Tutorial on Additive Lévy Processes - Lecture #2

Davar Khoshnevisan

Description

Tutorial on Additive Levy´ ProcessesLecture #2Davar KhoshnevisanDepartment of MathematicsUniversity of Utahhttp://www.math.utah.edu/˜davarInternational Conference on Stochastic Analysisand Its ApplicationsAugust 7–11, 2006D. Khoshnevisan (Salt Lake City, Utah) ICSAA, Seattle ’06 1 / 23åDefinition (Hausdorff, 1919)The s dimensional Hausdorff measure ofA iss sH (A) := limH (A).↓0Spherical measure (Besicovitch)Hausdorff Measure and DimensiondIf A∈ R and s > 0 then( )∞∞ [s sH (A) := inf (2r ) : A B(x , r ), 0 r .n n n nn=1 n=1D. Khoshnevisan (Salt Lake City, Utah) ICSAA, Seattle ’06 2 / 23åHausdorff Measure and DimensiondIf A∈ R and s > 0 then( )∞∞ [s sH (A) := inf (2r ) : A B(x , r ), 0 r .n n n nn=1 n=1Definition (Hausdorff, 1919)The s dimensional Hausdorff measure ofA iss sH (A) := limH (A).↓0Spherical measure (Besicovitch)D. Khoshnevisan (Salt Lake City, Utah) ICSAA, Seattle ’06 2 / 23Definition (“Hausdorff Dimension”; Hausdorff, 1919)s sdim A = sup{s : H (A) =∞} = inf{s > 0 : H (A) = 0}.HHausdorff Measure and DimensionTheorem (Hausdorff, 1919)s dFor all s > 0, the restriction of H to Borel sets in R is a measure.D. Khoshnevisan (Salt Lake City, Utah) ICSAA, Seattle ’06 3 / 23Hausdorff Measure and DimensionTheorem (Hausdorff, 1919)s dFor all s > 0, the restriction of H to Borel sets in R is a measure.Definition (“Hausdorff Dimension”; Hausdorff, 1919)s sdim A = sup{s : H (A) =∞} = inf{s > 0 : H (A) = 0}.HD. ...

Informations

• Publié par: Erkyo
• Nombre de lectures: 39
• Langue: English

Extrait

Tutorial on Additive Le´  vy Processes Lecture #2

Davar Khoshnevisan

Department of Mathematics University of Utah http://www.math.utah.edu/˜davar

International Conference on Stochastic Analysis and Its Applications August 7–11, 2006

D. Khoshnevisan (Salt Lake City, Utah)

ICSAA, Seattle ’06

1 / 23

Hausdorff Measure and Dimension

If

A∈R d ands>0 then ∞ H  s (A) :=inf ( n å ∞ =1 (2r n ) s :A⊂ [ B(x n n=1

D. Khoshnevisan (Salt Lake City, Utah)

,r n ),0≤r n

≤ ) .

ICSAA, Seattle ’06

2 / 23

Hausdorff Measure and Dimension

IfA∈R d ands>0 then f ( ∞ 1 (2r n ) s :A⊂ n ∞ [ =1 B(x n ,r n ),0≤r n H  s (A) :=in å n=

Denition (Hausdorff, 1919) Thes-dimensional Hausdorff measure ofAis

H s (A) :=l  i ↓ m 0 H  s (A).

Spherical measure (Besicovitch)

D. Khoshnevisan (Salt Lake City, Utah)

≤ ) .

ICSAA, Seattle ’06

2 / 23

Hausdorff Measure and Dimension

Theorem (Hausdorff, 1919) For all s>0, the restriction of H s to Borel sets inR d is a measure.

D. Khoshnevisan (Salt Lake City, Utah)

ICSAA, Seattle ’06

3 / 23

Hausdorff Measure and Dimension

Theorem (Hausdorff, 1919) For all s>0, the restriction of H s to Borel sets inR d is a measure.

Denition (“Hausdorff Dimension”; Hausdorff, 1919) dim H A=sup { s:H s (A) =∞}=inf { s>0:H s (

D. Khoshnevisan (Salt Lake City, Utah)

A) =0 } .

ICSAA, Seattle ’06

3 / 23

Hausdorff Measure and Dimension

Denition (s-dimensional energy ofµ∈P(A); M. Riesz)

I s

(µ) := Z

Z

D. Khoshnevisan (Salt

µ(dx)µ(dy) |x−y| s (s>

Lake City, Utah)

0) ,

ICSAA, Seattle ’06

4 / 23

Hausdorff Measure and Dimension

Denition (s-dimensional energy ofµ∈P(A); M. Riesz)

I s

(µ) := Z

Z

D. Khoshnevisan (Salt

µ(dx)µ(dy) |x−y| s (s>

Lake City, Utah)

0) ,

ICSAA, Seattle ’06

4 / 23

Hausdorff Measure and Dimension

Denition (s-dimensional energy ofµ∈P(A); M. Riesz)

I s

(µ) := ZZ

D. Khoshnevisan

µ(dx)µ(dy) |x−y| s (s>

(Salt Lake City, Utah)

0) ,

I 0

(µ) := Z

Z

log + |x−y| −1 µ(dx)µ(dy).

ICSAA, Seattle ’06

4 / 23

Hausdorff Measure and Dimension

Denition (s-dimensional energy ofµ∈P(A); M. Riesz)

I s (µ) := ZZ

µ(dx)µ(dy) |x−y| s (s>0) ,

I 0 (µ)

:= ZZ

log + |x−y| −1 µ(dx)µ(dy).

Denition (s-dimensional capacity ofA; C. Gauss, M. Riesz) 1 C s (A) :=  µ∈ i P nf (A) I s (µ)  − ,inf∅:=∞,1/∞:=0.

D. Khoshnevisan (Salt Lake City, Utah)

ICSAA, Seattle ’06

4 / 23

Hausdorff Measure and Dimension

Denition (s-dimensional energy ofµ∈P(A); M. Riesz)

I s (µ) := ZZ

µ(dx)µ(dy) |x−y| s (s>0) ,

I 0 (µ)

:= ZZ

log + |x−y| −1 µ(dx)µ(dy).

Denition (s-dimensional capacity ofA; C. Gauss, M. Riesz) C s (A) :=  µ∈ i P nf (A) I s )  −1 (µ ,inf∅:=∞,1/∞:=0.

Men are liars. We’ll lie about lying if we have to. I’m an algebra liar. I gure two good lies make a positive. –Tim Allen

D. Khoshnevisan (Salt Lake City, Utah)

ICSAA, Seattle ’06

4 / 23

Hausdorff Measure and Dimension

Theorem (Frostman, 1935)

dim H A=sup { s>0:C s (A)>0 } =inf {

D. Khoshnevisan (Salt Lake City, Utah)

s>0

:C s (A) =0 } .

ICSAA, Seattle ’06

5 / 23