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Outline Introduction Proof of the main results Main references

Global Hypoellipticity and Compactness of Resolvent for Fokker-Planck Operator

Wei-Xi Li wei-xi.li@whu.edu.cn

Wuhan University

January 9, 2010

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references

Outline

1

Introduction Problems considered Previous results and the approaches involved Our main result

2

Proof of the main results the approach Regularity in velocity variable Regularity in spacial variable

3

Main references

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Outline

1

Introduction Problems considered Previous results and the approaches involved Our main result

2

Proof of the main results the approach Regularity in velocity variable Regularity in spacial variable

3

Main references

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

In this talk we consider the Fokker-Planck operator P = y · ∂x − ∂xV (x) · ∂y − △y + |y|2 4 − n 2, (x, y) ∈ R2n, (1) where x denotes the space variable and y denotes the velocity vari- able, and V (x) is a real-valued potential defined in the whole space Rn

x.

Elementary Properties of Fokker-Planck Operator 1) Observe P is a non-selfadjoint operator: P = Re P + √ −1 Im P, where Re P = −△y + |y|2

4 − n 2 and Im P = y · Dx − ∂xV (x) · Dy

are operators with real symbols.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

2) If we rewrite P = √−1(y · Dx − ∂xV (x) · Dy) + n

j=1 L∗ j Lj with

Lj = ∂yj + yj

2 , then

∀ u ∈ C ∞

0 ,

Re Pu, uL2 ≥ 0. Hence P is accretive in C ∞

0 . And the spectral σ(P) of P is contained

in { z ∈ C; Re z ≥ 0 } . Moreover the closure ¯ P of P is maximally accretive. Hence −¯ P is a generator of a contractive semi-group

• e−t ¯

P t≥0.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

One question arising from statistical physics or the theory of kinetic equations is the exponential return to equilibrium. For a essential self-adjoint operator, it reduces to the estimate of its first nonzero

• eigenvalue. Unfortunately Fokker-Planck operate is only maximally

accretive but non-selfadjoint. However the hypoellipticity could give us a hand. This ideal has been used by H´ erau-Nier. It concerns the compactness of resolvent.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

The link between Fokker-Planck operator and Witten Laplace oper- ator The Witten Laplace operator △(0)

V /2 defined by

△(0)

V /2 = −△x + 1

4 |∂xV (x)|2 − 1 2△xV (x). Helffer-Nier’s Conjecture The Fokker-Planck operator has a compact resolvent if and only if the Witten Laplacian has a compact resolvent. The ”Only if” part can be deduced directly, while the ”if” part remains substantially open. Some positive answers have been given for some kind of potentials.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

In this talk we are mainly concerned with the conditions imposed on the potential V (x), so that the Fokker-Planck operator P admits a global hypoelliptic estimate and has a compact resolvent, and recall the works of Helffer,H´ erau and Nier on this problem.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Outline

1

Introduction Problems considered Previous results and the approaches involved Our main result

2

Proof of the main results the approach Regularity in velocity variable Regularity in spacial variable

3

Main references

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

The approach used by Helffer-Nier and H´ erau-Nier: Hypoelliptic techniques To analyze the compactness of resolvent of the operator P, the hypoellipticity techniques play an efficient role, one of which is re- ferred to Kohn’s method due to H¨

• rmander, Kohn,· · · , and another

is based on nilpotent Lie group technique developed by Rothschild- Stein.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Kohn’s method had been used by H´ erau-Nier 1 to study such a po- tential V (x) that behaves at infinity as a high-degree homogeneous function: ∂xV (x) ≈ x2M−1 and ∀ |γ| ≥ 1, |∂γ

x V (x)| ≤ Cγ x2M−|γ|

By developing the approach of H´ erau-Nier, Helffer-Nier 2 considered more general potential V (x).

• 1F. H´

erau and F. Nier,Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech.

• Anal. 171 (2004), no. 2, 151–218.
• 2B. Helffer and F. Nier,Hypoelliptic estimates and spectral theory for Fokker-

Planck operators and Witten Laplacians,Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Assumption I Suppose V is real-valued smooth function satisfying that with C, k some constants and ∂xV (x) =

• 1 + |∂xV (x)|2 1

2 , Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Assumption I Suppose V is real-valued smooth function satisfying that with C, k some constants and ∂xV (x) =

• 1 + |∂xV (x)|2 1

2 ,

1 C x

1 k ≤ ∂xV (x) ≤ C xk ; Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Assumption I Suppose V is real-valued smooth function satisfying that with C, k some constants and ∂xV (x) =

• 1 + |∂xV (x)|2 1

2 ,

1 C x

1 k ≤ ∂xV (x) ≤ C xk ;

∀ |γ| ≥ 2, |∂γ

x V (x)| ≤ Cγ ∂xV (x) .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Assumption I Suppose V is real-valued smooth function satisfying that with C, k some constants and ∂xV (x) =

• 1 + |∂xV (x)|2 1

2 ,

1 C x

1 k ≤ ∂xV (x) ≤ C xk ;

∀ |γ| ≥ 2, |∂γ

x V (x)| ≤ Cγ ∂xV (x) .

Assumption II Suppose V is real-valued smooth function satisfying that for some C, k ≥ 1,

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Assumption I Suppose V is real-valued smooth function satisfying that with C, k some constants and ∂xV (x) =

• 1 + |∂xV (x)|2 1

2 ,

1 C x

1 k ≤ ∂xV (x) ≤ C xk ;

∀ |γ| ≥ 2, |∂γ

x V (x)| ≤ Cγ ∂xV (x) .

Assumption II Suppose V is real-valued smooth function satisfying that for some C, k ≥ 1, ∂xV (x) ≤ C xk ;

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Assumption I Suppose V is real-valued smooth function satisfying that with C, k some constants and ∂xV (x) =

• 1 + |∂xV (x)|2 1

2 ,

1 C x

1 k ≤ ∂xV (x) ≤ C xk ;

∀ |γ| ≥ 2, |∂γ

x V (x)| ≤ Cγ ∂xV (x) .

Assumption II Suppose V is real-valued smooth function satisfying that for some C, k ≥ 1, ∂xV (x) ≤ C xk ; ∃ κ > 0, ∀ |α| = 2, |∂α

x V (x)| ≤ Cα ∂xV (x) x−κ .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Assumption I Suppose V is real-valued smooth function satisfying that with C, k some constants and ∂xV (x) =

• 1 + |∂xV (x)|2 1

2 ,

1 C x

1 k ≤ ∂xV (x) ≤ C xk ;

∀ |γ| ≥ 2, |∂γ

x V (x)| ≤ Cγ ∂xV (x) .

Assumption II Suppose V is real-valued smooth function satisfying that for some C, k ≥ 1, ∂xV (x) ≤ C xk ; ∃ κ > 0, ∀ |α| = 2, |∂α

x V (x)| ≤ Cα ∂xV (x) x−κ .

∀ |γ| ≥ 2, |∂γ

x V (x)| ≤ Cγ ∂xV (x) .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Let us denote Ψ(x, y; ξ, η) =

• 1 + |ξ|2 + η2 + |∂xV (x)|2 + |y|2 1

2

and g = |dx|2 + |dy|2 + |dξ|2+|dη|2

Ψ2

. If Assumption I or Assumption II is fulfilled, then Ψ is an admissible weight function and g is H¨

• rmander metric, that is, Ψ satisfies slowness condition and tem-

perance condition, and g satisfies slowness condition, temperance condition and the uncertainty principle. So global Weyl-H¨

• rmander

pseudo-differential calculus can be used. Now denote by Op(S(Ψs, g)) the set of the pseudo-differential operators whose symbols are in S(Ψs, g). Observe the pseudo-differential operator Λ2

x,y = 1 − △x − △y + 1

2 |∂xV (x)|2 + 1 2 |y|2 is elliptic in Op(S(Ψ2, g)). It is just the analog of ξ2 appearing in Kohn’s proof.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Theorem (Helffer-Nier,05) If V satisfies Assumption I or Assumption II, then ∀u ∈ S ,

• Λ

1 4

x,yu

• L2 ≤ C

Pu

• L2 +
• u
• L2
• (2)

with Λx,y =

• 1 − △x − △y + 1

2 |∂xV (x)|2 + 1 2 |y|2 1

2 . Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

The compactness of resolvent Write (1 + P)−1 = Λ

− 1

4

x,y ◦

• Λ

1 4

x,y ◦ (1 + P)−1

• .

The hypoelliptic estimate (2) shows that Λ

1 4

x,y ◦ (1 + P)−1 is bounded in L2(R2n).

Moreover Λ

− 1

4

x,y is compact in L2(R2n) if either lim|x|→+∞ |∂xV (x)| =

+∞ or the corresponding Witten Laplacian has a compact resolvent. Corollary (Helffer-Nier, 05) The Fokker-Planck operator has a compact resolvent if either V satisfies Assumption I or V satisfies Assumption II and the corre- sponding Witten Laplacian has a compact resolvent.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Nilpotent technique and optimal estimate As for the Kohn’s proof for the hypoellipticity, the exponent 1

4 in

(2) is not optimal. A better exponent 2

3 can be obtained via ex-

plicit method in the particular case when V (x) is a non-degenerate quadratic form, i.e., V is a polynomial real potential of degree less

• r equal to 2 with det (HessV ) = 0. For some kind of general poten-

tial, Helffer-Nier obtained the exponent 2

3, by virtue of the nilpotent

technique that initiated by Rothschild-Stein and then developed by Helffer-Nourrigat.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Theorem (Helffer-Nier,05) Suppose V (x) satisfies that ∀ |α| = 2, |∂α

x V (x)| ≤ Cα ∂xV (x)1−ρ

with ρ > 1 3. Then

• |∂xV (x)|

2 3 u

• L2 ≤ C

Pu

• L2 +
• u
• L2
• .

(3) Remark: Although the estimate (3) is better, the assumption is stronger than Assumption I,II for the second derivatives. Moreover in (3) some information on the Sobolev regularity in x is missing

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Outline

1

Introduction Problems considered Previous results and the approaches involved Our main result

2

Proof of the main results the approach Regularity in velocity variable Regularity in spacial variable

3

Main references

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Assumption on the potential: Let V (x) ∈ C 2(Rn) be a real-valued function satisfying that ∀ |α| = 2, ∃ Cα > 0, |∂α

x V (x)| ≤ Cα ∂xV (x)s ,

(4) where s < 4

3.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Theorem (Li,09) Under the above assumption (4), we can find a constant C, such that for any u ∈ C ∞

• R2n
• ne has
• |∂xV (x)|

2 3 u

• L2 ≤ C

Pu

• L2 +
• u
• L2
• ,

and with Λy =

• 1 − △y + |y|2 1

2 ,

• (1 − △x)

δ 2 u

• L2 +
• Λyu
• L2 ≤ C

Pu

• L2 +
• u
• L2
• ,

where δ equals to 2

3 if s ≤ 2 3, 4 3 − s if 2 3 < s ≤ 10 9 , and 2 3 − s 2 if 10 9 < s < 4 3.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Remark:In particular, if the assumption (4) is fulfilled with s = 2

3,

then we have the following hypoelliptic estimate which seems to be

• ptimal:
• |∂xV (x)|

2 3 u

• L2 +
• (1 − △x)

1 3 u

• L2 ≤ C
• Pu
• L2 +
• u
• L2
• .

Moreover one can deduce from the above estimate a better regularity in the velocity variable y, that is, ∀ u ∈ C ∞

• R2n

,

• 1 − △y + |y|2

u

• L2 ≤ C

Pu

• L2+
• u
• L2
• .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references Problems considered Previous results and the approaches involved Our main result

Using the functional calculus for self-adjoint operator instead of the pseudo-differential calculus used by Helffer-Nier, we have Corollary Let V (x) satisfy the condition (4). Then the Fokker-Planck operator P has a compact resolvent if either lim

|x|→+∞ |∂xV (x)| = +∞

• r the Witten Laplacian △(0)

V /2 has a compact resolvent.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Outline

1

Introduction Problems considered Previous results and the approaches involved Our main result

2

Proof of the main results the approach Regularity in velocity variable Regularity in spacial variable

3

Main references

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Our proof is inspired from the Poisson bracket analysis. Let (ξ, η) be the dual variables of (x, y). Rewrite P = y · ∂x − ∂xV (x) · ∂y − △y + |y|2

4 − n 2 as

P = p1(y; Dy) + √ −1 p2(x, y; Dx, Dy) with p1(y; η) = η2 + |y|2

4 − n 2 and P2(x, y; ξ, η) = y · ξ − ∂xV (x) · η.

{p2, {p2, p1}} = 2 |ξ|2+|∂xV (x)|2 2 −

• i,j
• ∂xixjV

2ηiηj + 1 2yiyj

• .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Outline

1

Introduction Problems considered Previous results and the approaches involved Our main result

2

Proof of the main results the approach Regularity in velocity variable Regularity in spacial variable

3

Main references

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

With Lj = ∂yj + yj

2 , j = 1, · · · n, we can write the operator P as

P =

n

• j=1

L∗

j Lj +

√ −1 p2(x, y; Dx, Dy). (5) As a result, we can find a constant C such that for any u ∈ C ∞

• R2n

,

• Λyu
• L2 ≤ C
• Pu
• L2 +
• u
• L2
• with Λy = (1 − △y + |y|2)

1 2 . Moreover

• Λ2

yu

• L2 ≤ C
• |∂xV (x)|

2 3 u

• L2 +
• (1 − △x)

1 3 u

• L2 +
• Pu
• L2
• .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Outline

1

Introduction Problems considered Previous results and the approaches involved Our main result

2

Proof of the main results the approach Regularity in velocity variable Regularity in spacial variable

3

Main references

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Proposition Suppose V satisfies the condition (4). Then ∃ C > 0, ∀ u ∈ C ∞

• R2n

, we have

• ∂xV (x)

2 3 u

• L2 ≤ C

Pu

• L2 +
• u
• L2
• .

Sketch of the proof. Consider the inner product Re Pu, RuL2 = Re √−1 p2u, Ru

• L2 + Re n

j=1

• L∗

j Lju, Ru

• L2

with R = R(x, y) = 2 ∂xV (x)− 2

3 ∂xV (x) · y. Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

The conclusion follows from the following estimates: ∀ u ∈ C ∞

• R2n

, Re √−1p2u, Ru

• L2 ≥ C −1

∂xV (x)

2 3 u

• 2

L2−C

Pu

• 2

L2 +

• u
• 2

L2

• Wei-Xi Li

Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

The conclusion follows from the following estimates: ∀ u ∈ C ∞

• R2n

, Re √−1p2u, Ru

• L2 ≥ C −1

∂xV (x)

2 3 u

• 2

L2−C

Pu

• 2

L2 +

• u
• 2

L2

• Ru
• L2 ≤ C
• |y| ∂xV (x)

1 3 u

• L2 ≤ C
• Λy ∂xV (x)

1 3 u

• L2;

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

The conclusion follows from the following estimates: ∀ u ∈ C ∞

• R2n

, Re √−1p2u, Ru

• L2 ≥ C −1

∂xV (x)

2 3 u

• 2

L2−C

Pu

• 2

L2 +

• u
• 2

L2

• Ru
• L2 ≤ C
• |y| ∂xV (x)

1 3 u

• L2 ≤ C
• Λy ∂xV (x)

1 3 u

• L2;
• Λyf (x)

1 3 u

• 2

L2 ≤ ε

• ∂xV (x)

2 3 u

• 2

L2+Cε

Pu

• 2

L2 +

• u
• 2

L2

• ;

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

The conclusion follows from the following estimates: ∀ u ∈ C ∞

• R2n

, Re √−1p2u, Ru

• L2 ≥ C −1

∂xV (x)

2 3 u

• 2

L2−C

Pu

• 2

L2 +

• u
• 2

L2

• Ru
• L2 ≤ C
• |y| ∂xV (x)

1 3 u

• L2 ≤ C
• Λy ∂xV (x)

1 3 u

• L2;
• Λyf (x)

1 3 u

• 2

L2 ≤ ε

• ∂xV (x)

2 3 u

• 2

L2+Cε

Pu

• 2

L2 +

• u
• 2

L2

• ;

n

• j=1
• L∗

j Lju, Ru

• L2
• ≤ ε
• ∂xV (x)

2 3 u

• 2

L2+Cε

Pu

• 2

L2 +

• u
• 2

L2

• .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Now let us consider the Sobolev regularity in x. To clarify our proof, we here only consider a simple case: ∀ |α| = 2, ∃Cα, |∂α

x V (x)| ≤ Cα ∂xV (x)

2 3

(6) Proposition Suppose V satisfies the condition (6). Then

• (1 − △x)

1 3 u

• L2 ≤ C
• Pu
• L2 +
• u
• L2
• .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Sketch of the proof. Take gx = ∂xV (x)

2 3 |dx|2 ,

x ∈ Rn. Then g is a slowly varying metric if V (x) satisfies the condition (6). As a result we can find some partitions of unity related to the metric: there exists a constant r0 > 0 and a sequence xµ ∈ Rn, µ ≥ 1, such that the union of the balls Ωµ,r0 =

• x ∈ Rn;

gxµ (x − xµ) < r2

• coves the whole space Rn. Moreover there exists a positive integer

N, depending only on r0, such that the intersection of more than N balls is always empty. One can choose a family of functions { ϕµ }µ≥1 such that supp ϕµ ⊂ Ωµ,r0,

• µ≥1

ϕ2

µ = 1 and

sup

µ≥1

|∂xϕµ(x)| ≤ C ∂xV (x)

1 3 . Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

We associate with each fixed xµ ∈ Rn the operator Pxµ = y · ∂x − ∂xV (xµ) · ∂y − △y + |y|2 4 − n 2. Then we have ϕ2

µPu = Pxµ ϕ2 µu + Rµu.

with Rµ given by Rµ = −2ϕµ(x) (y · ∂xϕµ(x)) − ϕµ(x)2 (∂xV (x) − ∂xV (xµ)) · ∂y.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Lemma Suppose V (x) satisfies the assumption (6). Then ∀ u ∈ C ∞

• R2n

,

• µ≥1
• Rµu
• 2

L2 ≤ C

Pu

• 2

L2 +

• u
• 2

L2

• .

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references the approach Regularity in velocity variable Regularity in spacial variable

Lemma Suppose V (x) satisfies the assumption (6). Then ∀ u ∈ C ∞

• R2n

,

• µ≥1
• Rµu
• 2

L2 ≤ C

Pu

• 2

L2 +

• u
• 2

L2

• .

Lemma Suppose V (x) satisfies the assumption (6). Then

• (1 − △x)

1 3 u

• 2

L2 ≤ C

Pxµu

• 2

L2 +

• u
• 2

L2

• .

And hence

• (1 − △x)

1 3 u

• 2

L2 ≤ C

• µ≥1
• Pxµϕ2

µu

• 2

L2 + C

• µ≥1
• ϕ2

µu

• 2

L2.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references

• B. Helffer and F. Nier,

Hypoelliptic estimates and spectral theory for Fokker-Planck op- erators and Witten Laplacians, Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005. à➡❧➜▼❻ô➜✜➅➂➜ ②➇➔❻➞➄➜ÚØ➜➱➬➀➷Ñ❻✖➜

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references

• B. Helffer and F. Nier,

Hypoelliptic estimates and spectral theory for Fokker-Planck op- erators and Witten Laplacians, Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005.

• F. H´

erau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker- Planck equation with a high-degree potential,

• Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218.

à➡❧➜▼❻ô➜✜➅➂➜ ②➇➔❻➞➄➜ÚØ➜➱➬➀➷Ñ❻✖➜

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references

• B. Helffer and F. Nier,

Hypoelliptic estimates and spectral theory for Fokker-Planck op- erators and Witten Laplacians, Lecture Notes in Mathematics, vol. 1862, Springer-Verlag, Berlin, 2005.

• F. H´

erau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker- Planck equation with a high-degree potential,

• Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151–218.

à➡❧➜▼❻ô➜✜➅➂➜ ②➇➔❻➞➄➜ÚØ➜➱➬➀➷Ñ❻✖➜1994.

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent

Outline Introduction Proof of the main results Main references

✻àP➇➭ ✮❋➥❲➜✜◆è①➐

Wei-Xi Li Global Hypoellipticity and Compactness of Resolvent