Spectral monotonicity under Gaussian convolution

We show that the Poincar\'e constant of a log-concave measure in Euclidean space is monotone increasing along the heat flow. In fact, the entire spectrum of the associated Laplace operator is monotone decreasing. Two proofs of these results are given. The first proof analyzes a curvature term of a certain time-dependent diffusion, and the second proof constructs a contracting transport map following the approach of Kim and Milman.


Introduction
The Poincaré constant C P (µ) of a Borel probability measure µ on R n is the smallest constant C ≥ 0 such that for any locally-Lipschitz function f ∈ L 2 (µ), where Var µ (f ) = R n f 2 dµ − R n f dµ 2 and | • | is the Euclidean norm.The Poincaré constant governs the rate of convergence to equilibrium of the Langevin dynamics in velocity space [29].
Suppose that µ admits a smooth, positive density ρ in R n .The Laplace operator associated with µ, defined a priori on smooth, compactly supported functions u : R n → R, is given by Lu = L µ u = ∆u + ∇(log ρ) • ∇u.
It satisfies ∇u, ∇v dµ for any two smooth functions u, v : R n → R, one of which is compactly supported.The operator L µ is essentially self-adjoint in L 2 (µ), negative semi-definite, with a simple eigenvalue at 0 corresponding to the constant eigenfunction (see [1,Corollary 3.2.2]).The Poincaré constant is given by where λ (µ) 1 is the spectral gap of L, the infimum over all positive λ > 0 that belong to the spectrum of −L.Under mild regularity assumptions the spectrum of L is discrete (e.g., when ρ is C 2 and ∆( √ ρ)/ √ ρ tends to infinity at infinity [1, Corollary 4.10.9],or when ρ is log-concave and | log ρ(x)|/|x| tends to infinity at infinity, as shown in Appendix A below).
In this case we write 0 = λ 3 ≤ . . .for the eigenvalues of −L, repeated according to their multiplicity.
A non-negative function ρ on R n is log-concave if K = {x ∈ R n ; ρ(x) > 0} is convex, and log ρ is concave in K.An absolutely continuous probability measure on R n is called log-concave if it has a log-concave density.An arbitrary probability measure on R n is called log-concave if it is the pushforward of some absolutely continuous log-concave probability measure on R k under an injective affine map.An example of a log-concave probability measure is γ s , the Gaussian probability measure on R n of mean zero and covariance s • Id.In a minor abuse of notation, we use γ s to denote also its density function γ s (x) = (2πs) −n/2 exp(−|x| 2 /(2s)).Another example of a log-concave probability measure is the uniform probability measure on any convex body in R n .The convolution of two log-concave probability measures is again log-concave, as follows from the Prékopa-Leindler inequality [6,Theorem 1.2.3] or from the earlier work by Davidovič, Korenbljum and Hacet [14].
The Poincaré constant is a particularly useful invariant in the class of log-concave probability measures.For example, when µ is absolutely-continuous and log-concave, its Poincaré constant is determined, up to a multiplicative universal constant, by the isoperimetric constant where the infimum runs over all open sets A ⊆ R n with smooth boundary.Indeed, the Cheeger [11] and Buser-Ledoux [7,24] inequalities state that for any absolutely-continuous, log-concave probability measure µ on R n , A well-known conjecture by Kannan, Lovász and Simonovits (KLS) states that the Poincaré constant of a log-concave probability measure is equivalent, up to a multiplicative universal constant, to the operator norm of the covariance matrix of µ.See the recent paper by Chen [12] for more background and for the best known result towards this conjecture.
Abbreviate γ = γ 1 , the standard Gaussian measure in R n , whose Poincaré constant is C P (γ) = 1 (e.g., [1,Proposition 4.1.1]).It was proven by Cattiaux and Guillin [9,Theorem 9.4.3] that when µ is a log-concave probability measure, where µ * γ is the convolution of µ and γ.The reverse inequality C P (µ) ≥ C P (µ * γ) − 1 is much easier to obtain and does not require log-concavity (see, e.g., [2, Proposition 1]).Our main result in this paper is an improvement upon (2): Theorem 1.1.Let µ be a log-concave probability measure on R n .Then, Moreover, assuming that µ admits a density that is smooth and positive in R n and that L µ has a discrete spectrum, we have Two proofs of Theorem 1.1 are presented here.One of these proofs utilizes a method from Kim and Milman [20] to construct a contraction transporting µ * γ to µ. Recall that a map Theorem 1.2.Let µ be a log-concave probability measure on R n .Then there exists a contraction T : R n → R n that pushes forward µ * γ to µ.
This result is reminiscent of Caffarelli's theorem [8], which states that there is a contraction pushing forward γ to µ in the case where the density of µ with respect to the measure γ is log-concave.As is well-known, Theorem 1.2 implies that the Poincaré constant of µ is not larger than that of µ * γ.Moreover, as explained e.g. in Ledoux [23,Proposition 1.2], it follows from Theorem 1.2 that when µ is an absolutely-continuous, log-concave probability measure on There is also a corresponding inequality between the log-Sobolev constants of µ and µ * γ, or any other quantity involving a Rayleigh-type quotient, see Caffarelli [8,Corollary 8].We explain the proof of Theorem 1.2 and its implications in §3.
We continue with a discussion of an additional proof of Theorem 1.1, which was chronologically the first proof that we found.For s > 0 denote µ s = µ * γ s , the evolution of the measure µ under the heat flow.The log-concavity of µ implies that µ s is log-concave as well.We will show that C P (µ s ) is nondecreasing in s.For a function f : R n → R we consider its evolution under the heat semigroup and again it is a contraction operator with Q s (1) = 1.It follows from ( 5) and ( 6) that the evolution equation for Q s is the parabolic equation We are thus led to define the "box operator" This operator resembles the Laplace operator L s := L µs .Indeed, we have The ✷ s operator obeys a Bochner-type formula, which is unsurprising as ✷ s equals half of the Laplace operator associated with the log-concave probability measure whose density is proportional to ρ 2 s .Indeed, we compute that for smooth u, v : R n → R, where ∇ 2 u HS is the Hilbert-Schmidt norm of the Hessian matrix ∇ 2 u.The expression in (11) is similar to the Bochner-type formula of the operator L s , the main difference being the factor 2 in front of the second summand in (11), which is the "curvature term."Moreover, setting Γ 0 (u, v) = uv and Γ 1 (u, v) = ∇u • ∇v, we have under some regularity assumptions to be explained below.It follows that the Rayleigh quotient This fact, formulated as Theorem 2.4 below, implies Theorem 1.1.More details, explanations and rigourous proofs are provided in §2.
In §4 we discuss conceptual aspects of the evolution (Q s ϕ) s≥0 , and explain how it is equivalent to Eldan's stochastic localization [16,26] and Föllmer's drift [28].We also provide a Bayesian interpretation of this evolution, and explore various connections between these points of view.

A dynamic variant of Γ-calculus
In this section we prove Theorem 1.1.Consider the linear differential operator ✷ s defined by formula (8) above.Similarly to the formalism from [1], for smooth functions u, v : R n → R we define Γ 0 (u, v) = uv, and for i ≥ 0 and s > 0, Thus Γ 1 (u, v) = ∇u • ∇v and Γ 2 (u, v) coincides with definition (10) above.The rationale for definition (12) is that, under regularity assumptions stated below, where ϕ s = Q s ϕ.If we were allowed to ignore all regularity issues, (13) could be proven as follows: differentiating under the integral sign and applying ( 5) and ( 7), Next we use (9) and the fact that R n (L s u)dµ s = 0 under regularity assumptions (e.g., when u is smooth and compactly supported).This yields This would be a rigorous proof for (13) had we worked in the context of a compact Riemannian manifold (which also has a heat kernel P s : L 2 (µ) → L 2 (µ s ) and a corresponding adjoint Q s = P * s ).However, in this paper we are interested in the non-compact situation of R n , since we rely on the fact that the heat flow preserves curvature conditions such as logconcavity, which is currently known to hold only for a Euclidean space [22].Nevertheless, the operators ✷ s and (Q s ) s≥0 seem rather natural also in the Riemannian setting.
Our first task in this section is to rigorously justify (13) for a fairly large class of functions ϕ.To do this, we shall express Q s explicitly as an integral operator.
Recall that we work with an absolutely-continuous, log-concave probability measure µ on R n having density ρ.As before, for s > 0 we write µ s = µ * γ s and ρ s = P s ρ = ρ * γ s , while the operator Q s is defined via formula (6).For s > 0 and y ∈ R n we define the probability density where is a normalizing constant (the "partition function").In the next lemma we express the value of Q s ϕ at the point y as the average of ϕ with respect to the density p s,y .
Lemma 2.1.Let y ∈ R n , s > 0 and suppose that ϕ ∈ L 1 (µ), or more generally, that ϕ : R n → R is such that ϕ(x)e −t|x| 2 ∈ L 1 (µ) for some t ∈ (0, s).Then Furthermore, Q s ϕ(y) is a smooth function of y ∈ R n and s > 0 which satisfies where Proof.According to ( 16) and (17), Now (18) follows from ( 6) and (21).The smoothness of Q s ϕ and equations (19) and (20) follow by differentiating ( 17) and ( 18) under the integral sign.This is legitimate, since any partial derivative in the (s, y)-variables of the function p s,y (x)ϕ(x) is seen to be bounded by an integrable function, and the bound is locally uniform in s and y.
By a multi-index (k, α) we mean a non-negative integer k and a vector α = (α 1 , . . ., α n ) of nonnegative integers.For a multi-index (k, α) and for a smooth function f (s, y) we abbreviate We denote |α| = i |α i |.We say that a measurable function ϕ : R n → R has subexponential decay relative to ρ if for any a > 0 there exists e −a|x| for all x ∈ R n for which ρ(x) > 0.Moreover, if s varies in an interval [s 0 , s 1 ] with s 0 > 0, then the implied constants in these two assertions may be chosen not to depend on s.
Proof.Using the heat equation, we can replace time derivatives of ρ s by space derivatives, so in (i) we only need consider space derivatives of log ρ s .By differentiating (17) with respect to y we see that Repeated differentiations show that conclusion (i) would follow once we prove the following claim: for any d > 0, the function Q s (|x| d )(y) grows at most polynomially at infinity as a function of y ∈ R n , with the implied constants not depending on s ∈ [s 0 , s 1 ].
Let us prove this claim.Since ρ is an integrable, log-concave function, there exist a, b > (see e.g., [6, Lemma 2.2.1]).In particular where the first inequality follows from the fact that |x − y| d is increasing in |x − y| while e −|x−y| 2 /(2s) is decreasing in |x − y|, and the second inequality for some coefficient Cd depending only on ρ and on d.This shows that Q s (|x| d )(y) grows at most polynomially, from which (i) follows.
We move on to the proof of (ii).Given a > 0, let C > 0 be such that ρ(x) • |ϕ(x)| ≤ Ce −a|x| for all x.From (6), where we have used the Cauchy-Schwarz inequality for P s .In order to conclude that ϕ s has subexponential decay relative to ρ s , it remains only to note the following: since P s is convolution with a Gaussian of covariance s • Id, there exists C = Ca,s,n > 0 such that We still need to bound the partial derivatives of ϕ s (y) with respect to the s-variable and y-variables.The first-order derivatives are given by formulas (19) and ( 20), and higherorder derivatives may be computed by repeated applications of these two formulas.Thus ∂ k s ∂ α y Q s ϕ(y) can be expressed as a sum with a fixed number of summands.Each of these summands is a product of a term of the form 1 s m Q s (f ϕ), where f is a polynomial of degree bounded by 2k+|α|, and terms of the form Q s (p) with p a polynomial in the space variables.For any such p, the function Q s (p) grows at most polynomially because Q s (|x| d ) does for all d.In addition, f ϕ has subexponential decay relative to ρ, so by the previous part of the proof, Q s (f ϕ) has subexponential decay relative to ρ s .Consequently, each of the summands in ∂ k s ∂ α y Q s ϕ(y) has subexponential decay relative to ρ s , so ∂ k s ∂ α y Q s ϕ(y) does as well.
Recall the definition (12) of Γ i (u, v).In the next proposition we rigorously justify the computations in ( 14) and (15).We discuss only the case i = 0, 1; while the extension to higher-order carrés des champs presents no particular difficulty, it has been omitted as it is unnecessary for our purposes.
According to Lemma 2.2 we may bound the expression in ( 24) by the integrable function Ce −a|y| for some C, a > 0, and the bound is locally uniform in s.This justifies interchanging differentiation and integration to obtain where we have used Lemma 2.1 and the heat equation ( 5).Next we need to carry out the integrations by parts of ( 14) and ( 15) and show that no boundary terms arise.When integrating the term Γ i (ϕ s , ϕ s )∆ρ s /2 by parts twice, we encounter the boundary integrands Γ i (ϕ s , ϕ s )∇ρ s and ∇Γ i (ϕ s , ϕ s ) • ρ s .Both of these decay exponentially at infinity, so the integration by parts over R n introduces no boundary terms, verifying (14).In (15), we use the integration by parts formula which is again justified by the exponential decay of ∇Γ i (ϕ s , ϕ s ) • ρ s at infinity.This completes the proof of (13).
We write H 1 (µ) for the space of all functions in L 2 (µ) whose weak derivatives belong to L 2 (µ), equipped with the norm See e.g. the appendix of [3] and the references therein for information about weak derivatives, the Sobolev space H 1 (µ), and for a proof of the fact that the space of smooth, compactly supported functions in R n is dense in H 1 (µ).
Theorem 2.4.Let µ be an absolutely-continuous, log-concave probability measure on R n and let 0 ≡ ϕ ∈ H 1 (µ).Then with ϕ s = Q s ϕ, the Rayleigh quotient For the proof of Theorem 2.4 we require the following technical lemma: Lemma 2.5.
For part (ii), note that ϕ → R ϕ (s) is locally uniformly continuous in H 1 (µ) \ {0}, being the quotient of two positive, 1-Lipschitz functions.Hence, it suffices to prove (26) for ϕ in a dense subset of H 1 (µ) \ {0}.We may thus assume that ϕ is smooth and compactly supported.We claim that for almost every y ∈ R n , In proving (30), we may thus assume that y ∈ ∂K, since the boundary of the convex set K has Lebesgue measure zero.If y ∈ K then ρ vanishes in a neighborhood of y, hence and (30) follows from the bounds ( 27) and ( 28).As ρ is log-concave, it is locally Lipschitz on K, so by the Rademacher theorem, ρ is differentiable almost everywhere in the interior of K.It thus suffices to prove (30) for y ∈ K such that ρ is differentiable at y. Differentiating ϕ s yields It is a property of the heat semigroup that if f is a bounded measurable function differentiable at a point y ∈ R n , then ∇P s (f )(y) → ∇f (y) as s → 0; this is easily shown by writing ∇P s f = f * ∇γ s and approximating f by its first-order Taylor polynomial.Applying this to the functions ρ and ϕρ which are bounded in R n and differentiable at y, we obtain ∇ρ s (y) → ∇ρ(y) and ∇P s (ϕρ)(y) → ∇(ϕρ)(y) as s → 0.Moreover, ϕ s (y) → ϕ(y), ρ s (y) → ρ(y) because ρ and ϕρ are continuous at y and bounded in R n .It follows that ∇ϕ s (y) → ∇ϕ(y), completing the proof of (30).
It follows from (12) and a straightforward computation that On the other hand, the Bochner formula for the differential operator L s = L µs states that for any smooth, compactly supported function u : See [1, §1.16.1] for a proof of (33).Formula (33) remains valid when u and its partial derivatives are smooth functions with subexponential decay relative to ρ s , since the integration by parts yield no boundary terms as in the proof of Proposition 2.3.Thanks to Lemma 2.2, we know that formula (33) is valid for u = Q s ϕ whenever ϕ has subexponential decay relative to ρ.
The integrand on the right-hand side of ( 33) is almost identical to the expression in (32), the only difference is the coefficient 2 in front of the second summand.
Proof of Theorem 2.4.When u is a smooth function such that u and its partial derivatives have subexponential decay relative to ρ s , we write for i = 1, 2, Thus u 2 Ḣ1 (µs) = R n |∇u| 2 dµ s .The operator L s is initially defined by the formula L s u = ∆u + ∇ log ρ s • ∇u assuming u and its partial derivatives have subexponential decay relative to ρ s .This operator is essentially self-adjoint and negative semi-definite in L 2 (µ s ) (e.g., [1, Corollary 3.2.2]).Hence, by the spectral theorem and the Cauchy-Schwarz inequality, Consider first the case where 0 ≡ ϕ ∈ H 1 (µ) has subexponential decay relative to ρ and s > 0. Thanks to Proposition 2.3 we may apply (13) and compute that .
By log-concavity ∇ 2 log ρ s ≤ 0. Hence we conclude from (34) that Therefore R ϕ (s) is non-increasing in s ∈ (0, ∞).It follows from (13) that and consequently log ϕ L 2 (µs) is convex in s ∈ (0, ∞).Lemma 2.5 now implies that R ϕ (s) Finally, compactly supported smooth functions, which certainly have subexponential decay relative to ρ, are dense in H 1 (µ).The Rayleigh quotient and ϕ s L 2 (µs) are continuous on H 1 (µ)\{0} by Lemma 2.5, hence we obtain that R ϕ (s) is non-increasing and Using the min-max characterization of eigenvalues, we derive our main result as a corollary to Theorem 2.4.
Proof of Theorem 1.1.We may set s = 1, since µ s = µ * γ for s = 1.We may assume that µ is absolutely continuous, as otherwise we may pass to a lower dimension thanks to the well-known fact that the Poincaré constant of a Cartesian product of two measures is the maximum of the Poincaré constants of the factors.The Poincaré constant of µ, which is finite and positive (see [4]), satisfies and similarly for µ s .For any ε > 0 there exists 0 ≡ ϕ ∈ H 1 (µ) with ϕdµ = 0 such that R ϕ (0) < C P (µ) −1 + ε.Since ϕ s dµ s = ϕdµ = 0, we deduce from Theorem 2.4 that, As ε > 0 was arbitrary, inequality (3) is proven.
Next, assume that L µ has discrete spectrum, and let k ≥ 1.There exists a (k + 1)- k for any 0 ≡ ϕ ∈ E. For s > 0 the linear operator Q s defined in ( 6) is one-to-one in L 1 (µ).(Indeed, given P s (ϕρ) we may recover the Fourier transform of ϕρ ∈ L 1 (R n ) which determines ϕ ∈ L 1 (µ).)Hence k for all ϕ ∈ E. In other words, there exists a (k + 1)-dimensional subspace E s ⊆ H 1 (µ s ) on which the Rayleigh quotient is at most λ The proof of Theorem 1.1 clearly shows that C P (µ * γ s ) ≥ C P (µ) for all s > 0, so by the semigroup property s → C P (µ * γ s ) is non-decreasing in s ∈ [0, ∞).Remark 2.7.Let µ be a log-concave probability measure in R n with density ρ = e −W , where W is a smooth function such that In this case, we have the strict inequality In order to prove (35), we first observe that ∇2 log ρ s (y) < 0 for all y ∈ R n as follows from the equality case of the Brascamp-Lieb inequality or from [13,15].Arguing as in the proof of Theorem 2.4 and using the fact that ∇ϕ s ≡ 0 as ϕ s is non-constant, we conclude that dR ϕ (s)/ds < 0 whenever 0 ≡ ϕ ∈ H 1 (µ) has subexponential decay relative to ρ.
Therefore (35) would follow from Theorem 2.4, as in the proof of Theorem 1.1 above, had we known that any eigenfunction ϕ of L µ has subexponential decay relative to ρ.
µ) be the isometry given by A(g) = e W 2 g.It is wellknown and easy to verify that A −1 L µ A is the Schrödinger operator which is of the form −∆ + V with V ≥ 0 and V → ∞ as x → ∞.By results on the decay of eigenfunctions of Schrödinger operators [32, Theorem XIII.70], the function A −1 ϕ has subexponential decay at infinity, and hence ϕ has subexponential decay relative to ρ.

A contraction transporting µ * γ to µ
In this section we prove Theorem 1.2 using the arguments of Kim and Milman [20].To begin with, we work with a log-concave probability measure µ with a smooth, strictly positive density ρ on R n .We furthermore make the regularity assumption that there exists ε > 0 We shall later remove these assumptions on ρ.As above, for s ≥ 0 we write µ s = µ * γ s and ρ s is the density of µ s .Thus ρ s is smooth, positive and log-concave in R n .For s ≥ 0 consider the advection field The "physical" interpretation of this vector field is as follows.One of the derivations of the heat equation is based on Fourier's law, according to which the flux of heat across a tiny surface in a short time interval is proportional to the temperature gradient across the surface.
If we think of the heat as carried by a fluid of particles with density ρ(x, t), this means that the current of heat is proportional to −∇ρ (we take 1 2 to be the constant of proportionality); since the current of heat is simply ρv, where v(x, t) is the bulk velocity of the fluid, we obtain v = − 1 With this point of view, the trajectory of a particle located at time s = 0 at the point y ∈ R n is the curve s → T s (y) where Lemma 3.1.Under the above assumptions on ρ, the ordinary differential equation ( 38) determines the family of maps (T s : R n → R n ) s≥0 .These maps are all diffeomorphisms, and Proof.Since ρ s = ρ * γ s , the function ρ s (y) is smooth and positive in (s, y) Once this is shown, the standard theory of ordinary differential equations implies the existence and uniqueness of solutions to (38) and their smooth dependence on initial conditions (e.g., [19,Chapter V]).The fact that the T s are diffeomorphisms follows from the theory of flows of time-dependent vector fields on manifolds (e.g., [25,Chapter 17]).
We need to compute the derivative of W s .As in the beginning of the proof of Lemma 2.2 above, by differentiating (17) we see that for any s > 0 and y ∈ R n , where Cov (p s,y ) ∈ R n×n is the covariance matrix of the probability density p s,y .Since ρ s is log-concave, the differential DW s is a symmetric positive semidefinite matrix.From ( 16) and the regularity assumption (36) we see that for s ≥ 0 and x ∈ R n , in the sense of symmetric matrices.It is well-known (see [5,Theorem 5.4]) that (40 From ( 39) and (41) we deduce the pointwise bound where • op is the operator norm.This bound clearly applies also for s = 0. Therefore W s : R n → R n is 1/(2ε)-Lipschitz for any s ≥ 0, completing the proof.
As explained in Kim and Milman [20], the diffeomorphism T s is an expansion, i.e., |T s (x) − T s (y)| ≥ |x − y| for all x, y and s.In order to prove this, we show that everywhere in R n , Inequality ( 42) is certainly true when s = 0, while the fact that DW s is positive semidefinite implies that ∂ ∂s Therefore (42) holds true.This implies that D T −1 s op ≤ 1, hence T −1 s is a contraction and T s is an expansion.Next, from (37) and the heat equation ∂ρ s /∂s = ∆ρ s /2 we obtain the linear transport equation (also known as the continuity equation), The continuity equation implies that ρ s is the density of the pushforward of µ under the diffeomorphism T s (see e.g.[34,Theorem 5.34]).Consequently, the map T −1 s is a contraction that pushes forward µ s to µ.
Proof of Theorem 1.2.Set s = 1 so that µ s = µ * γ.We have just established the existence of a contraction transporting µ s to µ under the additional requirement that µ admits a smooth, positive density satisfying the regularity assumption (36).
Consider now the case where µ is an arbitrary absolutely-continuous, log-concave probability measure in R n .For any ε > 0, the measure µ ε = µ * γ ε has a smooth, positive, log-concave density satisfying the regularity assumption (36), as follows from the computation in (39) above.Hence there exists a contraction transporting µ ε * γ to µ ε .By [20,Lemma 3.3], in order to show that there exists a contraction from µ * γ to µ, it suffices to show that in the total variation metric, and that µ ε * γ −→ µ * γ as ε → 0 in the weak topology.Since µ ε * γ = (µ * γ) ε and since convergence in total variation implies convergence in the weak topology, it suffices to prove (43).Thus we need to show that Arguing as in (22) and the paragraph following (23) above, we know that there exist a, b > 0 such that ρ ε (x) ≤ ae −b|x| for all x ∈ R n and 0 ≤ ε ≤ 1, with ρ 0 = ρ.Since ρ is continuous almost everywhere in R n , the integrand in (44) converges to zero almost everywhere, and (44) follows from the dominated convergence theorem.
Thus the conclusion of the theorem is valid when µ is an absolutely continuous, logconcave probability measure.Finally, if µ is not absolutely continuous, then we may project to a lower dimension using an orthogonal projection, which is a contraction, and reduce matters to the absolutely continuous case.Theorem 1.2 implies that for ν = µ * γ and 0 ≡ ϕ ∈ H 1 (ν) we have the following inequality between Rayleigh quotients: We may now repeat the proof of Theorem 1.1 from §2, with the linear map ϕ → ϕ • T playing the role of the linear map ϕ → Q 1 ϕ.This yields another proof of Theorem 1.1, relying on (45) in place of Theorem 2.4.

A Bayesian interpretation of Eldan's stochastic localization
Eldan's stochastic localization technique was introduced by Eldan in [16] and developed since then by several authors in different settings [12,17,21,26].The method has turned out to be useful in particular for the study of log-concave measures, culminating thus far in the breakthrough result of Chen [12] showing that the isotropic constant grows more slowly than any power of the dimension.In this section, we give a "Bayesian" interpretation of Eldan's stochastic localization relating it to the heat flow and to the operator Q s introduced above, as well as to the Föllmer drift in the theory of Wiener space.It was this line of development which led us to the results announced in the introduction; however, this section may be read independently.
We refer to [31] for background on stochastic processes.Let µ be an absolutely continuous probability measure on R n with density p 0 and with finite second moments.Let (W t ) t≥0 be a standard Brownian motion on R n with W 0 = 0.
The stochastic localization process, in the version introduced by [26], is a density-valued stochastic process p t driven by W t , defined as follows: for every x ∈ R n , the process (p t (x)) t≥0 is the solution to the stochastic differential equation with initial condition p 0 , where a t = R n x • p t (x) dx is the barycenter of p t .As this equation has no drift term, p t (x) is a martingale, and p t is almost surely a probability density.
In particular, E[p t (x)] = p 0 (x), and for any test function ϕ, we have The process (p t ) t≥0 has another description, as a stochastic "tilt" of p 0 .In this section, for t ≥ 0 and θ ∈ R n let p t,θ denote the probability density given by where Z(t, θ) = R n e θ,x − t|x| 2 2 p 0 (x) dx is a normalization constant.Let a(t, θ) denote the barycenter of p t,θ , and define the stochastic process θ t via the differential equation It turns out that when θ t and p t are driven by the same Brownian motion, p t is precisely equal to p t,θt .For proofs of these and other formulas relating to the stochastic localization process, and for the application to the KLS conjecture, see [26,27] or [12].
The Bayesian interpretation of the Eldan process is quite simple: let X be a random vector distributed according to µ, independent of the Brownian motion (W t ) t≥0 .Denote Our main observations are the following two claims: (i) The process ( θt ) t≥0 coincides in law with the process (θ t ) t≥0 which solves (48) above.
(ii) For any fixed t > 0 and θ ∈ R n , the probability density p t,θ on R n is precisely the conditional probability distribution of X given that θt = θ.
Thus, when we observe the tilt process (θ t ) t≥0 , we actually see a Brownian motion with a constant drift X which is unknown, but whose prior distribution is known to us.Moreover, the posterior probability density for the unknown drift X given the observation of the process (θ s ) 0≤s≤t until time t depends only on the state of the process at time t, and is equal to p t,θt .
In the following proposition we prove these two claims.For T > 0 let V T = C 0 ([0, T ], R n ) be the Wiener space of R n -valued continuous functions (W t ) 0≤t≤T with W 0 = 0. Slightly abusing notation, we write γ T for the Wiener probability measure on V T and {F t } 0≤t≤T for the natural filtration, i.e., F t is the σ-algebra generated by (W s ) 0≤s≤t .Proposition 4.1.Let µ be a probability measure on R n which is absolutely continuous with respect to the Lebesgue measure λ.Fix T > 0, and consider the space Ω = R n × V T and the transformation τ : Ω → Ω given by τ (x, (W t ) 0≤t≤T ) = (x, (W t + tx) 0≤t≤T ).
(ii) The measure ν is absolutely continuous with respect to λ ⊗ γ T on Ω with density for x ∈ R n and θ = ( θt ) 0≤t≤T ∈ V T .Consequently, when (X, ( θt ) 0≤t≤T ) is the stochastic process described in (49), the conditional distribution of X with respect to θ = ( θt ) 0≤t≤T is given by the probability density Proof.We first prove (ii).For x ∈ R n , let τ x : V T → V T be defined by τ x ((W t ) t≤T ) = (W t + tx) t≤T so that τ (x, ω) = (x, τ x (ω)).By Fubini's theorem, This means that for any test function g, Since τ x is just a translation in Wiener space by the deterministic function f x (t) = tx, the Cameron-Martin theorem [10] yields that the density of (τ x ) * γ T with respect to γ T at the point ( θt It follows from ( 51) and (52) that The probability measure ν is the joint distribution of the stochastic process (X, ( θt ) 0≤t≤T ) described in (49).Therefore, when conditioning on the entire stochastic process ( θt ) 0≤t≤T , it follows from (53) that the probability density function of X is proportional to x → p 0 (x)e θT ,x −T |x| 2 2 in R n .This completes the proof of (ii).
We move on to the proof of (i).We endow Ω with the probability measure µ ⊗ γ T , and assume that (X, (W t ) t≥0 ) is distributed according to this measure, while θt = tX + W t .Thus, Write N t for the σ-algebra generated by ( θs for the conditional expectation of X with respect to N t , which is a function of ( θs ) 0≤s≤t .According to (54) and [31,Theorem 8.4.3], the process ( θt ) 0≤t≤T coincides in law with the process (θ t ) 0≤t≤T defined by the initial condition θ 0 = θ0 = 0 and the stochastic differential equation (To be precise, the statement in [31,Theorem 8.4.3]only treats time-independent diffusions, but the proofs generalize almost verbatim to the time-dependent case which we need.)The random variable E[X| θ], viewed as an N t -measurable function on Ω, is the conditional expectation of X given θ = ( θs ) 0≤s≤t .According to (ii), the conditional distribution of X given θ is given by the probability density q t (x| θ) from (50).Hence for any 0 < t < T and θ ∈ V t , We have thus verified condition (55) with b(t, x) = a(t, x), completing the proof of (i).
Recalling that the tilt process (θ t ) t≥0 coincides in law with ( θt ) t≥0 , we conclude from (58) that the tilt process coincides in law with the time inversion of a Brownian motion with a starting point drawn from the distribution µ.
Applying this time inversion, we treat the time-inverted tilt process (Y s ) s≥0 as just a Brownian motion with a random starting point.Working with it requires nothing more than the explicit expression for the Euclidean heat kernel; for instance, the distribution of Y s is given by the probability density function ρ s = P s ρ with ρ = p 0 .Given a function ϕ on R n and t > 0, the random variable R n ϕp t associated to the tilt process coincides in law with the distribution of Q s ϕ under the measure µ s , for s = 1/t.Moreover, It is this elementary, "functional analytic" perspective on the measures p t,θ -or, in the new variables, p s,y -that is taken in §2, which makes no explicit use of stochastic localization, pathwise analysis, martingales or stochastic calculus at all.

Föllmer drift as a "time-compressed" version of stochastic localization
Föllmer drift is a well-known stochastic process which couples between an absolutely continuous measure µ and Wiener measure on path space over a finite time interval, without loss of generality [0, 1].It is the same process referred to as the "h-process" in Cattiaux and Guillin [9], because of its relation to Doob's h-transform.
In brief, the Föllmer drift associated to µ is a Brownian motion conditioned to have law µ at time t = 1.The measure P µ on V 1 = C 0 [0, 1] defining the Föllmer drift of µ is defined as the measure having Radon-Nikodym derivative where γ 1 on the left side of (60) is the Wiener measure on C 0 [0, 1], while γ on the right side of (60) is the standard Gaussian measure in R n .The Föllmer drift P µ turns out to have a certain energy-minimizing property, and its energy is precisely twice the relative entropy H(µ|γ), properties which make it quite useful for proving functional inequalities; see, e.g., [18,28].
For A ⊆ R n we write C ∞ c (A) for the class of smooth, compactly supported functions in R n that are supported in the set A. As explained in [1, §4.10], in order to prove Proposition A it suffices to show the following: (*) For any a > 0 there exists r > 0 such that for any f ∈ C ∞ c (R n \ B r ), Here B r = {x ∈ R n ; |x| ≤ r}.
So much for the one-dimensional case.Now let dµ = e −V dx be an n-dimensional log-concave measure, and consider the family of functions f R : S n−1 → R defined by f R (u) = V (Ru)−V (0) R .By convexity, f R is monotone increasing in R, and by assumption f R converges pointwise to infinity.Hence, applying Dini's theorem, we see that f R converges uniformly to ∞. Denote V u (r) = V (ru) for u ∈ S n−1 and r ≥ 0. By convexity,

Lemma 2 . 2 .
Fix s > 0 and let (k, α) be a multi-index.Then, (i) The function ∂ k s ∂ α y log ρ s (y) grows at most polynomially at infinity in y ∈ R n .(ii) Let ϕ have subexponential decay relative to ρ.Then the function ∂ k s ∂ α y Q s ϕ(y) has subexponential decay relative to ρ s .