In. J. Conemp. Mah. Scences, Vol. 8, 213, no. 4, 163-171 HIKARI Ld, www.m-har.com rac Propes of Normal Chan L Chen School of Mahemacs and Sascs, Zhengzhou Normal Unversy Zhengzhou Cy, Hennan Provnce, 4544, Chna Malng addss: No. 6, Yngca sr., Norh Unversy Cy, Zhengzhou Cy, Henan Provnce, Chna, 4544 cluu6697@sna.com Zhong-guang Fan School of Mahemacs and Sascs, Zhengzhou Normal Unversy. Zhengzhou Cy, Hennan Provnce, 4544, Chna Absrac In hs paper, search sae space srucu propes of normal chan from Ray-Kngh heory on a gven se : If S ( ω) whose probably s 1 a empy se,hen,he normal chan only has jumpng ype rac, a he me, he ransfer funcon mus be he leas and only deermned by desy mar; If S ( ω) whose probably s posve a no empy se, hen,he rac of normal chan s no jumpng ype rac, and s ransfer funcon canno deermne by densy mar. Keywords: Normal chan; Soppng me, Jumpng rac 1 Inroducon [1 3] In he classc boos of some Marov chan,for he ransfer funcon n a gven, he usuall pracce s consrucng canoncal chan on U. he sae space of model chan s very smply. Bu s rac jus has rgh lower sem connuy,
164 L Chen and Zhong-guang Fan and jus eep par of he srong Marov. As applcaon of Ray-ngh heory, he [4 5] normal chan can overcome hese defecs,bu he srucu of sae space may be very comple. So we wll search he propes of he normal chan. 2 Plmnary nowledge [3] Le = {1, 2, L }, P () = ( p()), and P () s he hones ransfer j, j funcon on, Q = ( q ), Q s densy mar of P (), R ( ) s a j, j j solven of P (), s Ray-Kngh gh for P () on, ( U α ) α > s a Ray solven for P (). And P () s Ray sem group for P (), ε s a Bol algebra on +.Le D [, ) = { ωω:[, ) }, F s a Bol algebra on D [, ), Coordnae process of F s { X } ( ), Le Ω denoe he subse { =, } c + = of D [, ). θ s he mappng: R θ : ω() a ω( + ), ω() D [, ),. P s he probably measu of on +. Defnon 1 [6] Le X = ( Ω, F, F, X, θ, P ), X s called normal chan on P (). Defnon 2 For a funcaon on { Ω, F}, : Ω [, ), f for any, he a { } F, hen we called s soppng me. Defnon 3 Le S ( ω) = { s X ( ω) = }, we call S ( ω ) s -consan value se ; he nerval of S ( ω ) s called -nerval of X. s
rac propes of normal chan 165 Defnon 4 If = nf{ s s, X X }, = nf{ s s >, X = X }, s called s s escape me. s called urn me. Defnon 5 For any +, P { = } = or 1, P { = } = or 1. If P { = } = 1, s called absorbng sae; If P { > } = 1, s called say sae; If P { = } = 1, s called gular sae; If P { = } = P { = } = 1, s called nsananeous sae. 3 Relevan heoms heom 1 Le (1) s Regular Sae., hen (2) On P, obey eponenal dsrbuon of parameer q. (3) On P, X and a ndependen. q j (4) If < q <, hen for any j, j, P { X = j} =. q Proof: (1) If s no Regular Sae, hen P { > } = 1, for any > s easy o see { X = } { } and when, { } { = }, so we have ha 1= lm P( ) = lm P { X = } lm P { } = hs s a conradcon sul. So s Regular Sae. (2) Refer o [2]. (3) If q = or, s clearly ha P { = } = 1 or P { = } = 1.,
166 L Chen and Zhong-guang Fan If < q <,for any A, A +, and s>,, P { > + s, X A} = P { >, oθ > s, X o θ A} = { P { oθ > s, X o θ A F}; > } X = { P { > s, X A}; > } = P { > } P { > s, X A} Le s, hen P { >, X A} = P { > } P { X A} So, on P, and X a ndependen. (4) If < q <, for any j, >, from (3) and he srong Marov of X, We ge ha R ( ) e = p ( ) d j = [ ( ) ] j e I X d { j} = [ e I ( X ) d] { j} = [ e [ e I ( X ) d] o θ ] { j} X = [ e [ e I ( X ) d]] { j} = [ e e P [ X = j] d] X = [ e e P( X,{ j}) d] = [ e U ( X,{ j})] = [ e ] [ U ( X,{ j})]
rac propes of normal chan 167 + For any, j, for connuous funcons f () on, so ha f ( ) =, f ( j) = 1, hen = f ( ) = lm U f ( ) lm U (,{ j}). Bu lm U ( j,{ j}) = lm R jj ( ) = 1, So q = R = e U X j 2 lm ( ) lm [ ]lm [ (,{ })] j j q = lm [lm U ( X,{ j})] + q = q P [ X = j] q j ha s P [ X = j] =. q s nsananeous sae(sable sae) f Noe1: From heom 1 we ge ha and only s ha s nsananeous sae(say sae) of normal chan X. heom 2 If ha a b and f q <, hen for any + a <, hen a b, he a a 1, b 1, a 2, b 2 L, so < ; If, 1 b b a + < <, for any, wll have S ( ω) = U [ a ( ω), b ( ω)). For any s <, le ξ (,) s denoe he number of [ a, b ) n [ s,], = 1, 2, L, hen { ξ ( s, )} q( s). Proof: Le
168 L Chen and Zhong-guang Fan a = nf{ u u, X = } 1 b = nf{ u u a, X } 1 1 a = nf{ u u b, X = } + 1 u b = nf{ u u a, X } + 1 u I s easy o now ha a, b, = 1,2, L a { F } soppng me. For any, f a <, because X s rgh connuous, and 1,we have ha u u X a P [ a <, b > a ] = P[ a <, o θ > ] a =, from (2) of heom = [ P [ o θ > F ]; a < ] = [ P [ > ]; a < ] = P [ a < ] a a hen, on { a < }, almos su have a < b, from he srong Marov of X and defne of b, for any, we ge ha = P [ b <, ε >, X [ b, b + ε), X = ] = P { b <, X =, o θ > } b b = [ P [ o θ > F ]; b <, X = ] = [ P [ > ]; b <, X = ] = P { X =, b < } b b b b b So, on { b < }, almos su have X b.for any < s <, s obvous ha ξ (,)} s = ξ (, s) o θ.(fer o [2](12-13)) s
rac propes of normal chan 169 { ξ ( s, )} { ξ (, s) o θ } s = P { X = } { ξ (, s)} q ( s) s For any >, have { ξ (, )} q.so,almos su ha he a only lmed [ a, b ), = 1,2, L on any lmed nerval. hen lm a =, we may ge ha S ( ω) = [ a ( ), b ( )) U ω ω heom 3 If q =, we have: (1) Almos su S ( ω ) does no conan any nerval. (2) Almos su S ( ω ) s own dense se. Proof:(1) I s clear ha S ( ω ) s a oponal se. Le A( ω) = sup{ s s <, s S ( ω)},, ω Ω }, s obvous { A } s monoone ncasng lgh connuous process, and suable { F }. Le B = lm A, hen { B } s s a rgh connuous process of oponal s suable { F }.Se U = {( ω, ) ε >, ( ε, + ε) S ( ω)}, ω Ω, hen U = {( ω, ) B ( ω) < }, so U s he oponal se suable { F }. We use D o denoe he opporuny me U, If P { D < } >, he ess u U { F } soppng me, so ha P { < } >, and on { < },( ω, ( ω)) U, From (2) of heom 1 we have ha P [ < ] = P [ <, X =, ε >, ε ( ε, + ε), X ]
17 L Chen and Zhong-guang Fan P [ <, X =, o θ > ] [ = [ P > ], < ] = hs s a conradcon sul! So P { D < } =.Namely, almos su S ( ω ) U does no conan any nerval. (2) Proof mehods s same o (1). 4 Man Resuls + Proposon1 If \,hen P { = } = 1. Proof: For any, q <, we use ( ) ( ) [, ] a b denoe he -h -nerval of S ( ω ), Le S ( ω) = { u X ( ω) }, S s he me of soppng he f U f vrualsae. For any +, >, + hen, meas S =, f \ f I = f s { S [, ]} { [1 I ( X )] ds} = [1 P { X }] ds = s, we have ha P { = } = 1. Proposon2 If Ω= ( q ) s all sably, For any ω Ω, Le, j j S ( ω) = { s s, ε > ; wll have nfne jumps}, hen S ( ω) s closed se. S ( ω) \ S f ( ω) conan counable pons, and meas S ( ω) almos su s. Proposon3 Le σ( ω) = nf{ ss>, s S ( ω)}, ω Ω, hen σ s { F } Soppng me. Proposon4 he campagn on (, ) \ S ( ω) mnmum ransfer funcon. Proof: For any n and,, j, we denoe: s deermned by he
rac propes of normal chan 171 I s easy ha: h () = P[ X = j.[,] have n jumps] ( n ) j () q h () e δ j j =, ( n) q n 1 s j j h () = e qh ( sdsn ), = 1,2, L P X j σ h p j ( n) mn [ =, < ] = ( ) ( ),,, n j Because (, ) \ S ( ω) s a open se, and all rac on he (, ) \ S ( ω) jumpng ype, for any { F } soppng me, P X j S ω X P a μ mn [ =,[, + ] I ( ) = = ] = ( ) + j So he campagn on (, ) \ S ( ω) funcon. Noe2: If S ( ω) s deermned by he mnmum ransfer whose probably s 1 s empy se, hen, he normal chan only has jumpng ype rac, a he me, he ransfer funcon mus be he leas and be only deermned by densy mar. If S ( ω) whose probably s posve s no empy se, hen he rac of normal chan s no jumpng ype rac, and s ransfer funcon canno deermne by densy mar. Refences [1]ZHONG Kala, Course n probably heory, Shangha scence and echnology Pss.1989 [2]WANG Zun; YANG Xangqun. Brh and deah processes and Marov chan, Bejng Scence Pss.25.55-57 [3]YANG Xangqun. Srucu heory of counable Marov processes.hunan scence and echnology Pss.1981.3-6 [4]HOU Zhengng;GUO qngfeng. Homogeneous counable Marov processes. Bejng scence and echnology Pss.1978. [5]HOU Zhenng. he only cron of Q-processes. Hunan scence and echnology Pss.1982. [6]WU Qunyng;ZHANG Hanjun;HOU Zhenng.An eended brh-deah Q-mar wh nsananeous Sae[J],Chnese J.conemp.Mah,23,24(2)159-168. Receved: November, 212