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Infinite divisibility of solutions to some self-similar

668 P Pierre 1 Introduction During the last decade, there has been a renewed interest in self-similar semigroups, something which seems to be attributed to their connections to several fields of mathematics and more generally to many area of the sciences



J´er ˆome Bolte Aris Daniilidis Adrian Lewis Masahiro Shiota

measure zero Our w ork relies on concepts of generalized critical p oin ts in the sense nonsmo oth analysis that w e no pro ceed to describ e W sa y x∗ is a limiting sub gr adient for the con tin uous function f on Rn at x, and w e write x∗ ∈ ∂f(x) (the limiting sub di er ential of f), if there exist sequences xn → x and x∗ n



Soumik Pal - University of Washington

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A B (Ton) Dieker Assistant Professor School for Industrial

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Annales de la Société Polonaise de Mathématique, Cracovie 6, 93-116 (in Johnson et al 1995) Geluk, J L and Peng, L (2000) An Adaptive Optimal Estimate of the Tail Index for MA(1) Time Series Statistics and Probability Letters 46, 217-227 Gnedenko, B V (1943) Sur la Distribution Limite du Terme d’une Serie Aleatoire



Weak quenched limiting distributions for transient one

probability measure P on Ω Given an environment ω ={ωx}x∈Z ∈Ω and an initial location x ∈Z,welet{Xn}n≥0 be the Markov chain with law Px ω defined by P ω x(X0 =x)=1, and Px ω(Xn+1 =zXn =y)= ωy,z=y +1, 1 −ωy,z=y −1, 0, otherwise Since the environment ω is random, Px ω(·) is a random probability measure and is called

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liminfy→xn,y?=xnf(y)-f(xn)- ?x?n,y-xn??y-xn?≥0, x??∂◦f(x) :=??∂f(x),

CR?0???

z?B(x,ε)? ∂f(z)? 1 ?1 ?∂f(x) ??f??x? ?∂f(x) =? x ??Rn: liminfy→x,y?=xf(y)-f(x)- ?x?,y-x??y-x?≥0? x??∂f(x)? ?xn?U,?x?n??∂f(xn) :xn→x, x?n→x???n→ ∞, ∂◦f(x) =??∂f(x), ?∂f(x)?∂f(x)?∂◦f(x). dom∂◦f f?? ∂◦f(a)?0, ?∂f=∂f, ?∂f=∂◦f. ????? ?x?V????? ?∂g(x)? ?G(x)T?∂f(G(x)), (4)∂g(x)=?G(x)T∂f(G(x)), ∂◦g(x)=?G(x)T∂◦f(G(x)),?x?V.

Δ:={t?[0,1] :z(t)?clF\F}

(∂f)-1(0) :={x?U:∂f(x)?0}. f:Rm×Rn×Rp?(x,y,z)?→ ?x? - ?y?, ∂f(0,0,z) =BRm(0,1)×Sn-1×{0}p

SL:={x?[0,1]e: 0?∂f(x)}??????∂f

0/?∂f(te), t?[0,1].

Γδ={x?[0,1]e:?x??∂f(x),|?x?,e?|> δ}. ?Γ+δ={x?clV:?x???∂f(x),?x?,e?> δ}, ??????∂f max{?x?,e?:x??∂f(x)}> δ???min{?x?,e?:x??∂f(x)}<-δ. v u???? ddtf(te)???? {g?(t)}=?∂g(t)? ?z(t),?∂f(z(t)? ? {?z(t),z?+(t)?}, g?(t) =?e,z?+(t)?+?z(t)-e,z?+(t)? ≥δ- ?z(t)-e?M≥δ-ηM, f(be)-f(ae) =b addtf(z(t))dt≥ [a,b]\Δg ?(t)dt- ddtf(z(t))???? dt ≥(l-η)(δ-ηM)-ηM.

Γ0={x?[0,1]e:?x??∂f(te),?x?,e?= 0}.

[0,1]e=Γ0?? i≥1Γ 1/i. i≥1Γ1/i i≥1Γ1/i ?Γε0:={x?clV:?x???∂f(x),|?x?,e?|< ε}. ?Γε0 a???? ddtf(z(t))???? [a,b]\Δ|h?(t)|dt+ ddtf(z(t))dt????

S:={x?U:∂◦f(x)?0}.

U?? {h?(t)}=?∂h(t)?z?(t)?∂f(z(t))? {0} ∂◦f(x) =??{∂f(x) +∂∞f(x)} x?Rn??????

Tf(x) =?

ε>0??

x?B(x0,ε)? ∂f(x)? f:U→R?????? ∂◦f(x)?Tf(x)????? ?x?U. ∂◦f(x) =Tf(x)????? ?x?U.

θ0: [0,π)→[0,π/2]

θ0(z) :=?z

z?→?θ0(z) :=θ0(z(modπ)).

σ(θ,z) :=?

1 ??θ≥?θ0(z)?

Φ1(?,θ,z) =?

(2/π)?θ0(z) +σ(θ,z)? (2/π)θ??? >(2/π)|θ-?θ0(z)|??????? (?,θ,z)?R?+×[0,π)×R?????

Φ2(?,θ,z) =?Φ1(?,θ,z)

Φ(?,θ,z) =?Φ2(?,θ,z)

Φ2(?,θ-π,z)??π < θ <2π??????

f(x,y,z) =Φ(?x2+y2,arctan(y/x),z). Fact arctan(1/t) =π/2-arctant??????t >0?? T f(u)?0} f= 1/2f= 1/2f= 1y x f= 0f= 0f= 1/2 f= 1/2xy f= 1 f= 1f= 0xf= 1/2y f= 1 f= 0 f= 1f= 0f= 1/2

θn=θ0+π2n+2

xn=12n?1 +a2n ?n=?x2n+y2n= (?1 +a2n)xn=12n. un:= (xn,yn,z0)???un= (-xn,-yn,z0). f(un) =f(un) =Φ(?n,θn,z0) = (2/π)θn. un ∂?(un) =∂Φ∂z(un) = 0 ∂θ(un) =2π, ?f(un) =2π? -ynx2n+y2n,xnx2n+y2n,0? {un}n≥1 ?f(un) =-?f(un), 0??

ε>0co{?f(u) :u?B(u0,ε)∩Df}

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