Averaging along irregular curves and regularization of ODEs Rmi - - PowerPoint PPT Presentation

Averaging along irregular curves and regularization of ODEs

Rémi Catellier

Centre Henri Lebesgue - IRMAR

October 24, 2014

joint work with Massimiliano Gubinelli (CEREMADE)

Rémi Catellier (CHL - IRMAR) October 24, 2014 1 / 21

1

Regularization of ODEs

2

The operator T w : (ρ, γ)-irregular functions

3

Stochastic processes

Rémi Catellier (CHL - IRMAR) October 24, 2014 2 / 21

Perturbed differential equations ?

Standard ODEs dxt = b(xt)dt

◮ For non Lipschitz b, the equation is ill posed ◮ Example b(x) = 2sign(x)

• |x|, x0 = 0.

Pertubed ODEs, Brownian case dxt = b(xt)dt + dBt

◮ When B is a Brownian motion, for measurable bounded b it is known

(Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution.

◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ?

Standard ODEs dxt = b(xt)dt

◮ For non Lipschitz b, the equation is ill posed ◮ Example b(x) = 2sign(x)

• |x|, x0 = 0.

Pertubed ODEs, Brownian case dxt = b(xt)dt + dBt

◮ When B is a Brownian motion, for measurable bounded b it is known

(Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution.

◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ?

Standard ODEs dxt = b(xt)dt

◮ For non Lipschitz b, the equation is ill posed ◮ Example b(x) = 2sign(x)

• |x|, x0 = 0.

Pertubed ODEs, Brownian case dxt = b(xt)dt + dBt

◮ When B is a Brownian motion, for measurable bounded b it is known

(Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution.

◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ?

Standard ODEs dxt = b(xt)dt

◮ For non Lipschitz b, the equation is ill posed ◮ Example b(x) = 2sign(x)

• |x|, x0 = 0.

Pertubed ODEs, Brownian case dxt = b(xt)dt + dBt

◮ When B is a Brownian motion, for measurable bounded b it is known

(Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution.

◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ?

Standard ODEs dxt = b(xt)dt

◮ For non Lipschitz b, the equation is ill posed ◮ Example b(x) = 2sign(x)

• |x|, x0 = 0.

Pertubed ODEs, Brownian case dxt = b(xt)dt + dBt

◮ When B is a Brownian motion, for measurable bounded b it is known

(Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution.

◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ?

Standard ODEs dxt = b(xt)dt

◮ For non Lipschitz b, the equation is ill posed ◮ Example b(x) = 2sign(x)

• |x|, x0 = 0.

Pertubed ODEs, Brownian case dxt = b(xt)dt + dBt

◮ When B is a Brownian motion, for measurable bounded b it is known

(Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution.

◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

For b : Rd → Rd et w : [0, T] → Rd we want to study the following equation xt = x0 + t b(xu)du + ws

Definition

For continuous vector fields b from Rd to Rd we define the averaging

• perator along w, T w by

T w

s,tb(x) =

t

s

b(x + wr)dr By a change of variable θt = xt − wt, we have the following equivalent equation θu = θ0 + t (∂uT w

u b)(θu)du =: θ0 +

t T w

dub(θu)

Rémi Catellier (CHL - IRMAR) October 24, 2014 4 / 21

For b : Rd → Rd et w : [0, T] → Rd we want to study the following equation xt = x0 + t b(xu)du + ws

Definition

For continuous vector fields b from Rd to Rd we define the averaging

• perator along w, T w by

T w

s,tb(x) =

t

s

b(x + wr)dr By a change of variable θt = xt − wt, we have the following equivalent equation θu = θ0 + t (∂uT w

u b)(θu)du =: θ0 +

t T w

dub(θu)

Rémi Catellier (CHL - IRMAR) October 24, 2014 4 / 21

For b : Rd → Rd et w : [0, T] → Rd we want to study the following equation xt = x0 + t b(xu)du + ws

Definition

For continuous vector fields b from Rd to Rd we define the averaging

• perator along w, T w by

T w

s,tb(x) =

t

s

b(x + wr)dr By a change of variable θt = xt − wt, we have the following equivalent equation θu = θ0 + t (∂uT w

u b)(θu)du =: θ0 +

t T w

dub(θu)

Rémi Catellier (CHL - IRMAR) October 24, 2014 4 / 21

T w

s,tb(x) =

t

s

b(x + wr)dr dθt = T w

dtb(θt)

Questions

What is the needed regularity of T wb for existence and uniqueness in the previous equation ? What are the admissible w for such a goal ? Is it possible to enhanced the averaging operator T w for more irregular vector fields ?

Rémi Catellier (CHL - IRMAR) October 24, 2014 5 / 21

T w

s,tb(x) =

t

s

b(x + wr)dr dθt = T w

dtb(θt)

Questions

What is the needed regularity of T wb for existence and uniqueness in the previous equation ? What are the admissible w for such a goal ? Is it possible to enhanced the averaging operator T w for more irregular vector fields ?

Rémi Catellier (CHL - IRMAR) October 24, 2014 5 / 21

T w

s,tb(x) =

t

s

b(x + wr)dr dθt = T w

dtb(θt)

Questions

What is the needed regularity of T wb for existence and uniqueness in the previous equation ? What are the admissible w for such a goal ? Is it possible to enhanced the averaging operator T w for more irregular vector fields ?

Rémi Catellier (CHL - IRMAR) October 24, 2014 5 / 21

T w

s,tb(x) =

t b(x + wu)du, θt − θ0 = t T w

drb(θr)

Theorem (Catellier, Gubinelli)

Let ψ(z) = 1 + log1/2(1 + z), z > 0. Let us suppose that there exists C > 0 and γ > 1

2 such that

i T wb has the following regularity |T w

s,tb(x) − T w s,tb(y)| ≤ C|t − s|γ|x − y|ψ(|x| + |y|)

then the equation has a solution in Cγ([0, T], Rd). ii If b ∈ Cb(Rd; Rd) it is enough to ask that |∇T w

s,tb(x) − ∇T w s,tb(y)| ≤ C|t − s|γ|x − y|1/2 ˜

ψ(|x| + |y|) iii Furthermore, when the solution is unique, it is locally Lipschitz continuous in the initial value, uniformly in time.

Rémi Catellier (CHL - IRMAR) October 24, 2014 6 / 21

T w

s,tb(x) =

t b(x + wu)du, θt − θ0 = t T w

drb(θr)

Theorem (Catellier, Gubinelli)

Let ψ(z) = 1 + log1/2(1 + z), z > 0. Let us suppose that there exists C > 0 and γ > 1

2 such that

i T wb has the following regularity |∇T w

s,tb(x) − ∇T w s,tb(y)| ≤ C|t − s|γ|x − y| ˜

ψ(|x| + |y|) then the equation has a unique solution in Cγ([0, T], Rd). ii If b ∈ Cb(Rd; Rd) it is enough to ask that |∇T w

s,tb(x) − ∇T w s,tb(y)| ≤ C|t − s|γ|x − y|1/2 ˜

ψ(|x| + |y|) iii Furthermore, when the solution is unique, it is locally Lipschitz continuous in the initial value, uniformly in time.

Rémi Catellier (CHL - IRMAR) October 24, 2014 6 / 21

T w

s,tb(x) =

t b(x + wu)du, θt − θ0 = t T w

drb(θr)

Theorem (Catellier, Gubinelli)

Let ψ(z) = 1 + log1/2(1 + z), z > 0. Let us suppose that there exists C > 0 and γ > 1

2 such that

i T wb has the following regularity |∇T w

s,tb(x) − ∇T w s,tb(y)| ≤ C|t − s|γ|x − y| ˜

ψ(|x| + |y|) then the equation has a unique solution in Cγ([0, T], Rd). ii If b ∈ Cb(Rd; Rd) it is enough to ask that |∇T w

s,tb(x) − ∇T w s,tb(y)| ≤ C|t − s|γ|x − y|1/2 ˜

ψ(|x| + |y|) iii Furthermore, when the solution is unique, it is locally Lipschitz continuous in the initial value, uniformly in time.

Rémi Catellier (CHL - IRMAR) October 24, 2014 6 / 21

T w

s,tb(x) =

t b(x + wu)du, θt − θ0 = t T w

drb(θr)

Theorem (Catellier, Gubinelli)

Let ψ(z) = 1 + log1/2(1 + z), z > 0. Let us suppose that there exists C > 0 and γ > 1

2 such that

i T wb has the following regularity |∇T w

s,tb(x) − ∇T w s,tb(y)| ≤ C|t − s|γ|x − y| ˜

ψ(|x| + |y|) then the equation has a unique solution in Cγ([0, T], Rd). ii If b ∈ Cb(Rd; Rd) it is enough to ask that |∇T w

s,tb(x) − ∇T w s,tb(y)| ≤ C|t − s|γ|x − y|1/2 ˜

ψ(|x| + |y|) iii Furthermore, when the solution is unique, it is locally Lipschitz continuous in the initial value, uniformly in time.

Rémi Catellier (CHL - IRMAR) October 24, 2014 6 / 21

1

Regularization of ODEs

2

The operator T w : (ρ, γ)-irregular functions

3

Stochastic processes

Rémi Catellier (CHL - IRMAR) October 24, 2014 7 / 21

(ρ, γ)-irregular functions

Let us define Φw

s,t(k) =

t

s eiwu.kdk, then

• T w

s,tb(x) − T w s,tb(y)

• =
• Rd dkˆ

b(k)

• eik.x − eik.y

Φw

t (k)

• |x − y|ν|t − s|γ
• Rd

|ˆ b(k)| (1 + |k|)ρ−ν dkΦwWρ,γ

T

Averaging constant : ΦwWρ,γ

T

= sup

s=t∈[0,T]; ξ∈Rd

(1 + |k|)ρ |t − s|γ

• Φw

s,t(k)

• Definition

Let ρ > 0 and γ ∈ [0, 1]. The function w from [0, T] to Rd is (ρ, γ)-irregular if ΦwWρ,γ

T

< +∞.

Rémi Catellier (CHL - IRMAR) October 24, 2014 8 / 21

(ρ, γ)-irregular functions

Let us define Φw

s,t(k) =

t

s eiwu.kdk, then

• T w

s,tb(x) − T w s,tb(y)

• =
• Rd dkˆ

b(k)

• eik.x − eik.y

Φw

t (k)

• |x − y|ν|t − s|γ
• Rd

|ˆ b(k)| (1 + |k|)ρ−ν dkΦwWρ,γ

T

Averaging constant : ΦwWρ,γ

T

= sup

s=t∈[0,T]; ξ∈Rd

(1 + |k|)ρ |t − s|γ

• Φw

s,t(k)

• Definition

Let ρ > 0 and γ ∈ [0, 1]. The function w from [0, T] to Rd is (ρ, γ)-irregular if ΦwWρ,γ

T

< +∞.

Rémi Catellier (CHL - IRMAR) October 24, 2014 8 / 21

Property

Let w a (ρ, γ)-irregular function, α ∈ R and FLα :=

• b ∈ S′ : bFLα :=
• Rd |ˆ

b(k)|(1 + |k|)αdk < +∞

• .

Then T w is a bounded operator from FLα into Cγ([0, T]; FLα+ρ) and T w

s,tbCγFLα+ρ bFLαΦwWρ,γ

T .

Remarks

When α > 0 we have FLα ⊂ Cα

b ⊂ Cb(Rd; Rd). When α < 0,

b ∈ FLα is only a Schwarz distribution. The same property is true for the Sobolev spaces Hα, α ∈ R.

Rémi Catellier (CHL - IRMAR) October 24, 2014 9 / 21

Property

Let w a (ρ, γ)-irregular function, α ∈ R and FLα :=

• b ∈ S′ : bFLα :=
• Rd |ˆ

b(k)|(1 + |k|)αdk < +∞

• .

Then T w is a bounded operator from FLα into Cγ([0, T]; FLα+ρ) and T w

s,tbCγFLα+ρ bFLαΦwWρ,γ

T .

Remarks

When α > 0 we have FLα ⊂ Cα

b ⊂ Cb(Rd; Rd). When α < 0,

b ∈ FLα is only a Schwarz distribution. The same property is true for the Sobolev spaces Hα, α ∈ R.

Rémi Catellier (CHL - IRMAR) October 24, 2014 9 / 21

Property

Let w a (ρ, γ)-irregular function, α ∈ R and FLα :=

• b ∈ S′ : bFLα :=
• Rd |ˆ

b(k)|(1 + |k|)αdk < +∞

• .

Then T w is a bounded operator from FLα into Cγ([0, T]; FLα+ρ) and T w

s,tbCγFLα+ρ bFLαΦwWρ,γ

T .

Remarks

When α > 0 we have FLα ⊂ Cα

b ⊂ Cb(Rd; Rd). When α < 0,

b ∈ FLα is only a Schwarz distribution. The same property is true for the Sobolev spaces Hα, α ∈ R.

Rémi Catellier (CHL - IRMAR) October 24, 2014 9 / 21

The space of (ρ, γ)-irregular functions

The name "irregular" is justify by the following proposition, due to Chouk and Gubinelli

Proposition (Chouk, Gubinelli)

If w is (ρ, γ)-irregular and w ∈ Cν([0, T]) with ν + γ > 1 then ν < 1−γ

ρ .

If we add a smooth function to a (ρ, γ)-irregular one, one can prove the following property

Proposition

If ϕ is Lipschitz continuous and w is (ρ, γ)-irregular, then w + ϕ is (ρ − δ, γ)-irregular for all δ > 0 such that δ + γ > 1.

Rémi Catellier (CHL - IRMAR) October 24, 2014 10 / 21

The space of (ρ, γ)-irregular functions

The name "irregular" is justify by the following proposition, due to Chouk and Gubinelli

Proposition (Chouk, Gubinelli)

If w is (ρ, γ)-irregular and w ∈ Cν([0, T]) with ν + γ > 1 then ν < 1−γ

ρ .

If we add a smooth function to a (ρ, γ)-irregular one, one can prove the following property

Proposition

If ϕ is Lipschitz continuous and w is (ρ, γ)-irregular, then w + ϕ is (ρ − δ, γ)-irregular for all δ > 0 such that δ + γ > 1.

Rémi Catellier (CHL - IRMAR) October 24, 2014 10 / 21

0.5 1 2 3 4

ρ(=

1 2H )

−3 −1 −1 .5 1

α H = 1

2

H = 1

4

H = 1

4

fractional Brownian motion C 1 functions functions distributions α = 3 / 2 − ρ α = 2 − ρ α = − ρ α = 1 − ρ E x i s t e n c e

• f

T

w

b E x i s t e n c e

• f

s

• l

u t i

• n

s U n i q u e n e s s

• f

s

• l

u t i

• n

s

Rémi Catellier (CHL - IRMAR) October 24, 2014 11 / 21

1

Regularization of ODEs

2

The operator T w : (ρ, γ)-irregular functions

3

Stochastic processes

Rémi Catellier (CHL - IRMAR) October 24, 2014 12 / 21

Fractional Brownian motion

Definition

Soit H ∈ (0, 1). The fractional Brownian motion of Hurst parameter H is the centered Gaussian process, BH, with stationary increments and with co-variance function E[BH

t BH s ] = 1

2(|t|2H + |s|2H − |t − s|2H)

Proposition

The fractional Brownian motion is self-similar of order H, BH

λt =L λHBH t .

Almost surely, for all 0 < ε < H the fractional Brownian motion is H − ε Hölder-continuous. For H = 1/2 we have the standard Brownian motion.

Rémi Catellier (CHL - IRMAR) October 24, 2014 13 / 21

Fractional Brownian motion

Definition

Soit H ∈ (0, 1). The fractional Brownian motion of Hurst parameter H is the centered Gaussian process, BH, with stationary increments and with co-variance function E[BH

t BH s ] = 1

2(|t|2H + |s|2H − |t − s|2H)

Proposition

The fractional Brownian motion is self-similar of order H, BH

λt =L λHBH t .

Almost surely, for all 0 < ε < H the fractional Brownian motion is H − ε Hölder-continuous. For H = 1/2 we have the standard Brownian motion.

Rémi Catellier (CHL - IRMAR) October 24, 2014 13 / 21

Rémi Catellier (CHL - IRMAR) October 24, 2014 14 / 21

Non-degenerate α-stable Lévy processes

Definition

let α ∈ (0, 2]. An α-stable Lévy process is a stochastic process X with independent and stationary increments, almost surely right continuous and self-similar of order 1/α. It is non-degenerate if there exists a constant C > 0 such that E[eik.Xt] ≤ e−Ct|k|α

Rémi Catellier (CHL - IRMAR) October 24, 2014 15 / 21

Processus Stochastiques

Theorem

Let BH a d-dimensional fractional Brownian motion of Hurst parameter H ∈ (0, 1). For all ρ <

1 2H there exists a γ > 1 2 such that BH is almost

surely (ρ, γ)-irregular. Furthermore there exists λ > 0 such that E[exp(λΦBH2

Wρ,γ

T )] < +∞.

Theorem

Let α ∈ (0, 2], ρ < α/2 and X a d-dimensional non-degenerate α-stable Lévy process. There exists γ > 1/2 such that X is almost surely (ρ, γ)-irregular. The integrability of ΦXWρ,γ

T

is comparable to the one of supt∈[0,T] |Xt|.

indices optimaux

Rémi Catellier (CHL - IRMAR) October 24, 2014 16 / 21

Processus Stochastiques

Theorem

Let BH a d-dimensional fractional Brownian motion of Hurst parameter H ∈ (0, 1). For all ρ <

1 2H there exists a γ > 1 2 such that BH is almost

surely (ρ, γ)-irregular. Furthermore there exists λ > 0 such that E[exp(λΦBH2

Wρ,γ

T )] < +∞.

Theorem

Let α ∈ (0, 2], ρ < α/2 and X a d-dimensional non-degenerate α-stable Lévy process. There exists γ > 1/2 such that X is almost surely (ρ, γ)-irregular. The integrability of ΦXWρ,γ

T

is comparable to the one of supt∈[0,T] |Xt|.

indices optimaux

Rémi Catellier (CHL - IRMAR) October 24, 2014 16 / 21

Heuristic of the proof

Let w a self-similar process of order H with stationary increments. Then Φw

s,t(k) =L eik.ws|k|−1/H

(t−s)|k|1/H eiwv dv ∼

• t − s

|k|1/H For large ξ, the result is almost a CLT, which explains the value of γ > 1/2.

Rémi Catellier (CHL - IRMAR) October 24, 2014 17 / 21

1/4 1/3 1/2 1

H

−2 −1 1

α C 1 functions Existence of the flow, random drifts Existence of T wb Existence, uniqueness for deterministic drifts Functions Distributions

Brownian Motion

Rémi Catellier (CHL - IRMAR) October 24, 2014 18 / 21

Generalization : Besov-Hölder sapces

We similar proofs, it is possible to extend the results for function or distributions in the Besov-Hölder spaces.

Definition

For n ∈ N and α ∈ (0, 1] we define the Besov-Hölder space of order α − n as Cα−n =

• f ∈ S′ : ∃g ∈ Cα such that f = Dng
• .

Rémi Catellier (CHL - IRMAR) October 24, 2014 19 / 21

Besov-Hölder vector fields

We are able to study the function T BHb when b is in the Besov-Hölder space b ∈ Cα with α > −1/2H. For α > −1/2H, ν ∈ [0, 1] et b ∈ Cα+ν, there exists γ > 1/2 and a random variable Cb,BH such that

• T BH

s,t b(x) − T BH s,t b(y)

• ≤ Cb,BHbCα|t − s|γ|x − y|νψ(|x| + |y|).

where ψ(z) = 1 + log1/2(1 + z). The random variable Cb,BH has Gaussiens moments independent of b. The full probability set where Cb,BH is finite depends on b.

Rémi Catellier (CHL - IRMAR) October 24, 2014 20 / 21

Besov-Hölder vector fields

We are able to study the function T BHb when b is in the Besov-Hölder space b ∈ Cα with α > −1/2H. For α > −1/2H, ν ∈ [0, 1] et b ∈ Cα+ν, there exists γ > 1/2 and a random variable Cb,BH such that

• T BH

s,t b(x) − T BH s,t b(y)

• ≤ Cb,BHbCα|t − s|γ|x − y|νψ(|x| + |y|).

where ψ(z) = 1 + log1/2(1 + z). The random variable Cb,BH has Gaussiens moments independent of b. The full probability set where Cb,BH is finite depends on b.

Rémi Catellier (CHL - IRMAR) October 24, 2014 20 / 21

Besov-Hölder vector fields

We are able to study the function T BHb when b is in the Besov-Hölder space b ∈ Cα with α > −1/2H. For α > −1/2H, ν ∈ [0, 1] et b ∈ Cα+ν, there exists γ > 1/2 and a random variable Cb,BH such that

• T BH

s,t b(x) − T BH s,t b(y)

• ≤ Cb,BHbCα|t − s|γ|x − y|νψ(|x| + |y|).

where ψ(z) = 1 + log1/2(1 + z). The random variable Cb,BH has Gaussiens moments independent of b. The full probability set where Cb,BH is finite depends on b.

Rémi Catellier (CHL - IRMAR) October 24, 2014 20 / 21

Besov-Hölder vector fields

We are able to study the function T BHb when b is in the Besov-Hölder space b ∈ Cα with α > −1/2H. For α > −1/2H, ν ∈ [0, 1] et b ∈ Cα+ν, there exists γ > 1/2 and a random variable Cb,BH such that

• T BH

s,t b(x) − T BH s,t b(y)

• ≤ Cb,BHbCα|t − s|γ|x − y|νψ(|x| + |y|).

where ψ(z) = 1 + log1/2(1 + z). The random variable Cb,BH has Gaussiens moments independent of b. The full probability set where Cb,BH is finite depends on b.

Rémi Catellier (CHL - IRMAR) October 24, 2014 20 / 21

Thank you for your attention.

Rémi Catellier (CHL - IRMAR) October 24, 2014 21 / 21