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Presentation Weak solutions Open problems

Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Alessandra Lunardi in collaboration with G. Da Prato Modena, Sept. 9th, 2010

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

A family of OU operators in infinite dimensions

Lαϕ(x) = 1

2 Tr [Q1−αD2ϕ(x)]− 1 2x,Q−αDϕ(x)

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

A family of OU operators in infinite dimensions

Lαϕ(x) = 1

2 Tr [Q1−αD2ϕ(x)]− 1 2x,Q−αDϕ(x) x ∈ H = infinite dimensional separable Hilbert space, with norm | · |;

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

A family of OU operators in infinite dimensions

Lαϕ(x) = 1

2 Tr [Q1−αD2ϕ(x)]− 1 2x,Q−αDϕ(x) x ∈ H = infinite dimensional separable Hilbert space, with norm | · |; Q ∈ L(H) self-adjoint positive operator with finite trace;

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

A family of OU operators in infinite dimensions

Lαϕ(x) = 1

2 Tr [Q1−αD2ϕ(x)]− 1 2x,Q−αDϕ(x) x ∈ H = infinite dimensional separable Hilbert space, with norm | · |; Q ∈ L(H) self-adjoint positive operator with finite trace; 0 ≤ α ≤ 1.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

A family of OU operators in infinite dimensions

Lαϕ(x) = 1

2 Tr [Q1−αD2ϕ(x)]− 1 2x,Q−αDϕ(x) x ∈ H = infinite dimensional separable Hilbert space, with norm | · |; Q ∈ L(H) self-adjoint positive operator with finite trace; 0 ≤ α ≤ 1.

{ek : k ∈ N} = orthonormal basis in H such that Qek = λkek,

Dk = derivative in the direction of ek, xk = x,ek,

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

A family of OU operators in infinite dimensions

Lαϕ(x) = 1

2 Tr [Q1−αD2ϕ(x)]− 1 2x,Q−αDϕ(x) x ∈ H = infinite dimensional separable Hilbert space, with norm | · |; Q ∈ L(H) self-adjoint positive operator with finite trace; 0 ≤ α ≤ 1.

{ek : k ∈ N} = orthonormal basis in H such that Qek = λkek,

Dk = derivative in the direction of ek, xk = x,ek,

Lαϕ(x) = 1

2

∞

∑

k=1

λ1−α

k

Dkkϕ(x)− 1 2

∞

∑

k=1

λ−α

k

xkDkϕ(x).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Lα is the Kolmogorov operator of the stochastic problem

dXα(t,x) = −1 2Q−αXα(t,x)dt + Q(1−α)/2dW(t), Xα(0,x) = x, where W(t) is a standard cylindrical Wiener process in H.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Lα is the Kolmogorov operator of the stochastic problem

dXα(t,x) = −1 2Q−αXα(t,x)dt + Q(1−α)/2dW(t), Xα(0,x) = x, where W(t) is a standard cylindrical Wiener process in H. The associated transition Markov semigroups Tα(t) are the Ornstein-Uhlenbeck semigroups Tα(t)ϕ(x) = E[ϕ(Xα(t,x))] =

• H ϕ(y +e−tAα/2x)N Qt(dy),

t > 0,

ϕ ∈ Cb(H),

with A = Q−1, Qt : Q(I − e−tAα), N Qt = the Gaussian measure with mean 0 and covariance Qt.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Lα is the Kolmogorov operator of the stochastic problem

dXα(t,x) = −1 2Q−αXα(t,x)dt + Q(1−α)/2dW(t), Xα(0,x) = x, where W(t) is a standard cylindrical Wiener process in H. The associated transition Markov semigroups Tα(t) are the Ornstein-Uhlenbeck semigroups Tα(t)ϕ(x) = E[ϕ(Xα(t,x))] =

• H ϕ(y +e−tAα/2x)N Qt(dy),

t > 0,

ϕ ∈ Cb(H),

with A = Q−1, Qt : Q(I − e−tAα), N Qt = the Gaussian measure with mean 0 and covariance Qt. Important common feature. For 0 ≤ α ≤ 1, Tα(t) have the same invariant measure µ := NQ. That is,

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Lα is the Kolmogorov operator of the stochastic problem

dXα(t,x) = −1 2Q−αXα(t,x)dt + Q(1−α)/2dW(t), Xα(0,x) = x, where W(t) is a standard cylindrical Wiener process in H. The associated transition Markov semigroups Tα(t) are the Ornstein-Uhlenbeck semigroups Tα(t)ϕ(x) = E[ϕ(Xα(t,x))] =

• H ϕ(y +e−tAα/2x)N Qt(dy),

t > 0,

ϕ ∈ Cb(H),

with A = Q−1, Qt : Q(I − e−tAα), N Qt = the Gaussian measure with mean 0 and covariance Qt. Important common feature. For 0 ≤ α ≤ 1, Tα(t) have the same invariant measure µ := NQ. That is,

• H

Tα(t)ϕdµ =

• H ϕdµ,

t > 0, ϕ ∈ Cb(H).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The Gaussian measure µ = NQ is defined first on the “cylindrical subsets” of H, i.e. subsets of the type

A = {x ∈ H : (x1,...,xn) ∈ A},

n ∈ N, A ∈ B(Rn)

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The Gaussian measure µ = NQ is defined first on the “cylindrical subsets” of H, i.e. subsets of the type

A = {x ∈ H : (x1,...,xn) ∈ A},

n ∈ N, A ∈ B(Rn) by

µ(A) =

1

• (2π)nλ1 ·...·λn
• A

exp

• −

n

∑

k=1

x2

k

2λk

• dx,

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The Gaussian measure µ = NQ is defined first on the “cylindrical subsets” of H, i.e. subsets of the type

A = {x ∈ H : (x1,...,xn) ∈ A},

n ∈ N, A ∈ B(Rn) by

µ(A) =

1

• (2π)nλ1 ·...·λn
• A

exp

• −

n

∑

k=1

x2

k

2λk

• dx,

then it is extended to all Borel sets B(H) by the Caratheodory teorem.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The most popular among the operators Lα are L0 and L1:

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The most popular among the operators Lα are L0 and L1:

L0ϕ(x) = 1

2 Tr [QD2ϕ(x)]− 1 2x,Dϕ(x), is the operator that arises in the Malliavin calculus,

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The most popular among the operators Lα are L0 and L1:

L0ϕ(x) = 1

2 Tr [QD2ϕ(x)]− 1 2x,Dϕ(x), is the operator that arises in the Malliavin calculus, and T0(t)ϕ(x) :=

• H ϕ(y + e−t/2x)N (1−e−t)Q(dy),

t > 0,

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The most popular among the operators Lα are L0 and L1:

L0ϕ(x) = 1

2 Tr [QD2ϕ(x)]− 1 2x,Dϕ(x), is the operator that arises in the Malliavin calculus, and T0(t)ϕ(x) :=

• H ϕ(y + e−t/2x)N (1−e−t)Q(dy),

t > 0, while

L1ϕ(x) = 1

2 Tr [D2ϕ(x)]− 1 2x,ADϕ(x), (with A = Q−1) is the generator of the Ornstein-Uhlenbeck semigroup T1(t) with the best smoothing properties (e.g., books of Da Prato-Zabczyk)

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The most popular among the operators Lα are L0 and L1:

L0ϕ(x) = 1

2 Tr [QD2ϕ(x)]− 1 2x,Dϕ(x), is the operator that arises in the Malliavin calculus, and T0(t)ϕ(x) :=

• H ϕ(y + e−t/2x)N (1−e−t)Q(dy),

t > 0, while

L1ϕ(x) = 1

2 Tr [D2ϕ(x)]− 1 2x,ADϕ(x), (with A = Q−1) is the generator of the Ornstein-Uhlenbeck semigroup T1(t) with the best smoothing properties (e.g., books of Da Prato-Zabczyk) and T1(t)ϕ(x) :=

• H ϕ(y + e−tA/2x)N Q(I−e−tA)(dy),

t > 0.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

The most popular among the operators Lα are L0 and L1:

L0ϕ(x) = 1

2 Tr [QD2ϕ(x)]− 1 2x,Dϕ(x), is the operator that arises in the Malliavin calculus, and T0(t)ϕ(x) :=

• H ϕ(y + e−t/2x)N (1−e−t)Q(dy),

t > 0, while

L1ϕ(x) = 1

2 Tr [D2ϕ(x)]− 1 2x,ADϕ(x), (with A = Q−1) is the generator of the Ornstein-Uhlenbeck semigroup T1(t) with the best smoothing properties (e.g., books of Da Prato-Zabczyk) and T1(t)ϕ(x) :=

• H ϕ(y + e−tA/2x)N Q(I−e−tA)(dy),

t > 0. For instance, T1(t)(Bb(H)) ⊂ C∞

b (H) for t > 0 while T0(t) is not strong Feller.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Some literature about elliptic and parabolic equations in Hilbert spaces.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Some literature about elliptic and parabolic equations in Hilbert spaces. Pioneering contributions: Yu. Daleckij, Dokl. Akad. Nauk SSSR, 1966, L. Gross, JFA, 1965

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Some literature about elliptic and parabolic equations in Hilbert spaces. Pioneering contributions: Yu. Daleckij, Dokl. Akad. Nauk SSSR, 1966, L. Gross, JFA, 1965 Monographs: Yu. Daleckij and S. V. Fomin, Kluwer 1991, Z. M. Ma and M. R¨

• ckner, Springer–Verlag, 1992, S. Cerrai, Springer–Verlag, 2001, G. Da

Prato and J. Zabczyk, London Mathematical Society, 2002.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Some literature about elliptic and parabolic equations in Hilbert spaces. Pioneering contributions: Yu. Daleckij, Dokl. Akad. Nauk SSSR, 1966, L. Gross, JFA, 1965 Monographs: Yu. Daleckij and S. V. Fomin, Kluwer 1991, Z. M. Ma and M. R¨

• ckner, Springer–Verlag, 1992, S. Cerrai, Springer–Verlag, 2001, G. Da

Prato and J. Zabczyk, London Mathematical Society, 2002. Besides their own mathematical interests these equations are useful to get informations about the corresponding stochastic PDEs. They arise in several applications, see for instance S. Albeverio and M. R¨

• ckner, PTRF 1991 for

the Dirichlet forms approach, G. Da Prato and A. Debussche, Journal Math. Pures Appl. 2003, for 3D Navier–Stokes equations, L. Zambotti, PTRF 2000,

• L. Ambrosio, G. Savar´

e, and L. Zambotti, PTRF 2009, V. Barbu, G. Da Prato, and L. Tubaro, Ann. Probab. 2009, for reflection problems.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Most of the quoted contributions concern elliptic and parabolic equations on the whole H.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Most of the quoted contributions concern elliptic and parabolic equations on the whole H. A few papers concern Dirichlet or Neumann problems in subsets of H, such as G. Da Prato, B.Goldys and J. Zabczyk, CRAS 1997, A. Talarczyk, Studia

• Math. 2000, for Dirichlet type problems and the quoted papers by L.

Zambotti, L. Ambrosio, G. Savar´ e and L. Zambotti and V. Barbu, G. Da Prato, and L. Tubaro, for Neumann type problems.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Literature

Most of the quoted contributions concern elliptic and parabolic equations on the whole H. A few papers concern Dirichlet or Neumann problems in subsets of H, such as G. Da Prato, B.Goldys and J. Zabczyk, CRAS 1997, A. Talarczyk, Studia

• Math. 2000, for Dirichlet type problems and the quoted papers by L.

Zambotti, L. Ambrosio, G. Savar´ e and L. Zambotti and V. Barbu, G. Da Prato, and L. Tubaro, for Neumann type problems. The papers by G. Da Prato, B.Goldys and J. Zabczyk, A. Talarczyk, concern Dirichlet problems in spaces of continuous and bounded functions. Here we consider L2 spaces.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Associated Sobolev spaces

Eα(H) := the linear span of the real and imaginary parts of the functions

x → eix,h, with h ∈ D(Q−α), 0 ≤ α ≤ 1.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Associated Sobolev spaces

Eα(H) := the linear span of the real and imaginary parts of the functions

x → eix,h, with h ∈ D(Q−α), 0 ≤ α ≤ 1. Integration formula

• H ϕLαψdµ = −1

2

• HQ(1−α)/2Dϕ,Q(1−α)/2Dψdµ,

ϕ,ψ ∈ Eα(H).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Associated Sobolev spaces

Eα(H) := the linear span of the real and imaginary parts of the functions

x → eix,h, with h ∈ D(Q−α), 0 ≤ α ≤ 1. Integration formula

• H ϕLαψdµ = −1

2

• HQ(1−α)/2Dϕ,Q(1−α)/2Dψdµ,

ϕ,ψ ∈ Eα(H).

W 1,2

α (H,µ) := the completion of Eα(H) in the norm associated to the scalar

product

ϕ,ψW 1,2

α (H,µ)

:=

• H ϕψdµ+
• HQ(1−α)/2Dϕ,Q(1−α)/2Dψdµ

=

• H ϕψdµ+

∞

∑

k=1

• H λ1−α

k

DkϕDkψdµ.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Associated Sobolev spaces

Eα(H) := the linear span of the real and imaginary parts of the functions

x → eix,h, with h ∈ D(Q−α), 0 ≤ α ≤ 1. Integration formula

• H ϕLαψdµ = −1

2

• HQ(1−α)/2Dϕ,Q(1−α)/2Dψdµ,

ϕ,ψ ∈ Eα(H).

W 1,2

α (H,µ) := the completion of Eα(H) in the norm associated to the scalar

product

ϕ,ψW 1,2

α (H,µ)

:=

• H ϕψdµ+
• HQ(1−α)/2Dϕ,Q(1−α)/2Dψdµ

=

• H ϕψdµ+

∞

∑

k=1

• H λ1−α

k

DkϕDkψdµ. For α = 0, W 1,2

0 (H,µ) is the domain of the Malliavin derivative in L2(H,µ).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

W 2,2

α (H,µ) := the completion of Eα(H) in the norm associated to the scalar

product

ϕ,ψW 2,2

α (H,µ) := ϕ,ψW 1,2 α (H,µ) +

∞

∑

h,k=1

• H λ1−α

h

λ1−α

k

Dh,kϕDh,kψdµ.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

W 2,2

α (H,µ) := the completion of Eα(H) in the norm associated to the scalar

product

ϕ,ψW 2,2

α (H,µ) := ϕ,ψW 1,2 α (H,µ) +

∞

∑

h,k=1

• H λ1−α

h

λ1−α

k

Dh,kϕDh,kψdµ. If K ⊂ H is a Borel set, we define W 1,2

α (K,µ) and W 2,2 α (K,µ) in a similar way.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Weak solutions to Dirichlet problems

Let K ⊂ H be a closed set, with ˚ K = /

0, ˚

K c = /

0.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Weak solutions to Dirichlet problems

Let K ⊂ H be a closed set, with ˚ K = /

0, ˚

K c = /

• 0. We study weak solutions to

the Dirichlet problem (with λ > 0 and f ∈ L2(K,µ))

   λϕ(x)−Lαϕ(x) = f(x), in K, ϕ(x) = 0,

• n ∂K.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Weak solutions to Dirichlet problems

Let K ⊂ H be a closed set, with ˚ K = /

0, ˚

K c = /

• 0. We study weak solutions to

the Dirichlet problem (with λ > 0 and f ∈ L2(K,µ))

   λϕ(x)−Lαϕ(x) = f(x), in K, ϕ(x) = 0,

• n ∂K.

The setting is the Sobolev space ˚ W 1,2

α (K,µ) of the functions ϕ : K → R

whose null extension to the whole H belongs to W 1,2

α (H,µ).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Weak solutions to Dirichlet problems

Let K ⊂ H be a closed set, with ˚ K = /

0, ˚

K c = /

• 0. We study weak solutions to

the Dirichlet problem (with λ > 0 and f ∈ L2(K,µ))

   λϕ(x)−Lαϕ(x) = f(x), in K, ϕ(x) = 0,

• n ∂K.

The setting is the Sobolev space ˚ W 1,2

α (K,µ) of the functions ϕ : K → R

whose null extension to the whole H belongs to W 1,2

α (H,µ).

A weak solution is a function ϕ ∈ ˚ W 1,2

α (K,µ) such that

λ

• K ϕv dµ+ 1

2

• KQ(1−α)/2Dϕ,Q(1−α)/2Dvdµ =
• K

f v dµ,

∀v ∈ ˚

W 1,2

α (K,µ).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Weak solutions to Dirichlet problems

Let K ⊂ H be a closed set, with ˚ K = /

0, ˚

K c = /

• 0. We study weak solutions to

the Dirichlet problem (with λ > 0 and f ∈ L2(K,µ))

   λϕ(x)−Lαϕ(x) = f(x), in K, ϕ(x) = 0,

• n ∂K.

The setting is the Sobolev space ˚ W 1,2

α (K,µ) of the functions ϕ : K → R

whose null extension to the whole H belongs to W 1,2

α (H,µ).

A weak solution is a function ϕ ∈ ˚ W 1,2

α (K,µ) such that

λ

• K ϕv dµ+ 1

2

• KQ(1−α)/2Dϕ,Q(1−α)/2Dvdµ =
• K

f v dµ,

∀v ∈ ˚

W 1,2

α (K,µ).

The existence of a unique weak solution follows easily from the Lax-Milgram Lemma.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

The operator LK

α associated to the quadratic form

Q α(u,v) := 1

2

• KQ(1−α)/2Du,Q(1−α)/2Dvdµ,

u,v ∈ ˚ W 1,2

α (K,µ),

is dissipative and self-adjoint in L2(K,µ), so it generates an analytic semigroup T K

α (t).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

The operator LK

α associated to the quadratic form

Q α(u,v) := 1

2

• KQ(1−α)/2Du,Q(1−α)/2Dvdµ,

u,v ∈ ˚ W 1,2

α (K,µ),

is dissipative and self-adjoint in L2(K,µ), so it generates an analytic semigroup T K

α (t).

Approximation of T K

α (t)

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

The operator LK

α associated to the quadratic form

Q α(u,v) := 1

2

• KQ(1−α)/2Du,Q(1−α)/2Dvdµ,

u,v ∈ ˚ W 1,2

α (K,µ),

is dissipative and self-adjoint in L2(K,µ), so it generates an analytic semigroup T K

α (t).

Approximation of T K

α (t)

Representation formula for T K

α (t)

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

The operator LK

α associated to the quadratic form

Q α(u,v) := 1

2

• KQ(1−α)/2Du,Q(1−α)/2Dvdµ,

u,v ∈ ˚ W 1,2

α (K,µ),

is dissipative and self-adjoint in L2(K,µ), so it generates an analytic semigroup T K

α (t).

Approximation of T K

α (t)

Representation formula for T K

α (t)

Meaning of the Dirichlet boundary condition

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

The operator LK

α associated to the quadratic form

Q α(u,v) := 1

2

• KQ(1−α)/2Du,Q(1−α)/2Dvdµ,

u,v ∈ ˚ W 1,2

α (K,µ),

is dissipative and self-adjoint in L2(K,µ), so it generates an analytic semigroup T K

α (t).

Approximation of T K

α (t)

Representation formula for T K

α (t)

Meaning of the Dirichlet boundary condition Regularity of the weak solution: interior regularity, regularity up to the boundary if ∂K is nice

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Approximation of T K

α (t)

Let Lα be the generator of the O-U semigroup Tα(t) in L2(H,µ).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Approximation of T K

α (t)

Let Lα be the generator of the O-U semigroup Tα(t) in L2(H,µ). Fix ε > 0 and a bounded nonnegative continuous function V that vanishes in K and has positive values in K c, and set

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Approximation of T K

α (t)

Let Lα be the generator of the O-U semigroup Tα(t) in L2(H,µ). Fix ε > 0 and a bounded nonnegative continuous function V that vanishes in K and has positive values in K c, and set Mε

α : D(Mε α) = D(Lα) → L2(H,µ),

Mε

αϕ = Lαϕ− 1

ε Vϕ.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Approximation of T K

α (t)

Let Lα be the generator of the O-U semigroup Tα(t) in L2(H,µ). Fix ε > 0 and a bounded nonnegative continuous function V that vanishes in K and has positive values in K c, and set Mε

α : D(Mε α) = D(Lα) → L2(H,µ),

Mε

αϕ = Lαϕ− 1

ε Vϕ.

Let T ε

α(t) be the semigroup generated by Mε α.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Approximation of T K

α (t)

Let Lα be the generator of the O-U semigroup Tα(t) in L2(H,µ). Fix ε > 0 and a bounded nonnegative continuous function V that vanishes in K and has positive values in K c, and set Mε

α : D(Mε α) = D(Lα) → L2(H,µ),

Mε

αϕ = Lαϕ− 1

ε Vϕ.

Let T ε

α(t) be the semigroup generated by Mε α.

• Proposition. Let either α = 0 or ∃s ∈ (0,1) such that Tr Qs < ∞.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Approximation of T K

α (t)

Let Lα be the generator of the O-U semigroup Tα(t) in L2(H,µ). Fix ε > 0 and a bounded nonnegative continuous function V that vanishes in K and has positive values in K c, and set Mε

α : D(Mε α) = D(Lα) → L2(H,µ),

Mε

αϕ = Lαϕ− 1

ε Vϕ.

Let T ε

α(t) be the semigroup generated by Mε α.

• Proposition. Let either α = 0 or ∃s ∈ (0,1) such that Tr Qs < ∞.

Then for every ϕ in L2(K,µ) and t > 0, λ > 0, T K

α (t)ϕ = lim ε→0T ε α(t)

ϕ|K,

R(λ,LK

α)ϕ = lim ε→0R(λ,Mε α)

ϕ|K

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Approximation of T K

α (t)

Let Lα be the generator of the O-U semigroup Tα(t) in L2(H,µ). Fix ε > 0 and a bounded nonnegative continuous function V that vanishes in K and has positive values in K c, and set Mε

α : D(Mε α) = D(Lα) → L2(H,µ),

Mε

αϕ = Lαϕ− 1

ε Vϕ.

Let T ε

α(t) be the semigroup generated by Mε α.

• Proposition. Let either α = 0 or ∃s ∈ (0,1) such that Tr Qs < ∞.

Then for every ϕ in L2(K,µ) and t > 0, λ > 0, T K

α (t)ϕ = lim ε→0T ε α(t)

ϕ|K,

R(λ,LK

α)ϕ = lim ε→0R(λ,Mε α)

ϕ|K

where

• ϕ(x) = ϕ(x), x ∈ K,
• ϕ(x) = 0, x ∈ K c.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Representation formula for T K

α (t)

Consider the stochastic equation dXα(t,x) = −1 2Q−αXα(t,x)dt + Q(1−α)/2dW(t), X(0,x) = x. Here W(t) is a standard cylindrical Wiener process in H, defined in a filtered probability space (Ω,F ,(Ft)t≥0,P).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Representation formula for T K

α (t)

Consider the stochastic equation dXα(t,x) = −1 2Q−αXα(t,x)dt + Q(1−α)/2dW(t), X(0,x) = x. Here W(t) is a standard cylindrical Wiener process in H, defined in a filtered probability space (Ω,F ,(Ft)t≥0,P). Xα has a.s. continuous paths for α = 0, and also for α ∈ (0,1] if Tr Qs < ∞ for some s ∈ (0,1), that we assume here.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Representation formula for T K

α (t)

Consider the stochastic equation dXα(t,x) = −1 2Q−αXα(t,x)dt + Q(1−α)/2dW(t), X(0,x) = x. Here W(t) is a standard cylindrical Wiener process in H, defined in a filtered probability space (Ω,F ,(Ft)t≥0,P). Xα has a.s. continuous paths for α = 0, and also for α ∈ (0,1] if Tr Qs < ∞ for some s ∈ (0,1), that we assume here. Stopped semigroup. The stopped semigroup is defined on Bb(K) by Pα(t)ϕ(x) := E[ϕ(Xα(t,x))1lτx≥t] =

• {τx≥t} ϕ(Xα(t,x))dP,

∀ x ∈ K,

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Representation formula for T K

α (t)

Consider the stochastic equation dXα(t,x) = −1 2Q−αXα(t,x)dt + Q(1−α)/2dW(t), X(0,x) = x. Here W(t) is a standard cylindrical Wiener process in H, defined in a filtered probability space (Ω,F ,(Ft)t≥0,P). Xα has a.s. continuous paths for α = 0, and also for α ∈ (0,1] if Tr Qs < ∞ for some s ∈ (0,1), that we assume here. Stopped semigroup. The stopped semigroup is defined on Bb(K) by Pα(t)ϕ(x) := E[ϕ(Xα(t,x))1lτx≥t] =

• {τx≥t} ϕ(Xα(t,x))dP,

∀ x ∈ K,

where τx is the entrance time in the complement of K,

τx := inf{t ≥ 0 : Xα(t,x) ∈ K c}, ∀ x ∈ K.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

• Proposition. The measure µ is subinvariant for Pα(t), that is,
• K

Pα(t)ϕdµ ≤

• K ϕdµ,

t > 0, ϕ ∈ Bb(K), and T K

α (t) is the extension to L2(K,µ) of Pα(t).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

• Proposition. The measure µ is subinvariant for Pα(t), that is,
• K

Pα(t)ϕdµ ≤

• K ϕdµ,

t > 0, ϕ ∈ Bb(K), and T K

α (t) is the extension to L2(K,µ) of Pα(t).

Proof: by approximation with Feynman–Kac semigroups in the whole space H, Pε

α(t)ϕ(x) = E

• ϕ(Xα(t,x))e− 1

ε

t

0 V(Xα(s,x))ds

, ϕ ∈ L2(H,µ),

where V is a (fixed) bounded continuous function that vanishes in K and has positive values in H \ K.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

• Proposition. The measure µ is subinvariant for Pα(t), that is,
• K

Pα(t)ϕdµ ≤

• K ϕdµ,

t > 0, ϕ ∈ Bb(K), and T K

α (t) is the extension to L2(K,µ) of Pα(t).

Proof: by approximation with Feynman–Kac semigroups in the whole space H, Pε

α(t)ϕ(x) = E

• ϕ(Xα(t,x))e− 1

ε

t

0 V(Xα(s,x))ds

, ϕ ∈ L2(H,µ),

where V is a (fixed) bounded continuous function that vanishes in K and has positive values in H \ K. The generator of Pε

α(t) in L2(H,µ) is the operator

Mε

α : D(Mε α) = D(Lα) → L2(H,µ),

Mε

αϕ = Lαϕ− 1

ε Vϕ,

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

• Proposition. The measure µ is subinvariant for Pα(t), that is,
• K

Pα(t)ϕdµ ≤

• K ϕdµ,

t > 0, ϕ ∈ Bb(K), and T K

α (t) is the extension to L2(K,µ) of Pα(t).

Proof: by approximation with Feynman–Kac semigroups in the whole space H, Pε

α(t)ϕ(x) = E

• ϕ(Xα(t,x))e− 1

ε

t

0 V(Xα(s,x))ds

, ϕ ∈ L2(H,µ),

where V is a (fixed) bounded continuous function that vanishes in K and has positive values in H \ K. The generator of Pε

α(t) in L2(H,µ) is the operator

Mε

α : D(Mε α) = D(Lα) → L2(H,µ),

Mε

αϕ = Lαϕ− 1

ε Vϕ,

where Lα is the generator of Tα(t) in L2(H,µ).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Interior regularity

Regularity in the whole space

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Interior regularity

Regularity in the whole space In the whole H we have maximal Sobolev regularity.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Interior regularity

Regularity in the whole space In the whole H we have maximal Sobolev regularity.

• Proposition. For λ > 0 and f ∈ L2(H,µ), the unique weak solution ϕ to

λϕ−Lαϕ = f

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Interior regularity

Regularity in the whole space In the whole H we have maximal Sobolev regularity.

• Proposition. For λ > 0 and f ∈ L2(H,µ), the unique weak solution ϕ to

λϕ−Lαϕ = f

(i.e., the unique ϕ ∈ W 1,2

α (H,µ) such that

λ

• H ϕv dµ+ 1

2

• HQ(1−α)/2Dϕ,Q(1−α)/2Dvdµ =
• H

f v dµ,

∀v ∈ W 1,2

α (H,µ),

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Interior regularity

Regularity in the whole space In the whole H we have maximal Sobolev regularity.

• Proposition. For λ > 0 and f ∈ L2(H,µ), the unique weak solution ϕ to

λϕ−Lαϕ = f

(i.e., the unique ϕ ∈ W 1,2

α (H,µ) such that

λ

• H ϕv dµ+ 1

2

• HQ(1−α)/2Dϕ,Q(1−α)/2Dvdµ =
• H

f v dµ,

∀v ∈ W 1,2

α (H,µ),

belongs to W 2,2

α (H,µ), and ϕW 2,2

α

≤ CfL2.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Interior regularity

Regularity in the whole space In the whole H we have maximal Sobolev regularity.

• Proposition. For λ > 0 and f ∈ L2(H,µ), the unique weak solution ϕ to

λϕ−Lαϕ = f

(i.e., the unique ϕ ∈ W 1,2

α (H,µ) such that

λ

• H ϕv dµ+ 1

2

• HQ(1−α)/2Dϕ,Q(1−α)/2Dvdµ =
• H

f v dµ,

∀v ∈ W 1,2

α (H,µ),

belongs to W 2,2

α (H,µ), and ϕW 2,2

α

≤ CfL2.

(Meyer inequalities (1980’s) for α = 0, Da Prato and Goldys (1991) for

α ∈ (0,1]).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity in the interior of K

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity in the interior of K Here we assume that

Tr Q1−α =

∞

∑

k=1

λ1−α

k

< ∞.

(so, α = 1 is excluded).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity in the interior of K Here we assume that

Tr Q1−α =

∞

∑

k=1

λ1−α

k

< ∞.

(so, α = 1 is excluded). For λ > 0 and f ∈ L2(K,µ), consider the weak solution ϕ to λϕ−Lαϕ = f in K, ϕ = 0 at ∂K.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity in the interior of K Here we assume that

Tr Q1−α =

∞

∑

k=1

λ1−α

k

< ∞.

(so, α = 1 is excluded). For λ > 0 and f ∈ L2(K,µ), consider the weak solution ϕ to λϕ−Lαϕ = f in K, ϕ = 0 at ∂K.

• Proposition. For every closed ball B(y,r) ⊂ ˚

K, the restriction of ϕ to B(y,r) belongs to W 2,2

α (B(y,r),µ).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity in the interior of K Here we assume that

Tr Q1−α =

∞

∑

k=1

λ1−α

k

< ∞.

(so, α = 1 is excluded). For λ > 0 and f ∈ L2(K,µ), consider the weak solution ϕ to λϕ−Lαϕ = f in K, ϕ = 0 at ∂K.

• Proposition. For every closed ball B(y,r) ⊂ ˚

K, the restriction of ϕ to B(y,r) belongs to W 2,2

α (B(y,r),µ).

Proof: by localization and by approximation with the operators Mε

α.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regular boundaries and traces

Assume that, for a “good” C1 function g : H → R, K = {x ∈ H : g(x) ≤ 1}

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regular boundaries and traces

Assume that, for a “good” C1 function g : H → R, K = {x ∈ H : g(x) ≤ 1} (e.g., g ∈ ∩k∈N,p>1W k,p

0 (H,µ), |Q1/2Dg|−1 ∈ ∩p>1Lp(H,µ)).

Then the Malliavin surface measure dσ is well defined on ∂K.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regular boundaries and traces

Assume that, for a “good” C1 function g : H → R, K = {x ∈ H : g(x) ≤ 1} (e.g., g ∈ ∩k∈N,p>1W k,p

0 (H,µ), |Q1/2Dg|−1 ∈ ∩p>1Lp(H,µ)).

Then the Malliavin surface measure dσ is well defined on ∂K. Integration by parts formula.

• K

Dhϕdµ = 1

λh

• K

xhϕdµ+

• ∂K

Dhg

|Q1/2Dg|ϕdσ,

h ∈ N, ϕ ∈ E0(H).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Replacing ϕ by λhϕ2Dhg and summing over h we get

• ∂K ϕ2|Q1/2Dg|dσ =
• K ϕQ1/2Dϕ,Q1/2Dgdµ+
• K

L0g ϕ2 dµ.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Replacing ϕ by λhϕ2Dhg and summing over h we get

• ∂K ϕ2|Q1/2Dg|dσ =
• K ϕQ1/2Dϕ,Q1/2Dgdµ+
• K

L0g ϕ2 dµ. So, if |Q1/2Dg| is bounded in K, |Q1/2Dg| ≥ a > 0 in ∂K and L0g(x) has at most linear growth as |x| → ∞, x ∈ K, we have

• ∂K ϕ2 dσ ≤ Cϕ2

W 1,2 (K,µ)

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Replacing ϕ by λhϕ2Dhg and summing over h we get

• ∂K ϕ2|Q1/2Dg|dσ =
• K ϕQ1/2Dϕ,Q1/2Dgdµ+
• K

L0g ϕ2 dµ. So, if |Q1/2Dg| is bounded in K, |Q1/2Dg| ≥ a > 0 in ∂K and L0g(x) has at most linear growth as |x| → ∞, x ∈ K, we have

• ∂K ϕ2 dσ ≤ Cϕ2

W 1,2 (K,µ)

and we can define the trace at ∂K of any ϕ ∈ W 1,2

0 (K,µ) (hence, of any

ϕ ∈ W 1,2

α (K,µ)) by density, as an element of L2(∂K,σ).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Replacing ϕ by λhϕ2Dhg and summing over h we get

• ∂K ϕ2|Q1/2Dg|dσ =
• K ϕQ1/2Dϕ,Q1/2Dgdµ+
• K

L0g ϕ2 dµ. So, if |Q1/2Dg| is bounded in K, |Q1/2Dg| ≥ a > 0 in ∂K and L0g(x) has at most linear growth as |x| → ∞, x ∈ K, we have

• ∂K ϕ2 dσ ≤ Cϕ2

W 1,2 (K,µ)

and we can define the trace at ∂K of any ϕ ∈ W 1,2

0 (K,µ) (hence, of any

ϕ ∈ W 1,2

α (K,µ)) by density, as an element of L2(∂K,σ).

• Proposition. Let ϕ ∈ ˚

W 1,2

α (K,µ), 0 ≤ α ≤ 1. Then the trace of ϕ at ∂K

vanishes.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Replacing ϕ by λhϕ2Dhg and summing over h we get

• ∂K ϕ2|Q1/2Dg|dσ =
• K ϕQ1/2Dϕ,Q1/2Dgdµ+
• K

L0g ϕ2 dµ. So, if |Q1/2Dg| is bounded in K, |Q1/2Dg| ≥ a > 0 in ∂K and L0g(x) has at most linear growth as |x| → ∞, x ∈ K, we have

• ∂K ϕ2 dσ ≤ Cϕ2

W 1,2 (K,µ)

and we can define the trace at ∂K of any ϕ ∈ W 1,2

0 (K,µ) (hence, of any

ϕ ∈ W 1,2

α (K,µ)) by density, as an element of L2(∂K,σ).

• Proposition. Let ϕ ∈ ˚

W 1,2

α (K,µ), 0 ≤ α ≤ 1. Then the trace of ϕ at ∂K

vanishes. Consequence: our weak solutions have null trace at the boundary ∂K.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity up to the boundary

A finite dimensional result. H = RN (L.-Metafune-Pallara 2005)

Lϕ = ∆ϕ−DU,Dϕ,

µ(dx) = e−U(x)dx.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity up to the boundary

A finite dimensional result. H = RN (L.-Metafune-Pallara 2005)

Lϕ = ∆ϕ−DU,Dϕ,

µ(dx) = e−U(x)dx.

• Proposition. Let ∂K, U be smooth and

U is quasi-convex, i.e. ∃c ∈ R such that D2U(x)+ cI ≥ 0 in K;

∂U/∂n + H ≤ 0 at ∂K (H = mean curvature).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity up to the boundary

A finite dimensional result. H = RN (L.-Metafune-Pallara 2005)

Lϕ = ∆ϕ−DU,Dϕ,

µ(dx) = e−U(x)dx.

• Proposition. Let ∂K, U be smooth and

U is quasi-convex, i.e. ∃c ∈ R such that D2U(x)+ cI ≥ 0 in K;

∂U/∂n + H ≤ 0 at ∂K (H = mean curvature).

Then for each λ > 0 and f ∈ L2(K,µ) the weak solution to λϕ−Lϕ = f in K,

ϕ = 0 at ∂K, belongs to W 2,2(K,µ), and ϕW 2,2 ≤ CfL2.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Regularity up to the boundary

A finite dimensional result. H = RN (L.-Metafune-Pallara 2005)

Lϕ = ∆ϕ−DU,Dϕ,

µ(dx) = e−U(x)dx.

• Proposition. Let ∂K, U be smooth and

U is quasi-convex, i.e. ∃c ∈ R such that D2U(x)+ cI ≥ 0 in K;

∂U/∂n + H ≤ 0 at ∂K (H = mean curvature).

Then for each λ > 0 and f ∈ L2(K,µ) the weak solution to λϕ−Lϕ = f in K,

ϕ = 0 at ∂K, belongs to W 2,2(K,µ), and ϕW 2,2 ≤ CfL2.

We have L = 2L0 for U(x) = |x|2/2, Q = I. If K = {x ∈ RN : g(x) ≤ 1},

∂U/∂n + H ≤ 0 at ∂K iff −2L0g + D2g · Dg,Dg |Dg|2 ≤ 0 at ∂K.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

In infinite dimensions, let K = {x ∈ H : g(x) ≤ 1} with g as before, α = 0.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

In infinite dimensions, let K = {x ∈ H : g(x) ≤ 1} with g as before, α = 0.

• Theorem. Assume that the function

h := −2 L0g

|Q1/2Dg|2 + Q1/2D2gQ1/2 · Q1/2Dg,Q1/2Dg |Q1/2Dg|4

is bounded from above in ∂K.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

In infinite dimensions, let K = {x ∈ H : g(x) ≤ 1} with g as before, α = 0.

• Theorem. Assume that the function

h := −2 L0g

|Q1/2Dg|2 + Q1/2D2gQ1/2 · Q1/2Dg,Q1/2Dg |Q1/2Dg|4

is bounded from above in ∂K. Then for every λ > 0 and f ∈ L2(K,µ) the weak solution of

   λϕ(x)−L0ϕ(x) = f(x), in K, ϕ(x) = 0,

• n ∂K,

belongs to W 2,2

0 (K,µ), and ϕW 2,2 (K,µ) ≤ CfL2(K,µ).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

In infinite dimensions, let K = {x ∈ H : g(x) ≤ 1} with g as before, α = 0.

• Theorem. Assume that the function

h := −2 L0g

|Q1/2Dg|2 + Q1/2D2gQ1/2 · Q1/2Dg,Q1/2Dg |Q1/2Dg|4

is bounded from above in ∂K. Then for every λ > 0 and f ∈ L2(K,µ) the weak solution of

   λϕ(x)−L0ϕ(x) = f(x), in K, ϕ(x) = 0,

• n ∂K,

belongs to W 2,2

0 (K,µ), and ϕW 2,2 (K,µ) ≤ CfL2(K,µ).

Proof: by approximation with finite-dimensional problems.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Examples

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Examples K = {x ∈ H : x,b ≤ 1} halfspace, g(x) = x,b, with b = 0. h(x) = 1

|Q1/2b|2 ,

x ∈ H.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Examples K = {x ∈ H : x,b ≤ 1} halfspace, g(x) = x,b, with b = 0. h(x) = 1

|Q1/2b|2 ,

x ∈ H. h is constant =

⇒ maximal Sobolev regularity.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Examples K = {x ∈ H : x,b ≤ 1} halfspace, g(x) = x,b, with b = 0. h(x) = 1

|Q1/2b|2 ,

x ∈ H. h is constant =

⇒ maximal Sobolev regularity.

K = B(0,1), g(x) = |x|2. h(x) = 1−TrQ

|Q1/2x|2 + |Qx|2 |Q1/2x|4

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Examples K = {x ∈ H : x,b ≤ 1} halfspace, g(x) = x,b, with b = 0. h(x) = 1

|Q1/2b|2 ,

x ∈ H. h is constant =

⇒ maximal Sobolev regularity.

K = B(0,1), g(x) = |x|2. h(x) = 1−TrQ

|Q1/2x|2 + |Qx|2 |Q1/2x|4 ≤

1

|Q1/2x|2

• 1−TrQ +λ1
• ,

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Examples K = {x ∈ H : x,b ≤ 1} halfspace, g(x) = x,b, with b = 0. h(x) = 1

|Q1/2b|2 ,

x ∈ H. h is constant =

⇒ maximal Sobolev regularity.

K = B(0,1), g(x) = |x|2. h(x) = 1−TrQ

|Q1/2x|2 + |Qx|2 |Q1/2x|4 ≤

1

|Q1/2x|2

• 1−TrQ +λ1
• ,

λ1 = maximum eigenvalue of Q.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Examples K = {x ∈ H : x,b ≤ 1} halfspace, g(x) = x,b, with b = 0. h(x) = 1

|Q1/2b|2 ,

x ∈ H. h is constant =

⇒ maximal Sobolev regularity.

K = B(0,1), g(x) = |x|2. h(x) = 1−TrQ

|Q1/2x|2 + |Qx|2 |Q1/2x|4 ≤

1

|Q1/2x|2

• 1−TrQ +λ1
• ,

λ1 = maximum eigenvalue of Q. h is upperly bounded if TrQ ≥ 1+λ1.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems Sobolev spaces Approximation of TK α (t) Representation formula for TK α (t) Interior regularity Regular boundaries and traces Regularity up to the boundary

Examples K = {x ∈ H : x,b ≤ 1} halfspace, g(x) = x,b, with b = 0. h(x) = 1

|Q1/2b|2 ,

x ∈ H. h is constant =

⇒ maximal Sobolev regularity.

K = B(0,1), g(x) = |x|2. h(x) = 1−TrQ

|Q1/2x|2 + |Qx|2 |Q1/2x|4 ≤

1

|Q1/2x|2

• 1−TrQ +λ1
• ,

λ1 = maximum eigenvalue of Q. h is upperly bounded if TrQ ≥ 1+λ1.

In this case we have maximal Sobolev regularity.

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems

Open questions

About Sobolev spaces: if ∂K is good, does ˚ W 1,2

α (K,µ) coincide with the

space of the functions f ∈ W 1,2

α (K,µ) whose trace at the boundary

vanishes?

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems

Open questions

About Sobolev spaces: if ∂K is good, does ˚ W 1,2

α (K,µ) coincide with the

space of the functions f ∈ W 1,2

α (K,µ) whose trace at the boundary

vanishes? About traces: if ∂K is good, which is the range of the trace operator W 1,2

α (K,µ) → L2(∂K,dσ)?

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems

Open questions

About Sobolev spaces: if ∂K is good, does ˚ W 1,2

α (K,µ) coincide with the

space of the functions f ∈ W 1,2

α (K,µ) whose trace at the boundary

vanishes? About traces: if ∂K is good, which is the range of the trace operator W 1,2

α (K,µ) → L2(∂K,dσ)?

About the Dirichlet semigroups T K

α (t): are they strong Feller? I.e., do

they map Bb(K) into Cb(K)?

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems

Open questions

About Sobolev spaces: if ∂K is good, does ˚ W 1,2

α (K,µ) coincide with the

space of the functions f ∈ W 1,2

α (K,µ) whose trace at the boundary

vanishes? About traces: if ∂K is good, which is the range of the trace operator W 1,2

α (K,µ) → L2(∂K,dσ)?

About the Dirichlet semigroups T K

α (t): are they strong Feller? I.e., do

they map Bb(K) into Cb(K)? About interior Sobolev regularity: What happens if our condition

∑∞

k=1 λ1−α k

< ∞ is violated? Are there counterexamples to W 2,2

α

local regularity?

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

Presentation Weak solutions Open problems

Open questions

About Sobolev spaces: if ∂K is good, does ˚ W 1,2

α (K,µ) coincide with the

space of the functions f ∈ W 1,2

α (K,µ) whose trace at the boundary

vanishes? About traces: if ∂K is good, which is the range of the trace operator W 1,2

α (K,µ) → L2(∂K,dσ)?

About the Dirichlet semigroups T K

α (t): are they strong Feller? I.e., do

they map Bb(K) into Cb(K)? About interior Sobolev regularity: What happens if our condition

∑∞

k=1 λ1−α k

< ∞ is violated? Are there counterexamples to W 2,2

α

local regularity? About Sobolev regularity up to the boundary: What happens if the function h is unbounded from above on ∂K? Are there counterexamples to W 2,2

α

regularity? (open even in finite dimensions).

Alessandra Lunardi in collaboration with G. Da Prato Dirichlet problems for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces