Long time asymptotic behaviour of the solutions to the Fokker-Planck equation 4 2.2

CHAPTER 4 - 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 3

2. Fokker-Planck equation and Poincar´e inequality 4

2.1. Long time asymptotic behaviour of the solutions to the Fokker-Planck equation 4

2.2. Proof of the Poincar´e inequality 5

2.3. A strong version of the Poincar´e inequality 7

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

3.1. Fisher information. 8

3.2. Entropy and Log-Sobolev inequality. 10

3.3. From log-Sobolev to Poincar´e. 12

4. WeightedL¹ semigroup spectral gap 13

5. Coming back to local in time estimates 15

6. Exercises and Complements 16

7. Bibliographic discussion 17

References 17

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,∞)×R^d, f(0,·) =f₀ in R^d. 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 constantC_p,d the following rate of decay holds:

(1.2) kf(t, .)kL^p ≤ Cp,d

t^d2(1−p1)

kf0k_L1 ∀t >0.

1

We aim to give a second proof of (1.2) in the case p= 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.

Nash inequality. There exists a constant C_d such that for any f ∈ L¹(R^d)∩H¹(R^d), there holds

(1.3) kfk^1+2/d_L2 ≤C_dkfk^2/d_L1 k∇fk_L2. Proof of Nash inequality. We write for anyR >0

kfk²_L2 = kfˆk²_L2 = Z

|ξ|≤R

|fˆ|²+ Z

|ξ|≥R

|fˆ|²

≤ cdR^dkfkˆ ²_L∞+ 1 R²

Z

|ξ|≥R

|ξ|²|f|ˆ²

≤ cdR^dkfk²_L1+ 1

R²k∇fk²_L2,

and we take the optimal choice forRby settingR:= (k∇fk²_L2/cdkfk²_L1)^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, .)k_L1 = d dt

Z

R^d

f(t, x)dx=1 2

Z

R^d

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

R^d

f(t, x)²dx= Z

R^d

f∆f dx=− Z

R^d

|∇f|²dx.

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

d dt

Z

R^d

f(t, x)²dx≤ −KZ

R^d

f(t, x)²dx^d+2_d

, K=Cdkf0k^−4/d_L1 . We last observe that for any solution uof the ordinary differential inequality

u⁰ ≤ −K u^1+α, α= 2/d >0, some elementary computations lead to the inequality

u^−α(t)≥α K t+u^α₀ ≥α K t, from which we conclude that

(1.4)

Z

R^d

f²(t, x)dx≤C

kf0k^4/d_L1

d/2

t^d/2 =Ckf0k²_L1

t^d/2 . That is nothing but the announced estimate (1.2) for p= 2.

In order to prove the estimate for the full range of exponent p∈(1,∞] we use a duality and an interpolation arguments as follow. We introduce the heat semigroupS(f)f0=f(t) associated to the heat equation as well as the dual semigroupS^∗(t). We clearly haveS^∗=S because the Laplacian opeartor is symetric in L²(R^d). As a consequence, thanks to (1.4) and for any f0∈L²(R^d), there holds

kS(t)f0kL^∞ = sup

φ∈B_L1

hS(t)f0, φi=supφ∈B_L1hf0, S(t)φi

≤ sup

φ∈B_L1

kf0kL²kS(t)φkL² ≤ kf0kL²

C t^d/4,

which exacly means that S(t) :L² →L^∞ for positive times with norm bounded by C t^−d/4. We deduce

kS(t)k_L1→L^∞ ≤ kS(t/2)k_L2→L^∞kS(t/2)k_L1→L² ≤ C t^d/2,

which establishes (1.2) forp=∞. Finally, for anyp∈(1,∞) and using the interpolation inequality kS(t)f₀k_Lp≤ kS(t)f₀k^θ_L1kS(t)f₀k^1−θ_L∞ ≤ kS(t)k^1−θ_L1→L^∞kf₀k_L1 ∀t >0,

withθ= 1/p, and that is nothing but (1.2) in the general case.

It is worth emphasasing that by differentiating the heat equation, we can easily establish some estimates on its smoothing effect. For example, for f₀ ∈H¹(R^d), the associated solution to the heat equation satisfies

∂_tf =1

2∆f and ∂_t∇f = 1 2∆∇f from what we deduce

d

dtkfk²_L2 =−k∇fk²_L2 and d

dtk∇fk²_L2=−kD²fk²_L2

and then

d dt

kfk²_L2+tk∇fk²_L2 =−tkD²fk²_L2≤0, ∀t >0.

Integrating in time this differential inequality, we readilly obtain that the solution to heat equation satifies

k∇f(t)kL² ≤ 1

t^1/2kf0kL², ∀t >0.

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 solutionF with particular form

F(t, x) =t^αG(t^βx),

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

R^d

F(t, x)dx= Z

R^d

F(0, x)dx=t^α Z

R^d

G(t^βx)dx,

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

∂tF =α t^α−1G(t^βx) +β t^α−1(t^βx)·(∇G)(t^βx), ∆F =t^αt^2β(∆G)(t^βx).

In order that (1.1) is satisfied, we have to take 2β+ 1 = 0. We conclude with (1.5) 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(R^d)∩P(R^d) to (1.5) 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)

(1.6) ∂

∂tg=1

2L g=1

2∇ ·(∇g+g x) in (0,∞)×R^d.

The link between the heat equation (1.1) and the Fokker-Planck equation (1.6) is as follows. Iff is a solution to the Fokker-Planck equation (1.6), 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) :=e^{d t/2}f(e^t−1, e^t/2x)

solves the Fokker-Planck equation (1.6). The last expression also gives the existence of a solutionin the sense of distributionsto the Fokker-Planck equation (1.6) for any initial datumf0=ϕ∈L¹(R^d) 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,∞)×R^d (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

R^d

f(t, x)dx= Z

R^d

∇x·(∇xf +f∇xV)dx= 0,

so that the mass (of the solution) is conserved. Moreover, the functionG=e^−V ∈L¹(R^d)∩P(R^d) 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∈L^p(R^d),1≤p <∞.

(1) There exists a unique global solutionf ∈C([0,∞);L^p(R^d))to the Fokker-Planck equation (2.1).

This solution is mass conservative (2.3) hf(t, .)i:=

Z

R^d

f(t, x)dx= Z

R^d

f0(x)dx=:hf0i, iff0∈L¹(R^d), and the following maximum principle holds

f₀≥0 ⇒ f(t, .)≥0 ∀t≥0.

(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

kfk²_E :=

Z

R^d

f²G⁻¹dx

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

More generally, for any weight function m: R^d → R+, we denote byL^p(m) the Lebesgue space associated to the norm kfk_Lp(m):=kf mk_Lp and we will just writeL^p_k:=L^p(hxi^k).

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 = divx ∇xf +G f∇xG⁻¹

= div_x G∇_x(f G⁻¹) .

We then compute 1 2

d dt

Z

f²G⁻¹ = Z

R^d

(∂_tf)f G⁻¹dx= Z

R^d

div_x

G∇_x f

G f

Gdx

= −

Z

R^d

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, .)/G and observing thathgiG= 0, we obtain

1 2

d dt

Z

f²G⁻¹≤ −λP

Z

R^d

G f

G 2

dx=−λP

Z

R^d

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

Theorem 2.2 (Poincar´e inequality). There exists a constantλP >0 (which only depends on the dimension) such that for anyg∈L²(G^1/2), there holds

(2.5)

Z

R^d

|∇g|²G dx≥λP

Z

R^d

|g− hgiG|²G dx, where we have defined

hgiµ :=

Z

R^d

g(x)µ(dx)

for any given (probability) measureµ∈P(R^d)and any function g∈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 setΩ).

Lemma 2.3. Let us denoteΩ = B_R the ball of R^d 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 constantκ∈(0,∞), such that for any (smooth) function f, there holds

(2.6) κ

Z

Ω

|f− hfiν|²ν≤ Z

Ω

|∇f|²ν, hfiν :=

Z

Ω

f ν,

and therefore

Z

Ω

f²ν ≤ hfi²_ν+1 κ

Z

Ω

|∇f|²ν.

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

Ω

Z 1

0

∇f(z_t)·(x−y)dt ν(y)dy.

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

Z

Ω

(f(x)− hfiν)²ν(x)dx≤ Z

Ω

Z

Ω

Z 1

0

|∇f(zt)|²|x−y|²dt ν(y)ν(x)dydx

≤C1

Z

Ω

Z

Ω

Z 1/2

0

|∇f(zt)|²dtdx ν(y)dy+C1

Z

Ω

Z

Ω

Z 1

1/2

|∇f(zt)|²dtdy ν(x)dx

=C1

Z

Ω

Z 1/2

0

Z

Ω

|∇f(z)|² dz

(1−t)^ddt ν(y)dy+C1

Z

Ω

Z 1

1/2

Z

Ω

|∇f(z)|²dz

t^d dt ν(x)dx

≤2C1

Z

Ω

|∇f(z)|²dz,

withC₁:=kνk_L^∞diam(Ω)². We immediately deduce the Poincar´e-Wirtinger inequality with the

constantκ⁻¹:= 2C₁k1/νk_L^∞.

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) (L^∗W)(x) := ∆W(x)− ∇V · ∇W(x)≤ −θ W(x) +b1B_R(x), ∀x∈R^d,

where BR =B(0, R) denotes the centered ball of radiusR. The proof is elementary. We look for W as W(x) :=e^γhxi. We then compute

∇W =γ x

hxie^γhxi and ∆W =

γ²+γd−1 hxi

e^γhxi,

and then

L^∗W = ∆W−x· ∇W = γd−1 hxi W +

γ²−γ|x|² hxi

W

≤ −θ W +b1B_R

with the choiceθ=γ= 1 and thenRandb large enough.

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

θ W(x) + b

θ W(x)1_BR(x) ∀x∈R^d. For anyf ∈C_b²(R^d), we deduce

Z

R^d

f²G ≤ − Z

R^d

f²L^∗W(x) θ W(x) G+b

θ Z

BR

f² 1

W G =: T1+T2. 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

B_R

G dx, νR:=G(BR)⁻¹G_|BR, hfiR= Z

B_R

f νR, we have

θ bT2 =

Z

BR

f² 1

W G≤G(BR) Z

BR

f²νR

≤ G(BR)

hfi²_R+CR

Z

B_R

|∇f|²νR

.

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

Z

R^d

f²G≤C hfi²_R+

Z

R^d

|∇f|²G .

Consider nowg∈C_b². We know that for anyc∈R, there holds (2.9)

Z

R^d

(g− hgiG)²G≤φ(c) :=

Z

R^d

(g−c)²G,

with hgi_G defined in (2.6), becauseφ is a polynomial function of second degree which reaches is minimum value inc_g :=hgi_G. We last definef :=g− hgi_R, so thathfi_R = 0, ∇f =∇g. Using first (2.9) and next (2.8), we obtain

Z

R^d

(g− hgiG)²G ≤ Z

R^d

f²G

≤ C hfi²_R+

Z

R^d

|∇f|²G

= C

Z

R^d

|∇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 anyλ < λ_P, there existsε >0 so that the following stronger version Z

R^d

∇ f

G

2

Gdx ≥ λ Z

R^d

f²G⁻¹dx +ε

Z

R^d

f²|x|²+|∇f|² G⁻¹dx

holds for any f ∈ D(R^d)with hfi= 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

R^d

∇ f

G

2

Gdx= Z

R^d

|∇f|² G⁻¹dx− Z

R^d

f²(∆Φ)G⁻¹dx.

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

T =

Z

R^d

∇(f G^−1/2)G^1/2+ (f G^−1/2)∇G^1/2

2

Gdx

= Z

R^d

∇(f G^−1/2)

2

dx+ Z

R^d

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

R^d

f²G⁻¹dx+θ Z

R^d

f² 1

16|∇Φ|²−3 4∆Φ

G⁻¹dx +θ

16 Z

R^d

f²|∇Φ|²G⁻¹dx+θ 2

Z

R^d

|∇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(R^d); f ≥0, Z

f = 1, Z

f x= 0, Z

f|x|²=d

and

D_≤:=

f ∈L¹(R^d); f ≥0, Z

f = 1, Z

f x= 0, Z

f|x|²≤d

.

We observe that D (and D≤) are invariant set for the flow of Fokker-Planck equation (2.1). We also observe that Gis the unique stationary solution which belongs to D. 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 any f ∈D_≤, there holds

(3.1) I(f|G)≥0,

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

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∈R^d 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) =e^C−|x|2/2 onOfor some constantC∈R. By continuity off, we deduce thatO=R^d, 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

f²∂if ∂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

f²∂_if ∂_jf− 1

f∂_ijf−δ_ij² 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 thex_i variable, we get 1

2I⁰(f)·∆f = Z 1

f∂jf ∂iijf− Z 1

2f²∂iif(∂jf)²

=

Z ∂if

f² ∂jf ∂ijf−1

f∂ijf ∂ijf

+ Z 1

f²∂if ∂jf ∂ijf−∂if

f³ ∂if(∂jf)²

= −X

ij

Z 1

f²∂_if ∂_jf− 1 f∂_ijf²

f.

Proof of (3.3). We write 1

2I⁰(f)·(∇ ·(f x)) = Z ∂jf

f ∂ij(f xi)−(∂jf)²

2f² ∂i(f xi).

We observe that

∂ij(f xi)−(∂jf)

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

2f −d 2∂jf

= (∂_ijf)x_i+ (d

2 + 1)∂_jf−∂_if ∂_jf x_i 2f. Gathering the two preceding equalities, we obtain

1

2I⁰(f)·(∇ ·(f x)) = (d

2+ 1)I(f) + Z ∂_jf

f ∂_ijf x_i− Z ∂_jf

f ∂_if ∂_jf x_i 2f. Last, we remark that

−d

2I(f) =1 2

Z

∂_i

(∂jf)² f

x_i=

Z ∂jf ∂ijf f x_i−1

2 (∂jf)²

f² ∂_if x_i, 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

f²∂if ∂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 L^q∩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

kfk_L2∗/2 =kp

fk²_L2∗ ≤Ck∇p

fk²_L2 =C I(f)≤C I(ϕ).

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

f(t_nk) which converges a.e. and strongly in L^2q and weakly in ˙H¹to a limit denoted by√

g. That implies thatf(t_nk) converges togstrongly in L^q∩L¹_k for any q∈[1,2^∗/2), k∈[0,2), and thatI(g)≤lim supI(f(tn_k))<∞, so thatg∈V. Finally, since 2∇p

f(tn_k)−xp

f(tn_k)→2∇√ g−x√

g weakly inL²_loc (for instance) we have 0≤J(g)≤lim inf

k→∞ J(f(t_nk, .) = lim inf

k→∞ I(f(t_nk, .)|G) = 0.

From J(g) = 0 andg∈V ∩D_≤ we get g=Gas a consequence of Lemma 3.1, and it is then the all family (f(t))_t≥0 which converges toGast → ∞. TheL¹₂ convergence is a consequence of the fact that the sequence (f(tn)|v|²)n is tight becausehf(t)|v|²i=hG|v|²ifor any timet≥0 . Exercise 3.4. Prove that0≤fn →f inL^q∩L¹_k,q >1,k >0, implies that H(fn)→H(f).

(Hint. Use the splitting s|logs| ≤√

s1_0≤s≤e−|x|k +s|x|^k1_e−|x|k

≤s≤1+s(logs)+1_s≥1 ∀s≥0 and the dominated convergence theorem).

Exercise 3.5. Prove the convergence (3.5)for anyϕ∈P(R^d)∩L¹₂(R^d)such thatI(ϕ)<∞.

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

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

R^d

f logf dx, H(f|G) =H(f)−H(G) = Z

R^d

j(f /G)G dx, where j(s) =slogs−s+ 1.

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

Z

R^d

(1 + logf) [∆f+∇ ·(x f)]

= −

Z

R^d

∇f· ∇logf− Z

R^d

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

R^d

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

R^d

flnf− Z

R^d

GlnG≤ 1 2

Z

R^d

∇f∇f f −d or also as

Z

R^d

u²log(u²)G(dx)≤2 Z

R^d

|∇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(f_T)] = 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(f_T|G)] = 1

2I(ϕ|G),

thanks to (3.5).

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

(3.10) kµ−νk²_{V T} :=kg−1k²_L1(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).

Thanks to the Cauchy-Schwarz inequality one deduces Z

|g−1|dµ≤ s

1 3

Z

(2g+ 4)dµ sZ

(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−Gk_L1 ≤√

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)khk²_L2(G^−1/2)≤ Z

|∇h|²G⁻¹ ∀h∈ D(R^d), hh[1, x,|x|²]i= 0, also holds (for the same constant λ >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(R^d) 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

2ε²I(f|G) =I(G+ε h)−I(G)

2ε² .

Expending up to order 2 the two functionals, we have

flogf =GlogG+ε h(1 + logG) +ε² 2

h²

G +O(ε³),

|∇f|²

f = |∇G|² G +ε

2∇G

G · ∇h−|∇G|² G² h

+ε²

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

R^d

(logG+ 1)h= 0,

I⁰(G)·h= Z

R^d

n|∇G|²

G² −2∆G G

o h= 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 weightedL¹ semigroup spectral gap estimate holds.

Theorem 4.1. For any a∈(−λ_P,0) and for any k > k^∗:=λ_P+d/2 there exists C_k,a such that for anyϕ∈L¹_k, the associated solutionf to the Fokker-Planck equation (2.1)-(2.2) satisfies

kf− hϕiGk_L1

k≤Ck,ae^{a t}kϕ− hϕiGk_L1 k.

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

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

where χR(x) =χ(x/R),χ ∈ D(R^d), 0≤χ ≤1,χ ≡1 on B1, 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 ∈L^p kAfk_Lp(G^1/p)≤C_p,R,Mkfk_Lp.

Step 2. For anyk, ε >0 and for anyM, R >0 large enough (which may depend onkandε) the operatorBis dissipative inL¹_k in the sense that

(4.1) ∀f ∈ D(R^d)

Z

R^d

(Bf) (signf)hxi^k≤(ε−k)kfk_L¹

k.

We set β(s) = |s| (and more rigorously we must take a smooth version of that function) and m=hxi^k, 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}, 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(R^d) ke^Btϕk_L2

k≤ Ca,k

t^d/2 e^{a t}kϕk_L1 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=e^Btϕthe 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∈(−λ_P,0), there holds

(4.3) k(AS_B)^(∗n)ϕk_L2(G^−1/2)≤Ck,n,ae^atkϕk_L1

k ∀t≥0,

forn=d+1 for instance. We just establish (4.3) whend= 1. We denoteE:=L¹_k,E:=L²(G^−1/2).

Observing that

kASB(t)kE→E≤ C₁

t^1/2e^a0t and kASB(t)kE→E≤C₂e^a0t, we compute

k(ASB)^(∗2)kE→E ≤ Z t

0

kASB(t−s)kE→EkASB(s)kE→Eds

≤ e^a0t Z t

0

C1

(t−s)^1/2C₂ds

= e^a0tC1C2t^1/2 Z 1

0

du u^1/2, from which we immediately conclude by taking a⁰∈(−λP, a).

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

We denote byL the differential Fokker-Planck operator in E and still byL the same operator in E. We also denote by S_L 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

S_L= ΠS_L+ (I−Π)S_L(I−Π) and by iterating once the Duhamel formula

S_L(t) = S_B(t) + Z t

0

S_L(t−s)AS_B(s)ds

= S_B(t) +S_L∗ AS_B(t), we have

S_L=S_B+S_B∗(AS_B) + S_L∗(AS_B)^(∗2). These two identities together, we have

S_L = ΠS_L+ (I−Π){S_B+S_B∗(AS_B) + S_L∗(AS_B)^(∗2)}(I−Π) or in other words

S_L−ΠS_L = (I−Π){S_B+S_B∗(AS_B)}(I−Π) +{(I−Π)S_L} ∗(AS_B)^(∗2)(I−Π).

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

and thanks to steps 2 and 4 above.

5. Coming back to local in time estimates

We consider a smooth and fast decaying initial datumf₀, the solutionfto associated heat equation, and for a givenα∈R^d, we defineg:=f e^ψ,ψ(x) :=α·x. The equation satisfied bygis

∂tg = 1

2e^ψ∆(g e^−ψ) = 1

2∆g− ∇ψ· ∇g+1 2|∇ψ|²g

= 1

2∆g−α· ∇g+1 2|α|²g For theL¹norm, we have

d

dtkgkL¹ =1

2α²kgkL¹

and then kg(t, .)kL¹ = e^α2t/2kg0kL¹ for any t ≥ 0. For the L² norm and thanks to the Nash inequality (1.3), we have

d

dtkgk²_L2 = −k∇gk²_L2+α²kgk²_L2

≤ −K0e^−2α2t/dkgk^2(1+2/d)_L2 +α²kgk²_L2

withK0:=CNkg0k^−4/d_L1 . We see that the function u(t) :=e^−α2tkg(t)k²_L2 satisfies the differential inequality

u⁰ ≤ −K₀u^1+2/d, from what, exactly as in the Section 1.1, we deduce

kg(t)k²_L2e^−α2t≤ kg0k²_L1

(2/d CNt)^d/2, ∀t >0.

Denoting by T(t) the semigroup associated to the parabolic equation satisfies by g, the above estimate writes

kT(t)g0k_L2 ≤C e^α2t/2

t^d/4 kg0k_L1, ∀t >0.

Because the equation associated to the dual operator is

∂th=1

2∆h+α· ∇h+1

2|α|²h, h(0) =h0, the same estimate holds onT^∗(t)h₀=h(t), and we thus deduce

kT(t)g0kL^∞ ≤ C e^α2t/2

t^d/4 kg0k_L2, ∀t >0.

Using the trickT(t) =T(t/2)T(t/2), both estimates together give an accurate time depend estimate on the mappingT(t) :L¹→L^∞ for anyt >0. More precisely and in other words, we have proved tha the heat semigroupS satisfies

k(S(t)f0)e^ψkL^∞ ≤ C

t^d/2e^α2t/2kf0e^ψk_L1, ∀t >0.

DenotingF(t, x, y) := (S(t)δx)(y) the fundamental solution associated to the heat equation when starting from the Dirac function inx∈R^d, the above estimate rewrites as

F(t, x, y)≤ C

t^d/2e^{α·(x−y)−α2t/2}, ∀t >0,∀x, y, α∈R^d. Choosingα:= (x−y)/t, we end with

F(t, x, y)≤ C

t^d/2e^−|x−y|

2

2t , ∀t >0,∀x, y∈R^d.

6. Exercises and Complements

Exercise 6.1. 1. Give another proof of the Nash inequality by using the Sobolev inequality in dimension d≥3. (Hint. Write the interpolation estimate

kfk_L2 ≤ kfk^θ_L1kfk^1−θ_L2∗

and then use the Sobolev inequality associated to the Lebesgue exponent p= 2).

2. Give another proof of the Nash inequality by using the Sobolev inequality in dimension d= 2.

(Hint. Prove the interpolation estimate

kfk_L2 ≤ kfk^1/4_L1 kf^3/2k^1/2_L2,

then use the Sobolev inequality associated to the Lebesgue exponent p= 1and p^∗ := 2 and finally the Cauchy-Schwartz inequality in order to bound the second term).

3. Give another proof of the Nash inequality by using the Sobolev inequality in dimension d= 1.

(Hint. Prove the interpolation estimate

kfkL²≤ kfk^1/2_L1 kf^3/2k^1/3_L∞,

then use the Sobolev inequality associated to the Lebesgue exponent p= 1 andp^∗:=∞and finally the Cauchy-Schwartz inequality in order to bound the second term).

We propose now a third proof based on the Poincar´e-Wirtinger inequality. We write kfk²_L2= (f, f−fr) + (f, fr), with fr(x) := 1

|B(x, r)|

Z

B(x,r)

f(y)dy.

We have

kfk²_L2 ≤ kfk_L2kf −frk_L2+kfk_L1kfrkL^∞. On the one hand,

kfrkL^∞ ≤ C r^d kfkL¹. On the other hand,

kf−frk²_L2 = Z

R^d

C_d r^d

Z

B(x,r)

(f(y)−f(x))dy

2

dy

≤ Z

R^d

Z

R^d

1_|y−x|≤r|f(y)−f(x)|²dxdy

≤ r² Z 1

0

Z

R^d

Z

R^d

1_|y−x|≤r|∇f((1−t)x+ty)|²dxdydt

≤ r² Z 1/2

0

Z

R^d

Z

R^d

1_|y−x|≤r|∇f((1−t)x+ty)|²dxdydt +r²

Z 1

1/2

Z

R^d

Z

R^d

1_|y−x|≤r|∇f((1−t)x+ty)|²dxdydt

≤ r² Z 1/2

0

Z

R^d

Z

R^d

1_|y−x|≤r|∇f(z)|²dzdydt +r²

Z 1

1/2

Z

R^d

Z

R^d

1_|y−x|≤r|∇f(z)|²dxdzdt

≤ r²C Z

R^d

|∇f(z)|²dz.

All together, we get

kfk²_L2 ≤ C₁rkfk_L2k∇fk_L2+C₂r^−dkfk²_L1

≤ 1

2kfk²_L2+C₁

2 r²k∇fk²_L2+C₂r^−dkfk²_L1

and we obtain the Nash inequality by choosingr:= (kfk²_L1/k∇fk²_L2)^1/(d+2). Exercise 6.2. Establish (2.7)in the following situations: