Harnack Chains and Control Problems in Hypoelliptic Partial - - PowerPoint PPT Presentation

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Harnack Chains and Control Problems in Hypoelliptic Partial Differential Equations

Sergio Polidoro

Universit` a di Modena e Reggio Emilia

Paris - September 29 - October 3, 2014

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Degenerate equations

Consider the PDE in Ω ⊆ RN × R: L u(x, t) :=

m

• k=1

X 2

k u(x, t) + X0u(x, t) − ∂tu(x, t) = 0,

Xk(x) :=

N

• j=1

ak

j (x)∂xj

ak

j ∈ C ∞(Ω),

k = 0, . . . , m.

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Degenerate equations

Consider the PDE in Ω ⊆ RN × R: L u(x, t) :=

m

• k=1

X 2

k u(x, t) + X0u(x, t) − ∂tu(x, t) = 0,

Xk(x) :=

N

• j=1

ak

j (x)∂xj

ak

j ∈ C ∞(Ω),

k = 0, . . . , m. Control problem for

◮ γ′(t) = m j=1 ωj(t)Xj(γ(t)) +

• X0(γ(t)) − ∂t
• Harnack Chains and Control Problems in Hypoelliptic PDEs

Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Focus on:

◮ Harnack inequalities

◮ [Cinti, Nystr¨

• m, P.] (2010)

◮ [Cinti, Menozzi, P.] (2014) ◮ [Kogoj, Pinchover, P.] (submitted)

◮ Asymptotic bounds for positive solutions

◮ [Boscain, P.] (2007), [Cinti, P.] (2008) ◮ [Cinti, Menozzi, P.] (2014) ◮ [Garofalo, P.] (in progress)

◮ Boundary Harnack inequality for Kolmogorov equations

◮ [Cinti, Nystr¨

• m, P.] (2012)

◮ [Cinti, Nystr¨

• m, P.] (2013)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Plan of the talk

◮ Harnack inequalities for Parabolic Equations ◮ Harnack inequalities for Degenerate PDEs ◮ Asymptotic bounds for Degenerate PDEs ◮ Boundary Harnack inequality for Kolmogorov Equations

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Euclidean setting

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

1 - Parabolic equations

Theorem ([Pini] - 1954, [Hadamard] - 1954)

Let Qr(x, t) = B(x, r)×]t − r 2, t[⊂ Rn+1, and let α, β, γ, δ ∈]0, 1[ with α + β + γ < 1. Then there exists C = C(α, β, γ, δ, n) such that sup

Q−

r (x,t)

u ≤ C inf

Q+

r (x,t)

u for every u : Qr(x, t) → R, u ≥ 0, satisfying ut = ∆u.

b

(x, t) r 2 βr 2 δr γr 2 δr αr 2

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Equivalent formulation

Theorem

For any compact set K ⊂ Q1(0, 0) there exists CK > 0: sup

(x,t)+δr K

u ≤ CKu(x, t) for every non-negative caloric function u : Qr(x, t) → R.

b

(x, t) (x, t) + δrK ≈ r 2

δr(x, t) = (rx, r 2t)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

2 - Bounds of the fundamental solution

Theorem ([Nash - 1958] [Moser - 1964] [Aronson, Serrin - 1967])

Let Γ be the fundamental solution of ∂t −

N

• i,j=1

∂xi

• aij(x, t)∂xj
• in

RN × R. Then there exist two positive constants c, C such that c (t − τ)

N 2

e−C |x−ξ|2

t−τ

≤ Γ(x, t, ξ, τ) ≤ C (t − τ)

N 2

e−c |x−ξ|2

t−τ . Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Uniqueness in the Cauchy problem

Corollary

Let u be a solution to

• ∂tu(x, t) − N

i,j=1 ∂xi

• aij(x, t)∂xju(x, t)
• (x, t) ∈ RN×]0, T[,

u(x, 0) = 0 x ∈ RN.

◮ Upper bound ⇒ If |u(x, t)| ≤ MeC|x|2 in RN×]0, T[, then

u ≡ 0.

◮ Lower bound ⇒ If u(x, t) ≥ 0 in RN×]0, T[, then u ≡ 0.

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

3 - Boundary behavior

◮ Divergence form Parabolic Equations

◮ [Fabes and Kenig] (1981) ◮ [Fabes and Stroock] (1986) ◮ [Garofalo] (1984) ◮ [Krylov and Safonov] (1980) ◮ [Fabes, Safonov and Yuan] (1999)

◮ Non Divergence form

◮ Fabes, Garofalo and Salsa (1986) ◮ [Fabes, Safonov (1997) ◮ [Nystr¨

• m] (1997)

◮ Divergence and non Divergence

◮ [Bauman] (1984) ◮ [Caffarelli, Fabes, Mortola and Salsa] (1981) ◮ [Fabes, Garofalo, Marin-Malave and Salsa] (1988) ◮ [Jerison and Kenig] (1982) Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Boundary Harnack inequality

Theorem ([Salsa] - 1981)

Let Σ be a Lipschitz subset of the parabolic boundary of Q1(0, 0), and let K be a compact subset of Q1(0, 0) such that K ∩ ∂Q1(0, 0) ⊂ Σ. Then there exists CK,Σ > 0: sup

Kr(x,t)

u ≤ CK,Σu(x, t) for every non-negative solution u : Qr(x, t) → R to ∆u = ut vanishing at Σr(x, t).

b

(x, t) Kr(x, t) Σr(x, t)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Harnack inequalities for Degenerate Partial Differential Equations

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Degenerate equations

Consider the PDE in RN × R: L u(x, t) =

m

• k=1

X 2

k u(x, t) + X0u(x, t) − ∂tu(x, t),

Xk(x) =

N

• j=1

ak

j (x)∂xj

ak

j ∈ C ∞,

k = 0, . . . , m. Examples:

◮ L :=

• ∂x + 2y∂s

2 +

• ∂y − 2x∂s

2 − ∂t, (x, y, s) ∈ R3

◮ L := ∂2 x + x∂y − ∂t,

(x, y) ∈ R2

◮ L := ∂2 x + x2∂y − ∂t,

(x, y) ∈ R2

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Regularity

Theorem ([H¨

• rmander] - 1967)

Let u be a (distributional) solution to L u = f in Ω ⊂ RN × R. If span

• X0 − ∂t, X1, . . . , Xm, [Xi, Xj], . . . , [Xi, . . . , [Xj, Xl]]
• = RN+1

Then f ∈ C ∞(Ω) ⇒ u ∈ C ∞(Ω). Commutators: [Xi, Xj]f := XiXjf − XjXif

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Example

Kolmogorov operator: L := ∂2

x + x∂y − ∂t ◮ X1 = ∂x,

X0 = x∂y, X0 − ∂t ∼   x −1   X1 ∼   1   [X1, X0 − ∂t] ∼   1  

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Question

It is possible to prove the extend the parabolic result to degenerate Kolmogorov equations?

Theorem

For any compact set K ⊂ Q1(0, 0) there exists CK > 0: sup

Kr (x,t)

u ≤ CKu(x, t) for every non-negative solution u : Qr(x, t) → R to L u = 0?

b

(x, t) Kr(x, t) ≈ r 2

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Strong maximum principle

Theorem ([Bony] - 1969)

Let u : Q → R be a non-positive solution to uxx + xuy = ut. If u(0, 0, 0) = 0, then... t x y

b

(0, 0, 0)

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Strong maximum principle

Theorem ([Bony] - 1969)

Let u : Q → R be a non-positive solution to uxx + xuy = ut. If u(0, 0, 0) = 0, then... t x y

b

(0, 0, 0)

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Strong maximum principle

Theorem ([Bony] - 1969)

Let u : Q → R be a non-positive solution to uxx + xuy = ut. If u(0, 0, 0) = 0, then... t x y

b

(0, 0, 0)

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Strong maximum principle

Theorem ([Bony] - 1969)

Let u : Q → R be a non-positive solution to uxx + xuy = ut. If u(0, 0, 0) = 0, then... t x y

b

(0, 0, 0) y

b

(0, 0, 0) ...u ≡ 0 in the Propagation set A(0,0).

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Answer

It is possible to prove the extend the parabolic result to degenerate Kolmogorov equations?

Theorem

For any compact set K ⊂ Q1(0, 0) there exists CK > 0: sup

Kr (x,t)

u ≤ CKu(x, t) for every non-negative solution u : Qr(x, t) → R to L u = 0?

b

(x, t) Kr(x, t)

Harnack Chains and Control Problems in Hypoelliptic PDEs

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Answer

It is possible to prove the extend the parabolic result to degenerate Kolmogorov equations?

Theorem

For any compact set K ⊂ Q1(0, 0) there exists CK > 0: sup

Kr (x,t)

u ≤ CKu(x, t) for every non-negative solution u : Qr(x, t) → R to L u = 0?

b

(x, t) Kr(x, t)

b

(x, t) δrK(x, t)

Answer: No! if Kr(x, t) ⊆ A(x,t).

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Harnack inequality

Theorem ([Cinti, Nystr¨

• m, P.], [Cinti, Menozzi, P.])

For any compact set K ⊂ Int

• A(x,t)
• there exists CK > 0:

sup

Kr (x,t)

u ≤ CKu(x, t) for every non-negative solution u : Qr(x, t) → R to L u = 0.

b

(x, t) Kr(x, t) A(x, t)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Asymptotic bounds for Degenerate Partial Differential Equations

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Bounds of the fundamental solution

Theorem ([Nash - 1958] [Moser - 1964] [Aronson, Serrin - 1967])

Let Γ be the fundamental solution of ∂t −

N

• i,j=1

∂xi

• aij(x, t)∂xj
• in

RN × R. Then there exist two positive constants c, C such that c (t − τ)

N 2

e−C |x−ξ|2

t−τ

≤ Γ(x, t, ξ, τ) ≤ C (t − τ)

N 2

e−c |x−ξ|2

t−τ . Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Sub-Riemannian opeators

Theorem

Let Γ be the fundamental solution of L =

m

• j=1

X 2

j − ∂t,

X0 = 0 Then there exist two positive constants c, C such that c (t − τ)

Q−2 2

e−C dc (x,ξ)2

t−τ

≤ Γ(x, t, ξ, τ) ≤ C (t − τ)

Q−2 2

e−c dc (x,ξ)2

t−τ .

◮ [Jerison, S´

anchez-Calle] (1986)

◮ [Kusuoka, Stroock] (1987) ◮ [Varopoulos, Saloff-Coste, Coulhon] (1992)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Harnack chains

Proof of the lower bound [Moser] (1964) [Aronson, Serrin] (1967) Γ(x, t, 0, 0) ≥

c t

N 2 e−C |x|2 t

b

(x0, t0)

b

(x, t)

b b b

(x1, t1)

b

(x2, t2)

b

(x3, t3)

b b b

Harnack chains require

◮ dilations, ◮ translations, ◮ “lines”

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Kolmogorov equations

If uxx(x, y, t) + xuy(x, y, t) = ut(x, y, t), then:

◮ dilations: v(x, y, t) = u(rx, r 3y, r 2t)

satisfies vxx + xvy = vt

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Kolmogorov equations

If uxx(x, y, t) + xuy(x, y, t) = ut(x, y, t), then:

◮ dilations: v(x, y, t) = u(rx, r 3y, r 2t)

satisfies vxx + xvy = vt

◮ translations: w(x, y, t) = u(x + ξ, y + η − ξt, t + τ)

satisfies wxx + xwy = wt

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Kolmogorov equations

If uxx(x, y, t) + xuy(x, y, t) = ut(x, y, t), then:

◮ dilations: v(x, y, t) = u(rx, r 3y, r 2t)

satisfies vxx + xvy = vt

◮ translations: w(x, y, t) = u(x + ξ, y + η − ξt, t + τ)

satisfies wxx + xwy = wt Invariant Harnack inequality

b

(0, 0, 0) r 2

b b

(ξ, η, τ)

b

(x0, y0, t0)

b b

(x0 + ξ, y0 + η − τx0, t0 + τ)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Kolmogorov equations

If uxx(x, y, t) + xuy(x, y, t) = ut(x, y, t), then:

◮ dilations: v(x, y, t) = u(rx, r 3y, r 2t)

satisfies vxx + xvy = vt

◮ translations: w(x, y, t) = u(x + ξ, y + η − ξt, t + τ)

satisfies wxx + xwy = wt Invariant Harnack inequality

b

(0, 0, 0) r 2

b b

(ξ, η, τ)

b

(x0, y0, t0)

b b

(x0 + ξ, y0 + η − τx0, t0 + τ)

◮ “lines”: L := m j=1 X 2 j + X0 − ∂t

γ′(t) = m

j=1 ωj(t)Xj(γ(t)) +

• X0(γ(t)) − ∂t
• Harnack Chains and Control Problems in Hypoelliptic PDEs

Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Control problem 1/2

Theorem ([Boscain, P.] (2007))

Assume that L := m

j=1 X 2 j + X0 − ∂t is hypoelliptic, dilation and

translation invariant.

◮ Let u be a positive solution to L u = 0 in RN×]T0, T1[, and

T0 < t < t0 < T1

◮ Let γ be a path such that γ(0) = (x0, t0), γ(t0 − t) = (x, t)

and γ′(s) = m

j=1 ωj(s)Xj(γ(s)) +

• X0(γ(s)) − ∂t
• ◮ Then u(x, t) ≤ C0 exp
• C

t0−t |ω(s)|2ds

• u(x0, t0)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Control problem 2/2

Value function. Find an L -admissible path minimizing the “cost” V (x, t, x0, t0) := inf

ω

t0−t |ω(s)|2ds

Theorem ([Boscain, P.] (2007))

◮ Let Γ be the fundamental solution of L ◮ Then Γ(x, t, x0, t0) ≥ C − t

Q−2 2 e−c−V (x,t,x0,t0) Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Control problem 2/2

Value function. Find an L -admissible path minimizing the “cost” V (x, t, x0, t0) := inf

ω

t0−t |ω(s)|2ds

Theorem ([Boscain, P.] (2007))

◮ Let Γ be the fundamental solution of L ◮ Then Γ(x, t, x0, t0) ≥ C − t

Q−2 2 e−c−V (x,t,x0,t0)

Theorem ([Cinti, P.] (2008))

◮ Let Γ be the fundamental solution of L . Assume that the

• ptimal control problem has no abnormal extremal

◮ Then Γ(x, t, x0, t0) ≤ C + t

Q−2 2 e−c+V (x,t,x0,t0) Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Upper bound

◮ Moser’s iteration. ◮ Value function V is a viscosity solution to the

Hamilton-Jacobi-Bellman equation:

• X0 − ∂t
• V +

m

• j=1
• XjV

2 = 0

◮ The Aronson’s methods requires that V is a distributional

solution of the above PDE. We use a theorem in [Cannarsa, Rifford] (2008).

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

General Kolmogorov equations 1/2

Let Wt be an m-dimensional Wiener process, σ a constant N × m matrix, and B a constant N × N matrix. Consider the linear SDE dXt = σdWt + BXtdt, Xt ∈ RN. The linear PDEs (x, t) ∈ RN+1 L = 1

2div

• σσT ∇
• + Bx, ∇ − ∂t

is the Kolmogorov equation of the SDE.

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

General Kolmogorov equations 2/2

L = div (A∇) + Bx, ∇ − ∂t, A = 1

2σσT ≥ 0

The following statements are equivalent

◮ H¨

• rmarder condition:

rank Lie

• ∂x1, . . . , ∂xm, Bx, ∇ − ∂t
• = N + 1

◮ Kalman condition: rank

• A, BA, . . . , BN−1A
• = N

◮

C(t) = t e−sBAe−sBT ds> > > 0

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

General Kolmogorov equations 2/2

L = div (A∇) + Bx, ∇ − ∂t, A = 1

2σσT ≥ 0

The following statements are equivalent

◮ H¨

• rmarder condition:

rank Lie

• ∂x1, . . . , ∂xm, Bx, ∇ − ∂t
• = N + 1

◮ Kalman condition: rank

• A, BA, . . . , BN−1A
• = N

◮

C(t) = t e−sBAe−sBT ds> > > 0 Γ(x, t) = (4π)−N/2

• det C(t)

e− 1

4 C−1(t)x,x

is the fundamental solution of L

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Equation x2uxx + xuy = ut

Arithmetic Average American Options in the Black & Scholes setting (In preparation, with Cibelli and Rossi)      ˙ x(s) = ωx(s) ˙ y(s) = x(s) ˙ t(s) = −1

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Equation uxx + x2uy = ut

[Cinti, Menozzi, P.] (2014) Consider Xt = Wt, Yt = t |Xs|2ds The joint density of (Xt, Yt) ∈ Rn+1 satisfies ∆xu + |x|2uy = ut Yt is a χ2 random variable We prove upper and lower bounds for the density of (Xt, Yt).

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Equation uxx + x2uy = ut

[Cinti, Menozzi, P.] (2014) Consider Xt = Wt, Yt = t |Xs|2ds The joint density of (Xt, Yt) ∈ Rn+1 satisfies ∆xu + |x|2uy = ut Yt is a χ2 random variable We prove upper and lower bounds for the density of (Xt, Yt). Known results

◮ [Kac] (1949) Laplace transform, ◮ [Borodin, Salminen] (2002) special functions (for n = 1) ◮ [Smirnov] (1936), [Tolmatz] (2000) related results

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Asymptotic estimates

Theorem ([Cinti, Menozzi, P.] (2014))

Let Γ denote the fundamental solution of ∂xx + x2∂y − ∂t.

◮ If η − y ≤ 0, then Γ(x, y, t, ξ, η, τ) = 0;

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Asymptotic estimates

Theorem ([Cinti, Menozzi, P.] (2014))

Let Γ denote the fundamental solution of ∂xx + x2∂y − ∂t.

◮ If η − y ≤ 0, then Γ(x, y, t, ξ, η, τ) = 0; ◮ if η−y (t−τ)2 > x2+ξ2 t−τ + 1, then

Γ(x, y, t, ξ, η, τ) ≈ 1 (t − τ)5/2 exp

• −C

(x − ξ)2 t − τ + η − y (t − τ)2

• ;

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Asymptotic estimates

Theorem ([Cinti, Menozzi, P.] (2014))

Let Γ denote the fundamental solution of ∂xx + x2∂y − ∂t.

◮ If η − y ≤ 0, then Γ(x, y, t, ξ, η, τ) = 0; ◮ if η−y (t−τ)2 > x2+ξ2 t−τ + 1, then

Γ(x, y, t, ξ, η, τ) ≈ 1 (t − τ)5/2 exp

• −C

(x − ξ)2 t − τ + η − y (t − τ)2

• ;

◮ if 0 < η−y (t−τ)2 < 1 2, then

Γ(x, y, t, ξ, η, τ) ≈ 1 (t − τ)5/2 exp

• −C

x4 + ξ4 + (t − τ)2 η − y

• .

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Method of the proof

◮ Diagonal estimates: Malliavin Calculus ◮ Upper bounds: Malliavin Calculus ◮ Lower bounds: Harnack Chains

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Method of the proof

◮ Diagonal estimates: Malliavin Calculus ◮ Upper bounds: Malliavin Calculus ◮ Lower bounds: Harnack Chains

New difficulties:

◮ Lifting: u satisfies uxx + x2uy = ut

⇐ ⇒ v(x, y, w, t) = u(x, y, t) satisfies vxx + xvw + x2vy = vt

◮ Dilations: δλ(x, y, w, t) = (λx, λ4y, λ3w, λ2t) ◮ Harnack inequality

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

uxx + xuy = ut

Kolmogorov equation: A(0,0,0)

t x y

b

(0, 0, 0)

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

uxx + xuy = ut

Kolmogorov equation: A(0,0,0)

t x y

b

(0, 0, 0) y t

b

• x = 0
• Harnack Chains and Control Problems in Hypoelliptic PDEs

Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

uxx + xuw + x2uy = ut

Equation uxx + xuw + x2uy = ut A(0,0,0,0) =

• (x, w, y, t) ∈] − 1, 1[4 | 0 ≤ y ≤ −t, w2 ≤ −ty
• t

y w

b

(0, 0, 0)

• x = 0
• Harnack Chains and Control Problems in Hypoelliptic PDEs

Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

More general operators

Our method applies to: L = ∆x + |x|k∂y − ∂t, k even, x ∈ Rn L = ∆x +

n

• j=1

xk

j ∂y − ∂t,

k ∈ N, x ∈ Rn

◮ Lifting: [Bonfiglioli, Lanconelli] (2009) Note that we don’t

need the explicit formula for the group law when using L -admissible paths;

◮ Interior points of Az0: we can easily find points such that the

differential of the end point map is surjective.

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Remarks

◮ Abnormal extremal [Tr´

elat] (2000)      ˙ x(s) = ωx(s) ˙ y(s) = x2(s) ˙ t(s) = −1

◮ [Ben Arous, L´

eandre] (1991) X 1

t = x1 + W 1 t

X 2

t = x2 +

t (X 1

s )mdW 2 s +

t (X 1

s )kds

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Equation uxx + x2uy + xuw = ut

Semiflexible polymers (In progress, with March and Garofalo)            ˙ x(s) = ωx(s) ˙ y(s) = x2(s) ˙ w(s) = x(s) ˙ t(s) = −1 Implies (w1 − w0)2 ≤ (y1 − y0)(t0 − t1)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Boundary regularity for Kolmogorov Equations

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Obstacle problem and Boundary regularity

◮ Obstacle problem: weak solutions [Di Francesco, Pascucci, P.]

(2008), [Pascucci] (2008)

◮ Optimal local regularity [Frentz, Nystrom, Pascucci, P.]

(2010), [Nystr¨

• m, Pascucci, P.] (2010)

◮ Boundary Harnack inequalities, Carleson Estimates [Cinti,

Nystr¨

• m, P.] (2012-13)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Boundary Harnack inequality

Theorem ([Cinti, Nystr¨

• m, P.] - 2012)

Let Σ be a Lipschitz subset of the parabolic boundary of Q1(0, 0), and let K be a compact subset of Q1(0, 0) such that K ∩ ∂Q1(0, 0) ⊂ Σ. Then there exists CK,Σ > 0: sup

Kr(x,t)

u ≤ CK,Σu(x, t) for every non-negative solution u : Qr(x, t) → R to uxx + xuy = ut vanishing at Σr(x, t).

b

(x, t) Kr(x, t) Σr(x, t)

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Elliptic equation

b

(0, 0, 0)

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Elliptic equation

b

(0, 0, 0)

b

(0, 0, 0)

b x0

Barriers u(x) ≤ Cd(x, ∂Ω)α maxΩ∩Br u Harnack chains u(x) ≤ C

u(x0) d(x,∂Ω)β

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Kolmogorov equation

t x y

b

(0, 0, 0)

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Kolmogorov equation

t x y

b

(0, 0, 0)

b

(0, 0, 0)

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Kolmogorov equation

t x y

b

(0, 0, 0)

b

(0, 0, 0) y

b

(0, 0, 0)

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Kolmogorov equation

t x y

b

(0, 0, 0)

b

(0, 0, 0) y

b

(0, 0, 0)

b

(0, 0, 0) γ(s) = δs(1, 0, −1) = (s, 0, −s2) doesn’t belong to A(0,0). γ(s) = δs(1, 2/3, −1) = (s, 2/3s3, −s2) is L -admissible.

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Next steps

◮ Backward Harnack inequality; ◮ Quotient of solutions vanishing at the boundary; ◮ Green function, Harmonic measure; ◮ ...

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia

Introduction Harnack inequalities Asymptotic behavior Boundary regularity

Many thanks for your attention!

Harnack Chains and Control Problems in Hypoelliptic PDEs Universit` a di Modena e Reggio Emilia