Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Surface measures in Banach spaces
Alessandra Lunardi
Dipartimento di Scienze Matematiche, Fisiche e Informatiche
Università di Parma
Operator Semigroups in Analysis: Modern Developments,
Bȩdlewo April 24-28, 2017
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
X = Banach space, γ = Borel probability measure in X
AIM: define reasonable surface measures, such that there exist integration by
parts formulae for good functions on domains Ω ⊂ X with good boundary.
Finite dimensional case:
2
X = Rd , γ = standard Gaussian measure (2π)−d /2 e−|x | /2 dx.
For any open set Ω with smooth boundary, h ∈ Rd , f ∈ Cb1 (Rd , R) we have
Z
Z
∂h f d γ =
Ω
Ω
hh, x if d γ +
Z
2 /2
hn, hif (2π)−d /2 e−|x |
∂Ω
n(x ) = exterior normal vector to ∂Ω at x.
Alessandra Lunardi
Surface measures in Banach spaces
dHd −1 ,
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
X = Banach space, γ = Borel probability measure in X
AIM: define reasonable surface measures, such that there exist integration by
parts formulae for good functions on domains Ω ⊂ X with good boundary.
Finite dimensional case:
2
X = Rd , γ = standard Gaussian measure (2π)−d /2 e−|x | /2 dx.
For any open set Ω with smooth boundary, h ∈ Rd , f ∈ Cb1 (Rd , R) we have
Z
Z
∂h f d γ =
Ω
Ω
hh, x if d γ +
Z
2 /2
hn, hif (2π)−d /2 e−|x |
∂Ω
n(x ) = exterior normal vector to ∂Ω at x.
Alessandra Lunardi
Surface measures in Banach spaces
dHd −1 ,
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
X = Banach space, γ = Borel probability measure in X
AIM: define reasonable surface measures, such that there exist integration by
parts formulae for good functions on domains Ω ⊂ X with good boundary.
Finite dimensional case:
2
X = Rd , γ = standard Gaussian measure (2π)−d /2 e−|x | /2 dx.
For any open set Ω with smooth boundary, h ∈ Rd , f ∈ Cb1 (Rd , R) we have
Z
Z
∂h f d γ =
Ω
Ω
hh, x if d γ +
Z
2 /2
hn, hif (2π)−d /2 e−|x |
∂Ω
n(x ) = exterior normal vector to ∂Ω at x.
Alessandra Lunardi
Surface measures in Banach spaces
dHd −1 ,
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
X = Banach space, γ = Borel probability measure in X
AIM: define reasonable surface measures, such that there exist integration by
parts formulae for good functions on domains Ω ⊂ X with good boundary.
Finite dimensional case:
2
X = Rd , γ = standard Gaussian measure (2π)−d /2 e−|x | /2 dx.
For any open set Ω with smooth boundary, h ∈ Rd , f ∈ Cb1 (Rd , R) we have
Z
Z
∂h f d γ =
Ω
Ω
hh, x if d γ +
Z
2 /2
hn, hif (2π)−d /2 e−|x |
∂Ω
n(x ) = exterior normal vector to ∂Ω at x.
Alessandra Lunardi
Surface measures in Banach spaces
dHd −1 ,
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
X = Banach space, γ = Borel probability measure in X
AIM: define reasonable surface measures, such that there exist integration by
parts formulae for good functions on domains Ω ⊂ X with good boundary.
Finite dimensional case:
2
X = Rd , γ = standard Gaussian measure (2π)−d /2 e−|x | /2 dx.
For any open set Ω with smooth boundary, h ∈ Rd , f ∈ Cb1 (Rd , R) we have
Z
Z
∂h f d γ =
Ω
Ω
hh, x if d γ +
Z
2 /2
hn, hif (2π)−d /2 e−|x |
∂Ω
n(x ) = exterior normal vector to ∂Ω at x.
Alessandra Lunardi
Surface measures in Banach spaces
dHd −1 ,
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Different constructions, in infinite dimension.
Skorohod (1974), Uglanov (1981–2001), Airault–Malliavin (1988), Feyel-De la
Pradelle (1992), Pugachev (1998–2009), Da Prato, L., Tubaro (2014),
Bogachev, Malofeev (2016), . . .
Airault–Malliavin (1988): X = {f ∈ C ([0, 1]; R) : f (0) = 0}, γ = Wiener
measure. Construction of surface measures on the level sets of good
functions, through the image measures approach;
Feyel–De la Pradelle (1992): X = separable Banach space, γ = any
non degenerate centered Gaussian measure. Construction of an
(∞ − 1)-dimensional Hausdorff measure on the Borel sets of X , through
finite dimensional approximations.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Different constructions, in infinite dimension.
Skorohod (1974), Uglanov (1981–2001), Airault–Malliavin (1988), Feyel-De la
Pradelle (1992), Pugachev (1998–2009), Da Prato, L., Tubaro (2014),
Bogachev, Malofeev (2016), . . .
Airault–Malliavin (1988): X = {f ∈ C ([0, 1]; R) : f (0) = 0}, γ = Wiener
measure. Construction of surface measures on the level sets of good
functions, through the image measures approach;
Feyel–De la Pradelle (1992): X = separable Banach space, γ = any
non degenerate centered Gaussian measure. Construction of an
(∞ − 1)-dimensional Hausdorff measure on the Borel sets of X , through
finite dimensional approximations.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Different constructions, in infinite dimension.
Skorohod (1974), Uglanov (1981–2001), Airault–Malliavin (1988), Feyel-De la
Pradelle (1992), Pugachev (1998–2009), Da Prato, L., Tubaro (2014),
Bogachev, Malofeev (2016), . . .
Airault–Malliavin (1988): X = {f ∈ C ([0, 1]; R) : f (0) = 0}, γ = Wiener
measure. Construction of surface measures on the level sets of good
functions, through the image measures approach;
Feyel–De la Pradelle (1992): X = separable Banach space, γ = any
non degenerate centered Gaussian measure. Construction of an
(∞ − 1)-dimensional Hausdorff measure on the Borel sets of X , through
finite dimensional approximations.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Different constructions, in infinite dimension.
Skorohod (1974), Uglanov (1981–2001), Airault–Malliavin (1988), Feyel-De la
Pradelle (1992), Pugachev (1998–2009), Da Prato, L., Tubaro (2014),
Bogachev, Malofeev (2016), . . .
Airault–Malliavin (1988): X = {f ∈ C ([0, 1]; R) : f (0) = 0}, γ = Wiener
measure. Construction of surface measures on the level sets of good
functions, through the image measures approach;
Feyel–De la Pradelle (1992): X = separable Banach space, γ = any
non degenerate centered Gaussian measure. Construction of an
(∞ − 1)-dimensional Hausdorff measure on the Borel sets of X , through
finite dimensional approximations.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Gaussian measures
X = separable Banach space, with norm k · k,
γ = centered non-degenerate Gaussian measure in X , with Cameron-Martin
space H.
a Gaussian measure γ is a Borel probability measure such that for each
f ∈ X ∗ , B 7→ γ(f −1 (B )) is a Gaussian measure in R;
R
γ is called centered if X f (x )γ(dx ) = 0 for every f ∈ X ∗ ; non-degenerate
if supp γ = X ;
H = {h ∈ X : the measure γh (A) := γ(A − h) is absolutely continuous
with respect to γ};
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Gaussian measures
X = separable Banach space, with norm k · k,
γ = centered non-degenerate Gaussian measure in X , with Cameron-Martin
space H.
a Gaussian measure γ is a Borel probability measure such that for each
f ∈ X ∗ , B 7→ γ(f −1 (B )) is a Gaussian measure in R;
R
γ is called centered if X f (x )γ(dx ) = 0 for every f ∈ X ∗ ; non-degenerate
if supp γ = X ;
H = {h ∈ X : the measure γh (A) := γ(A − h) is absolutely continuous
with respect to γ};
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Gaussian measures
X = separable Banach space, with norm k · k,
γ = centered non-degenerate Gaussian measure in X , with Cameron-Martin
space H.
a Gaussian measure γ is a Borel probability measure such that for each
f ∈ X ∗ , B 7→ γ(f −1 (B )) is a Gaussian measure in R;
R
γ is called centered if X f (x )γ(dx ) = 0 for every f ∈ X ∗ ; non-degenerate
if supp γ = X ;
H = {h ∈ X : the measure γh (A) := γ(A − h) is absolutely continuous
with respect to γ};
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Gaussian measures
X = separable Banach space, with norm k · k,
γ = centered non-degenerate Gaussian measure in X , with Cameron-Martin
space H.
a Gaussian measure γ is a Borel probability measure such that for each
f ∈ X ∗ , B 7→ γ(f −1 (B )) is a Gaussian measure in R;
R
γ is called centered if X f (x )γ(dx ) = 0 for every f ∈ X ∗ ; non-degenerate
if supp γ = X ;
H = {h ∈ X : the measure γh (A) := γ(A − h) is absolutely continuous
with respect to γ};
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Gaussian measures
X = separable Banach space, with norm k · k,
γ = centered non-degenerate Gaussian measure in X , with Cameron-Martin
space H.
a Gaussian measure γ is a Borel probability measure such that for each
f ∈ X ∗ , B 7→ γ(f −1 (B )) is a Gaussian measure in R;
R
γ is called centered if X f (x )γ(dx ) = 0 for every f ∈ X ∗ ; non-degenerate
if supp γ = X ;
H = {h ∈ X : the measure γh (A) := γ(A − h) is absolutely continuous
with respect to γ};
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Cameron-Martin space
H is the subspace of all h ∈ X such that there exists (a unique) ĥ ∈ L1 (X , γ)
which satisfies
Z
Z
∂h f d γ = f ĥ d γ, f ∈ Cb1 (X ; R).
X
X
The functions ĥ belong in fact to Lp (X , γ) for every p ∈ [1, +∞), and to the
closure of X ∗ in L2 (X , γ).
(Recall that for X = Rd , γ = standard Gaussian measure, we have
ĥ(x ) = hh, x i. )
H is a Hilbert space with the scalar product
hh, k iH := hĥ, k̂ iL2 (X ,γ) .
It is continuously embedded in X , and the embedding is compact.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Cameron-Martin space
H is the subspace of all h ∈ X such that there exists (a unique) ĥ ∈ L1 (X , γ)
which satisfies
Z
Z
∂h f d γ = f ĥ d γ, f ∈ Cb1 (X ; R).
X
X
The functions ĥ belong in fact to Lp (X , γ) for every p ∈ [1, +∞), and to the
closure of X ∗ in L2 (X , γ).
(Recall that for X = Rd , γ = standard Gaussian measure, we have
ĥ(x ) = hh, x i. )
H is a Hilbert space with the scalar product
hh, k iH := hĥ, k̂ iL2 (X ,γ) .
It is continuously embedded in X , and the embedding is compact.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Cameron-Martin space
H is the subspace of all h ∈ X such that there exists (a unique) ĥ ∈ L1 (X , γ)
which satisfies
Z
Z
∂h f d γ = f ĥ d γ, f ∈ Cb1 (X ; R).
X
X
The functions ĥ belong in fact to Lp (X , γ) for every p ∈ [1, +∞), and to the
closure of X ∗ in L2 (X , γ).
(Recall that for X = Rd , γ = standard Gaussian measure, we have
ĥ(x ) = hh, x i. )
H is a Hilbert space with the scalar product
hh, k iH := hĥ, k̂ iL2 (X ,γ) .
It is continuously embedded in X , and the embedding is compact.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Cameron-Martin space
H is the subspace of all h ∈ X such that there exists (a unique) ĥ ∈ L1 (X , γ)
which satisfies
Z
Z
∂h f d γ = f ĥ d γ, f ∈ Cb1 (X ; R).
X
X
The functions ĥ belong in fact to Lp (X , γ) for every p ∈ [1, +∞), and to the
closure of X ∗ in L2 (X , γ).
(Recall that for X = Rd , γ = standard Gaussian measure, we have
ĥ(x ) = hh, x i. )
H is a Hilbert space with the scalar product
hh, k iH := hĥ, k̂ iL2 (X ,γ) .
It is continuously embedded in X , and the embedding is compact.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Sobolev spaces
For f ∈ Cb1 (X ; R) and x ∈ X , f 0 (x ) ∈ X ∗ ⊂ H ∗ ⇒ ∃! v ∈ H: f 0 (x )h = hv , hiH .
We set v =: ∇H f (x ).
So, ∇H f (x ) is the unique v ∈ H such that
f (x + h) − f (x ) = hv , hiH + o(|h|H ), for all h ∈ H.
Proposition. For each p ∈ [1, +∞), ∇H : Cb1 (X , R) 7→ Lp (X , γ; H ) is closable
in Lp (X , γ).
Definition. W 1,p (X , γ) is the domain of ∇H , still denoted by ∇H .
So, W 1,p (X , γ) is a Banach space with the graph norm
kf kW 1,p (X ,γ) = kf kLp (X ,γ) +
Z
X
1/p
|∇H f (x )|pH γ(dx )
,
and Cb1 (X , R) is dense in W 1,p (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Sobolev spaces
For f ∈ Cb1 (X ; R) and x ∈ X , f 0 (x ) ∈ X ∗ ⊂ H ∗ ⇒ ∃! v ∈ H: f 0 (x )h = hv , hiH .
We set v =: ∇H f (x ).
So, ∇H f (x ) is the unique v ∈ H such that
f (x + h) − f (x ) = hv , hiH + o(|h|H ), for all h ∈ H.
Proposition. For each p ∈ [1, +∞), ∇H : Cb1 (X , R) 7→ Lp (X , γ; H ) is closable
in Lp (X , γ).
Definition. W 1,p (X , γ) is the domain of ∇H , still denoted by ∇H .
So, W 1,p (X , γ) is a Banach space with the graph norm
kf kW 1,p (X ,γ) = kf kLp (X ,γ) +
Z
X
1/p
|∇H f (x )|pH γ(dx )
,
and Cb1 (X , R) is dense in W 1,p (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Sobolev spaces
For f ∈ Cb1 (X ; R) and x ∈ X , f 0 (x ) ∈ X ∗ ⊂ H ∗ ⇒ ∃! v ∈ H: f 0 (x )h = hv , hiH .
We set v =: ∇H f (x ).
So, ∇H f (x ) is the unique v ∈ H such that
f (x + h) − f (x ) = hv , hiH + o(|h|H ), for all h ∈ H.
Proposition. For each p ∈ [1, +∞), ∇H : Cb1 (X , R) 7→ Lp (X , γ; H ) is closable
in Lp (X , γ).
Definition. W 1,p (X , γ) is the domain of ∇H , still denoted by ∇H .
So, W 1,p (X , γ) is a Banach space with the graph norm
kf kW 1,p (X ,γ) = kf kLp (X ,γ) +
Z
X
1/p
|∇H f (x )|pH γ(dx )
,
and Cb1 (X , R) is dense in W 1,p (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Sobolev spaces
For f ∈ Cb1 (X ; R) and x ∈ X , f 0 (x ) ∈ X ∗ ⊂ H ∗ ⇒ ∃! v ∈ H: f 0 (x )h = hv , hiH .
We set v =: ∇H f (x ).
So, ∇H f (x ) is the unique v ∈ H such that
f (x + h) − f (x ) = hv , hiH + o(|h|H ), for all h ∈ H.
Proposition. For each p ∈ [1, +∞), ∇H : Cb1 (X , R) 7→ Lp (X , γ; H ) is closable
in Lp (X , γ).
Definition. W 1,p (X , γ) is the domain of ∇H , still denoted by ∇H .
So, W 1,p (X , γ) is a Banach space with the graph norm
kf kW 1,p (X ,γ) = kf kLp (X ,γ) +
Z
X
1/p
|∇H f (x )|pH γ(dx )
,
and Cb1 (X , R) is dense in W 1,p (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Sobolev spaces
For f ∈ Cb1 (X ; R) and x ∈ X , f 0 (x ) ∈ X ∗ ⊂ H ∗ ⇒ ∃! v ∈ H: f 0 (x )h = hv , hiH .
We set v =: ∇H f (x ).
So, ∇H f (x ) is the unique v ∈ H such that
f (x + h) − f (x ) = hv , hiH + o(|h|H ), for all h ∈ H.
Proposition. For each p ∈ [1, +∞), ∇H : Cb1 (X , R) 7→ Lp (X , γ; H ) is closable
in Lp (X , γ).
Definition. W 1,p (X , γ) is the domain of ∇H , still denoted by ∇H .
So, W 1,p (X , γ) is a Banach space with the graph norm
kf kW 1,p (X ,γ) = kf kLp (X ,γ) +
Z
X
1/p
|∇H f (x )|pH γ(dx )
,
and Cb1 (X , R) is dense in W 1,p (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Sobolev spaces
For f ∈ Cb1 (X ; R) and x ∈ X , f 0 (x ) ∈ X ∗ ⊂ H ∗ ⇒ ∃! v ∈ H: f 0 (x )h = hv , hiH .
We set v =: ∇H f (x ).
So, ∇H f (x ) is the unique v ∈ H such that
f (x + h) − f (x ) = hv , hiH + o(|h|H ), for all h ∈ H.
Proposition. For each p ∈ [1, +∞), ∇H : Cb1 (X , R) 7→ Lp (X , γ; H ) is closable
in Lp (X , γ).
Definition. W 1,p (X , γ) is the domain of ∇H , still denoted by ∇H .
So, W 1,p (X , γ) is a Banach space with the graph norm
kf kW 1,p (X ,γ) = kf kLp (X ,γ) +
Z
X
1/p
|∇H f (x )|pH γ(dx )
,
and Cb1 (X , R) is dense in W 1,p (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Sobolev spaces
For f ∈ Cb1 (X ; R) and x ∈ X , f 0 (x ) ∈ X ∗ ⊂ H ∗ ⇒ ∃! v ∈ H: f 0 (x )h = hv , hiH .
We set v =: ∇H f (x ).
So, ∇H f (x ) is the unique v ∈ H such that
f (x + h) − f (x ) = hv , hiH + o(|h|H ), for all h ∈ H.
Proposition. For each p ∈ [1, +∞), ∇H : Cb1 (X , R) 7→ Lp (X , γ; H ) is closable
in Lp (X , γ).
Definition. W 1,p (X , γ) is the domain of ∇H , still denoted by ∇H .
So, W 1,p (X , γ) is a Banach space with the graph norm
kf kW 1,p (X ,γ) = kf kLp (X ,γ) +
Z
X
1/p
|∇H f (x )|pH γ(dx )
,
and Cb1 (X , R) is dense in W 1,p (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The spaces W k ,p (X , γ), k = 2, 3, . . . are defined similarly.
For h ∈ H and f ∈ W 1,p (X , γ) we set ∂h f (x ) := h∇H f (x ), hiH .
The integration by parts formula holds also for Sobolev functions:
Z
Z
∂h f d γ = ĥf d γ, h ∈ H , f ∈ W 1,p (X , γ).
X
X
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The spaces W k ,p (X , γ), k = 2, 3, . . . are defined similarly.
For h ∈ H and f ∈ W 1,p (X , γ) we set ∂h f (x ) := h∇H f (x ), hiH .
The integration by parts formula holds also for Sobolev functions:
Z
Z
∂h f d γ = ĥf d γ, h ∈ H , f ∈ W 1,p (X , γ).
X
X
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The spaces W k ,p (X , γ), k = 2, 3, . . . are defined similarly.
For h ∈ H and f ∈ W 1,p (X , γ) we set ∂h f (x ) := h∇H f (x ), hiH .
The integration by parts formula holds also for Sobolev functions:
Z
Z
∂h f d γ = ĥf d γ, h ∈ H , f ∈ W 1,p (X , γ).
X
X
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Hausdorff-Gauss surface measure
The finite dimensional case. If F = Rm , m ≥ 2, we set
θF (dx ) :=
1
(2π)m/2
exp(−|x |2 /2)Hm−1 (dx ),
Hm−1 being the spherical m − 1 dimensional Hausdorff measure in Rm ,
Hm−1 (B ) = limδ→0 inf
∑i ∈N ωm−1 rim−1 : ∪i ∈N B (xi , ri ) ⊃ B , ri < δ ∀i
where ωm−1 = π(m−1)/2 /Γ(1 + (m − 1)/2) is the Lebesgue measure of the
unit sphere in Rm−1 .
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Hausdorff-Gauss surface measure
The finite dimensional case. If F = Rm , m ≥ 2, we set
θF (dx ) :=
1
(2π)m/2
exp(−|x |2 /2)Hm−1 (dx ),
Hm−1 being the spherical m − 1 dimensional Hausdorff measure in Rm ,
Hm−1 (B ) = limδ→0 inf
∑i ∈N ωm−1 rim−1 : ∪i ∈N B (xi , ri ) ⊃ B , ri < δ ∀i
where ωm−1 = π(m−1)/2 /Γ(1 + (m − 1)/2) is the Lebesgue measure of the
unit sphere in Rm−1 .
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Hausdorff-Gauss surface measure
The finite dimensional case. If F = Rm , m ≥ 2, we set
θF (dx ) :=
1
(2π)m/2
exp(−|x |2 /2)Hm−1 (dx ),
Hm−1 being the spherical m − 1 dimensional Hausdorff measure in Rm ,
Hm−1 (B ) = limδ→0 inf
∑i ∈N ωm−1 rim−1 : ∪i ∈N B (xi , ri ) ⊃ B , ri < δ ∀i
where ωm−1 = π(m−1)/2 /Γ(1 + (m − 1)/2) is the Lebesgue measure of the
unit sphere in Rm−1 .
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Hausdorff-Gauss surface measure
The finite dimensional case. If F = Rm , m ≥ 2, we set
θF (dx ) :=
1
(2π)m/2
exp(−|x |2 /2)Hm−1 (dx ),
Hm−1 being the spherical m − 1 dimensional Hausdorff measure in Rm ,
Hm−1 (B ) = limδ→0 inf
∑i ∈N ωm−1 rim−1 : ∪i ∈N B (xi , ri ) ⊃ B , ri < δ ∀i
where ωm−1 = π(m−1)/2 /Γ(1 + (m − 1)/2) is the Lebesgue measure of the
unit sphere in Rm−1 .
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The infinite dimensional case. We fix a finite dimensional subspace F ⊂ H,
identified with Rm for some m. The orthogonal projection from H to F is
e = Ker ΠF we have
extended to a projection ΠF defined in X ; setting F
e,
X = F ⊕F
γ = γ ◦ (ΠF )−1 ⊗ γ ◦ (I − ΠF )−1 := γF ⊗ γFe
where γF is the standard Gaussian measure in Rm .
For every Borel set A ⊂ X we set
Z
ΘF (A) = θF (Ax ) γFe (dx ),
e
F
e.
Ax = {y ∈ F : x + y ∈ A}, x ∈ F
ΘF is increasing: if F2 ⊃ F1 then ΘF2 (A) ≥ ΘF1 (A).
The Hausdorff-Gauss measure of Feyel-de La Pradelle (1992) is defined by
ρ(A) = sup{ΘF (A) : F finite dimensional subspace of H }
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The infinite dimensional case. We fix a finite dimensional subspace F ⊂ H,
identified with Rm for some m. The orthogonal projection from H to F is
e = Ker ΠF we have
extended to a projection ΠF defined in X ; setting F
e,
X = F ⊕F
γ = γ ◦ (ΠF )−1 ⊗ γ ◦ (I − ΠF )−1 := γF ⊗ γFe
where γF is the standard Gaussian measure in Rm .
For every Borel set A ⊂ X we set
Z
ΘF (A) = θF (Ax ) γFe (dx ),
e
F
e.
Ax = {y ∈ F : x + y ∈ A}, x ∈ F
ΘF is increasing: if F2 ⊃ F1 then ΘF2 (A) ≥ ΘF1 (A).
The Hausdorff-Gauss measure of Feyel-de La Pradelle (1992) is defined by
ρ(A) = sup{ΘF (A) : F finite dimensional subspace of H }
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The infinite dimensional case. We fix a finite dimensional subspace F ⊂ H,
identified with Rm for some m. The orthogonal projection from H to F is
e = Ker ΠF we have
extended to a projection ΠF defined in X ; setting F
e,
X = F ⊕F
γ = γ ◦ (ΠF )−1 ⊗ γ ◦ (I − ΠF )−1 := γF ⊗ γFe
where γF is the standard Gaussian measure in Rm .
For every Borel set A ⊂ X we set
Z
ΘF (A) = θF (Ax ) γFe (dx ),
e
F
e.
Ax = {y ∈ F : x + y ∈ A}, x ∈ F
ΘF is increasing: if F2 ⊃ F1 then ΘF2 (A) ≥ ΘF1 (A).
The Hausdorff-Gauss measure of Feyel-de La Pradelle (1992) is defined by
ρ(A) = sup{ΘF (A) : F finite dimensional subspace of H }
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The infinite dimensional case. We fix a finite dimensional subspace F ⊂ H,
identified with Rm for some m. The orthogonal projection from H to F is
e = Ker ΠF we have
extended to a projection ΠF defined in X ; setting F
e,
X = F ⊕F
γ = γ ◦ (ΠF )−1 ⊗ γ ◦ (I − ΠF )−1 := γF ⊗ γFe
where γF is the standard Gaussian measure in Rm .
For every Borel set A ⊂ X we set
Z
ΘF (A) = θF (Ax ) γFe (dx ),
e
F
e.
Ax = {y ∈ F : x + y ∈ A}, x ∈ F
ΘF is increasing: if F2 ⊃ F1 then ΘF2 (A) ≥ ΘF1 (A).
The Hausdorff-Gauss measure of Feyel-de La Pradelle (1992) is defined by
ρ(A) = sup{ΘF (A) : F finite dimensional subspace of H }
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The infinite dimensional case. We fix a finite dimensional subspace F ⊂ H,
identified with Rm for some m. The orthogonal projection from H to F is
e = Ker ΠF we have
extended to a projection ΠF defined in X ; setting F
e,
X = F ⊕F
γ = γ ◦ (ΠF )−1 ⊗ γ ◦ (I − ΠF )−1 := γF ⊗ γFe
where γF is the standard Gaussian measure in Rm .
For every Borel set A ⊂ X we set
Z
ΘF (A) = θF (Ax ) γFe (dx ),
e
F
e.
Ax = {y ∈ F : x + y ∈ A}, x ∈ F
ΘF is increasing: if F2 ⊃ F1 then ΘF2 (A) ≥ ΘF1 (A).
The Hausdorff-Gauss measure of Feyel-de La Pradelle (1992) is defined by
ρ(A) = sup{ΘF (A) : F finite dimensional subspace of H }
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The infinite dimensional case. We fix a finite dimensional subspace F ⊂ H,
identified with Rm for some m. The orthogonal projection from H to F is
e = Ker ΠF we have
extended to a projection ΠF defined in X ; setting F
e,
X = F ⊕F
γ = γ ◦ (ΠF )−1 ⊗ γ ◦ (I − ΠF )−1 := γF ⊗ γFe
where γF is the standard Gaussian measure in Rm .
For every Borel set A ⊂ X we set
Z
ΘF (A) = θF (Ax ) γFe (dx ),
e
F
e.
Ax = {y ∈ F : x + y ∈ A}, x ∈ F
ΘF is increasing: if F2 ⊃ F1 then ΘF2 (A) ≥ ΘF1 (A).
The Hausdorff-Gauss measure of Feyel-de La Pradelle (1992) is defined by
ρ(A) = sup{ΘF (A) : F finite dimensional subspace of H }
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The infinite dimensional case. We fix a finite dimensional subspace F ⊂ H,
identified with Rm for some m. The orthogonal projection from H to F is
e = Ker ΠF we have
extended to a projection ΠF defined in X ; setting F
e,
X = F ⊕F
γ = γ ◦ (ΠF )−1 ⊗ γ ◦ (I − ΠF )−1 := γF ⊗ γFe
where γF is the standard Gaussian measure in Rm .
For every Borel set A ⊂ X we set
Z
ΘF (A) = θF (Ax ) γFe (dx ),
e
F
e.
Ax = {y ∈ F : x + y ∈ A}, x ∈ F
ΘF is increasing: if F2 ⊃ F1 then ΘF2 (A) ≥ ΘF1 (A).
The Hausdorff-Gauss measure of Feyel-de La Pradelle (1992) is defined by
ρ(A) = sup{ΘF (A) : F finite dimensional subspace of H }
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Integration by parts formula, on sublevel sets Ω = {x : g (x ) < r } of good (for
simplicity: continuous) functions g.
(i) g ∈ W 2,p (X , γ), for all p > 1 (regularity),
(ii) x 7→ |∇ g1(x )| ∈ Lp (X , γ), for all p > 1 (non degeneracy).
H
H
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
{x : g (x )=r }
fh
∇H g
, hiH d ρ,
|∇H g |H
h ∈ H, f ∈ W 1,q (X , γ), q > 1.
Celada, L. (JFA, 2014): traces of Sobolev functions on regular hypersurfaces.
Examples of admissible Ω: halfspaces {x ∈ X : g (x ) < r } for any g ∈ X ∗ ,
subgraphs of smooth enough functions defined on 1-codimensional
subspaces, balls and ellipsoids in Hilbert spaces.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Integration by parts formula, on sublevel sets Ω = {x : g (x ) < r } of good (for
simplicity: continuous) functions g.
(i) g ∈ W 2,p (X , γ), for all p > 1 (regularity),
(ii) x 7→ |∇ g1(x )| ∈ Lp (X , γ), for all p > 1 (non degeneracy).
H
H
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
{x : g (x )=r }
fh
∇H g
, hiH d ρ,
|∇H g |H
h ∈ H, f ∈ W 1,q (X , γ), q > 1.
Celada, L. (JFA, 2014): traces of Sobolev functions on regular hypersurfaces.
Examples of admissible Ω: halfspaces {x ∈ X : g (x ) < r } for any g ∈ X ∗ ,
subgraphs of smooth enough functions defined on 1-codimensional
subspaces, balls and ellipsoids in Hilbert spaces.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Integration by parts formula, on sublevel sets Ω = {x : g (x ) < r } of good (for
simplicity: continuous) functions g.
(i) g ∈ W 2,p (X , γ), for all p > 1 (regularity),
(ii) x 7→ |∇ g1(x )| ∈ Lp (X , γ), for all p > 1 (non degeneracy).
H
H
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
{x : g (x )=r }
fh
∇H g
, hiH d ρ,
|∇H g |H
h ∈ H, f ∈ W 1,q (X , γ), q > 1.
Celada, L. (JFA, 2014): traces of Sobolev functions on regular hypersurfaces.
Examples of admissible Ω: halfspaces {x ∈ X : g (x ) < r } for any g ∈ X ∗ ,
subgraphs of smooth enough functions defined on 1-codimensional
subspaces, balls and ellipsoids in Hilbert spaces.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Integration by parts formula, on sublevel sets Ω = {x : g (x ) < r } of good (for
simplicity: continuous) functions g.
(i) g ∈ W 2,p (X , γ), for all p > 1 (regularity),
(ii) x 7→ |∇ g1(x )| ∈ Lp (X , γ), for all p > 1 (non degeneracy).
H
H
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
{x : g (x )=r }
fh
∇H g
, hiH d ρ,
|∇H g |H
h ∈ H, f ∈ W 1,q (X , γ), q > 1.
Celada, L. (JFA, 2014): traces of Sobolev functions on regular hypersurfaces.
Examples of admissible Ω: halfspaces {x ∈ X : g (x ) < r } for any g ∈ X ∗ ,
subgraphs of smooth enough functions defined on 1-codimensional
subspaces, balls and ellipsoids in Hilbert spaces.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Integration by parts formula, on sublevel sets Ω = {x : g (x ) < r } of good (for
simplicity: continuous) functions g.
(i) g ∈ W 2,p (X , γ), for all p > 1 (regularity),
(ii) x 7→ |∇ g1(x )| ∈ Lp (X , γ), for all p > 1 (non degeneracy).
H
H
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
{x : g (x )=r }
fh
∇H g
, hiH d ρ,
|∇H g |H
h ∈ H, f ∈ W 1,q (X , γ), q > 1.
Celada, L. (JFA, 2014): traces of Sobolev functions on regular hypersurfaces.
Examples of admissible Ω: halfspaces {x ∈ X : g (x ) < r } for any g ∈ X ∗ ,
subgraphs of smooth enough functions defined on 1-codimensional
subspaces, balls and ellipsoids in Hilbert spaces.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Integration by parts formula, on sublevel sets Ω = {x : g (x ) < r } of good (for
simplicity: continuous) functions g.
(i) g ∈ W 2,p (X , γ), for all p > 1 (regularity),
(ii) x 7→ |∇ g1(x )| ∈ Lp (X , γ), for all p > 1 (non degeneracy).
H
H
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
{x : g (x )=r }
fh
∇H g
, hiH d ρ,
|∇H g |H
h ∈ H, f ∈ W 1,q (X , γ), q > 1.
Celada, L. (JFA, 2014): traces of Sobolev functions on regular hypersurfaces.
Examples of admissible Ω: halfspaces {x ∈ X : g (x ) < r } for any g ∈ X ∗ ,
subgraphs of smooth enough functions defined on 1-codimensional
subspaces, balls and ellipsoids in Hilbert spaces.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Integration by parts formula, on sublevel sets Ω = {x : g (x ) < r } of good (for
simplicity: continuous) functions g.
(i) g ∈ W 2,p (X , γ), for all p > 1 (regularity),
(ii) x 7→ |∇ g1(x )| ∈ Lp (X , γ), for all p > 1 (non degeneracy).
H
H
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
{x : g (x )=r }
fh
∇H g
, hiH d ρ,
|∇H g |H
h ∈ H, f ∈ W 1,q (X , γ), q > 1.
Celada, L. (JFA, 2014): traces of Sobolev functions on regular hypersurfaces.
Examples of admissible Ω: halfspaces {x ∈ X : g (x ) < r } for any g ∈ X ∗ ,
subgraphs of smooth enough functions defined on 1-codimensional
subspaces, balls and ellipsoids in Hilbert spaces.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Malliavin approach: surface measures on level sets of
good functions.
X = abstract Wiener space := a separable Banach space with a centered
nondegenerate Gaussian measure γ.
g : X 7→ R is a continuous function such that
(i) g ∈ W k ,p (X , γ), for all k ∈ N, p > 1,
(ii) x →
7 |∇ g1(x )| ∈ Lp (X , γ), for all p > 1.
H
H
Step 1. The Borel measure in R, B 7→ γ(g −1 (B )) is absolutely continuous
with respect to the Lebesgue measure, with smooth density q1 ∈ L1 (R)
Z
Z
γ(g −1 (B )) =
1dγ =
q1 (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
More generally, for every good ϕ : X → R, ∃ a smooth qϕ ∈ L1 (R) such that
Z
Z
ϕ d γ = qϕ (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Malliavin approach: surface measures on level sets of
good functions.
X = abstract Wiener space := a separable Banach space with a centered
nondegenerate Gaussian measure γ.
g : X 7→ R is a continuous function such that
(i) g ∈ W k ,p (X , γ), for all k ∈ N, p > 1,
(ii) x →
7 |∇ g1(x )| ∈ Lp (X , γ), for all p > 1.
H
H
Step 1. The Borel measure in R, B 7→ γ(g −1 (B )) is absolutely continuous
with respect to the Lebesgue measure, with smooth density q1 ∈ L1 (R)
Z
Z
γ(g −1 (B )) =
1dγ =
q1 (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
More generally, for every good ϕ : X → R, ∃ a smooth qϕ ∈ L1 (R) such that
Z
Z
ϕ d γ = qϕ (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Malliavin approach: surface measures on level sets of
good functions.
X = abstract Wiener space := a separable Banach space with a centered
nondegenerate Gaussian measure γ.
g : X 7→ R is a continuous function such that
(i) g ∈ W k ,p (X , γ), for all k ∈ N, p > 1,
(ii) x →
7 |∇ g1(x )| ∈ Lp (X , γ), for all p > 1.
H
H
Step 1. The Borel measure in R, B 7→ γ(g −1 (B )) is absolutely continuous
with respect to the Lebesgue measure, with smooth density q1 ∈ L1 (R)
Z
Z
γ(g −1 (B )) =
1dγ =
q1 (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
More generally, for every good ϕ : X → R, ∃ a smooth qϕ ∈ L1 (R) such that
Z
Z
ϕ d γ = qϕ (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Malliavin approach: surface measures on level sets of
good functions.
X = abstract Wiener space := a separable Banach space with a centered
nondegenerate Gaussian measure γ.
g : X 7→ R is a continuous function such that
(i) g ∈ W k ,p (X , γ), for all k ∈ N, p > 1,
(ii) x →
7 |∇ g1(x )| ∈ Lp (X , γ), for all p > 1.
H
H
Step 1. The Borel measure in R, B 7→ γ(g −1 (B )) is absolutely continuous
with respect to the Lebesgue measure, with smooth density q1 ∈ L1 (R)
Z
Z
γ(g −1 (B )) =
1dγ =
q1 (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
More generally, for every good ϕ : X → R, ∃ a smooth qϕ ∈ L1 (R) such that
Z
Z
ϕ d γ = qϕ (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Malliavin approach: surface measures on level sets of
good functions.
X = abstract Wiener space := a separable Banach space with a centered
nondegenerate Gaussian measure γ.
g : X 7→ R is a continuous function such that
(i) g ∈ W k ,p (X , γ), for all k ∈ N, p > 1,
(ii) x →
7 |∇ g1(x )| ∈ Lp (X , γ), for all p > 1.
H
H
Step 1. The Borel measure in R, B 7→ γ(g −1 (B )) is absolutely continuous
with respect to the Lebesgue measure, with smooth density q1 ∈ L1 (R)
Z
Z
γ(g −1 (B )) =
1dγ =
q1 (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
More generally, for every good ϕ : X → R, ∃ a smooth qϕ ∈ L1 (R) such that
Z
Z
ϕ d γ = qϕ (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Malliavin approach: surface measures on level sets of
good functions.
X = abstract Wiener space := a separable Banach space with a centered
nondegenerate Gaussian measure γ.
g : X 7→ R is a continuous function such that
(i) g ∈ W k ,p (X , γ), for all k ∈ N, p > 1,
(ii) x →
7 |∇ g1(x )| ∈ Lp (X , γ), for all p > 1.
H
H
Step 1. The Borel measure in R, B 7→ γ(g −1 (B )) is absolutely continuous
with respect to the Lebesgue measure, with smooth density q1 ∈ L1 (R)
Z
Z
γ(g −1 (B )) =
1dγ =
q1 (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
More generally, for every good ϕ : X → R, ∃ a smooth qϕ ∈ L1 (R) such that
Z
Z
ϕ d γ = qϕ (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Malliavin approach: surface measures on level sets of
good functions.
X = abstract Wiener space := a separable Banach space with a centered
nondegenerate Gaussian measure γ.
g : X 7→ R is a continuous function such that
(i) g ∈ W k ,p (X , γ), for all k ∈ N, p > 1,
(ii) x →
7 |∇ g1(x )| ∈ Lp (X , γ), for all p > 1.
H
H
Step 1. The Borel measure in R, B 7→ γ(g −1 (B )) is absolutely continuous
with respect to the Lebesgue measure, with smooth density q1 ∈ L1 (R)
Z
Z
γ(g −1 (B )) =
1dγ =
q1 (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
More generally, for every good ϕ : X → R, ∃ a smooth qϕ ∈ L1 (R) such that
Z
Z
ϕ d γ = qϕ (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The Malliavin approach: surface measures on level sets of
good functions.
X = abstract Wiener space := a separable Banach space with a centered
nondegenerate Gaussian measure γ.
g : X 7→ R is a continuous function such that
(i) g ∈ W k ,p (X , γ), for all k ∈ N, p > 1,
(ii) x →
7 |∇ g1(x )| ∈ Lp (X , γ), for all p > 1.
H
H
Step 1. The Borel measure in R, B 7→ γ(g −1 (B )) is absolutely continuous
with respect to the Lebesgue measure, with smooth density q1 ∈ L1 (R)
Z
Z
γ(g −1 (B )) =
1dγ =
q1 (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
More generally, for every good ϕ : X → R, ∃ a smooth qϕ ∈ L1 (R) such that
Z
Z
ϕ d γ = qϕ (ξ)d ξ, B ∈ B (R).
g −1 (B )
B
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
A candidate to be a surface integral is
lim
ε→0
1
Z
2ε {x : r −ε<g (x )<r +ε}
ϕ d γ = lim
ε→0
1
2ε
Z
r +ε
r −ε
qϕ (ξ)d ξ = qϕ (r ).
Notice that if ϕ ≡ 1, g (x ) = dist(x , Σ), with γ(Σ) = 0, r = 0, this is the
Minkowski content of Σ.
g
Step 2. Show that there exists a measure σr , concentrated on g −1 (r ), such
R
g
g
that qϕ (r ) = X ϕ d σr for smooth ϕ. However, σr depends on g.
Step 3. Show that
σr : σgr |∇H g |H
is independent of g.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
A candidate to be a surface integral is
lim
ε→0
1
Z
2ε {x : r −ε<g (x )<r +ε}
ϕ d γ = lim
ε→0
1
2ε
Z
r +ε
r −ε
qϕ (ξ)d ξ = qϕ (r ).
Notice that if ϕ ≡ 1, g (x ) = dist(x , Σ), with γ(Σ) = 0, r = 0, this is the
Minkowski content of Σ.
g
Step 2. Show that there exists a measure σr , concentrated on g −1 (r ), such
R
g
g
that qϕ (r ) = X ϕ d σr for smooth ϕ. However, σr depends on g.
Step 3. Show that
σr : σgr |∇H g |H
is independent of g.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
A candidate to be a surface integral is
lim
ε→0
1
Z
2ε {x : r −ε<g (x )<r +ε}
ϕ d γ = lim
ε→0
1
2ε
Z
r +ε
r −ε
qϕ (ξ)d ξ = qϕ (r ).
Notice that if ϕ ≡ 1, g (x ) = dist(x , Σ), with γ(Σ) = 0, r = 0, this is the
Minkowski content of Σ.
g
Step 2. Show that there exists a measure σr , concentrated on g −1 (r ), such
R
g
g
that qϕ (r ) = X ϕ d σr for smooth ϕ. However, σr depends on g.
Step 3. Show that
σr : σgr |∇H g |H
is independent of g.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
A candidate to be a surface integral is
lim
ε→0
1
Z
2ε {x : r −ε<g (x )<r +ε}
ϕ d γ = lim
ε→0
1
2ε
Z
r +ε
r −ε
qϕ (ξ)d ξ = qϕ (r ).
Notice that if ϕ ≡ 1, g (x ) = dist(x , Σ), with γ(Σ) = 0, r = 0, this is the
Minkowski content of Σ.
g
Step 2. Show that there exists a measure σr , concentrated on g −1 (r ), such
R
g
g
that qϕ (r ) = X ϕ d σr for smooth ϕ. However, σr depends on g.
Step 3. Show that
σr : σgr |∇H g |H
is independent of g.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
A candidate to be a surface integral is
lim
ε→0
1
Z
2ε {x : r −ε<g (x )<r +ε}
ϕ d γ = lim
ε→0
1
2ε
Z
r +ε
r −ε
qϕ (ξ)d ξ = qϕ (r ).
Notice that if ϕ ≡ 1, g (x ) = dist(x , Σ), with γ(Σ) = 0, r = 0, this is the
Minkowski content of Σ.
g
Step 2. Show that there exists a measure σr , concentrated on g −1 (r ), such
R
g
g
that qϕ (r ) = X ϕ d σr for smooth ϕ. However, σr depends on g.
Step 3. Show that
σr : σgr |∇H g |H
is independent of g.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
A candidate to be a surface integral is
lim
ε→0
1
Z
2ε {x : r −ε<g (x )<r +ε}
ϕ d γ = lim
ε→0
1
2ε
Z
r +ε
r −ε
qϕ (ξ)d ξ = qϕ (r ).
Notice that if ϕ ≡ 1, g (x ) = dist(x , Σ), with γ(Σ) = 0, r = 0, this is the
Minkowski content of Σ.
g
Step 2. Show that there exists a measure σr , concentrated on g −1 (r ), such
R
g
g
that qϕ (r ) = X ϕ d σr for smooth ϕ. However, σr depends on g.
Step 3. Show that
σr : σgr |∇H g |H
is independent of g.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
A candidate to be a surface integral is
lim
ε→0
1
Z
2ε {x : r −ε<g (x )<r +ε}
ϕ d γ = lim
ε→0
1
2ε
Z
r +ε
r −ε
qϕ (ξ)d ξ = qϕ (r ).
Notice that if ϕ ≡ 1, g (x ) = dist(x , Σ), with γ(Σ) = 0, r = 0, this is the
Minkowski content of Σ.
g
Step 2. Show that there exists a measure σr , concentrated on g −1 (r ), such
R
g
g
that qϕ (r ) = X ϕ d σr for smooth ϕ. However, σr depends on g.
Step 3. Show that
σr : σgr |∇H g |H
is independent of g.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Step 4. Integration by parts formula, for smooth ϕ, and h ∈ H,
Z
Z
{x : g (x )<r }
∂h ϕ d γ =
Recall that
Z
{x : g (x )<r }
Z
ĥ ϕ d γ +
{x : g (x )<r }
{x : g (x )=r }
Z
∂h ϕ d γ =
ϕ
Z
ĥ ϕ d γ +
{x : g (x )<r }
∂h g
d σr ,
|∇H g |H
ϕ
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
ρ = Hausdorff–Gauss surface measure. Therefore, σr = ρ|g −1 (r ) .
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Step 4. Integration by parts formula, for smooth ϕ, and h ∈ H,
Z
Z
{x : g (x )<r }
∂h ϕ d γ =
Recall that
Z
{x : g (x )<r }
Z
ĥ ϕ d γ +
{x : g (x )<r }
{x : g (x )=r }
Z
∂h ϕ d γ =
ϕ
Z
ĥ ϕ d γ +
{x : g (x )<r }
∂h g
d σr ,
|∇H g |H
ϕ
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
ρ = Hausdorff–Gauss surface measure. Therefore, σr = ρ|g −1 (r ) .
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Step 4. Integration by parts formula, for smooth ϕ, and h ∈ H,
Z
Z
{x : g (x )<r }
∂h ϕ d γ =
Recall that
Z
{x : g (x )<r }
Z
ĥ ϕ d γ +
{x : g (x )<r }
{x : g (x )=r }
Z
∂h ϕ d γ =
ϕ
Z
ĥ ϕ d γ +
{x : g (x )<r }
∂h g
d σr ,
|∇H g |H
ϕ
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
ρ = Hausdorff–Gauss surface measure. Therefore, σr = ρ|g −1 (r ) .
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Step 4. Integration by parts formula, for smooth ϕ, and h ∈ H,
Z
Z
{x : g (x )<r }
∂h ϕ d γ =
Recall that
Z
{x : g (x )<r }
Z
ĥ ϕ d γ +
{x : g (x )<r }
{x : g (x )=r }
Z
∂h ϕ d γ =
ϕ
Z
ĥ ϕ d γ +
{x : g (x )<r }
∂h g
d σr ,
|∇H g |H
ϕ
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
ρ = Hausdorff–Gauss surface measure. Therefore, σr = ρ|g −1 (r ) .
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The geometric measure theoretic approach
Fukushima (2000), Fukushima, Hino (2001), Ambrosio, Maniglia, Miranda,
Pallara (2010): BV functions in abstract Wiener spaces.
Definition. We say that Ω ⊂ X has finite perimeter ⇔ 1lΩ ∈ BV (X , γ) ⇔ ∃ an
H-valued measure m such that
Z
Z
Z
(1)
∂h f d γ = ĥf d γ + f d hh, mi, f ∈ Cb1 (X , R), h ∈ H
Ω
Ω
⇔ sup
Z
Ω
X
divγ Φ d γ : Φ ∈
Cb1 (X ; H ),
|Φ(x )|H ≤ 1 ∀x
< +∞
The Gaussian divergence divγ is −∇∗H (adjoint in L2 ). For any orthonormal
basis {hj : j ∈ N} of H, setting Φ(x ) = ∑∞
j =1 ϕj (x )hj we have
divγ Φ = ∑∞
j =1 (∂hj ϕj − ĥj ϕj ).
Alessandra Lunardi
Surface measures in Banach spaces
(2)
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The geometric measure theoretic approach
Fukushima (2000), Fukushima, Hino (2001), Ambrosio, Maniglia, Miranda,
Pallara (2010): BV functions in abstract Wiener spaces.
Definition. We say that Ω ⊂ X has finite perimeter ⇔ 1lΩ ∈ BV (X , γ) ⇔ ∃ an
H-valued measure m such that
Z
Z
Z
(1)
∂h f d γ = ĥf d γ + f d hh, mi, f ∈ Cb1 (X , R), h ∈ H
Ω
Ω
⇔ sup
Z
Ω
X
divγ Φ d γ : Φ ∈
Cb1 (X ; H ),
|Φ(x )|H ≤ 1 ∀x
< +∞
The Gaussian divergence divγ is −∇∗H (adjoint in L2 ). For any orthonormal
basis {hj : j ∈ N} of H, setting Φ(x ) = ∑∞
j =1 ϕj (x )hj we have
divγ Φ = ∑∞
j =1 (∂hj ϕj − ĥj ϕj ).
Alessandra Lunardi
Surface measures in Banach spaces
(2)
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The geometric measure theoretic approach
Fukushima (2000), Fukushima, Hino (2001), Ambrosio, Maniglia, Miranda,
Pallara (2010): BV functions in abstract Wiener spaces.
Definition. We say that Ω ⊂ X has finite perimeter ⇔ 1lΩ ∈ BV (X , γ) ⇔ ∃ an
H-valued measure m such that
Z
Z
Z
(1)
∂h f d γ = ĥf d γ + f d hh, mi, f ∈ Cb1 (X , R), h ∈ H
Ω
Ω
⇔ sup
Z
Ω
X
divγ Φ d γ : Φ ∈
Cb1 (X ; H ),
|Φ(x )|H ≤ 1 ∀x
< +∞
The Gaussian divergence divγ is −∇∗H (adjoint in L2 ). For any orthonormal
basis {hj : j ∈ N} of H, setting Φ(x ) = ∑∞
j =1 ϕj (x )hj we have
divγ Φ = ∑∞
j =1 (∂hj ϕj − ĥj ϕj ).
Alessandra Lunardi
Surface measures in Banach spaces
(2)
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The geometric measure theoretic approach
Fukushima (2000), Fukushima, Hino (2001), Ambrosio, Maniglia, Miranda,
Pallara (2010): BV functions in abstract Wiener spaces.
Definition. We say that Ω ⊂ X has finite perimeter ⇔ 1lΩ ∈ BV (X , γ) ⇔ ∃ an
H-valued measure m such that
Z
Z
Z
(1)
∂h f d γ = ĥf d γ + f d hh, mi, f ∈ Cb1 (X , R), h ∈ H
Ω
Ω
⇔ sup
Z
Ω
X
divγ Φ d γ : Φ ∈
Cb1 (X ; H ),
|Φ(x )|H ≤ 1 ∀x
< +∞
The Gaussian divergence divγ is −∇∗H (adjoint in L2 ). For any orthonormal
basis {hj : j ∈ N} of H, setting Φ(x ) = ∑∞
j =1 ϕj (x )hj we have
divγ Φ = ∑∞
j =1 (∂hj ϕj − ĥj ϕj ).
Alessandra Lunardi
Surface measures in Banach spaces
(2)
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The geometric measure theoretic approach
Fukushima (2000), Fukushima, Hino (2001), Ambrosio, Maniglia, Miranda,
Pallara (2010): BV functions in abstract Wiener spaces.
Definition. We say that Ω ⊂ X has finite perimeter ⇔ 1lΩ ∈ BV (X , γ) ⇔ ∃ an
H-valued measure m such that
Z
Z
Z
(1)
∂h f d γ = ĥf d γ + f d hh, mi, f ∈ Cb1 (X , R), h ∈ H
Ω
Ω
⇔ sup
Z
Ω
X
divγ Φ d γ : Φ ∈
Cb1 (X ; H ),
|Φ(x )|H ≤ 1 ∀x
< +∞
The Gaussian divergence divγ is −∇∗H (adjoint in L2 ). For any orthonormal
basis {hj : j ∈ N} of H, setting Φ(x ) = ∑∞
j =1 ϕj (x )hj we have
divγ Φ = ∑∞
j =1 (∂hj ϕj − ĥj ϕj ).
Alessandra Lunardi
Surface measures in Banach spaces
(2)
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
The geometric measure theoretic approach
Fukushima (2000), Fukushima, Hino (2001), Ambrosio, Maniglia, Miranda,
Pallara (2010): BV functions in abstract Wiener spaces.
Definition. We say that Ω ⊂ X has finite perimeter ⇔ 1lΩ ∈ BV (X , γ) ⇔ ∃ an
H-valued measure m such that
Z
Z
Z
(1)
∂h f d γ = ĥf d γ + f d hh, mi, f ∈ Cb1 (X , R), h ∈ H
Ω
Ω
⇔ sup
Z
Ω
X
divγ Φ d γ : Φ ∈
Cb1 (X ; H ),
|Φ(x )|H ≤ 1 ∀x
< +∞
The Gaussian divergence divγ is −∇∗H (adjoint in L2 ). For any orthonormal
basis {hj : j ∈ N} of H, setting Φ(x ) = ∑∞
j =1 ϕj (x )hj we have
divγ Φ = ∑∞
j =1 (∂hj ϕj − ĥj ϕj ).
Alessandra Lunardi
Surface measures in Banach spaces
(2)
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
If Ω has finite perimeter, the H-valued measure m is rewritten as
m = n(x )|dm| where n is an H-valued unit vector field, |dm| is the total
variation measure of m, called the perimeter measure.
Examples.
1. Ω = {x : g (x ) < r }, with r ∈ R and g ∈ ∩p>1 W 2,p (X , γ),
1/|∇H g |H ∈ ∩p>1 Lp (X , γ). In this case we know that
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
f
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
for every f ∈ Cb1 (X , R), h ∈ H; therefore (1) holds, so that
1l{g <r } ∈ BV (X , γ), n(x ) = ∇H g (x )/|∇H g (x )|H , |dm| = ρ|{g =r } .
2. Another example (Caselles, L. , Miranda, Novaga 2012): every open
convex set Ω ⊂ X has finite perimeter. We proved that (2) holds, so that
1lΩ ∈ BV (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
If Ω has finite perimeter, the H-valued measure m is rewritten as
m = n(x )|dm| where n is an H-valued unit vector field, |dm| is the total
variation measure of m, called the perimeter measure.
Examples.
1. Ω = {x : g (x ) < r }, with r ∈ R and g ∈ ∩p>1 W 2,p (X , γ),
1/|∇H g |H ∈ ∩p>1 Lp (X , γ). In this case we know that
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
f
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
for every f ∈ Cb1 (X , R), h ∈ H; therefore (1) holds, so that
1l{g <r } ∈ BV (X , γ), n(x ) = ∇H g (x )/|∇H g (x )|H , |dm| = ρ|{g =r } .
2. Another example (Caselles, L. , Miranda, Novaga 2012): every open
convex set Ω ⊂ X has finite perimeter. We proved that (2) holds, so that
1lΩ ∈ BV (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
If Ω has finite perimeter, the H-valued measure m is rewritten as
m = n(x )|dm| where n is an H-valued unit vector field, |dm| is the total
variation measure of m, called the perimeter measure.
Examples.
1. Ω = {x : g (x ) < r }, with r ∈ R and g ∈ ∩p>1 W 2,p (X , γ),
1/|∇H g |H ∈ ∩p>1 Lp (X , γ). In this case we know that
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
f
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
for every f ∈ Cb1 (X , R), h ∈ H; therefore (1) holds, so that
1l{g <r } ∈ BV (X , γ), n(x ) = ∇H g (x )/|∇H g (x )|H , |dm| = ρ|{g =r } .
2. Another example (Caselles, L. , Miranda, Novaga 2012): every open
convex set Ω ⊂ X has finite perimeter. We proved that (2) holds, so that
1lΩ ∈ BV (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
If Ω has finite perimeter, the H-valued measure m is rewritten as
m = n(x )|dm| where n is an H-valued unit vector field, |dm| is the total
variation measure of m, called the perimeter measure.
Examples.
1. Ω = {x : g (x ) < r }, with r ∈ R and g ∈ ∩p>1 W 2,p (X , γ),
1/|∇H g |H ∈ ∩p>1 Lp (X , γ). In this case we know that
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
f
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
for every f ∈ Cb1 (X , R), h ∈ H; therefore (1) holds, so that
1l{g <r } ∈ BV (X , γ), n(x ) = ∇H g (x )/|∇H g (x )|H , |dm| = ρ|{g =r } .
2. Another example (Caselles, L. , Miranda, Novaga 2012): every open
convex set Ω ⊂ X has finite perimeter. We proved that (2) holds, so that
1lΩ ∈ BV (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
If Ω has finite perimeter, the H-valued measure m is rewritten as
m = n(x )|dm| where n is an H-valued unit vector field, |dm| is the total
variation measure of m, called the perimeter measure.
Examples.
1. Ω = {x : g (x ) < r }, with r ∈ R and g ∈ ∩p>1 W 2,p (X , γ),
1/|∇H g |H ∈ ∩p>1 Lp (X , γ). In this case we know that
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
f
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
for every f ∈ Cb1 (X , R), h ∈ H; therefore (1) holds, so that
1l{g <r } ∈ BV (X , γ), n(x ) = ∇H g (x )/|∇H g (x )|H , |dm| = ρ|{g =r } .
2. Another example (Caselles, L. , Miranda, Novaga 2012): every open
convex set Ω ⊂ X has finite perimeter. We proved that (2) holds, so that
1lΩ ∈ BV (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
If Ω has finite perimeter, the H-valued measure m is rewritten as
m = n(x )|dm| where n is an H-valued unit vector field, |dm| is the total
variation measure of m, called the perimeter measure.
Examples.
1. Ω = {x : g (x ) < r }, with r ∈ R and g ∈ ∩p>1 W 2,p (X , γ),
1/|∇H g |H ∈ ∩p>1 Lp (X , γ). In this case we know that
Z
{x : g (x )<r }
Z
∂h f d γ =
Z
ĥf d γ +
{x : g (x )<r }
f
{x : g (x )=r }
∂h g
d ρ,
|∇H g |H
for every f ∈ Cb1 (X , R), h ∈ H; therefore (1) holds, so that
1l{g <r } ∈ BV (X , γ), n(x ) = ∇H g (x )/|∇H g (x )|H , |dm| = ρ|{g =r } .
2. Another example (Caselles, L. , Miranda, Novaga 2012): every open
convex set Ω ⊂ X has finite perimeter. We proved that (2) holds, so that
1lΩ ∈ BV (X , γ).
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Da Prato, L. , Tubaro (2014): in the case of Gaussian measures in Banach
spaces, construction of surface measures on the level sets of g by the
Airault-Malliavin approach, with different proofs that needed less restrictive
assumptions on g
0
(g ∈ W 1,p (X , γ), ∇g, ∇H g /|∇H g |2H ∈ W 1,p (X , γ; H ) for some p > 1.)
Da Prato, L. , Tubaro (TAMS, to appear): extension of this approach for
non-Gaussian measures ν in Hilbert spaces.
From now on, X is a separable Hilbert space endowed with a nondegenerate
Borel probability measure ν.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Da Prato, L. , Tubaro (2014): in the case of Gaussian measures in Banach
spaces, construction of surface measures on the level sets of g by the
Airault-Malliavin approach, with different proofs that needed less restrictive
assumptions on g
0
(g ∈ W 1,p (X , γ), ∇g, ∇H g /|∇H g |2H ∈ W 1,p (X , γ; H ) for some p > 1.)
Da Prato, L. , Tubaro (TAMS, to appear): extension of this approach for
non-Gaussian measures ν in Hilbert spaces.
From now on, X is a separable Hilbert space endowed with a nondegenerate
Borel probability measure ν.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Da Prato, L. , Tubaro (2014): in the case of Gaussian measures in Banach
spaces, construction of surface measures on the level sets of g by the
Airault-Malliavin approach, with different proofs that needed less restrictive
assumptions on g
0
(g ∈ W 1,p (X , γ), ∇g, ∇H g /|∇H g |2H ∈ W 1,p (X , γ; H ) for some p > 1.)
Da Prato, L. , Tubaro (TAMS, to appear): extension of this approach for
non-Gaussian measures ν in Hilbert spaces.
From now on, X is a separable Hilbert space endowed with a nondegenerate
Borel probability measure ν.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Assumptions.
1. There is R ∈ L (X ) such that the operator
R ∇ : Cb1 (X , R) ⊂ Lp (X , ν) 7→ Lp (X , ν; X )
is closable in Lp (X , ν), for every p > 1. The closure is denoted by Mp , its
domain by W 1,p (X , ν).
2. For every p > 1 and z ∈ X , there is Cp,z > 0 such that
Z
hR ∇ϕ, z i d ν ≤ Cp,z kϕkLp (X ,ν) ,
X
ϕ ∈ Cb1 (X , R).
Equivalently: constant vector fields F (x ) ≡ z belong to D (Mp∗ ), for every
0
p > 1. So, for every z ∈ X there exists Mp∗ z =: vz ∈ Lp (X , ν) for every p,
such that
Z
Z
hMp ϕ, z i d ν = vz ϕ d ν, ϕ ∈ W 1,p (X , ν).
X
X
3. g ∈ ∩p>1 W 1,p (X , ν) is continuous and such that Ψ := Mp g /kMp g k2
belongs to D (Mp∗ ), for every p > 1.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Assumptions.
1. There is R ∈ L (X ) such that the operator
R ∇ : Cb1 (X , R) ⊂ Lp (X , ν) 7→ Lp (X , ν; X )
is closable in Lp (X , ν), for every p > 1. The closure is denoted by Mp , its
domain by W 1,p (X , ν).
2. For every p > 1 and z ∈ X , there is Cp,z > 0 such that
Z
hR ∇ϕ, z i d ν ≤ Cp,z kϕkLp (X ,ν) ,
X
ϕ ∈ Cb1 (X , R).
Equivalently: constant vector fields F (x ) ≡ z belong to D (Mp∗ ), for every
0
p > 1. So, for every z ∈ X there exists Mp∗ z =: vz ∈ Lp (X , ν) for every p,
such that
Z
Z
hMp ϕ, z i d ν = vz ϕ d ν, ϕ ∈ W 1,p (X , ν).
X
X
3. g ∈ ∩p>1 W 1,p (X , ν) is continuous and such that Ψ := Mp g /kMp g k2
belongs to D (Mp∗ ), for every p > 1.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Assumptions.
1. There is R ∈ L (X ) such that the operator
R ∇ : Cb1 (X , R) ⊂ Lp (X , ν) 7→ Lp (X , ν; X )
is closable in Lp (X , ν), for every p > 1. The closure is denoted by Mp , its
domain by W 1,p (X , ν).
2. For every p > 1 and z ∈ X , there is Cp,z > 0 such that
Z
hR ∇ϕ, z i d ν ≤ Cp,z kϕkLp (X ,ν) ,
X
ϕ ∈ Cb1 (X , R).
Equivalently: constant vector fields F (x ) ≡ z belong to D (Mp∗ ), for every
0
p > 1. So, for every z ∈ X there exists Mp∗ z =: vz ∈ Lp (X , ν) for every p,
such that
Z
Z
hMp ϕ, z i d ν = vz ϕ d ν, ϕ ∈ W 1,p (X , ν).
X
X
3. g ∈ ∩p>1 W 1,p (X , ν) is continuous and such that Ψ := Mp g /kMp g k2
belongs to D (Mp∗ ), for every p > 1.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Assumptions.
1. There is R ∈ L (X ) such that the operator
R ∇ : Cb1 (X , R) ⊂ Lp (X , ν) 7→ Lp (X , ν; X )
is closable in Lp (X , ν), for every p > 1. The closure is denoted by Mp , its
domain by W 1,p (X , ν).
2. For every p > 1 and z ∈ X , there is Cp,z > 0 such that
Z
hR ∇ϕ, z i d ν ≤ Cp,z kϕkLp (X ,ν) ,
X
ϕ ∈ Cb1 (X , R).
Equivalently: constant vector fields F (x ) ≡ z belong to D (Mp∗ ), for every
0
p > 1. So, for every z ∈ X there exists Mp∗ z =: vz ∈ Lp (X , ν) for every p,
such that
Z
Z
hMp ϕ, z i d ν = vz ϕ d ν, ϕ ∈ W 1,p (X , ν).
X
X
3. g ∈ ∩p>1 W 1,p (X , ν) is continuous and such that Ψ := Mp g /kMp g k2
belongs to D (Mp∗ ), for every p > 1.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Assumptions.
1. There is R ∈ L (X ) such that the operator
R ∇ : Cb1 (X , R) ⊂ Lp (X , ν) 7→ Lp (X , ν; X )
is closable in Lp (X , ν), for every p > 1. The closure is denoted by Mp , its
domain by W 1,p (X , ν).
2. For every p > 1 and z ∈ X , there is Cp,z > 0 such that
Z
hR ∇ϕ, z i d ν ≤ Cp,z kϕkLp (X ,ν) ,
X
ϕ ∈ Cb1 (X , R).
Equivalently: constant vector fields F (x ) ≡ z belong to D (Mp∗ ), for every
0
p > 1. So, for every z ∈ X there exists Mp∗ z =: vz ∈ Lp (X , ν) for every p,
such that
Z
Z
hMp ϕ, z i d ν = vz ϕ d ν, ϕ ∈ W 1,p (X , ν).
X
X
3. g ∈ ∩p>1 W 1,p (X , ν) is continuous and such that Ψ := Mp g /kMp g k2
belongs to D (Mp∗ ), for every p > 1.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
g
Then for every r ∈ R there are surface measures σr such that for every
ϕ ∈ Cb1 (X , R) and z ∈ X
Z
{x : g (x )<r }
Z
Z
hMp ϕ, z i d ν =
{x : g (x )<r }
vz ϕ d ν +
{x : g (x )=r }
ϕhMp g , z id σgr .
g
The normalized surface measures ρr := σr kMg k are independent of g and
the integration formula reads as
Z
{x : g (x )<r }
hMp ϕ, z i d ν =
Z
{x : g (x )<r }
Alessandra Lunardi
Z
vz ϕ d ν +
ϕh
{x : g (x )=r }
Surface measures in Banach spaces
Mp g
kMp g k
, z id ρr .
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
g
Then for every r ∈ R there are surface measures σr such that for every
ϕ ∈ Cb1 (X , R) and z ∈ X
Z
{x : g (x )<r }
Z
Z
hMp ϕ, z i d ν =
{x : g (x )<r }
vz ϕ d ν +
{x : g (x )=r }
ϕhMp g , z id σgr .
g
The normalized surface measures ρr := σr kMg k are independent of g and
the integration formula reads as
Z
{x : g (x )<r }
hMp ϕ, z i d ν =
Z
{x : g (x )<r }
Alessandra Lunardi
Z
vz ϕ d ν +
ϕh
{x : g (x )=r }
Surface measures in Banach spaces
Mp g
kMp g k
, z id ρr .
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Remark: if ν is a centered nondegenerate Gaussian measure with covariance
Q, then H = Q 1/2 (X ). So, 1 and 2 hold with R = Q 1/2 (no novelties).
Examples.
Weighted Gaussian measures, ν(dx ) = w (x )µ(dx ) with w,
log w ∈ ∩p>1 W 1,p (X , µ), µ Gaussian, R = Q 1/2 , Q = covariance of µ.
No surprises: we get ρr = w (x )ρ|{g =r } .
Non Gaussian product measures in X = L2 (0, 1)
Invariant measures of stochastic PDEs: (i) reaction-diffusion equations
with polynomial nonlinearity and (2) stochastic Burgers equation,
R = (−∆D )−β , β > 0.
In these cases, g (x ) = hx , bi with b 6= 0, g (x ) = kx k2 satisfy our
assumptions.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Remark: if ν is a centered nondegenerate Gaussian measure with covariance
Q, then H = Q 1/2 (X ). So, 1 and 2 hold with R = Q 1/2 (no novelties).
Examples.
Weighted Gaussian measures, ν(dx ) = w (x )µ(dx ) with w,
log w ∈ ∩p>1 W 1,p (X , µ), µ Gaussian, R = Q 1/2 , Q = covariance of µ.
No surprises: we get ρr = w (x )ρ|{g =r } .
Non Gaussian product measures in X = L2 (0, 1)
Invariant measures of stochastic PDEs: (i) reaction-diffusion equations
with polynomial nonlinearity and (2) stochastic Burgers equation,
R = (−∆D )−β , β > 0.
In these cases, g (x ) = hx , bi with b 6= 0, g (x ) = kx k2 satisfy our
assumptions.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Remark: if ν is a centered nondegenerate Gaussian measure with covariance
Q, then H = Q 1/2 (X ). So, 1 and 2 hold with R = Q 1/2 (no novelties).
Examples.
Weighted Gaussian measures, ν(dx ) = w (x )µ(dx ) with w,
log w ∈ ∩p>1 W 1,p (X , µ), µ Gaussian, R = Q 1/2 , Q = covariance of µ.
No surprises: we get ρr = w (x )ρ|{g =r } .
Non Gaussian product measures in X = L2 (0, 1)
Invariant measures of stochastic PDEs: (i) reaction-diffusion equations
with polynomial nonlinearity and (2) stochastic Burgers equation,
R = (−∆D )−β , β > 0.
In these cases, g (x ) = hx , bi with b 6= 0, g (x ) = kx k2 satisfy our
assumptions.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Remark: if ν is a centered nondegenerate Gaussian measure with covariance
Q, then H = Q 1/2 (X ). So, 1 and 2 hold with R = Q 1/2 (no novelties).
Examples.
Weighted Gaussian measures, ν(dx ) = w (x )µ(dx ) with w,
log w ∈ ∩p>1 W 1,p (X , µ), µ Gaussian, R = Q 1/2 , Q = covariance of µ.
No surprises: we get ρr = w (x )ρ|{g =r } .
Non Gaussian product measures in X = L2 (0, 1)
Invariant measures of stochastic PDEs: (i) reaction-diffusion equations
with polynomial nonlinearity and (2) stochastic Burgers equation,
R = (−∆D )−β , β > 0.
In these cases, g (x ) = hx , bi with b 6= 0, g (x ) = kx k2 satisfy our
assumptions.
Alessandra Lunardi
Surface measures in Banach spaces
Presentation
The Hausdorff-Gauss surface measure
The Malliavin approach
The geometric measure theoretic approach
Remark: if ν is a centered nondegenerate Gaussian measure with covariance
Q, then H = Q 1/2 (X ). So, 1 and 2 hold with R = Q 1/2 (no novelties).
Examples.
Weighted Gaussian measures, ν(dx ) = w (x )µ(dx ) with w,
log w ∈ ∩p>1 W 1,p (X , µ), µ Gaussian, R = Q 1/2 , Q = covariance of µ.
No surprises: we get ρr = w (x )ρ|{g =r } .
Non Gaussian product measures in X = L2 (0, 1)
Invariant measures of stochastic PDEs: (i) reaction-diffusion equations
with polynomial nonlinearity and (2) stochastic Burgers equation,
R = (−∆D )−β , β > 0.
In these cases, g (x ) = hx , bi with b 6= 0, g (x ) = kx k2 satisfy our
assumptions.
Alessandra Lunardi
Surface measures in Banach spaces