Tutorial on Additive Lévy Processes - Lecture #1
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Tutorial on Additive Lévy Processes - Lecture #1

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Tutorial on Additive Levy´ ProcessesLecture #1Davar 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 / 20N NCould have R in place of R , etc.+dX ,..., X independent Brownian motions in R .1 N“Additive Brownian motion”:NX(t) := X (t )+···+ X (t ) for all t = (t ,..., t )∈ R .1 1 N N 1 N +Likewise, can have “additive stable,” “additive Levy´ ,” etc.IntroductionSome TerminologyAn “(N, d) random field” X has N parameters and takes values indR (Adler, 1981);d Ni.e., X(t)∈ R for all t := (t ,..., t )∈ R .1 ND. Khoshnevisan (Salt Lake City, Utah) ICSAA, Seattle ’06 2 / 20dX ,..., X independent Brownian motions in R .1 N“Additive Brownian motion”:NX(t) := X (t )+···+ X (t ) for all t = (t ,..., t )∈ R .1 1 N N 1 N +Likewise, can have “additive stable,” “additive Levy´ ,” etc.IntroductionSome TerminologyAn “(N, d) random field” X has N parameters and takes values indR (Adler, 1981);d Ni.e., X(t)∈ R for all t := (t ,..., t )∈ R .1 NN NCould have R in place of R , etc.+D. Khoshnevisan (Salt Lake City, Utah) ICSAA, Seattle ’06 2 / 20Likewise, can have “additive stable,” “additive Levy´ ,” etc.IntroductionSome TerminologyAn “(N, d) random field” X has N parameters and takes values indR (Adler, 1981);d Ni.e., X(t)∈ R for all t := (t ,..., t )∈ R .1 NN NCould have R in ...

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International Conference on Stochastic Analysis and Its Applications August 7–11, 2006
20
Tutorial on Additive Le´ vy Processes Lecture #1
Department of Mathematics University of Utah http://www.math.utah.edu/˜davar
Davar Khoshnevisan
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An “(N,d)random eld”XhasNparameters and takes values in Rd(Adler, 1981); i.e.,X(t)Rdfor allt:= (t1, . . . ,tN)RN.
Some Terminology
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Some Terminology
An “(N,d)random eld”XhasNparameters and takes values in Rd(Adler, 1981); i.e.,X(t)Rdfor allt:= (t1, . . . ,tN)RN. Could haveR+Nin place ofRN, etc.
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An “(N,d)random eld”XhasNparameters and takes values in Rd(Adler, 1981); i.e.,X(t)Rdfor allt:= (t1, . . . ,tN)RN. Could haveRN+in place ofRN, etc. X1, . . . ,XNindependent Brownian motions inRd. “Additive Brownian motion”:
X(t) :=X1(t1) +∙ ∙ ∙+XN(tN)
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Likewise,canhaveadditivestable,”additiveL´evy,”etc.
X(t) :=X1(t1) +∙ ∙ ∙+XN(tN)for allt= (t1, . . . ,tN)R+N.
An “(N,d)random eld”XhasNparameters and takes values in Rd(Adler, 1981); i.e.,X(t)Rdfor allt:= (t1, . . . ,tN)RN. Could haveRN+in place ofRN, etc. X1, . . . ,XNindependent Brownian motions inRd. “Additive Brownian motion”:
Some Terminology
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ABM and the Local Dynamics
of
Brownian
Sheet
˙ W
:=white noise onR+.
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ABM and the Local Dynamics of Brownian Sheet
˙ W:=white noise onR+. I.e., Gaussian, and ˙ ˙ IfAB=thenW(A)andW(B)are indept.
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˙ W:=white noise onR+. I.e., Gaussian, and ˙ ˙ IfAB=thenW(A)andW(B)are indept. ˙ ˙ E W(A) =0 and VarW(A) =meas(A).
ABM and the Local Dynamics of Brownian Sheet
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˙ W:=white noise onR+. I.e., Gaussian, and ˙ ˙ IfAB=thenW(A)andW(B)are indept. ˙ ˙ E W(A) =0 and VarW(A) =meas(A).
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ABM and the Local Dynamics of Brownian Sheet
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as long as meas(An)<for all n.
ABM and the Local Dynamics of Brownian Sheet
Lemma If A1,A2, . . .are nonrandom and disjoint then a.s., W˙n=[1An!=n=å1 ˙ W(An),
˙ W:=white noise onR+. I.e., Gaussian, and ˙ ˙ IfAB=thenW(A)andW(B)are indept. ˙ ˙ E W(A) =0 and VarW(A) =meas(A).
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˙ (N,1)“Brownian sheet”B:=the dF ofW; i.e.,
for allt= (t1, . . . ,tN)R+N.
˙ B(t) :=W([0,t1]× ∙ ∙ ∙ ×[0,tN])
ABM and the Local Dynamics of Brownian Sheet
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