A DIMENSION-FREE LOGARITHMIC SOBOLEV INEQUALITY
FOR SUBGAUSSIAN LOG-CONCAVE MEASURES
An expository rewrite: one theorem, every step, in plain language.
0. HOW TO READ THIS
This document proves a single theorem and explains every step of its
proof. You need single-variable calculus, basic linear algebra, and a
first course in probability. Anything beyond that is defined the moment
it is used. Long arguments are split into labelled lemmas (Lemma,
Proposition, Theorem) so you can jump around. The proof is a proof by
contradiction that runs through six sections; Section 4 gives the whole
roadmap before any heavy machinery starts, so read it early.
Notation used throughout (all ASCII, so it copies cleanly):
R^n n-dimensional Euclidean space (n can be any positive
integer; nothing below depends on how big n is).
|x| length of the vector x; x.y = dot product.
D gradient (the vector of first partial derivatives).
D^2 Hessian (the matrix of second partial derivatives).
Delta Laplacian (the trace of the Hessian).
int f dmu the integral of f against the measure mu.
mu(A) probability that a mu-random point lands in the set A.
A measure mu on R^n is a probability measure when mu(R^n)=1. It has a
density rho when mu(A) = int_A rho(x) dx for every set A; then rho is a
nonnegative function integrating to 1. "Full-dimensional support" means
the set where rho>0 is not flat (it has volume in every direction).
1. THE CLAIM, IN PLAIN WORDS
1.1 Entropy.
For a nonnegative function g and a probability measure mu, the entropy
of g with respect to mu is
Ent_mu(g) = int g log g dmu - (int g dmu) log( int g dmu ),
with the convention 0 log 0 = 0. Entropy measures how uneven g is: it
is zero exactly when g is constant, and it grows as g becomes spiky.
The subtraction term is there so that adding a constant to log g, or
rescaling g by a positive factor, changes nothing.
1.2 Dirichlet energy.
For a smooth function f, its Dirichlet energy with respect to mu is
int |Df|^2 dmu,
the average squared steepness of f. It is zero exactly when f is
constant. It measures how much f wiggles.
1.3 The logarithmic Sobolev inequality (LSI).
Saying mu satisfies an LSI with constant R means
Ent_mu(f^2) <= 2 R int |Df|^2 dmu for all smooth f,
again with constant functions giving equality at zero. In words: if f
wiggles only a little (small energy), then f^2 cannot be very uneven
(small entropy). The factor 2 is a convention; R is what carries the
information. The smallest such R is written R(mu).
1.4 Why "dimension-free" is the whole point.
Many inequalities in n dimensions carry constants that blow up like n,
sqrt(n), or worse as n grows. A constant that is completely independent
of n is called dimension-free, or absolute if it is a fixed number like
17. Bounds whose constants grow with n become useless for the
high-dimensional objects (convex bodies, machine-learning models,
statistical estimators) that motivate the question. We want one fixed
number that works in every dimension.
1.5 Log-concave measures.
A density rho on R^n is log-concave when rho = exp(-V) for a convex
function V, on the set where rho>0. Convexity of V means its graph
bends upward; equivalently the superlevel sets {rho >= t} are convex.
Examples include the uniform measure on any convex body (a ball, a
cube, a simplex), the Gaussian, the exponential, and products and
Gaussian convolutions of all of these. Log-concave measures are the
standard model for "a probability sitting on a convex region and
decaying outward without bumps."
1.6 Subgaussian linear functionals.
Fix a centered random vector X with law mu (centered means E X = 0).
For a unit vector theta, the scalar theta.X is one linear shadow of X.
We call X linearly a-subgaussian when
E exp( (theta.X)^2 / a^2 ) <= 2 for every unit vector theta.
This says every one-dimensional shadow has tails no heavier than a
Gaussian of variance about a^2: it controls all the linear functions of
X at once, but only the linear ones. The parameter a is exactly the
"width" of these shadows.
The question the paper answers: do these Gaussian tails of the LINEAR
shadows already force an LSI for ALL (including nonlinear) test
functions, with a constant independent of dimension?
2. THE THEOREM AND ITS TRANSPORT CONSEQUENCE
2.1 The main theorem.
Theorem 1 (Main). There is an absolute constant C < infinity with the
following property. For every n >= 1, every centered log-concave
probability measure mu on R^n that has a Lebesgue density, and every
a > 0 such that a random vector X with law mu is linearly a-subgaussian,
Ent_mu(f^2) <= C a^2 int |Df|^2 dmu for every smooth compactly
supported f on R^n.
The same C works in every dimension and for every such mu.
In the convention of Section 1.3 (Ent(f^2) <= 2 R(mu) int|Df|^2), the
theorem says R(mu) <= (C/2) a^2: the logarithmic Sobolev constant is
bounded by a universal multiple of the squared subgaussian width. Note
what is NOT assumed: the density need not be smooth, and no positive
curvature (no D^2 V >= kappa > 0) is required. So uniform measures on
convex bodies, whose potentials are flat and only convex, are included.
Before this result, only a dimension-dependent bound like R(mu) <= C n
a^2 or C sqrt(n) a^2 was known in general.
2.2 Why the converse is free (so the hypothesis is sharp).
The LSI already forces Gaussian tails for 1-Lipschitz functions by the
classical Herbst argument; in particular it forces exactly the
subgaussian bound on linear functions that Theorem 1 assumes. So the
hypothesis "linear functions are a-subgaussian" is, up to universal
constants, also necessary for an LSI with constant of order a^2. The
theorem is therefore not wasting any assumption: it upgrades a
necessary condition to a sufficient one, at the same scale.
2.3 A consequence for transport.
Two standard objects. For two probability measures alpha, beta on R^n
with finite second moment, the squared 2-Wasserstein distance is
W2(alpha,beta)^2 = inf over couplings pi of int |x-y|^2 dpi(x,y),
where a coupling is a joint law with the two given marginals. It is the
cheapest way to move the mass of alpha onto beta, paying squared
distance. The relative entropy of alpha with respect to beta is
H(alpha | beta) = int p log p dbeta when alpha = p beta,
= +infinity when alpha is not absolutely
continuous with respect to beta.
Corollary 1 (Quadratic transport-entropy inequality). Under the
hypotheses of Theorem 1, every probability nu with finite second moment
satisfies
W2(nu,mu)^2 <= C a^2 H(nu | mu),
with the same absolute constant C. This follows from the LSI through the
Otto-Villani theorem, which converts any LSI into a
transport-entropy inequality with matching constants; the proof at the
level of this document is in Section 5.7 and only needs an
approximation argument to remove the smoothness of the density.
The corollary is the useful form in applications: it says it costs very
little mass-movement energy (order a^2 times a bit of relative entropy)
to reach mu from any competitor nu. This is what turns "control of
linear functions" into concrete concentration statements.
3. HISTORY AND WHY IT MATTERS
3.1 The inequality and the Gaussian.
The logarithmic Sobolev inequality was introduced by Leonard Gross in
1975 in his study of hypercontractivity, the property that a semigroup
pushes functions into higher integrability. For the standard Gaussian
measure the LSI holds with R = 1, independent of dimension, and Gross
showed the constant cannot be improved. This paper asks how far the
"constant 1, no dimension" phenomenon survives beyond the Gaussian.
3.2 The curvature approach and its blind spot.
Bakry and Emery (1985) gave a powerful criterion. Write the density as
rho = exp(-V). Suppose the potential is strictly convex in the strong
sense D^2 V >= kappa Id with kappa > 0. Then mu satisfies the LSI with
R <= 1/kappa. The proof runs a diffusion whose drift is D V and shows
the flow contracts distances at rate kappa.
The catch: log-concavity says only D^2 V >= 0, i.e. kappa = 0. The
curvature criterion then gives nothing finite. A uniform measure on a
convex body is the extreme case: its potential is infinite outside the
body and flat inside, so it has zero curvature in the interior and
infinite curvature on the boundary. The Bakry-Emery machine is blind to
exactly the examples we care about. The whole difficulty is to replace a
local curvature hypothesis by the global tail hypothesis that linear
functions are subgaussian.
3.3 Necessity and the earlier dimension-dependent bounds.
The Herbst argument (see Ledoux's 1999 survey) shows that an LSI with
constant R gives Gaussian concentration: for every 1-Lipschitz u,
log E exp( t (u - E u) ) <= R t^2 / 2.
Applying this to linear functions, an LSI with constant of order a^2
forces the linear subgaussian bound with width of order a. So some
subgaussian control of linear functions is necessary. The open problem
was sufficiency: does it give the LSI back, dimension-free?
Sergey Bobkov (1999) gave a norm-based criterion: the entropy constant
is bounded by a universal multiple of the squared subgaussian norm of
the scalar |X|. Combining this with the linear subgaussian assumption,
which controls |X| only through a Holder step E exp(|X|^2/(n a^2)) <= 2,
yields an entropy constant of order n a^2, with a dimension factor.
Pierre Bizeul (conjectured in his 2023 preprint, cited here as the 2026
journal version) formulated precisely that linear subgaussian tails
should suffice for a dimension-free LSI within the log-concave class.
Bizeul proved an improvement with constant of order sqrt(n) a^2 in
general, and the fully dimension-free statement in the special case of
rotationally invariant log-concave measures. The survey of Klartag and
Lehec (2024/2025) restates the conjecture together with these bounds.
3.4 What is genuinely new.
The linear subgaussian hypothesis controls only one-dimensional shadows,
while the LSI conclusion controls the entropy of genuinely nonlinear
test functions. The gap between "linear control" and "nonlinear
conclusion" is where the entire argument lives. Emanuel Milman (2010)
had shown that for log-concave measures, Gaussian concentration for all
Lipschitz functions is equivalent to Gaussian isoperimetry, which in
turn gives an LSI; but "all Lipschitz functions" is much stronger than
"linear functions." The theorem closes the remaining gap: linear
functions turn out to already force the concentration of all Lipschitz
functions, hence the LSI. Theorem 1 is the first proof that the
conjecture is true in full generality.
4. THE PLAN OF THE PROOF (READ THIS FIRST)
The proof is a proof by contradiction. We assume Theorem 1 fails and
derive nonsense. Everything technical is in service of one idea: a
counterexample that violates the LSI would have to behave like a
Gaussian in a very rigid way, both from above (entropy contraction) and
from below (an information test), and those two descriptions cannot
both hold.
4.1 Assume a counterexample, then clean it up.
Failure of Theorem 1 gives a sequence of centered, compactly supported,
full-dimensional log-concave laws mu such that the LSI constant R(mu)
is normalized to 1 while the linear subgaussian width a tends to 0. The
ratio R(mu)/a^2 then blows up; that is the failure. Two harmless
operations keep the failure: restricting to a large ball and tilting by
exp(-epsilon |x|^2 / 2). They keep the law log-concave, compactly
supported, and smooth enough to work with.
4.2 Gaussian regularization.
Convolve mu with a tiny Gaussian of variance s. Write
mu_s = law(X + sqrt(s) G), with G standard Gaussian independent of X,
and fix s = 1/8 once and for all. Convolution makes the density smooth
and strictly positive and keeps it log-concave (a Gaussian is
log-concave and a convolution of log-concave densities is
log-concave). Its subgaussian width a_s still tends to 0. Crucially,
the regularized LSI constants R_s tend to 1, because s is fixed and
small. So from now on we have a smooth, positive, log-concave law mu_s
sitting at the edge: it is one unit of entropy-constant away from a
counterexample.
4.3 The critical law and a near-first-eigenfunction.
From mu_s we build a second probability eta and a vector field w. There
are two cases, and we must handle both because we cannot predict which
occurs:
Strict branch. The entropy quotient is attained at a positive
optimizer f solves the logarithmic Euler equation. Put
eta = f^2 mu_s and w = D log f^2. Then eta satisfies its own LSI
with constant <= 1/lambda, and w is an approximate first
eigenfunction of the diffusion A_eta = -Delta - D log(eta).D:
(A_eta - 1) w is small in L^2, E_eta|w|^2 = I > 0.
Gap branch. The LSI and Poincare constants coincide. Then w = D phi,
where phi is an eigenfunction of A_{mu_s} with eigenvalue lambda, and
eta = mu_s. The same near-eigenfunction relation holds.
In both cases eta satisfies a logarithmic Sobolev inequality with a
constant close to 1, and w is close to a first eigenfunction of its
diffusion. That is the single object the rest of the proof works with.
4.4 The prediction curve.
Add fresh Gaussian noise. For r > 0 form U_r = Y + sqrt(r) G with
Y ~ eta and G standard, and let m(r) be the fraction of w that the best
predictor can recover from U_r:
m(r) = (1/I) int |E[w(Y) | U_r]|^2 deta.
If w were an exact first eigenfunction, the semigroup would give
m(r) = e^{-r} exactly. We prove this is essentially true anyway:
Theorem (prediction curve). For every fixed 0 < r < 1/4,
m(r) -> e^{-r} along the counterexample sequence.
The proof pairs two facts. Upper bound: entropy contraction of the
diffusion (the LSI of eta) caps how much information about w can
survive the noise. Lower bound: an entropy-loss comparison lemma
tracks the loss under repeated Gaussian observations; observing many
independent copies lets us throw away at most one direction per copy,
accumulating negligible cost. Together they pin m(r) to e^{-r}.
4.5 From the curve to tensors.
The exact identity m(r) = e^{-r}, integrated against Gaussian Hermite
polynomials, says the prediction is Bessel-tight at every derivative
order. This is the bridge from a statement about the vector w to a
statement about all its derivatives. It produces symmetric tensor
fields T_l of order l+1 with
||T_l||^2 = (1+o(1)) I, ||DT_l - T_{l+1}|| = o(sqrt I),
||(A-1) T_l|| = o(sqrt I), <T_l, K T_l> = o(I),
where K = D^2(-log mu_s) is the curvature. So derivatives keep the
same size, stay nearly eigenfunctions, and avoid the curved directions
of mu_s. This holds simultaneously for any fixed finite list of levels.
4.6 Matrices and the frozen law.
Pack the tensors of a band of levels into a big column Psi, separate
one slot, and let
rho = Psi Psi^*, R = rho^{1/2}
be a positive matrix field. Many symmetric slots let us average the
action over slots with only O(1/k) error, and shifting the band costs
only its endpoints; this yields sharp energy: ||DR||^2 -> 1. From the
derivative-action estimate one builds a bounded symmetric matrix column
H_i with a Sylvester-type relation D_i R ~ (H_i R + R H_i)/2 and almost
commuting H_i. Then, running the stationary diffusion for time t and
freezing H_i at the initial point, R(Y_t) is described by a matrix
lognormal:
sigma = direct-sum over spectral blocks beta^2 exp( sqrt(8t) H[N] - 4t ),
with N standard Gaussian. This is the frozen diffusion law. It is the
engine of the final contradiction, and it is where the many symmetric
tensor slots earn their keep: they make H[N] behave like a scalar
Gaussian even though the H_i need not commute.
4.7 The information contradiction.
We now observe a Gaussian vector whose covariance is a bounded
increasing function of R(Y_t)^2, with a scale T:
Z_T = ( Id + f_L(T R(Y_t)^2) )^{1/2} G, f_L(q) = q/(1+q/L).
Upper bound (from the LSI): the mutual information between the state
and the observation, integrated against dT/T^2, is at most a universal
constant. The factor dT/T^2 is the load-bearing trick. In a spectral
block of size beta^2 the substitution x = T beta^2 gives
dT/T^2 = beta^2 dx/x^2,
assigning to each block exactly the weight beta^2 that the matrix
(frozen) trace uses, even when the spectrum is arbitrarily diffuse.
Lower bound (from the frozen law): on the same event, the observation
reproduces a scalar Gaussian variance-mixture experiment. The entropy
of that experiment, integrated over the same scales, grows without bound
once the diffusion time t and the covariance cap L are taken large
enough.
Upper bound is a constant, lower bound is unbounded. Contradiction.
4.8 The order of limits.
Throughout, every auxiliary parameter (time t, cap L, truncation m,
band size k) is fixed BEFORE the counterexample index is sent to
infinity. Only after that sequence limit do we let t and L grow. No
uniformity in the growing parameters across the sequence is ever
required. This order is repeated where it matters so it is never in
doubt.
4.9 Reading order for the technical part.
Section 5 analytic toolkit: entropy, information, curvature,
diffusion, entropy dissipation, Gaussian channels.
Section 6 normalization and construction of the critical law
(both branches).
Section 7 the prediction curve.
Section 8 the tensor hierarchy.
Section 9 the matrix square root and the frozen law.
Section 10 the covariance observations and the final information
contradiction, then the assembly of Theorem 1.
Figure 1 (description): two arrows from "strict branch" and "gap
branch" merge into "common critical law and field." From there the
prediction curve feeds the tensor hierarchy, the hierarchy feeds the
matrix frozen law, and the frozen law plus the LSI both feed the
information contradiction. Only Theorem 1 needs the two branches to
rejoin at the end.
5. TOOLKIT: ENTROPY, DIFFUSION, AND GAUSSIAN CHANNELS
This section sets up the analytic language used in every proof that
follows. Nothing here is specific to the counterexample; it is the common
vocabulary.
5.1 Two time conventions that set all factors.
We will run two different "time" evolutions and must not confuse them.
(a) Gaussian convolution. Convolving a law with a centered Gaussian of
covariance r Id has generator (1/2) Delta. Its time parameter is r,
a variance.
(b) Stationary diffusion. A reversible diffusion with invariant law pi
and generator -A_pi, with noise coefficient sqrt(2). Its time
parameter is a genuine clock time.
The two conventions differ by factors of 2 throughout. We fix them once
and state every dissipation identity in the convention used, so no
factor can drift.
5.2 Relative entropy, Fisher information, and the two constants.
For probabilities p, pi with p absolutely continuous relative to pi,
H(p | pi) = int log(dp/dpi) dp, H(p | pi) = +infinity otherwise.
The Fisher information of p relative to pi is
Ical(p | pi) = 4 int |D sqrt(dp/dpi)|^2 dpi.
When the density is positive and smooth this equals
int |D log(dp/dpi)|^2 dp: the average squared score. The value is
+infinity unless the square root of the density lies in the weighted
Sobolev space.
The logarithmic Sobolev constant R(pi) and the Poincare constant
R^P(pi) are defined by the best constants in
Ent_pi(f^2) <= 2 R(pi) int |Df|^2 dpi,
Var_pi(f) <= R^P(pi) int |Df|^2 dpi, Var_pi(f)=int f^2 dpi - (int f dpi)^2.
Equivalently, in "relative" form, the LSI constant R = R(pi) is the
smallest number with
2 H(p | pi) <= R Ical(p | pi) for all p.
The Dirichlet form is int |Df|^2 dpi. Constant functions have zero
entropy and zero energy, which is why the entropy is measured relative
to the mean.
Linearization: expanding Ent about f = 1 + epsilon g to second order in
epsilon gives the variance bound, so R^P(pi) <= R(pi). The Poincare
constant is never larger than the log-Sobolev constant, as one expects:
a variance bound is weaker than an entropy bound.
5.3 Sobolev space and why approximations are allowed.
For a smooth positive density the weighted Sobolev space is the
completion of the smooth compactly supported functions in the norm with
squared norm int (f^2 + |Df|^2) dpi. Cutoffs, truncation of values, and
smoothing on compact sets show this is also the space of functions whose
weak gradient is square-integrable. A compact-test LSI extends to the
whole space by lower semicontinuity of the entropy and the Fisher
information, and by approximation. Every inequality below stated for
smooth compact tests therefore holds on this space; we use this silently.
5.4 The entropy-duality formula.
For a probability pi, one exact identity underpins most of the analysis:
H(p | pi) = sup over bounded g of { int g dp - log int e^g dpi }.
Jensen's inequality gives the upper bound (log int e^g dpi is an
exponential-moment), and bounded truncations of log(dp/dpi) give
equality. Three consequences we will use.
(i) Data processing. Pushing a probability through a measurable map,
or averaging through a Markov kernel, never increases relative
entropy. This is conditional Jensen.
(ii) Chain rule. Disintegrating a joint density and splitting its
logarithm gives
H(P_{U,Z} | Q_U x nu) = H(P_U | Q_U) + E_{P_U} H(P_{Z|U} | nu),
with conditional entropies averaged over U. Mutual information is
I(U;Z) = H(P_{U,Z} | P_U x P_Z).
(iii) Mean lower bound. Taking g(z)=v.z in the duality formula and
optimizing over v gives H(p | gamma) >= |E_p Z|^2/2 for a Gaussian
gamma of unit covariance: relative to a Gaussian, an entropy
budget bounds how far the mean can be pushed.
5.5 Positive curvature: the Bakry-Emery lemma.
Lemma 1 (Positive curvature). Let S be a positive definite matrix on R^n,
and let U be a proper lower semicontinuous convex function, allowed to be
+infinity (so the support may be a convex set with boundary). Suppose
pi(x) = Z^{-1} exp( -(1/2) x.Sx - U(x) ) dx
is a probability with full-dimensional support. Then
Ent_pi(f^2) <= 2 E_pi < Df, S^{-1} Df >,
Var_pi(f) <= E_pi < Df, S^{-1} Df >,
first for compact smooth f, then for all f in the form domain.
Proof. A linear change of variables reduces to S = Id. First suppose
V(x) = |x|^2/2 + U(x) is smooth with bounded Hessian. Run the diffusion
with generator L = Delta - DV.D, drifting with -DV; it is reversible
with invariant law pi because the weight exp(-V) makes DV the score.
Let J_t be the derivative of the flow with respect to the starting
point. Differentiating the flow equation gives J'_t = -D^2 V(X_t) J_t,
so with D^2 V >= Id we get ||J_t|| <= e^{-t}: the flow contracts
distances exponentially. Weighted Cauchy-Schwarz then gives, for a
smooth positive q bounded away from zero,
|D P_t q|^2 / (P_t q) <= e^{-2t} P_t ( |Dq|^2 / q ),
where P_t = e^{tL} is the semigroup. Integration by parts yields
d/dt int (P_t q) log(P_t q) dpi = - int |D P_t q|^2 / (P_t q) dpi.
Couple a trajectory started at x to a stationary trajectory driven by
the same Brownian motion; the contraction bound says the distance
shrinks like e^{-t}. Since pi has Gaussian tails, P_t q -> E_pi q and
its entropy -> 0. Integrating the two displayed identities from 0 to
infinity gives Ent_pi(q) <= (1/2) int |Dq|^2/q dpi. Put q = f^2+eps and
send eps -> 0.
For general U, replace U by its quadratic infimal convolutions
U_j(x) = inf_z { U(z) + j|x-z|^2 }. These are convex, increase to U,
and have Lipschitz gradients, so U_j is smooth enough after mollifying
at radii -> 0. The affine minorant of U gives a uniform Gaussian-times-
linear domination, the LSI passes to the limit, and the boundary of a
full-dimensional convex support has zero Lebesgue measure so boundary
values do not matter. Inserting f = 1 + eps g and comparing eps^2
coefficients yields the variance inequality. QED.
The lemma is the entire content of the curvature method: curvature
D^2 V >= Id buys an LSI with constant 1. A "minimum curvature kappa"
version rescales to constant 1/kappa. And it is exactly this hypothesis
that log-concavity (D^2 V >= 0) fails to provide.
5.6 Products and Gaussian convolution add the constants.
Suppose independent pieces pi_1, pi_2 satisfy LSIs with energy forms
Q_1, Q_2. Their product satisfies the LSI with energy Q_1 + Q_2. Indeed
decompose the entropy of f on the product into conditional entropy on
the fiber plus entropy of the fiber integral, and use
|D_x ( int f(x,y)^2 dpi_2(y) )^{1/2}|^2 <= int |D_x f(x,y)|^2 dpi_2(y).
The variance version uses conditional variance and differentiation of
the fiber mean. In particular, for independent X and Z,
R( law(X+Z) ) <= R( law(X) ) + R( law(Z) ).
The Gaussian of variance s Id has R = s; adding Gaussian noise adds s to
the constant. This is why the regularized constants R_s in Section 4.2
obey R_s <= 1 + s.
5.7 Herbst's argument: entropy control forces concentration.
Lemma 2 (Herbst). If pi satisfies the LSI with constant R, then every
integrable 1-Lipschitz function u satisfies
log E_pi exp( t (u - E_pi u) ) <= R t^2 / 2 for all real t.
Proof. Apply the LSI to e^{tu/2}, which has D(e^{tu/2}) = (t/2) e^{tu/2}
Du. Writing psi(t) = log E_pi e^{tu}, the LSI reads
t psi'(t) - psi(t) <= R t^2 / 2. That is the derivative of psi(t)/t:
integrate from 0 to t using psi(0)=0 and continuity. Bounded Lipschitz
truncations and Fatou extend to all integrable 1-Lipschitz u. QED.
Distance to the mean under pi therefore has a subgaussian tail of
variance R. Conversely, if every 1-Lipschitz function has a subgaussian
tail of variance R, this is "concentration," and it is one route back to
the LSI (Milman's theorem, cited in Section 3). Applied to linear
functions u(x) = theta.x, Herbst says an LSI of constant R forces the
linear subgaussian bound of Section 1.6 with a of order sqrt(R). This is
the sharpness statement of Section 2.2.
5.8 The Dirichlet operator and the stationary diffusion.
Lemma 3 (Dirichlet operator). Let pi be a smooth strictly positive
probability density on R^n. On smooth compactly supported functions
define
A_pi = - Delta - D log(pi) . D.
This nonnegative symmetric operator is essentially self-adjoint in
L^2(pi), and smooth compact functions are a core for its closure. In
particular, if f in L^2(pi) has A_pi f (in the distributional sense)
in L^2(pi), then f is in the domain of the closed operator. The
semigroup P_u = e^{-u A_pi} preserves constants and is realized by the
stationary diffusion
dY_u = sqrt(2) dB_u + D log pi(Y_u) du, Y_0 ~ pi.
For f in the domain, the stationary Ito formula holds in L^2, with drift
- A_pi f and martingale integrand sqrt(2) Df.
Proof sketch, complete in its logical steps. Integration by parts gives
<f, A_pi g> = int Df . Dg dpi, so A_pi is symmetric and nonnegative. If
v in L^2(pi) is orthogonal to (1 + A_pi) of all smooth compact
functions, then (1 + A_pi) v = 0 distributionally. Conjugating by
sqrt(pi) reduces this to a Poisson equation with a smooth local
potential, and local elliptic bootstrap makes v smooth. Testing against
chi_R^2 v with a cutoff chi_R equal to 1 on a ball of radius R and
|D chi_R| <= C/R, and using 2ab <= a^2/2 + 2b^2, gives
int chi_R^2 v^2 dpi + (1/2) int chi_R^2 |Dv|^2 dpi <= 2 int |D chi_R|^2 v^2 dpi,
which forces v = 0. Hence the range of 1 + A_pi is dense. The bound
||(1+A_pi)f||_2 >= ||f||_2 says the closure's range is also closed, so
it is all of L^2; the closure is self-adjoint (solve the adjoint
equation and use the orthogonal complement argument). This proves the
domain and self-adjointness claims. The cutoffs converge to 1 while
their energy tends to 0, so 1 is in the kernel and P_u 1 = 1. Solving
the stochastic equation with killing on balls, whose kernels are
symmetric sub-Markov by the Dirichlet heat equation, and increasing the
balls, identifies the minimal semigroup with P_u; essential
self-adjointness shows no mass is lost, so the diffusion is stationary
and non-exploding. QED.
The point of the lemma: it gives a genuine reversible Markov process
with known generator and a well-defined energy form, on which "time"
entropy gradients are computable. Everything downstream is a statement
about this process.
5.9 Entropy dissipation (de Bruijn identities).
Lemma 4 (Entropy dissipation). For the semigroup of Lemma 3, if
p_0 = q_0 pi has finite relative entropy and p_u = (P_u q_0) pi, then
for 0 <= a <= b,
H(p_a | pi) - H(p_b | pi) >= int_a^b Ical(p_u | pi) du.
If instead two probabilities both evolve by Gaussian convolution,
p_r = p_0 * gamma_r and q_r = q_0 * gamma_r, then
H(p_a | q_a) - H(p_b | q_b) >= (1/2) int_a^b Ical(p_r | q_r) dr
whenever the left-hand entropy at time a is finite; here gamma_r is the
centered Gaussian of covariance r Id, and a = 0 is allowed. For the
standard Gaussian Ornstein-Uhlenbeck semigroup the first inequality is
an equality, including at time 0 when the initial entropy is finite.
All assertions hold after averaging over conditioning labels whose
noises are independent given those labels.
Proof. For a bounded density q bounded away from zero, the form chain
rule, first for smooth functions and then by form approximation, gives
(d/du) int (P_u q) log(P_u q) dpi = - int |D P_u q|^2 / (P_u q) dpi.
Spectral smoothing puts P_u q in the operator domain for u > 0.
Approximating an arbitrary finite-entropy density from above and below
and normalizing, the initial entropies converge; L^1 contraction keeps
later densities close; lower semicontinuity of entropy and of the
Fisher information, plus Fatou in time, gives the first display from
time 0. Restarting at a gives arbitrary a <= b.
For simultaneous heat evolution put z = p_r / q_r. Direct
differentiation with (d/dr) p_r = Delta p_r / 2 gives, for convex
Phi_k increasing to z log z - z + 1,
(d/dr - (1/2) Delta) { q_r Phi_k(z) } = -(1/2) q_r Phi_k''(z) |Dz|^2.
Integrate against spatial cutoffs, let the cutoff go to infinity, then
k -> infinity, and use monotone convergence. Data processing gives
H(p_eps | q_eps) <= H(p_0|q_0), so the initial time is included. For
Ornstein-Uhlenbeck evolution relative to a standard Gaussian,
generator Delta - x.D, the same convex truncation stacked with expanding
radial cutoffs gives equality on positive time; relative entropy is
continuous at time 0 by data processing plus lower semicontinuity.
Tonelli handles conditional versions. QED.
In words: relative entropy is nonincreasing under both evolutions, and
its rate of decrease equals the Fisher information. This is the "engine"
that converts an entropy budget into an integral information budget.
5.10 The Gaussian channel identity (Guo-Shamai-Verdu).
This is the standard "I-MMSE" identity: the derivative of entropy
through a Gaussian channel is half the power of the minimum-mean-square
predictor. We need its conditional, time-integrated form in our
conventions.
Lemma 5 (Gaussian signals). Let S be an R^d-valued random vector with
finite second moment, let U be any side information, and let G be a
standard Gaussian independent of (S,U). Write Z_t = sqrt(t) S + G and
d(t) = E_U H( Law(Z_t | U) | gamma ),
the conditional relative entropy against the standard Gaussian gamma,
averaged over the side information. Then d(0) = 0, d(t) <= t E|S|^2/2,
and
d(t) = (1/2) int_0^t E |E[S | Z_v, U]|^2 dv.
The integrand is nondecreasing in v. Taking U constant gives the
unconditional identity.
Proof. Conditional convexity of relative entropy and the shift formula
H( gamma(. - sqrt(t) s) | gamma ) = t|s|^2/2 give the upper bound.
Gaussian kernel differentiation gives the score of the output,
D_z log( dLaw(Z_t|U) / dgamma )(z) = sqrt(t) E[S | Z_t = z, U].
This is justified at every t > 0 by the Gaussian kernel and the finite
second moment, or by truncating S. Running Ornstein-Uhlenbeck evolution
for time u replaces t by t e^{-2u}. Its entropy dissipation identity
together with dt/du = -2t gives d'(t) = (1/2) E |E[S|Z_t,U]|^2 almost
everywhere on (0,infinity). The upper bound gives d(t) -> 0 at zero,
so integrating from 0 proves the identity. Monotonicity of the
prediction energy: the channel at v < t is obtained from the channel at
t by multiplying the output by sqrt(v/t) and adding fresh Gaussian noise
of covariance (1 - v/t) Id, and conditional Jensen gives
nondecreasingness. QED.
The lemma turns a Gaussian observation into a computing device: the
information it carries is exactly the integral of the squared best
prediction of the signal. We will use it both directions - to bound
information from a predictor, and to read off a predictor from
information.
5.11 The transport consequence of the LSI.
Proof of Corollary 1. Jensen in each coordinate gives
int |x|^2 dmu <= n a^2 log 2, so mu has finite second moment. Only the
case H(nu|mu) < infinity needs proof. For s > 0 put gamma_s =
N(0, s Id), mu_s = mu * gamma_s, and nu_s = nu * gamma_s. The density of
mu_s is smooth and strictly positive.
Claim: mu_s is log-concave with convex potential V_s. For X ~ mu and an
independent standard Gaussian G, the posterior potential of X given
X + sqrt(s) G = y is the original convex potential plus
|x - y|^2/(2s), so Lemma 1 bounds the posterior covariance by s Id.
Hence the Hessian of V_s satisfies
D^2 V_s(y) = s^{-1} Id - s^{-2} Cov( X | X + sqrt(s) G = y ) >= 0,
and Gaussian differentiation is valid without compact support. So mu_s
is centered and log-concave, smooth.
Next, mu_s is a_s-subgaussian with a_s = a + beta_s and
beta_s = sqrt(8s/3). For a scalar standard Gaussian Z,
E exp( z^2 / beta_s^2 ) = 2. The elementary inequality
(u+t)^2/(a+beta_s)^2 <= (a/(a+beta_s)) u^2/a^2 + (beta_s/(a+beta_s)) t^2/beta_s^2
and Holder show that if X is a-subgaussian and sqrt(s)G has width
beta_s, then X + sqrt(s)G is a_s-subgaussian. Theorem 1 with the Sobolev
extension gives the LSI with constant C a_s^2.
In the conventions of Otto-Villani, Ent(f^2) <= L int|Df|^2 is LSI(2/L),
whose transport conclusion is W2^2 <= L H. Their theorem applies to the
smooth convex potential of mu_s, hence
W2(nu_s, mu_s)^2 <= C a_s^2 H(nu_s | mu_s) <= C a_s^2 H(nu | mu).
The last step is data processing for the common Gaussian kernel, as in
Section 5.4. Coupling a vector with itself plus independent Gaussian
noise gives W2(mu_s, mu) <= sqrt(n s) and the same for nu. Let
s -> 0 at fixed n, with a_s -> a. QED.
The whole proof of the corollary is therefore three moves: regularize
by Gaussian convolution (which buys smoothness and curvature), invoke
the main inequality plus Otto-Villani on the regularized law, and let
the regularization vanish. The only inputs are Lemmas 1 and the data
processing bound.
6. THE CRITICAL LAW AND A NEAR-FIRST-EIGENFUNCTION
Suppose Theorem 1 is false. This section converts "a counterexample
exists" into a single smooth object: a probability eta satisfying an LSI
with constant close to 1, plus a vector field w that is almost a first
eigenfunction of eta's diffusion. There are two constructions of (eta,w);
we keep both alive because we cannot predict which occurs. Every limit
in this section is along the counterexample sequence; we suppress its
index.
6.1 Normalized counterexamples.
Lemma 6 (Normalized counterexamples). Failure of Theorem 1 produces a
sequence of centered, compactly supported, full-dimensional log-concave
probabilities mu such that
R(mu) = 1, sup_{|theta|=1} E_mu exp( (X.theta)^2 / a^2 ) <= 2, a -> 0.
For an absolute constant C they also satisfy
log E_mu exp( v.X ) <= C a^2 |v|^2, Cov_mu(X) <= C a^2 Id.
Proof. Start from laws and compact smooth tests for which the ratio
(entropy)/(energy)/a^2 tends to infinity. Choose the tests with positive
energy: a smooth test of zero energy is constant on the connected
interior of the convex support, and its entropy then vanishes. Restrict
the law to a large ball; this preserves each test quotient arbitrarily
well. Multiply the restricted density by exp(-eps |x|^2/2) and
normalize, with eps > 0 small enough to preserve the quotient again.
Both operations keep log-concavity, the combined density relative to the
original law is bounded by 2, and Lemma 1 gives a finite LSI constant
for the resulting compact law.
A bounded density increase changes the exponential-square expectation by
at most that factor. If E exp(Z^2/A^2) <= 4, Jensen gives
E exp(Z^2/(2A^2)) <= 2. Centering costs only another absolute factor:
E Z^2 <= A^2 log 4 and (Z - E Z)^2 <= 2Z^2 + 2(E Z)^2. Apply this to
each unit linear form with the same constants. Translation preserves
the test quotient. Thus the centered compact laws still have finite
R with R/a^2 -> infinity. Dilation by R^{-1/2} makes R(mu) = 1.
For the moment bound: if a centered scalar Z obeys
E exp(Z^2/a^2) <= 2, then E|Z|^k <= 2 a^k Gamma(k/2+1) by integrating
its tail. Expansion of exp(tZ), with the linear term absent, gives
log E exp(tZ) <= C a^2 t^2 for a|t| <= c; for a|t| >= c use
tZ <= Z^2/a^2 + a^2 t^2/4. Apply to Z = X.theta. Differentiating the
centered moment generating function at zero gives the covariance bound.
QED.
6.2 Gaussian regularization and posterior identities.
Fix s = 1/8. Let G be a fresh standard Gaussian and set
mu_s = law( X + sqrt(s) G ), X ~ mu,
R_s = R(mu_s),
b_s(y) = E[ X | X + sqrt(s) G = y ] (posterior mean),
V_s(y) = - log mu_s(y), (potential of mu_s).
Gaussian differentiation and the posterior covariance give the three
basic identities
D V_s(y) = ( y - b_s(y) ) / s,
M_s(y) = D b_s(y) = Cov( X | X + sqrt(s) G = y ) / s,
K_s(y) = D^2 V_s(y) = ( Id - M_s(y) ) / s.
The posterior of X given y is the original convex potential plus
|x - y|^2/(2s); Lemma 1 (applied to its linear forms) bounds its
covariance by s Id. Hence
0 <= M_s <= Id, 0 <= K_s <= s^{-1} Id.
All these functions are smooth. Compact support of X also gives
|b_s| <= sup over the support of mu of |x|, so b_s is bounded.
The matrix field K_s is the curvature of mu_s. Much of the proof is
about showing that the near-eigenfield w avoids K_s; K_s is only
nonnegative and can be huge near the boundary of the support, which is
exactly why a uniform radius eigenvalue argument is unavailable and the
whole argument must be structured around "w sees almost no curvature."
6.3 The regularized constants barely change.
Lemma 7 (Bounds for regularized constants). The function s -> R_s is
nondecreasing and 1-Lipschitz on (0, infinity), and
1 <= R_s <= 1 + s.
Proof. The heat equation for mu_s is a continuity equation with
velocity D V_s / 2. On a compact interval of positive times this
velocity is globally Lipschitz and grows at most linearly, so its smooth
flow is a diffeomorphism carrying mu_s to mu_t. Its Jacobian J satisfies
J' = K_t J / 2. Since K_t >= 0 the flow expands distances, so its
inverse is 1-Lipschitz for t >= s. Transport a test function by that
inverse: its entropy is unchanged and its Dirichlet energy cannot
increase. Hence R_t >= R_s: monotone.
The convolution bound R_{s+h} <= R_s + h from Section 5.6 and
R_s <= R_0 + s = 1 + s give the upper bounds. Finally, for every compact
smooth test the entropy and the energy converge to those under mu as
s -> 0, so taking the supremum of the quotient gives
liminf_{s->0} R_s >= R(mu) = 1; monotonicity then forces the whole
range to start at >= 1. QED.
So for the fixed tiny regularization s = 1/8 we have R_s very close to
1, while each mu_s is smooth, positive, and log-concave with a
well-behaved diffusion. That is our working law.
6.4 Compactness and the Gaussian entropy defect.
To extract an optimizer we need two facts about the regularized law
nu = mu * gamma_s: its form domain is compactly embedded in L^2(nu),
and the LSI holds with a small additive defect that lets maximizing
sequences be compactified.
Lemma 8 (Compactness and Gaussian entropy defect). Fix s > 0 and a
compactly supported probability mu on R^n, and put nu = mu * gamma_s.
Its form domain embeds compactly into L^2(nu). For every eps > 0 there
is a finite C_eps, depending on this fixed law, such that
Ent_nu(u^2) <= 2(s+eps) ||Du||_2^2 + C_eps ||u||_2^2.
If u_j -> u in L^2(nu), ||u_j||_2 = 1, and ||D u_j||_2^2 -> E_* <
infinity, then
limsup_j Ent_nu(u_j^2) <= Ent_nu(u^2) + 2s ( E_* - ||Du||_2^2 ).
The sharp coefficient 2s governs escape into the Gaussian tails; the
additive constant governs maximizing sequences that would collapse to a
constant.
Proof. Write r = dnu/dgamma_s. The Gaussian kernel formula gives
D log r = b_s / s, which is bounded, and |log r(y)| <= C(1+|y|). For
v = u sqrt(r),
Ent_nu(u^2) = Ent_{gamma_s}(v^2) - int v^2 log r dgamma_s,
Dv = sqrt(r) ( Du + (1/2) u D log r ).
Gaussian integration by parts followed by Cauchy-Schwarz gives
int |y|^2 v^2 dgamma_s = s n ||v||_2^2 + 2s int v (y.Dv) dgamma_s
<= 2 s n ||v||_2^2 + 4 s^2 ||Dv||_2^2.
Hence the linear-growth entropy correction is bounded by
delta ||Dv||_2^2 + C_delta ||v||_2^2 for every delta > 0. Gaussian LSI
and the derivative formula, with another small Cauchy-Schwarz
parameter, give the defective inequality.
The same moment estimate shows tightness of the L^2(nu) mass of any
bounded form sequence. On a fixed ball the density is bounded above and
away from zero, so the ordinary local Rellich theorem gives L^2
precompactness; a diagonal extraction plus tightness give global
compactness.
For the sharp coefficient 2s in the entropy-defect display, set
v_{j,L} = (|u_j| - L)_+ and m_{j,L} = ||v_{j,L}||_2^2. Strong L^2
convergence gives lim_{L} sup_j m_{j,L} = 0. The energies of v_{j,L}
and |u_j| wedge L sum to ||D u_j||_2^2, so lower semicontinuity for the
clipped functions gives
lim_{L} limsup_j ||D v_{j,L}||_2^2 <= E_* - ||Du||_2^2.
For a nonnegative v with squared norm m <= 1, the negative part of
v^2 log v^2 has integral at most C sqrt(m) (use -z log z <= C sqrt(z) on
0 <= z <= 1), and m log m <= 0, so the defective inequality gives
int v^2 log_+ v^2 dnu <= 2(s+eps)||Dv||_2^2 + C_eps m + C sqrt(m).
If |u_j| > M L with M > 1 fixed then |u_j| <= c_M v_{j,L} with
c_M = M/(M-1). Passing bounded truncations of z^2 log z^2 to the
limit by strong L^2 convergence, and the negative part likewise,
assembles the lost positive entropy as at most 2s(E_* - ||Du||_2^2).
The normalization terms vanish because ||u||_2 = ||u_j||_2 = 1. QED.
The coefficient 2s is where "escape into the Gaussian tails costs
entropy at rate at most 2s" is made precise. The next subsection uses
this defect to prove that a maximizer of the entropy quotient exists.
6.5 Existence of an optimizer (strict branch).
The strict branch is the case where the entropy constant strictly beats
both the Poincare constant and the Gaussian-regularization scale. There
the entropy quotient is attained.
Lemma 9 (Existence of an optimizer). Let nu = mu * gamma_s, with s > 0
and mu compactly supported. Suppose R = R(nu) and P = R^P(nu) satisfy
infinity > R > max(P, s). Then there is a smooth strictly positive
nonconstant f with
int f^2 dnu = 1, Ent_nu(f^2) = 2 R int |Df|^2 dnu,
and f satisfies the logarithmic Euler equation
A_nu f = lambda f log f, lambda = R^{-1},
first distributionally and then classically.
Proof of existence and the Euler equation. Take nonnegative unit-norm
maximizing tests f_j. Replacing a test by its absolute value preserves
entropy and energy, so this loses nothing. The defective inequality of
Lemma 8, applied with s + eps < R, bounds the energies ||D f_j||_2.
First exclude energies tending to zero; this is the only place the
strict inequality R > P is used. Suppose eps_j^2 = ||D f_j||_2^2 -> 0,
and write c_j = E_nu f_j, h_j = (f_j - c_j)/eps_j. Poincare gives
||h_j||_2^2 <= P, while ||D h_j||_2^2 = 1 and 1 - c_j^2 = eps_j^2
||h_j||_2^2, so c_j -> 1. By compactness, after extracting, h_j -> h in
L^2 with E h = 0, ||Dh||_2^2 <= 1, and ||h||_2^2 <= P ||Dh||_2^2.
For c > 0, eps > 0 and z >= -c/eps define
Q_{c,eps}(z) = eps^{-2} [ (c+eps z)^2 log((c+eps z)^2/c^2)
- (c+eps z)^2 + c^2 ].
Normalization and -log c_j^2 - 1 + c_j^2 >= 0 give
eps_j^{-2} Ent(f_j^2) <= int Q_{c_j,eps_j}(h_j) dnu.
The scalar estimate we need: for 1/2 <= c <= 1 and 0 < eps <= 1/2,
0 <= Q_{c,eps}(z) <= (2+2log2) z^2 + z^2 log_+ z^2,
Q_{c,eps}(z) -> 2 z^2 uniformly on bounded z-intervals.
To verify, put t = eps z / c, so Q = z^2 B(t) with
B(t) = 2 + 4 int_0^1 (1-r) log(1 + r t) dr, -1 <= t < infinity. For
t <= 0, B(t) <= 2; for t >= 0, t <= z gives
B(t) <= 2 + 2 log(1+z) <= 2 + 2log2 + log_+ z^2.
Now use the general tail bound: the tail argument in Lemma 8 never
used unit norm until its last line, so for any bounded form sequence
u_j -> u in L^2 with energies -> E_*,
lim_{K} limsup_j int_{|u_j|>K} u_j^2 log_+ u_j^2 dnu
<= 2s (E_* - ||Du||_2^2).
Apply the compact convergence in the scalar estimate on |h_j| <= K and
the upper bound on the complement. Strong L^2 convergence removes the
quadratic tail as K -> infinity; the general tail bound controls the
entropy tail. Therefore
limsup_j eps_j^{-2} Ent(f_j^2)
<= 2||h||_2^2 + 2s(1 - ||Dh||_2^2) <= 2 max(P, s) < 2R,
contradicting the maximizing property. So energies do not vanish.
Hence we may extract f_j -> f strongly in L^2 with
||D f_j||_2^2 -> E_* > 0. Put E = ||Df||_2^2. The entropy defect and
the sharp inequality give
2R E_* <= Ent(f^2) + 2s(E_* - E) <= 2R E + 2s(E_* - E).
Since R > s and E <= E_*, equality forces E = E_* > 0. Thus f is a
nonnegative, nonconstant optimizer.
The homogeneous functional Ent(g^2) - 2R ||Dg||_2^2 is nonpositive and
is zero at f. Its derivative along every compact smooth variation zeta
is
4 int f log f zeta dnu - 4R int Df.Dzeta dnu = 0.
The entropy normalization term cancels the additional linear term, and
differentiation at zeros is legitimate because z^2 log z^2 has zero
derivative at zero. This is the Euler equation A_nu f = lambda f log f
in distributional form. Smoothness and strict positivity are handled
next. QED.
So in the strict branch we have a bona fide optimizer f, and w = D log
f^2 must be a nearly first eigenfunction. Turning the Euler equation
into that statement, and proving f is smooth and positive, is the rest
of the section.
6.6 Regularity and strict positivity of the optimizer.
We have a nonnegative optimizer satisfying the weak Euler equation. To
use its logarithmic derivative we need local regularity and strict
positivity.
Regularity. Write rho = exp(-V) for the smooth density of nu, set
v = exp(-V/2) f >= 0. Differentiating the weak equation directly gives
-Delta v = lambda v log v
+ ( lambda V/2 - |DV|^2/4 + Delta V/2 ) v.
The conjugating factor and its reciprocal are bounded on each fixed
ball. Since |z log z| <= C_delta (1 + z^{1+delta}) for z >= 0, local
L^p integrability of v puts the right side in L^q_loc with
q = p/(1+delta) > 1. Localizing the right side on a larger ball and
taking its Newtonian potential u solving -Delta u = F, the interior L^q
estimate gives u in W^{2,q}_loc; the difference v - u is
distributionally harmonic, hence smooth by Weyl lemma. Then Sobolev
embedding improves the integrability of v step by step: before
boundedness is reached, the reciprocal integrability exponent 1/p
improves to (1+delta)/p - 2/n, a gain. After finitely many steps q >
n/2, so v, hence f, is locally bounded and Holder continuous; the
local Schauder estimate then gives f in C^{2,alpha}_loc. In dimension
one the same conclusion follows by twice integrating. Every constant
here concerns one fixed law and dimension and never enters a uniform
estimate along the sequence.
Positivity. Suppose f(x_0) = 0. Choose a ball B about x_0 on which
0 <= f <= b < 1 on its closure. Start the nu-diffusion of Lemma 3 at
x_0 and stop it at the exit time tau from B. Put Z_t = f(Y_{t and tau}),
m(t) = E Z_t, g(z) = -lambda z log z. The stopped Ito formula gives,
for almost every t,
m'(t) = E[ 1_{t<tau} g(Z_t) ] <= E g(Z_t) <= g(m(t)), m(0) = 0,
because g is nonnegative and concave on [0,b] and g(0)=0. Also m' >= 0.
If m(t_1) > 0 with t_1 > 0, and t_a is the first time m = a, then for
0 < a < m(t_1),
int_a^{m(t_1)} dz/( -lambda z log z ) <= t_1 - t_a <= t_1,
contradicting the divergence of the integral as a -> 0. Hence m = 0.
The killed nondegenerate diffusion reaches every open subset of B with
positive probability (follow a smooth interior path and require the
Brownian motion to stay in a thin tube), so f vanishes on all of B, and
its zero set is both open and closed. Since R^n is connected and f is
nonzero, f must be strictly positive. Smooth elliptical bootstrap then
makes f smooth everywhere. QED.
6.7 Change of reference at an optimizer.
An optimizer supplies a second probability with its own LSI, and no
convexity of that second probability is needed.
Lemma 10 (Change of reference). Under Lemma 9, suppose in addition that
nu = exp(-V) is log-concave with bounded curvature K = D^2 V. Define
eta = f^2 nu, h = log f^2, w = D h, I = E_eta |w|^2.
Then I = 2 lambda H(eta | nu) > 0, R(eta) <= 1/lambda, each component
of w lies in the domain of A_eta, and
A_nu h = lambda h + (1/2)|w|^2, (i)
A_eta w = (lambda Id - K) w. (ii)
Moreover E_eta <w, K w> <= lambda |E_eta w|^2. (iii)
Proof. Since w = 2 Df/f, I = 4 ||Df||_{L^2(nu)}^2; optimizer equality
gives I = 2 lambda H(eta|nu) > 0. The chain rule applied to the Euler
equation gives (i). Commuting one derivative with A_nu gives
A_nu w = D(A_nu h) - K w = lambda w + (Dw) w - K w.
Since A_eta = A_nu - w.D and Dw is symmetric, (ii) follows
distributionally. Its right side is in L^2(eta) because K is bounded and
w in L^2(eta), so Lemma 3 puts w in the operator domain; no unproved
higher integrability has been used.
For the inequality, let p = g^2 eta have unit mass with g smooth compact.
Direct expansion and integration by parts give
Ical(p|nu) = Ical(p|eta) + int g^2 |w|^2 deta + 2 int D(g^2).w deta
= Ical(p|eta) + int g^2 (|w|^2 + 2 A_eta h) deta
= Ical(p|eta) + 2 lambda int h dp.
On the other hand H(p|nu) = H(p|eta) + int h dp. Subtracting
2 lambda times this gives the exact deficit identity
Ical(p|nu) - 2 lambda H(p|nu) = Ical(p|eta) - 2 lambda H(p|eta).
Apply the nu-inequality to the compact smooth function g f: its deficit
is nonnegative, so the right side is nonnegative. Homogeneity and
Sobolev approximation give R(eta) <= 1/lambda. Finally, the domain
identity gives E_eta|Dw|^2 = lambda I - E_eta<w,Kw>, while Poincare for
eta gives the lower bound lambda(I - |E_eta w|^2). Subtract to obtain
(iii). QED.
The lemma does two things at once. It transfers a logarithmic Sobolev
inequality from nu to the tilted law eta = f^2 nu (so eta inherits the
good entropy constant), and it writes the score w as an approximate
first eigenfunction of eta's diffusion. Everything after this point
happens under eta.
6.8 Translation gain: uniform control of means.
The critical law eta has a near-first-eigenfield w, but we also need to
know that its mean and its curvature-action are small. The tool is a
stability estimate for the LSI under translating a probability.
Lemma 11 (Translation gain). Let mu satisfy the normalized
counterexample conditions of Lemma 6 with a <= 1. Fix a compact
interval J inside (0, 1/2). There is a constant C_J, independent of the
dimension and of the support radius of mu, with the following property.
For s in J, put lambda = R_s^{-1}. If sigma is any probability with
D_0 = H(sigma | mu_s) < infinity, F_0 = Ical(sigma | mu_s) < infinity,
and p = E_sigma Y is its mean, then
F_0 - 2 lambda D_0 >= ( s^{-2} - lambda/s ) |p|^2
- C_J sqrt(a) ( F_0 + D_0 ).
Proof. The centered linear moment generating functions under mu_s obey
log E_{mu_s} exp(v.Y) <= C_J |v|^2,
log E_{mu_s} exp(v.b_s(Y)) <= C a^2 |v|^2,
the first by independent Gaussian addition and the moment bound of
Lemma 6, the second by conditional Jensen. Entropy duality, optimized
over the scalar multiple of each unit vector, gives
|p| <= C_J sqrt(D_0), |E_sigma b_s(Y)| <= C a sqrt(D_0).
Compact support of X makes E_{mu_s} exp(c|Y|^2) finite for some c > 0,
so all these expectations exist; by duality E_sigma |Y|^2 < infinity.
The latter is used only for integrability, not for a dimension bound.
Let sigma_u be the translate of sigma by -u p, 0 <= u <= 1, and let
D_u = H(sigma_u | mu_s). Change of variables in entropy and the
Hessian bound give
D_u = D_0 + E_sigma[ V_s(Y - up) - V_s(Y) ]
<= D_0 - u|p|^2/s + u p.E_sigma b_s(Y)/s + u^2 |p|^2/(2s)
<= C_J D_0.
Thus the mean estimate applied to sigma_u gives
|E_sigma b_s(Y - up)| <= C_J a sqrt(D_0) uniformly in u. Differentiating
the exact entropy expression and integrating gives
D_1 = D_0 - |p|^2/(2s) + O_J(a D_0).
Its derivative is -(1-u)|p|^2/s + p.E_sigma b_s(Y - up)/s.
Write r(Y) = D log( dsigma / dmu_s )(Y) in the weak square-root sense.
Its mean is
E_sigma r = E_sigma D V_s = ( p - E_sigma b_s ) / s.
To justify this for finite Fisher information, the Lebesgue density q of
sigma has weak derivative Dq = q(r - D V_s) in L^1: r is square
integrable, and D V_s has linear growth. Integrating against expanding
cutoffs gives int Dq = 0.
In moving coordinates Y, the score of sigma_1 relative to mu_s is
r(Y) - p/s + d(Y)/s with d(Y) = b_s(Y) - b_s(Y - p). Since
0 <= D b_s <= Id, integration along the segment gives
|d(Y)|^2 <= int_0^1 |M_s(Y - up) p|^2 du <= p.d(Y). The mean estimates
at the two endpoints therefore imply
E_sigma |d|^2 <= C_J a D_0.
Expanding the score square and using the mean identity,
E_sigma |r - p/s|^2 = F_0 - |p|^2/s^2 + 2 p.E_sigma b_s/s^2.
The remaining cross term is bounded by C_J sqrt(a D_0)(sqrt(F_0) +
sqrt(D_0)), and the d-square term by C_J a D_0. Therefore the translated
information is finite and
F_1 = F_0 - |p|^2/s^2 + O_J( sqrt(a) (F_0 + D_0) ).
These calculations use only the weak derivatives already justified, so
they apply when the density of sigma vanishes. Apply the LSI to sigma_1,
F_1 >= 2 lambda D_1, and combine with the two translated displays. Since
2/3 <= lambda <= 1 on J, the result is the claim. QED.
The lemma says: any probability close to mu_s in entropy and Fisher
information cannot have a large mean. A nonzero mean would force a
strictly positive energy surplus, at least (s^{-2} - lambda/s)|p|^2, and
that surplus is controlled up to sqrt(a) times the sizes involved. Since
a -> 0, means are forced to zero.