Tutorial on Additive Lévy Processes - Lecture #2
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Tutorial on Additive Lévy Processes - Lecture #2

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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. ...

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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
ARdands>0 then Hs(A) :=inf(nå=1(2rn)s:A[B(xn n=1
D. Khoshnevisan (Salt Lake City, Utah)
,rn),0rn
).
ICSAA, Seattle ’06
2 / 23
Hausdorff Measure and Dimension
IfARdands>0 then f(1(2rn)s:An[=1B(xn,rn),0rn Hs(A) :=inå n=
Denition (Hausdorff, 1919) Thes-dimensional Hausdorff measure ofAis
Hs(A) :=lim0Hs(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 Hsto Borel sets inRdis 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 Hsto Borel sets inRdis a measure.
Denition (“Hausdorff Dimension”; Hausdorff, 1919) dimHA=sup{s:Hs(A) =∞}=inf{s>0:Hs(
D. Khoshnevisan (Salt Lake City, Utah)
A) =0}.
ICSAA, Seattle ’06
3 / 23
Hausdorff Measure and Dimension
Denition (s-dimensional energy ofµP(A); M. Riesz)
Is
(µ) :=Z
Z
D. Khoshnevisan (Salt
µ(dx)µ(dy) |xy|s(s>
Lake City, Utah)
0),
ICSAA, Seattle ’06
4 / 23
Hausdorff Measure and Dimension
Denition (s-dimensional energy ofµP(A); M. Riesz)
Is
(µ) :=Z
Z
D. Khoshnevisan (Salt
µ(dx)µ(dy) |xy|s(s>
Lake City, Utah)
0),
ICSAA, Seattle ’06
4 / 23
Hausdorff Measure and Dimension
Denition (s-dimensional energy ofµP(A); M. Riesz)
Is
(µ) :=ZZ
D. Khoshnevisan
µ(dx)µ(dy) |xy|s(s>
(Salt Lake City, Utah)
0),
I0
(µ) :=Z
Z
log+|xy|1µ(dx)µ(dy).
ICSAA, Seattle ’06
4 / 23
Hausdorff Measure and Dimension
Denition (s-dimensional energy ofµP(A); M. Riesz)
Is(µ) :=ZZ
µ(dx)µ(dy) |xy|s(s>0),
I0(µ)
:=ZZ
log+|xy|1µ(dx)µ(dy).
Denition (s-dimensional capacity ofA; C. Gauss, M. Riesz) 1 Cs(A) :=µiPnf(A)Is(µ),inf:=,1/:=0.
D. Khoshnevisan (Salt Lake City, Utah)
ICSAA, Seattle ’06
4 / 23
Hausdorff Measure and Dimension
Denition (s-dimensional energy ofµP(A); M. Riesz)
Is(µ) :=ZZ
µ(dx)µ(dy) |xy|s(s>0),
I0(µ)
:=ZZ
log+|xy|1µ(dx)µ(dy).
Denition (s-dimensional capacity ofA; C. Gauss, M. Riesz) Cs(A) :=µiPnf(A)Is)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)
dimHA=sup{s>0:Cs(A)>0}=inf{
D. Khoshnevisan (Salt Lake City, Utah)
s>0
:Cs(A) =0}.
ICSAA, Seattle ’06
5 / 23