Gradient estimates and Harnack inequality on non compact ... - HIM

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Gradient estimates and Harnack inequality on non compact ... - HIM

An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> ong>inequalityong> on

non compact manifold

Marc Arnaudon

Université de Poitiers, France

11 October 2007

Complex stochastic systems: Discrete vs continuous

W2 Stochastic calculus on manifolds, graphs ong>andong> rong>andong>om

structures

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

joint works with

Elton P. Hsu

Anton Thalmaier

Feng-Yu Wang

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

1 An elementary lemma

2 Horizontal diffusion in C 1 path space

3 ong>Gradientong> ong>estimatesong> using coupling

The main result

Case Ric −HessV bounded

The proof for P D t

f

The general case

4 ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

ong>Harnackong> ong>inequalityong>

Heat kernel estimate

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

An elementary lemma

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

Lemma 1

(E, F, ˜µ) measured space such that ˜µ(E) < ∞.

f ∈ L 1 (˜µ) positive satisfying ˜µ(f ) > 0

ψ mesurable satisfying ψf ∈ L 1 (˜µ).

Then


E


ψf d ˜µ ≤

E

f log

f

˜µ(f )

∫E

d ˜µ + ˜µ(f ) log e ψ d ˜µ.

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

Horizontal diffusion in C 1 path space

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

M a complete Riemannian manifold

L = 1 2 ∆ + Z , Z ∈ Γ(TM).

If Y is a L-diffusion, denote by W (Y ) t : T Y0 M → T Yt M the

deformed parallel translation along Y :

(

DW (Y ) = − 1 )

2 Ric♯ (W (Y )) + ∇ W (Y ) Z dt.

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

Theorem 1

u ↦→ ϕ(u) a C 1 path in M, defined on [0, ∞[

X 0 a L-diffusion started at ϕ(0).

There exists a unique family u ↦→ (X t (u)) t≥0 of L-diffusions,

a.s. continuous in (t, u) ong>andong> C 1 in u,

satisfying X(0) = X 0 , X 0 (u) = ϕ(u), ong>andong>

Moreover

∂ u X t (u) = W (X(u)) t ( ˙ϕ(u)),

dX t (u) = P X(·)

t,0,u d mXt

0 + Z Xt (u) dt

with v ↦→ P X(·)

t,0,v parallel transport along v ↦→ X t(v).

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

ong>Gradientong> ong>estimatesong> using coupling

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

M non compact Riemannian manifold

L = 1 2 ∆ + Z ,

Z = grad V , V ∈ C2 (M)

P t Dirichlet semigroup associated with L :

P t f (x) = E [ f (X t (x))1 {t


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

D a C 2 relatively compact domain in M

Pt D Dirichlet semigroup :

Pt D f (x) = E [ ]

f (X t (x))1 {t


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

[Qi S. Zhang 06] : f ∈ B + b

(nonnegative measurable functions)


∥grad Pt D f (x) ∥ ≤ C(x, t)Pt D f (x)

where C(x, t) is a locally bounded function on D×]0, ∞[.

Consequence :

If f ∈ B + b

, t > 0, x, y ∈ D,

P D t f (x) ≤ ˜C(x, y, t)P D t f (y)

where ˜C is locally bounded on D × D×]0, ∞[

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

Aim: obtain similar ong>estimatesong> on M

Remark: these ong>estimatesong> are wrong in general, take for

instance M = R d ong>andong> L = 1 2 ∆.

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

The main result

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


Theorem 2

An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

There exists a continuous function F :]0, 1] × M → R + such

that ∀δ > 0, x ∈ M, t > 0, f ∈ B + b ,

(

‖grad P t f (x)‖ ≤ δ

[P t f log

(

+ F(δ ∧ 1, x)

f

P t f (x)

(

)]

(x)

)

1

δ(t ∧ 1) + 1 + 2δ

e

)

P t f (x).

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

Case Ric −HessV bounded

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

Let h s = 1 − s t , v ∈ T xM, ϕ(u) = exp(uv). then for l > 0

d (X s (lh s )) = (dX s )(lh s ) + ∂X s (lh s )lḣs ds

= (dX s )(lh s ) + W (X(lh s )) ˙ϕ(lh s )lḣs ds.

The last term in the right is uniformly bounded. Let

∫ s 〈


Ns l = − W (X(lh r v)) ˙ϕ(lh r )lḣr , d m X r (lh r ) ,

0

R l s = E(N l ) s , Q l = R l · P.

Under Q l , X s (lh s ) is a L-diffusion started at exp(lv), satisfying

X t (lh t ) = X 0

t . Marc Arnaudon ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

Consequence

Choosing v =

one obtains

grad P t f (x) = E

[ ]

P t f (exp(lv)) = E f (Xt 0 )Rl t .

grad P tf (x)

, differentiating w.r. to l at l = 0,

‖ grad P t f (x)‖

[

f (X 0

t ) ∫ t

0


− W (X 0 )(vḣ), d mX 0〉] .

Then applying the elementary lemma to δf (Xt 0)

ong>andong>

− 1 ∫ t 〈

W (X 0

δ

)(vḣ), d mX 0〉 one obtains the wanted result

0

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

The proof for P D t

f

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

Let c ∈]0, 1[ ong>andong> h s = ( 1 − s )

ct + . Then

1

(

) [

Pt D f (exp(lv)) − Pt D f (x) ≤E f (X 0

l

Then we prove that for c small

( ) ]

Rt l − 1

t )1 1

{t


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

The general case

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


Lemma 2

An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

For x ∈ D, let h x (s, z) the density of (τ(x), X τ(x) (x)). Then


P t f (x) = Pt D f (x) + P t−s f (z)h x (s, z) dsν(dz).

]0,t]×∂D

Proof: strong Markov property

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

consequence :

grad P t f (x) = grad Pt D f (x)


+ P t−s f (z) grad log h·(s, z)h x (s, z)dsν(dz).

]0,t]×∂D

then apply the elementary lemma to the second integral, with

the measure on ]0, t] × ∂D

˜µ = h x (s, z)dsν(dz) with total mass P(τ(x) ≤ t < ξ(x))

ong>andong> the functions δP t−s f (z), 1 δ ‖ grad log h·(s, z)‖. We obtain for

the secon term

[

δE f log f (X ]

t)

1

I τ(x)≤t


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

The main result

Case Ric −HessV bounded

The proof for P D t f

The general case

ong>Harnackong> ong>inequalityong>

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


Corollary 1

An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

ong>Harnackong> ong>inequalityong>

Heat kernel estimate

,

There exists a continuous C :]1, ∞[×M × M → R + , such that

∀α > 1, t > 0, x, y ∈ M, f ∈ B + b

(P t f (x)) α

[ 2(α − 1)

≤ P t f α (y) exp

e

( αρ 2 )]

(x, y)

+ αC(α, x, y)

(α − 1)(t ∧ 1) + ρ(x, y)

where ρ(x, y) is the Riemannian distance from x to y.

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

ong>Harnackong> ong>inequalityong>

Heat kernel estimate

Proof

Let γ s be the minimal geodesic from x to y in time 1, Theorem

2 yields a bound below for

d

(

ds log P t f 1+s(α−1))

Integrating from 0 to 1 gives the result.

α

1+s(α−1)

(γ s ).

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

ong>Harnackong> ong>inequalityong>

Heat kernel estimate

Heat kernel estimate

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities


Corollary 2

An elementary lemma

Horizontal diffusion in C 1 path space

ong>Gradientong> ong>estimatesong> using coupling

ong>Harnackong> ong>inequalityong> ong>andong> heat kernel estimate

ong>Harnackong> ong>inequalityong>

Heat kernel estimate

,

∀δ > 2, ∃C δ : [0, ∞[×M → R + , ∀x, y ∈ M, t ∈]0, 1[,

[ ]

exp −ρ(x,y) 2

δt

+ C δ (t, x) + C δ (t, y)

p t (x, y) ≤ √

µ(B(x, √ t))µ(B(y, √ ,

t))

with µ(dx) = e V (x) dx.

Marc Arnaudon

ong>Gradientong> ong>estimatesong> ong>andong> ong>Harnackong> inequalities

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