincaré constant along the Polchinski flow. The Poincaré constant C P (ν t ) of ν t is defined as the smaller constant K t such that for all ν t -centered and compactly supported φ,

R d φ 2 dν t ≤ K t R d |∇φ| 2 C t dν t .
Contrary to the usual case of study of the Poincaré constant along a flow presented in Section 2, the metric used in the definition of the Poincaré constant now depends on time. This is because the Polchinski flow gets smaller and smaller until it reaches the Dirac delta function, so in order to be able to see anything at all, we need to expand the metric by an adequate factor. The adequate factor is C -1 t . In the case of the heat kernel decomposition,

C -1 t = e tC -1
∞ and in the case of the Pauli-Villars decomposition, C -1

t = t 2 A + 1 t 2
, which both tend towards +∞ when t goes to +∞. By starting the Polchinski flow from time s instead of time zero, we can see by imitating the argument for ν 0 that the measure ν s satisfies a Poincaré inequality with a sharp constant ( 16)

C P (ν s ) ≤ |C s | ∞ s e -2λt dt.
In particular, this implies that

C P (ν s ) -→ s→+∞ C P (ν ∞ ) = C P (δ 0 ) = 0.
To better understand the decay of the Poincaré constant along the flow (ν t ) t , we need to define the following quantity.

Definition 1. For all t ≥ 0, we define α t as the biggest eigenvalue of the matrix

(C t ) 1 2 ∇ 2 V t + (C ∞ -C t ) -1 (C t ) 1 2 ,
where (C t ) 1 2 denotes the positive semidefinite square root of the positive semidefinite matrix C t .

The quantity α t is defined in order to have the following inequality:

(C t ) 1 2 ∇ 2 V t + (C ∞ -C t ) -1 (C t ) 1 2 ≤ α t I d ,
where I d is the identity matrix on R d . Since C t increases from zero to C ∞ , the α t will be positive as soon as ∇ 2 V t is not too negative. In particular, this is so under the Bakry-Emery multiscale criterion when the λ t are not too often negative. But this is a necessary condition for the Bakry-Emery multiscale criterion to give a non-trivial Poincaré inequality, because of the necessity of the integrability of e -2λt at +∞ (see Formula ( 13)). In the log-concave case, i.e. when ∇V 0 ≥ 0, then with the heat-kernel covariance decomposition

C t = C ∞ -C ∞ e -tC -1 ∞ , we get that α t ≥ λ max (C -1 ∞ ) > 0.

Main result

We can now state the main theorem of this paper.

Theorem 2. If the distribution ν 0 given in (9) is such that its Polchinski flow (ν t ) t satisfies the multiscale Bakry-Emery criterion [START_REF] Klartag | Bourgain's slicing problem and KLS isoperimetry up to polylog[END_REF], then for all 0 ≤ s ≤ t, we have

C P (ν s ) ≤ e (αt-αs)-2(λt-λs) C P (ν t ), where λ t := t 0 λ s ds is defined from the multiscale curvatures λ t , and α t := t 0 α s ds are given in Definition 1. Moreover, if the distribution ν 0 has discrete spectrum (in the sense of improved Poincaré inequalities), then for all 0 ≤ s ≤ t, [START_REF] Polchinski | Renormalization and effective Lagrangians[END_REF] λ k (ν t ) ≤ e (αt-αs)-2(λt-λs) λ k (ν s ),

where λ k denotes the k-th non-zero eigenvalue.

Let us underline the following.

1. When ν 0 = γ (i.e. C ∞ = I d and V 0 = 0), then ν t = γ e -t I d , so C p (ν t ) = e -t , λ t = α t = t and ( 17) is trivially true.

When ν

0 = γ C∞ (i.e. V 0 = 0), then ν t = γ C∞e -tC -1 ∞ , so C p (ν t ) = λ max (C ∞ e -tC -1 ∞ ), λ t = λ min (C∞) 2 t, α t = λ max (C ∞ )t and (17) becomes λ max (C ∞ e -sC -1 ∞ ) ≤ e (t-s)(λmax(C∞)-λ min (C∞)) λ max (C ∞ e -tC -1 ∞ ).
3. When ν 0 is log-concave, then ( 17) is a monotonicity type result of the Poincaré constant along a flow of log-concave measures. Contrary to [12, Theorem 1.1] which shows the monotonicity along the heat flow, here it is not an exact monotonicity, and moreover the Polchinski semigroup (P 0t ) t never coincides with the heat semigroup. However, Theorem 2 has a broader range of application since it applies to probability distributions which are not log-concave, for instance the ϕ 4 measures (see Section 5).

The sequel is devoted to the proof of Theorem 2. This proof is inspired by the corresponding proof of [12, Theorem 1.1] by B. Klartag and E. Putterman. We begin with the following lemma.

Lemma 3. Let φ : R d → R be smooth and compactly supported. We denote by φ t := P 0,t φ.

Then the Raylegh quotient

R φ (t) := E νt |∇φ t | 2 C t E νt φ 2 t satisfies for all t > s, R φ (t) ≤ R φ (s) exp ((α t -α s ) -2(λ t -λ s )) ,
where the λ t are given by Assumption 1, and α t is the largest eigenvalue of the symmetric matrix

C t (∇ 2 V t + (C ∞ -C t ) -1 ).
Proof First, let us underline that E νt φ t = 0, justifying that E νt φ 2 t is the ν t -variance of φ t . We can compute that (see [START_REF] Bauerschmidt | Log-Sobolev inequality for the continuum sine-Gordon model[END_REF]Proposition 2.1])

∂ ∂t E νt |∇φ t | 2 C t = E νt 2 < ∇L t φ t , ∇φ t > C t -L t |∇φ t | 2 C t + |∇φ t | 2 C t and ∂ ∂t E νt φ 2 t = -E νt |∇φ t | 2 C t .
Hence

E νt φ 2 t 2 R (t) = E νt φ 2 t E νt 2 < ∇L t φ t , ∇φ t > C t -L t |∇φ t | 2 C t + |∇φ t | 2 C t + E νt |∇φ t | 2 C t 2 .
But the Bochner-type formula for the operator Γ 2,t associated to the diffusion operator L t gives

-2 Γ 2,t (φ t ) = 2 < ∇L t φ t , ∇φ t > C t -L t |∇φ t | 2 C t = -||∇ 2 φ t || 2 C t -2 < ∇ 2 V t C t ∇φ t , ∇φ t > C t .
So using following multiscale Bakry-Emery condition [START_REF] Klartag | Bourgain's slicing problem and KLS isoperimetry up to polylog[END_REF] (see Assumption 1),

C t ∇ 2 V t C t ≥ 1 2 C t + λ t C t , one gets -2 < ∇ 2 V t C t ∇φ t , ∇φ t > C t +|∇φ t | 2 C t ≤ -2λ t |∇φ t | 2 C t . Therefore we obtain R (t) ≤ E νt (φ 2 t ) E νt -||∇ 2 φ t || 2 C t -2λ t |∇φ t | 2 C t + E νt |∇φ t | 2 C t 2 (E νt φ 2 t ) 2 = E νt |∇φ t | 2 C t 2 -E νt (φ 2 t ) E νt ||∇ 2 φ t || 2 C t (E νt φ 2 t ) 2 -2λ t R φ (t)
Moreover, by denoting by Γ Lt 2 the Γ 2 -operator of L t , we have

E νt |∇φ t | 2 C t = -E νt (φ t L t φ t ) ≤ E νt φ 2 t 1 2 E νt (-L φ t ) 2 1 2 = E νt φ 2 t 1 2 E νt Γ Lt 2 (φ t ) 1 2 = E νt φ 2 t 1 2 E νt ||∇ 2 φ t || 2 C t + E νt < ∇ 2 V t + (C ∞ -C t ) -1 C t ∇φ t , ∇φ t > C t 1 2 .
Since by definition we have

C t (∇ 2 V t + (C ∞ -C t ) -1 ) C t ≤ α t C t , we obtain E νt |∇φ t | 2 C t 2 -E νt φ 2 t E νt ||∇ 2 φ t || 2 C t ≤ E νt φ 2 t E νt < ∇ 2 V t + (C ∞ -C t ) -1 C t ∇φ t , ∇φ t > C t ≤ α t E νt φ 2 t E νt |∇φ t | 2 C t ,
where α t := t 0 α s ds. Finally, the derivative of R φ satisfies

R φ (t) ≤ (α t -2λ t ) R φ (t),
and the Gronwall lemma gives

R φ (t) ≤ R φ (s) exp ((α t -α s ) -2(λ t -λ s )) ,
where α t := t 0 α s ds. The proof is complete. The Cauchy-Schwarz type inequality

∀u ∈ C 2 c (R d ), E νt |∇u| 2 C t 2 ≤ E νt u 2 E νt ||∇ 2 u|| 2 C t
never holds true. Indeed, coupled with the Poincaré inequality, it implies

E νt u 2 ≤ C P (ν t )E νt ||∇ 2 u|| 2 C t ,
which is easily seen to be false by taking u as an affine function. However, one can see in the proof of Lemma 3 that the Raylegh quotient satisfies the following non linear differential inequation

R φ (t) ≤ R φ (t) 2 -λ t R φ (t).
But it is not clear that this inequation gives more information about R φ than we got in the proof. We can now prove Theorem 2.

Proof of Theorem 2 By definition of the spectral gap, one has

1 C P (ν 0 ) = inf{R φ (0) | φ ∈ C ∞ c (R d )},
where C ∞ c (R d ) denotes the set of all smooth compactly supported functions on R d . Let then t ≥ 0. For all ε > 0, there exists φ

∈ C ∞ c (R d ) such that R φ (0) < 1 C P (ν 0 ) + ε.
By Lemma 3, it follows that

e 2λt-αt R φ (t) < 1 C P (ν 0 ) + ε.
But by the definition of the Poincaré constant, 1 C P (νt) ≤ R φ (t), so we get

∀ε > 0, e 2λt-αt C P (ν t ) < 1 C P (ν 0 ) + ε, from which we get C P (ν t ) ≥ e 2λt-αt C P (ν 0 )
by let ε go to zero. Inequality ( 17) is therefore proven thanks to the semigroup property: the idea is to start the flow from the time s instead of 0, and with the same reasoning, we obtain exactly [START_REF] Neeman | Lipschitz changes of variables via heat flow[END_REF]. Assume that the distribution ν 0 has discrete spectrum (in the sense of improved Poincaré inequalities), and let us denote by λ k (ν 0 ) its k-th non-zero eigenvalue. By the min-max theorem, there exists a

(k + 1)-dimensional subspace E ⊂ H 1 (ν 0 ) such that ∀φ ∈ E \ {0}, R φ (0) ≤ λ k (µ).
Let us then define the space E t := {Q t (φ) | φ ∈ E}. Since for all t ≥ 0, the operator P 0t (f ) = e Vt γ Ct * (f e -V 0 ) is a bijection from L 1 (ν 0 ) onto L 1 (ν 0 ), (we can recover the Fourier transform of f e -V 0 from the Gaussian convolution γ Ct * (f e -V 0 )), then E t is a (k + 1)-dimensional vector space in H 1 (µ t ). In addition, by using Lemma 3, we have that for all ψ = P 0t (φ) ∈ E t ,

E νt |∇ψ| 2 C t E νt [ψ 2 ] = R φ (t) ≤ e αt-2λt R φ (0) ≤ e αt-2λt λ k (ν 0 ).
Again by the min-max theorem, we can therefore conclude that λ k (ν t ) ≤ e αt-2λt λ k (ν 0 ), from which we can derive Inequality (18) by the semigroup property.

5 Application to the ϕ 4 models

In this section, we illustrate Theorem 2 on the general ϕ 4 models (see [5, Chapter 1]) for a more detailed introduction to this model from Quantum Field Theory). Let Λ be a finite set, A = (A x,y ) x,y∈Λ be a symmetric positive definite matrix, and let g > 0, ν ∈ R and h ∈ R Λ be constants. The ϕ 4 general measure is a probability distribution on R Λ and is given by the density

(19) dµ Λ,h A,g,ν (ϕ) = 1 Z exp - 1 2 (ϕ, Aϕ) -V 0 (ϕ) + (h, ϕ) dϕ
where Z is a normalization constant, (•, •) denotes the dot product on R Λ , and

V 0 (ϕ) = Λ 1 4 gϕ 4 x + 1 2 νϕ 2 x .
The probability distribution µ Λ,h A,g,ν is of the form (9) with C ∞ = A -1 . We use the Pauli-Villars covariance regularization C 0 = 0 and C t = A + 1 t -1 , t > 0. Then the multiscale Bakry-Emery criterion [START_REF] Klartag | Bourgain's slicing problem and KLS isoperimetry up to polylog[END_REF] is satisfied with [START_REF] Shenfeld | Exact renormalization groups and transportation of measures[END_REF] λ t = 1 t -χ t t 2 , t > 0, where the quantity χ t is the susceptibility, and is defined as the maximum over Λ of the correlations between two points:

χ t := max x∈Λ y∈Λ R Λ ϕ x ϕ y dµ Λ,0
A,g,ν+ 1 t (ϕ).

This was proved in [7, Proposition 2.5] and used to show that the ϕ 4 measures satisfy a logarithmic-Sobolev inequality. Therefore, Theorem 2 applies to the measure µ Λ,h A,g,ν , and we obtain the following.

∇ 2 V t (ϕ) + (C ∞ -C t ) -1 (C t ) 1 2 = (C t ) 1 2 C -1 t -C -1 t Σ t (ϕ)C -1 t + (C ∞ -C t ) -1 (C t ) 1 2 ≤ 1 t - λ min (Σ t (ϕ)) t 2
+ λ max (A)(tλ max (A) + 1),

where inequations are taken in the sense of quadratic forms.

In Formula (22), the term 1 t -1 t 2 inf ϕ∈R Λ λ min (Σ t (ϕ)) controls the variation of C P (ν t ) when t is small, and the term λ max (A)(tλ max (A) + 1) controls the variation of C P (ν t ) when t is large. If we take s = 0 and if we let t go to infinity in [START_REF] Tanana | Comparison of transport map generated by heat flow interpolation and the optimal transport Brenier map[END_REF], according to ( 16), we have that C P (ν t ) tends towards zero. However, the exponential term then behaves as its dominant term e tλmax(A) 2 and thus counterbalances the decrease of C P (ν t ).

Proposition 4 .

 4 Let ν t be the Polchinski flow associated to the distribution µ Λ,h A,g,ν . Then the Poincaré constant along this flow satisfies

( 21 )

 21 ∀0 ≤ s < t, C P (ν s ) ≤ e (αt-αs)-

  2(λt-λs) C P (ν t ), Proof It only remains to prove the bound for α t . On the one hand, we compute (C ∞ -C t ) -1 = A(tA+I d ) and moreover (C t ) I d . On the other hand, by using the same calculation as in [7, Section 2], we have∇ 2 V t (ϕ) = C -1 t -C -1 t Σ t (ϕ)C -1t , where Σ t (ϕ) denotes the covariance matrix of the probability distribution µ

	we obtain that for all ϕ ∈ R Λ ,	1 2 = Ct t decreases in t, so (C t )	1 2 ≤ (C 0 )	1 2 = Λ,C -1 t ϕ A,g,ν+ 1 t	. Therefore
	(C t )	1 2
	with λ t :=	t 0	1 s -χs s 2 ds, and α t :=	t 0 α s ds where
	(22)			α t =	1 t	-	1 t 2 inf
						Λ,C -1 t ϕ
						A,g,ν+ 1

ϕ∈R Λ λ min (Σ t (ϕ)) + λ max (A)(tλ max (A) + 1),

where Σ t (ϕ) denotes the covariance matrix of the probability distribution µ t .

Acknowledgments: This work was supported by the Labex Cimi of the University of Toulouse. I would like to thank Yair Shenfeld for our interesting conversation which enlightened me for Section 3 and for his comments on the first version of this paper.