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CHAPTER 3 - MORE ABOUT THE HEAT EQUATION

In this chapter we present some qualitative properties of the heat equation and more particularly we present several results on the self-similar behavior of the solutions in large time. These results are deduced from several functional inequalities, among them the Nash inequality, the Poincar´e inequality and the Log-Sobolev inequality.

Let us emphasize that the used methods lie on an interplay between evolution PDEs and functional inequalities and, although we only deal with (simple) linear situations, these methods are robust enough to be generalized to (some) nonlinear situations.

Contents

1. The heat equation 1

1.1. Nash inequality and heat equation 1

1.2. Self-similar solutions and the Fokker-Planck equation 2

2. Fokker-Planck equation and Poincar´e inequality 3

2.1. Long time asymptotic behaviour of the solutions to the Fokker-Planck equation 3

2.2. Proof of the Poincar´e inequality 4

2.3. A strong version of the Poincar´e inequality 6

3. Fokker-Planck equation and Log Sobolev inequality. 7

3.1. Fisher information. 7

3.2. Entropy and Log-Sobolev inequality. 10

3.3. From log-Sobolev to Poincar´e. 11

4. WeightedL1 semigroup spectral gap 12

5. Exercises and Complements 14

6. Bibliographic discussion 15

References 15

1. The heat equation

1.1. Nash inequality and heat equation. We consider the heat equation

(1.1) ∂f

∂t = 1

2∆f in (0,∞)×Rd, f(0,·) =f0 in Rd. One can classically prove thanks to the representation formula

f(t, .) =γt∗f0, γt(x) := 1

(2πt)d/2 exp

−|x|2 2t

and the H¨older inequality thatf(t, .)→0 ast → ∞, and more precisely, that for anyp∈(1,∞]

and a constantCp,d the following rate of decay holds:

(1.2) kf(t, .)kLp ≤ Cp,d

td2(1−p1)

kf0kL1 ∀t >0.

We aim to give a second proof of (1.2) in the casep= 2 which is not based on the above repre- sentation formula, which is clearly longer and more complicated, but which is also more robust in the sense that it applies to more general equations, even sometimes nonlinear.

1

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Nash inequality. There exists a constant Cd such that for any f ∈ L1(Rd)∩H1(Rd), there holds

kfk1+2/dL2 ≤Cdkfk2/dL1 k∇fkL2. Proof of Nash inequality. We write for anyR >0

kfk2L2 = kfˆk2L2 = Z

|ξ|≤R

|fˆ|2+ Z

|ξ|≥R

|fˆ|2

≤ cdRdkfkˆ 2L+ 1 R2

Z

|ξ|≥R

|ξ|2|f|ˆ2

≤ cdRdkfk2L1+ 1

R2k∇fk2L2,

and we take the optimal choice forRby settingR:= (k∇fk2L2/cdkfk2L1)d+21 so that the two terms

at the RHS pf the last line are equal.

We assume for the sake of simplicity that f0 ≥0, and thenf(t, .)≥ 0, thanks to the maximum principle. We then compute

d

dtkf(t, .)kL1 = d dt

Z

Rd

f(t, x)dx=1 2

Z

Rd

divx

xf(t, x) dx= 0, so that the mass is conserved (by the flow of the heat equation)

kf(t, .)kL1 =kf0kL1 ∀t≥0.

On the other hand, there holds d dt

Z

Rd

f(t, x)2dx= Z

Rd

f∆f dx=− Z

Rd

|∇f|2dx.

Putting together that last equation, the Nash inequality and the mass conservation, we obtain the following ordinary differential inequality

d dt

Z

Rd

f(t, x)2dx≤ −KZ

Rd

f(t, x)2dxd+2d

, K=Cdkf0k−4/dL1 . We last observe that for any solution uof the ordinary differential inequality

u0 ≤ −K u1+α, α= 2/d >0, some elementary computations lead to the inequality

u−α(t)≥α K t+uα0 ≥α K t, from which we conclude that

Z

Rd

f2(t, x)dx≤C

kf0k4/dL1d/2

td/2 =Ckf0k2L1

td/2 . That is nothing but the announced estimate.

1.2. Self-similar solutions and the Fokker-Planck equation. It is in fact possible to describe in a more accurate way that the mere estimate (1.2) how the heat equation solutionf(t, .) converges to 0 as time goes on. In order to do so, the first step consists in looking for particular solutions to the heat equation that we will discover by identifying some good change of scaling. We thus look for a self-similar solution to (1.2), namely we look for a solution F with particular form

F(t, x) =tαG(tβx),

for some α, β∈Rand a“self-similar profile”G. AsF must be mass conserving, we have Z

Rd

F(t, x)dx= Z

Rd

F(0, x)dx=tα Z

Rd

G(tβx)dx,

and we get from that the first equation α=β d. On the other hand, we easily compute

tF=α tα−1G(tβx) +β tα−1(tβx)·(∇G)(tβx), ∆F =tαt(∆G)(tβx).

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In order that (1.1) is satisfied, we have to take 2β+ 1 = 0. We conclude with (1.3) F(t, x) =t−d/2G(t−1/2x), 1

2∆G+1

2div(x G) = 0.

We observe (and that is not a surprise!) that a solutionG∈L1(Rd)∩P(Rd) to (1.3) will satisfy

∇G+x G= 0, it is thus unique and given by

G(x) :=c0e−|x|2/2, c−10 = (2π)d/2 (normalized Gaussian function).

To sum up, we have proved that F is our favorite solution to the heat equation: that is the fundamental solution to the heat equation.

Changing of point view, we may now considerGas a stationary solution to the harmonic Fokker- Planck equation (sometimes also called the Ornstein-Uhlenbeck equation)

(1.4) ∂

∂tg=1

2L g=1

2∇ ·(∇g+g x) in (0,∞)×Rd.

The link between the heat equation (1.1) and the Fokker-Planck equation (1.4) is as follows. Iff is a solution to the Fokker-Planck equation (1.4), some elementary computations permit to show that

f(t, x) = (1 +t)−d/2g(log(1 +t),(1 +t)−1/2x)

is a solution to the heat equation (1.1), with f(0, x) =g(0, x). Reciprocally, iff is a solution to the heat equation (1.1) then

g(t, x) :=ed t/2f(et−1, et/2x)

solves the Fokker-Planck equation (1.4). The last expression also gives the existence of a solutionin the sense of distributionsto the Fokker-Planck equation (1.4) for any initial datumf0=ϕ∈L1(Rd) as soon as we know the existence of a solution to the heat equation for the same initial datum (what we get thanks to the usual representation formula for instance).

2. Fokker-Planck equation and Poincar´e inequality

2.1. Long time asymptotic behaviour of the solutions to the Fokker-Planck equation.

We consider the Fokker-Planck equation

∂tf =L f = ∆f +∇ ·(f∇V) in (0,∞)×Rd (2.1)

f(0, x) =f0(x) =ϕ(x), (2.2)

and we assume that the“confinement potential”V is the harmonic potential V(x) := |x|2

2 +V0, V0:= d 2 log 2π.

We start observing that d dt

Z

Rd

f(t, x)dx= d dt

Z

Rd

x·(∇xf+f∇xV)dx= 0,

so that the mass (of the solution) is conserved. Moreover, the functionG=e−V ∈L1(Rd)∩P(Rd) is nothing but the normalized Gaussian function, and since ∇G = −G∇V, it is a stationary solution to the Fokker-Planck equation (2.1).

Theorem 2.1. Let us fixϕ∈Lp(Rd),1≤p <∞.

(1) There exists a unique global solutionf ∈C([0,∞);Lp(Rd))to the Fokker-Planck equation(2.1).

This solution is mass conservative

(2.3) hf(t, .)i:=

Z

Rd

f(t, x)dx= Z

Rd

f0(x)dx=:hf0i, if f0∈L1(Rd), and the following maximum principle holds

f0≥0 ⇒ f(t, .)≥0 ∀t≥0.

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(2) Asymptotically in large time the solution converges to the unique stationary solution with same mass, namely

(2.4) kf(t, .)− hf0iGkE≤e−λPtkf0− hf0iGkE as t→ ∞, wherek · kE stands for the norm of the Hilbert spaceE:=L2(G−1/2) defined by

kfk2E :=

Z

Rd

f2G−1dx

andλP is the best (larger) constant in the Poincar´e inequality.

More generally, for any weight function m: Rd → R+, we denote byLp(m) the Lebesgue space associated to the norm kfkLp(m):=kf mkLp and we will just writeLpk:=Lp(hxik).

For the proof of point (1) we refer to Chapter 1 as well as the final remark of Section 1. We are going to give the main lines of the proof of point 2. Because the equation is linear, we may assume in the sequel thathf0i= 0.

Using that G G−1 = 1, we deduce that∇V =−G−1∇G=G· ∇(G−1). We can then write the Fokker-Planck equation in the equivalent form

∂tf = divxxf +G f∇xG−1

= divx G∇x(f G−1) . We then compute

1 2

d dt

Z

f2G−1 = Z

Rd

(∂tf)f G−1dx= Z

Rd

divx

G∇x

f G

f Gdx

= −

Z

Rd

G

x

f G

2

dx.

Using the Poincar´e inequality established in the next Theorem 2.2 with the choice of function g:=f(t, .)/Gand observing thathgiG= 0, we obtain

1 2

d dt

Z

f2G−1≤ −λP

Z

Rd

G f

G 2

dx=−λP

Z

Rd

f2G−1dx, and we conclude using the Gronwall lemma.

Theorem 2.2 (Poincar´e inequality). There exists a constant λP >0 (which only depends on the dimension) such that for anyg∈L2(G1/2), there holds

(2.5)

Z

Rd

|∇g|2G dx≥λP

Z

Rd

|g− hgiG|2G dx, where we have defined

hgiµ:=

Z

Rd

g(x)µ(dx)

for any given (probability) measure µ∈P(Rd) and any functiong∈L1(µ).

2.2. Proof of the Poincar´e inequality. We split the proof into three steps.

2.2.1. Poincar´e-Wirtinger inequality (in an open and bounded set Ω).

Lemma 2.3. Let us denote Ω = BR the ball of Rd with center 0 and radius R >0, and let us consider ν ∈ P(Ω) a probability measure such that (abusing notations) ν,1/ν ∈ L(Ω). There exists a constant κ∈(0,∞), such that for any (smooth) function f, there holds

(2.6) κ

Z

|f− hfiν|2ν ≤ Z

|∇f|2ν, hfiν :=

Z

f ν,

and therefore

Z

f2ν≤ hfi2ν+ 1 κ

Z

|∇f|2ν.

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Proof of Lemma 2.3. We start with f(x)−f(y) =

Z 1

0

∇f(zt)·(x−y)dt, zt= (1−t)x+t y.

Multiplying that identity byν(y) and integrating in the variabley∈Ω the resulting equation, we get

f(x)− hfiν = Z

Z 1

0

∇f(zt)·(x−y)dt ν(y)dy.

Using the Cauchy-Schwarz inequality, the fact that zt ∈[x, y]⊂ Ω and the changes of variables (x, y)7→(z, y) and (x, y)7→(x, z), we deduce

Z

(f(x)− hfiν)2ν(x)dx≤ Z

Z

Z 1

0

|∇f(zt)|2|x−y|2dt ν(y)ν(x)dydx

≤C1 Z

Z

Z 1/2

0

|∇f(zt)|2dtdx ν(y)dy+C1 Z

Z

Z 1

1/2

|∇f(zt)|2dtdy ν(x)dx

=C1

Z

Z 1/2

0

Z

|∇f(z)|2 dz

(1−t)ddt ν(y)dy+C1

Z

Z 1

1/2

Z

|∇f(z)|2dz

td dt ν(x)dx

≤2C1

Z

|∇f(z)|2dz,

withC1:=kνkLdiam(Ω)2. We immediately deduce the Poincar´e-Wirtinger inequality with the

constantκ−1:= 2C1k1/νkL.

2.2.2. A Liapunov function. There exists a function W such that W ≥ 1 and there exist some constantsθ >0,b, R≥0 such that

(2.7) (LW)(x) := ∆W(x)− ∇V · ∇W(x)≤ −θ W(x) +b1BR(x), ∀x∈Rd,

whereBR=B(0, R) denotes the centered ball of radiusR. The proof is elementary. We look for W asW(x) :=eγhxi. We then compute

∇W =γ x

hxieγhxi and ∆W =

γ2+γd−1 hxi

eγhxi,

and then

LW = ∆W −x· ∇W = γd−1 hxi W+

γ2−γ|x|2 hxi

W

≤ −θ W+b1BR

with the choiceθ=γ= 1 and thenR andblarge enough.

2.2.3. End of the proof of the Poincar´e inequality. We write (2.7) as 1≤ −LW(x)

θ W(x) + b

θ W(x)1BR(x) ∀x∈Rd. For anyf ∈Cb2(Rd), we deduce

Z

Rd

f2G ≤ − Z

Rd

f2LW(x) θ W(x) G+b

θ Z

BR

f2 1

W G =: T1+T2.

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On the one hand, we have θ T1 =

Z

∇W ·n

∇ f2

W

G+f2 W ∇Go

+ Z f2

W ∇V · ∇W G

= Z

∇W · ∇ f2

W

G

= Z

2 f

W ∇W· ∇f G− Z f2

W2|∇W|2G

= Z

|∇f|2G− Z

f

W ∇W− ∇f

2

G

≤ Z

|∇f|2G.

On the other hand, using the Poincar´e-Wirtinger inequality inBR and the notation G(BR) :=

Z

BR

G dx, νR:=G(BR)−1G|BR, hfiR= Z

BR

f νR, we have

θ bT2 =

Z

BR

f2 1

W G≤G(BR) Z

BR

f2νR

≤ G(BR)

hfi2R+CR Z

BR

|∇f|2νR .

Gathering the two above estimates, we have shown (2.8)

Z

Rd

f2G≤C hfi2R+

Z

Rd

|∇f|2G .

Consider nowg∈Cb2. We know that for anyc∈R, there holds (2.9)

Z

Rd

(g− hgiG)2G≤φ(c) :=

Z

Rd

(g−c)2G,

with hgiG defined in (2.6), because φis a polynomial function of second degree which reaches is minimum value in cg :=hgiG. We last define f :=g− hgiR, so that hfiR= 0, ∇f =∇g. Using first (2.9) and next (2.8), we obtain

Z

Rd

(g− hgiG)2G ≤ Z

Rd

f2G

≤ C hfi2R+

Z

Rd

|∇f|2G

= C

Z

Rd

|∇g|2G.

That ends the proof of the Poincar´e inequality (2.5).

2.3. A strong version of the Poincar´e inequality.

Proposition 2.4. For any λ < λP, there existsε >0 so that the following stronger version Z

Rd

∇ f

G

2

Gdx ≥ λ Z

Rd

f2G−1dx +ε

Z

Rd

f2|x|2+|∇f|2 G−1dx holds for any f ∈ D(Rd)withhfi= 0.

Proof of Proposition 2.4. We define Φ := −logG= |x|2/2 + log(2π)d/2. On the one hand, by developing the LHS term, we find

T :=

Z

Rd

∇ f

G

2

Gdx= Z

Rd

|∇f|2G−1dx− Z

Rd

f2(∆Φ)G−1dx.

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On the other hand, a similar computation leads to the following identity

T =

Z

Rd

∇(f G−1/2)G1/2+ (f G−1/2)∇G1/2

2

Gdx

= Z

Rd

∇(f G−1/2)

2

dx+ Z

Rd

f2 1

4|∇Φ|2−1 2∆Φ

G−1dx.

The two above identities together with (2.5) imply that for anyθ∈(0,1) T ≥ (1−θ)λP

Z

Rd

f2G−1dx+θ Z

Rd

f2 1

16|∇Φ|2−3 4∆Φ

G−1dx +θ

16 Z

Rd

f2|∇Φ|2G−1dx+θ 2

Z

Rd

|∇f|2 G−1dx.

Observe that |∇Φ|2−12∆Φ ≥0 for xlarge enough, and we can choose θ > 0 small enough to

conclude the proof.

3. Fokker-Planck equation and Log Sobolev inequality.

The estimate (2.4) gives a satisfactory (optimal) answer to the convergence to the equilibrium issue for the Fokker-Planck equation (2.1). However, we may formulate two criticisms. The proof is “completely linear” (in the sense that it can not be generalized to a nonlinear equation) and the considered initial data are very confined/localized (in the sense that they belong to the strong weighted spaceE, and again that it is not always compatible with the well posedness theory for nonlinear equations).

We present now a series of results which apply to more general initial data but, above all, which can be adapted to nonlinear equations. On the way, we will establish several functional inequalities of their own interest, among them the famous Log-Sobolev (or logarithmic Sobolev) inequality.

3.1. Fisher information. We are still interested in the harmonic Fokker-Planck equation (2.1)- (2.2). We define

D:=

f ∈L1(Rd); f ≥0, Z

f = 1, Z

f x= 0, Z

f|x|2=d

and

D :=

f ∈L1(Rd); f ≥0, Z

f = 1, Z

f x= 0, Z

f|x|2≤d

.

We observe thatD (andD) are invariant set for the flow of Fokker-Planck equation (2.1). We also observe thatG is the unique stationary solution which belongs toD. Indeed, the equations for the first moments are

thfi= 0, ∂thf xi=−hf xi, ∂thf|x|2i= 2dhfi −2hf|x|2i.

It is therefore quite natural to think that any solution to the Fokker-Planck equation (2.1)-(2.2) with initial datumϕ∈D converges toG. It is what we will establish in the next paragraphs.

We define the Fisher information (or Linnik functional) I(f) and the relative Fisher information by

I(f) =

Z |∇f|2 f = 4

Z

|∇p

f|2, I(f|G) =I(f)−I(G) =I(f)−d.

Lemma 3.1. For anyf ∈D, there holds

(3.1) I(f|G)≥0,

with equality if, and only if, f =G.

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Proof of Lemma 3.1. We defineV :={f ∈D and∇√

f ∈L2}. We start with the proof of (3.1).

For anyf ∈V, we have 0≤J(f) :=

Z 2∇p

f +xp f

2

dx

= Z

4|∇p

f|2+ 2x· ∇f+|x|2f

dx=I(f) +hf|x|2i −2d

≤ I(f)−d=I(f)−I(G) =I(f|G).

We consider now the case of equality. IfI(f|G) = 0 thenJ(f) = 0 and 2∇√ f+x√

f = 0 a.e.. By a bootstrap argument, using Sobolev inequality, we deduce that√

f ∈C0. Considerx0∈Rd such thatf(x0)>0 (which exists becausef ∈V) and thenOthe open and connected tox0 component of the set {f >0}. We deduce from the preceding identity that∇(log√

f+|x|2/4) = 0 inOand thenf(x) =eC−|x|2/2 onOfor some constantC∈R. By continuity off, we deduce thatO=Rd, and thenC=−log(2π)d/2(because of the normalized condition imposed by the fact thatf ∈V).

Lemma 3.2. For any (smooth) function f, we have 1

2I0(f)·∆f =−X

ij

Z 1

f2if ∂jf−1 f∂ijf2

f, (3.2)

1

2I0(f)·(∇ ·(f x)) =I(f), (3.3)

1

2I0(f)·L f =−X

ij

Z 1

f2if ∂jf− 1

f∂ijf−δij

2 f−

I(f)−I(G) . (3.4)

As a consequence, there holds 1

2I0(f)·L(f)≤ −I(f|G)≤0.

Proof of Lemma 3.2. Proof of (3.2). First, we have I0(f)·h= 2

Z ∇f f ∇h−

Z |∇f|2 f2 h.

Integrating by part with respect to thexi variable, we get 1

2I0(f)·∆f = Z 1

f∂jf ∂iijf − Z 1

2f2iif(∂jf)2

=

Z ∂if

f2jf ∂ijf− 1

f∂ijf ∂ijf

+ Z 1

f2if ∂jf ∂ijf −∂if

f3if(∂jf)2

= −X

ij

Z 1

f2if ∂jf−1 f∂ijf2

f.

Proof of (3.3). We write 1

2I0(f)·(∇ ·(f x)) = Z ∂jf

f ∂ij(f xi)−(∂jf)2

2f2i(f xi).

We observe that

ij(f xi)−(∂jf)

2f ∂i(f xi) = (∂ijf)xi+d ∂jf +δijjf −∂if ∂jf xi

2f −d 2∂jf

= (∂ijf)xi+ (d

2+ 1)∂jf −∂if ∂jf xi 2f. Gathering the two preceding equalities, we obtain

1

2I0(f)·(∇ ·(f x)) = (d

2 + 1)I(f) + Z ∂jf

f ∂ijf xi− Z ∂jf

f ∂if ∂jf xi

2f. Last, we remark that

−d

2I(f) = 1 2

Z

i

(∂jf)2 f

xi=

Z ∂jf ∂ijf f xi−1

2 (∂jf)2

f2if xi,

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and we then conclude

1

2I0(f)·(∇ ·(f x)) =I(f).

Proof of (3.4). Developing the expression below and using (3.2), we have

0 ≤ X

ij

Z 1

f2if ∂jf−1

f∂ijf−δij

2

f

= −1

2I0(f)·∆f+ 2X

i

Z

iif− 1

f (∂if)2 +d

Z f.

FromR

f = 1,R

iif = 0 and (3.3), we then deduce 0 ≤ −1

2I0(f)·∆f−2I(f) +d=−1

2I0(f)·Lf +d−I(f),

which ends the proof of (3.4).

Theorem 3.3. The Fisher information I is decreasing along the flow of the Fokker-Planck equa- tion, i.e. I is a Liapunov functional, and more precisely

(3.5) I(f(t, .)|G)≤e−2tI(ϕ|G).

That implies the convergence in large time to G of any solution to the Fokker-Planck equation associated to any initial condition ϕ∈D∩V. More precisely,

(3.6) ∀ϕ∈D∩V f(t, .)→G in Lq∩L12 as t→ ∞, for anyq∈[1,2/2) where2= 2d/(d−2).

Proof of Theorem 3.3. On the one hand, thanks to (3.4), we have

(3.7) d

dtI(f|G)≤ −2I(f|G),

and we conclude to (3.5) thanks to the Gronwall lemma. On the other hand, thanks to the Sobolev inequality, we have

kfkL2/2 =kp

fk2L2 ≤Ck∇p

fk2L2 =C I(f)≤C I(ϕ).

Consider now an increasing sequence (tn) which converges to +∞. Thanks to estimate (3.5) and the Rellich Theorem, we may extract a subsequencep

f(tnk) which converges a.e. and strongly in L2q and weakly in ˙H1 to a limit denoted by√

g. That implies thatf(tnk) converges tog strongly inLq∩L1k for anyq∈[1,2/2), k∈[0,2), and thatI(g)≤lim supI(f(tnk))<∞, so that g∈V. Finally, since 2∇p

f(tnk)−xp

f(tnk)→2∇√ g−x√

gweakly inL2loc (for instance) we have 0≤J(g)≤lim inf

k→∞ J(f(tnk, .) = lim inf

k→∞ I(f(tnk, .)|G) = 0.

FromJ(g) = 0 andg ∈V ∩D we getg =Gas a consequence of Lemma 3.1, and it is then the all family (f(t))t≥0 which converges toGas t→ ∞. TheL12 convergence is a consequence of the fact that the sequence (f(tn)|v|2)n is tight because hf(t)|v|2i=hG|v|2ifor any timet≥0 . Exercise 3.4. Prove that0≤fn→f in Lq∩L1k,q >1,k >0, implies thatH(fn)→H(f).

(Hint. Use the splitting

s|logs| ≤√

s10≤s≤e−|x|k+s|x|k1e−|x|k≤s≤1+s(logs)+1s≥1 ∀s≥0 and the dominated convergence theorem).

Exercise 3.5. Prove the convergence (3.5)for any ϕ∈P(Rd)∩L12(Rd)such that I(ϕ)<∞.

(Hint. Compute the equations for the moments of order1 and2).

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3.2. Entropy and Log-Sobolev inequality. For a functionf ∈D, we define the entropyH(f)∈ R∪ {+∞}and the relative entropy H(f|G)∈R∪ {+∞} by

H(f) = Z

Rd

f logf dx, H(f|G) =H(f)−H(G) = Z

Rd

j(f /G)G dx, where j(s) =slogs−s+ 1.

We start observing that forf ∈P(Rd)∩ S(Rd), there holds H0(f)·L(f) :=

Z

Rd

(1 + logf) [∆f+∇ ·(x f)]

= −

Z

Rd

∇f· ∇logf− Z

Rd

x f· ∇logf

= −I(f) +dhfi=−I(f|G).

As a consequence, the entropy is a Liapunov functional for the Fokker-Planck equation and more precisely

(3.8) d

dtH(f) =−I(f|G)≤0.

Theorem 3.6. (Logarithmic Sobolev inequality). For any ϕ∈D, √

ϕ∈H˙1, the following Log-Sobolev inequality holds

(3.9) H(ϕ|G)≤ 1

2I(ϕ|G).

That one also writes equivalently as Z

Rd

f /Gln(f /G)G dx= Z

Rd

flnf− Z

Rd

GlnG≤ 1 2

Z

Rd

∇f∇f f −d or also as

Z

Rd

u2log(u2)G(dx)≤2 Z

Rd

|∇u|2G(dx).

For some applications, it is worth noticing that the constant in the Log-Sobolev inequality does not depend on the dimension, what it is not true for the Poincar´e inequality.

Proof of Theorem 3.6. On the one hand, from (3.6) (and more precisely the result of Exercise 3.4) and (3.8), we get

H(ϕ)−H(G) = lim

T→∞[H(ϕ)−H(fT)] = lim

T→∞

Z T

0

−d dtH(f)

dt

= lim

T→∞

Z T

0

[I(f|G)]dt.

From that identity and (3.7), we deduce H(ϕ)−H(G) ≤ lim

T→∞

Z T

0

−1 2

d dtI(f|G)

dt

= lim

T→∞

1

2[I(ϕ|G)−I(fT|G)] =1

2I(ϕ|G),

thanks to (3.5).

Lemma 3.7. (Csisz´ar-Kullback inequality). Considerµandν two probability measures such that ν=g µ for a given nonnegative measurable functiong. Then

(3.10) kµ−νk2V T :=kg−1k2L1(dµ)≤2 Z

glogg dµ.

Proof of Lemma 3.7. First proof. One easily checks (by differentiating three times both functions) that

∀u ≥0 3 (u−1)2≤(2u+ 4) (ulogu−u+ 1).

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Thanks to the Cauchy-Schwarz inequality one deduces Z

|g−1|dµ≤ s

1 3

Z

(2g+ 4)dµ s

Z

(glogg−g+ 1)dµ= s

2 Z

glogg dµ.

Second proof. Thanks to the Taylor-Laplace formula, there holds j(g) := glogg−g+ 1 =j(1) + (g−1)j0(1) + (g−1)2

Z 1

0

j00(1 +s(g−1)) (1−s)ds

= (g−1)2 Z 1

0

1−s 1 +s(g−1)ds.

Using Fubini theorem, we get H(g) :=

Z

(glogg−g+ 1)dµ= Z 1

0

(1−s)

Z (g−1)2

1 +s(g−1)dµ ds.

For any s ∈ [0,1], we use the Cauchy-Schwarz inequality and the fact that bothµ and g µ are probability measures in order to deduce

Z

|g−1|dµ 2

Z (g−1)2 1 +s(g−1)dµ

Z

[1 +s(g−1)]dµ

=

Z (g−1)2 1 +s(g−1)dµ.

As a conclusion, we obtain H(g)≥

Z 1

0

Z

|g−1|dµ 2

(1−s)ds= 1 2

Z

|g−1|dµ 2

,

which ends the proof of the Csisz´ar-Kullback inequality.

Putting together (3.8), (3.9) and (3.10), we immediately obtain the following convergence result.

Theorem 3.8. For anyϕ∈Dsuch thatH(ϕ)<∞the associated solutionf to the Fokker-Planck equation (2.1)-(2.2)satisfies

H(f|G)≤e−2tH(ϕ|G), and then

kf−GkL1 ≤√

2e−tH(ϕ|G)1/2.

3.3. From log-Sobolev to Poincar´e.

Lemma 3.9. If the log-Sobolev inequality λ H(f|G)≤ 1

2I(f|G) ∀f ∈D holds for some constant λ >0, then the Poincar´e inequality

(λ+d)khk2L2(G−1/2)≤ Z

|∇h|2G−1 ∀h∈ D(Rd), hh[1, x,|x|2]i= 0, also holds (for the same constantλ >0).

That lemma gives an alternative proof of the Poincar´e inequality. Of course that proof is not very

“cheap” in the sense that one needs to prove first the log-Sobolev inequality which is somewhat more difficult to prove than the Poincar´e inequality. Moreover, the log-Sobolev inequality is known to be true under more restrictive assumption on the confinement potential than the Poincar´e inequality.

However, that allows to compare the constants involved in the two inequalities and the proof is robust enough so that it can be adapted to nonlinear situations.

Proof of Lemma 3.9. Consider h∈ D(Rd) such that R

h(x) [1, x,|x|2]dv= [0,0,0]. Applying the Log-Sobolev inequality to the functionf =G+ε h∈D forε >0 small enough, we have

λH(G+ε h)−H(G)

ε2 = λ

ε2H(f|G)≤ 1

2I(f|G) = I(G+ε h)−I(G)

2 .

Expending up to order 2 the two functionals, we have

flogf =GlogG+ε h(1 + logG) +ε2 2

h2

G +O(ε3),

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|∇f|2

f = |∇G|2 G +ε

2∇G

G · ∇h−|∇G|2 G2 h

2

2

|∇h|2

G −2h∇G

G2 · ∇h+|∇G|2 G3 h2

+O(ε3).

Passing now to the limitε→0 in the first inequality and using that the zero and first order terms vanish because (performing one integration by parts)

H0(G)·h= Z

Rd

(logG+ 1)h= 0, I0(G)·h=

Z

Rd

n|∇G|2

G2 −2∆G G

oh= 0, we get

λ H00(G)·(h, h)≤I00(G)·(h, h).

More explicitly, we have λ

Z h2 G ≤

Z |∇h|2

G +∇∇G G2

h2+|∇G|2 G3 h2

, and then

(λ+d) Z h2

G = Z h2

G

λ−∆G

G +|∇G|2 G2

Z |∇h|2 G ,

which is nothing but the Poincar´e inequality.

4. WeightedL1 semigroup spectral gap

In that last section, we establish that as a consequence of the Poincar´e inequality, the following weighted L1semigroup spectral gap estimate holds.

Theorem 4.1. For anya∈(−λP,0) and for anyk > k:=λP +d/2 there existsCk,a such that for any ϕ∈L1k, the associated solution f to the Fokker-Planck equation (2.1)-(2.2)satisfies

kf− hϕiGkL1

k≤Ck,aea tkϕ− hϕiGkL1 k.

A refined version of the proof below shows that the same estimate holds witha:=−λP and for any k > k∗∗:=λP.

Proof of Theorem 4.1. We introduce the splittingL=A+Bwith Bf := ∆f+∇ ·(f x)−M f χR, Af :=M f χR,

where χR(x) =χ(x/R), χ ∈ D(Rd), 0 ≤χ≤1,χ ≡1 onB1, and where R, M >0 are two real constants to be chosen later. We splits the proof into several steps.

Step 1. The operatorAis clearly bounded in any Lebesgue space and more precisely

∀f ∈Lp kAfkLp(G1/p)≤Cp,R,MkfkLp.

Step 2. For anyk, ε >0 and for anyM, R >0 large enough (which may depend onkandε) the operator Bis dissipative inL1k in the sense that

(4.1) ∀f ∈ D(Rd)

Z

Rd

(Bf) (signf)hxik ≤(ε−k)kfkL1 k.

We set β(s) = |s| (and more rigorously we must take a smooth version of that function) and m=hxik, and we compute

Z

(L f)β0(f)m = Z

(∆f+d f+x· ∇f)β0(f)m

= Z

{−∇f∇(β0(f)m) +d|f|m+m x· ∇|f|}

= −

Z

|∇f|2β00(f)m+ Z

|f| {∆m+d− ∇(x m)}

≤ Z

|f| {∆m−x· ∇m},

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where we have used thatβ is a convex function. Defining ψ := ∆m−x· ∇m−M χRm

= (k2|x|2hxi−4−k|x|2hxi−2−M χR)m

we easily see that we can chooseM, R >0 large enough such that ψ≤(ε−k)m and then (4.1) follows.

Step 3. Fix nowk > k. For anya∈(−λP,0), there holds (4.2) ∀ϕ∈ D(Rd) keBtϕkL2

k≤ Ca,k

td/2 ea tkϕkL1 k

A similar computation as in step 2 shows Z

(Bf)f m2 = − Z

|∇(f m)|2+ Z

|f|2n|∇m|2 m2 +d

2−x· ∇m−M χR

o m2

= −

Z

|∇(f m)|2+ (d

2 +ε−k) Z

|f|2m2,

forM, R >0 chosen large enough. Denoting byf(t) =SB(t)ϕ=eBtϕthe solution to the evolution PDE

tf =Bf, f(0) =ϕ, we (formally) have

1 2

d dt

Z

f2m2= Z

(Bf)f m2≤ − Z

|∇(f m)|2+a Z

|f|2m2,

from which (4.2) follows by using the Nash inequality similarly as in the proof of estimate (1.2) in section 1.1.

Step 4. For anyk > k anda∈(−λP,0), there holds

(4.3) k(ASB)(∗n)ϕkL2(G−1/2)≤Ck,n,aeatkϕkL1

k ∀t≥0,

forn=d+1 for instance. We just establish (4.3) whend= 1. We denoteE:=L1k,E:=L2(G−1/2).

Observing that

kASB(t)kE→E ≤ C1

t1/2ea0t and kASB(t)kE→E ≤C2ea0t, we compute

k(ASB)(∗2)kE→E ≤ Z t

0

kASB(t−s)kE→EkASB(s)kE→Eds

≤ ea0t Z t

0

C1

(t−s)1/2C2ds

= ea0tC1C2t1/2 Z 1

0

du u1/2, from which we immediately conclude by takinga0 ∈(−λP, a).

Step 5. We define in both spacesE andE the projection operator Πf :=hfiG.

We denote byL the differential Fokker-Planck operator inE and still by Lthe same operator in E. We also denote by SL and SL the associated semigroups. Since G∈ E ⊂ E is a stationary solution to the Fokker-Planck equation and the mass is preserved by the associated flow, we have SL(I−Π) = (I−Π)SL as well as

(4.4) kSL(t)(I−Π)kE→E=k(I−Π)SL(t)kE→E≤e−λPt ∀t≥0, which is nothing but (2.4). Now, we decompose the semigroup on invariant spaces

SL= ΠSL+ (I−Π)SL(I−Π)

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and by iterating once the Duhamel formula SL(t) = SB(t) +

Z t

0

SL(t−s)ASB(s)ds

= SB(t) +SL∗ ASB(t), we have

SL=SB+SB∗(ASB) + SL∗(ASB)(∗2). These two identities together, we have

SL = ΠSL+ (I−Π){SB+SB∗(ASB) + SL∗(ASB)(∗2)}(I−Π) or in other words

SL−ΠSL = (I−Π){SB+SB∗(ASB)}(I−Π) +{(I−Π)SL} ∗(ASB)(∗2)(I−Π).

We conclude by observing that the RHS in the above expression isO(eat) thanks to estimate (4.4)

and thanks to steps 2 and 4 above.

5. Exercises and Complements

Exercise 5.1. Give another proof of the Nash inequality by using the Sobolev inequality in dimension d≥3. (Hint. Write the interpolation estimate

kfkL2≤ kfkθL1kfk1−θ

L2

and then use the Sobolev inequality).

We propose now a third proof based on the Poincar´e-Wirtinger inequality. We write kfk2L2 = (f, f−fr) + (f, fr), with fr(x) := 1

|B(x, r)|

Z

B(x,r)

f(y)dy.

We have

kfk2L2 ≤ kfkL2kf−frkL2+kfkL1kfrkL. On the one hand,

kfrkL ≤ C rdkfkL1. On the other hand,

kf−frk2L2 = Z

Rd

Cd

rd Z

B(x,r)

(f(y)−f(x))dy

2

dy

≤ Z

Rd

Z

Rd

1|y−x|≤r|f(y)−f(x)|2dxdy

≤ r2 Z 1

0

Z

Rd

Z

Rd

1|y−x|≤r|∇f((1−t)x+ty)|2dxdydt

≤ r2 Z 1/2

0

Z

Rd

Z

Rd

1|y−x|≤r|∇f((1−t)x+ty)|2dxdydt

+r2 Z 1

1/2

Z

Rd

Z

Rd

1|y−x|≤r|∇f((1−t)x+ty)|2dxdydt

≤ r2 Z 1/2

0

Z

Rd

Z

Rd

1|y−x|≤r|∇f(z)|2dzdydt +r2

Z 1

1/2

Z

Rd

Z

Rd

1|y−x|≤r|∇f(z)|2dxdzdt

≤ r2C Z

Rd

|∇f(z)|2dz.

All together, we get

kfk2L2 ≤ C1rkfkL2k∇fkL2+C2r−dkfk2L1

≤ 1

2kfk2L2+C1

2 r2k∇fk2L2+C2r−dkfk2L1

and we obtain the Nash inequality by choosingr:= (kfk2L1/k∇fk2L2)1/(d+2).

Exercise 5.2. Establish (2.7)in the following situations:

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(i) V(x) :=hxiα withα≥1;

(ii) there existα >0andR≥0such that

x· ∇V(x)≥α ∀x /∈BR; (iii) there exista∈(0,1),c >0andR≥0such that

a|∇V(x)|2−∆V(x)≥c ∀x /∈BR;

(iv) V is convex (or it is a compact supported perturbation of a convex function) and satisfiese−V ∈ L1(Rd).

Exercise 5.3. Generalize the Poincar´e inequality to a general superlinear potentialV(x) =hxiα/α+V0, α≥1, in the following strong (weighted) formulation

Z

|∇g|2G ≥κ Z

|g− hgiG|2(1 +|∇V|2)G ∀g∈ D(Rd),

where we have definedG:=e−V ∈P(Rd)(for an appropriate choice ofV0∈R).

Exercise 5.4. Generalize Theorem 3.6 and Theorem 3.8 to the case of a super-harmonic potentialV(x) = hxiα/α,α≥2, and to an initial datumϕ∈P(Rd)∩L12(Rd)such thatH(ϕ)<∞.

6. Bibliographic discussion

The Nash inequality and its application to the heat equation is due to Nash [12]. The Poincar´e inequality for gaussian measure can be proved thanks to the help to Hermit polynomial and it is quite hold (see again [12] for instance). The proof we present here is based on the use of Lyapunov fiunction and it is picked up from [2]. The strong version of the Poincar´e inequality belongs to folklore. The logarithmic Sobolev inequality is due to Stam [15], Blachman [4] and rediscoved by Gross [8]. It is related to the hypercontractivity property of Nelson [13] and the Γ2 calculus of Bakry and Emery [3]. We follow here the presentation given by Toscani [16]. The Csisz´ar-Kullback inequality is due to Kullback [10], Pinsker [14]

and Csisz´ar [5]. The proofs we present here are picked up (first proof) from [1] and (second proof) from some notes I read from C. Villani. The fact that the log-Sobolev implies the Poincar´e inequality (as stated in Proposition 2.4) is due to Gross [8]. The weightedL1convergence presented in Section 4 are taken from recent results due to Gualdani, Mouhot and myself [9, 11]. See also [7] for related previous results. The third proof of the Nash inequality presented in Section 5 is due to Diaconis and Saloff-Coste [6].

References

[1] An´e, C., Blach`ere, S., Chafa¨ı, D., Foug`eres, P., Gentil, I., Malrieu, F., Roberto, C., and Scheffer, G.Sur les in´egalit´es de Sobolev logarithmiques, vol. 10 ofPanoramas et Synth`eses [Panoramas and Syntheses].

Soci´et´e Math´ematique de France, Paris, 2000. With a preface by Dominique Bakry and Michel Ledoux.

[2] Bakry, D., Barthe, F., Cattiaux, P., and Guillin, A.A simple proof of the Poincar´e inequality for a large class of probability measures including the log-concave case.Electron. Commun. Probab. 13 (2008), 60–66.

[3] Bakry, D., and ´Emery, M.Diffusions hypercontractives. Ineminaire de probabilit´es, XIX, 1983/84, vol. 1123 ofLecture Notes in Math.Springer, Berlin, 1985, pp. 177–206.

[4] Blachman, N. M.The convolution inequality for entropy powers. IEEE Trans. Information Theory IT-11 (1965), 267–271.

[5] Csisz´ar, I.A note on Jensen’s inequality.Studia Sci. Math. Hungar. 1 (1966), 185–188.

[6] Diaconis, P., and Saloff-Coste, L. Nash inequalities for finite Markov chains. J. Theoret. Probab. 9, 2 (1996), 459–510.

[7] Gallay, T., and Wayne, C. E.Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations onR2.Arch. Ration. Mech. Anal. 163, 3 (2002), 209–258.

[8] Gross, L.Logarithmic Sobolev inequalities.Amer. J. Math. 97, 4 (1975), 1061–1083.

[9] Gualdani, M. P., Mischler, S., and Mouthot, C.Factorization of non-symmetric operators and exponential H-Theorem. hal-00495786.

[10] Kullback, S.Information theory and statistics. John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1959.

[11] Mischler, S., and Mouthot, C.Exponential stability of slowly decaying solutions to the kinetic-fokker-planck equation. work in progress.

[12] Nash, J.Continuity of solutions of parabolic and elliptic equations.Amer. J. Math. 80 (1958), 931–954.

[13] Nelson, E.The free Markoff field.J. Functional Analysis 12(1973), 211–227.

[14] Pinsker, M. S.Information and information stability of random variables and processes. Translated and edited by Amiel Feinstein. Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964.

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[15] Stam, A. J. Some inequalities satisfied by the quantities of information of Fisher and Shannon.Information and Control 2 (1959), 101–112.

[16] Toscani, G.Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation.

Quart. Appl. Math. 57, 3 (1999), 521–541.

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