Case Study of Markov Chains Ray-Knight Compactification

Transcription

1 Internatonal Journal of Contemporary Mathematcal Scences Vol. 9, 24, no. 6, HIKAI Ltd, Case Study of Marov Chans ay-knght Compactfcaton HaXa Du and YanLng Pan Department of Mathematcs and Statstcs Zhengzhou Normal Unversty Zhengzhou, 4544, Chna Copyrght 24 HaXa Du and YanLng Pan. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal wor s properly cted. Abstract It gves ay-knght compactfcatons of specfc examples to llustrate the complexty. Keywords: ay - Knght compactfcaton; Transfer functon;esolvent. Introducton Structural problems of Marov chan suffered for decades, s not completely resolved, the man reason s the general Marov chan has only locally strong Marov property. eference [] usng ay-knght compactfcaton method, construct strong Marov process correspondng to the transfer functon. eference [2] usng ay - Knght of the Marov chan method, proves that the Marov chan the exstence of the local tme of. eference [3], [4], [5] usng Marov chan ay-knght compactfcaton method, solves the structural problem of blateral brth- death process. eference [6] usng Marov chan ay-knght compactfcaton method, solves the structural problem of brth- death process. eference [7], [8] llustrates the relatonshp of Martn entrance boundary and ay-knght compactfcaton of mnmal Q-processes. Thus, ay-knght compactfcaton s the brdge to solve the problem of Marov chan structure. In ths artcle, we wll through specfc examples to llustrate the ay - Knght com-

2 754 HaXa Du and YanLng Pan pactfcaton of Marov chan, at the same tme, t shows that ay - Knght compactfcaton complexty. 2. Prelmnares Let E =,2,,we agree on the topology of E s dscrete topology, then E s a locally compact and has a countable topologcal space, and the functon of E s a contnuous functon. Called the state of the elements n E. All bounded functons on E denoted as M, wth bounded nonnegatve functons denoted as M, M and M topology s the topology of unform convergence. Sometmes we tae M and M elements as nfnte dmensonal,, column vector. The famly of functons Pt p t t on called the transfer functon on E, f p t p t 2 3 p t s p t p s If there s lm p t p t,, E, s, then called Ptt as a standard. Sad Ptt s honest (or not nterrupted), f establshed for E all equal sgn n (2). Known that f Ptt transfer functon s the standard, then there s a lmt: Matrx t p lm q,, E. and q, q q, q q. t t s called, E Q q t P t t of densty matrx on E. t Let e p t dt,, E,. ( ) s called the resolvent of

3 Case study of Marov chans ay-knght compactfcaton 755 Ptt ( ) s the resolvent of the transfer functon of only f the followng condtons (6) (9) holds ( ) (6) E P t t, f and ( ) ( ) ( ) ( ) ( ) (7) E lm ( ) = ( 8) lm ( ) = q ( 9) Let E =,2,,P (t) s the honesty transfer functon on E, ( ) s the resolvent of P (t), The norm of M s defned as: for arbtrary f M, f sup f. Let the functon f E s f. Obvously, f E s lnerator on M and. For any G M, let n u( G) = { u f n,, u, f G, n}, ( G) = { f f n, f, f G}, n n Tae H= H I E, then H s countable sudset of G, let () ( n) ( n) ( n) = u( H), = ( + u( )), ( n) =. let g m m= n s the dense subset of, d( x, y) = [ gm( x) gm( y) ], x, y S, ( ) m 2 m Then d, s the measure on E. E s the completon of the E under d, Apparently, E s compact metrc space, called ay-knght compactfcaton of E. ( ) can expand nto U x, dy whch s the ray resolvent on E, P x, dy t.

4 756 HaXa Du and YanLng Pan s the ray sem-group correspondng to U x, dy E,P, x. Let D x x x, then D s called non-branch pont sets, the pont of D s called non-branch pont. 3. Example of ay-knght compactfcaton Let E =,2,, q q q2 q2 Q q3 q3 q, q 2, are a lst of ntegers. The mnmum transfer functon of Q s named m n m n P t, the resolvent of P t defned by f mn ( q ) ( q ), f () f ( q ) and m n P t s honest f and only f - = q (references 9 ). m n Example. Let the mnmum transfer functon P t s honest, then q - =. mn to satsfy the followng assumptons: () E s topologcal space whch s the locally compact and has countable topologcal base (eferred to as L.C.C.B); (2) C E C E s Marov and, b b for arbtrary > ; mn (3) For arbtrary f C E and x E, f x f x Proof. For arbtrary b lm,lm. f, lm f. Usng mathematcal nducton, t s

5 Case study of Marov chans ay-knght compactfcaton 757 easy to prove that for arbtrary n, lm. By denotng E f f E=, then E s the one pont compactfcaton of E. For arbtrary,, lm. so U E E, D E. then U,. s a non-branch pont. Example 2. Let,, 2, s a probablty measure on E., for arbtrary E, m n the mnmum transfer functon mnum Ptt s transfer founcton of Doob process whch s Proof. From references 9, we now that - = contnue. The condtons dstrbuton of P X = P t s nterrupt, Q, type. q, the orbt of {X t } s rght s remembered to P, the correspondng condtonal expectaton s remembered to E.Let nf sx s =, then s the frst leap pont and the followng equaton, E, > ( 2), The resolvent of holds. E { e } ( q ) mn ( ) mn E follows: ( ) ( ) +E{ e },, E ( 3). [ E { e }] For any, E,,obvously, E mn ( ) mn E E e E E e lm { },lm ( ),lm ( ). [ { }] P t s as Modeled on the example one, t s easy to prove that for arbtrary n, lm s exst. By the defnton of E, then f f arbtrary, by the defnton of U,then E= E, For

6 758 HaXa Du and YanLng Pan mn ( ) E U (, ) lm ( ), E. [ E { e }] E Therefore, U E P non-branch pont.,,,, for arbtrary E. s a E =. E, D E emar: In the same way to prove the followng concluson, f there s E mae that, then the ay-knght compactfcaton of E s named E whch meets the followng equaton, E E. and s the lmt pont of the sequence pont of,2,3,. At the same tme, E E E D. Example 3. Let,, 2, s a probablty measure on E., for arbtrary E, The Marov chan m n the mnmum transfer functon mnum t X correspondng to P t s nterrupt, P t t s not Doob process. Proof. From references, we now that - = q, q lm E e lm. q Snce Q s sngle outflow zero nflow, accordng to the general conclusons of sngle outflow of Marov chan, so there s,,2,,such that Moreover, ( ), [ E { e }], > ( 4) s the resolvent of the transfer functon of mn ( ) mn E E E e P t t and ( ) ( ) +E{ e },, E, > ( 5) [ { }] Modeled on the example two, we can prove that E= E, For arbtrary

7 Case study of Marov chans ay-knght compactfcaton 759, mn ( ) E U (, ) lm ( ), E. [ E { e }] E (6) t Use the formula of (4), then lm U,, that s P E E D E { }. lm,. So For the transfer functon whch contans nstantaneous state, t s ay-knght compactfcaton structure s more complcated, please loo at the case. Example 4. If q,, 2, s a lst of postve number, consder the followng matrx Q-. t q2 q2 Q q3 q3 q q 4 4 wth q. 2 The matrx s called Kolmogorov matrx. It corresponds to the resolvent ( ) ( 2 ) q ( ) ( ), q 2 q ( ) ( ), 2 q q ( ) ( ),, 2 q q q Obvously, lm ( ) ( ), t s easy to prove E= E, and under the topology of ay-knght, s the lmt pont of the sequence pont of 2,3,.

8 76 HaXa Du and YanLng Pan eferences [] J. C. Dong, Marov chans ay-knght compactfcaton, Zhengzhou Unversty, 26. [2] Q. L. Zhao, Local tme of Marov chans on the S, Zhengzhou Unversty, 26. [3] L. Chen, Paths of B-lateral Brth-Death Processes and constructon theory, Journal of Henan Unversty (Natural Scence), 4 (22), [4] L. Chen, Paths of B-lateral Brth-Death Processes and constructon theory, Journal of, Henan Unversty (Natural Scence), (23), 5 -. [5] L. Chen, L. X. Xue, Paths of B-lateral Brth-Death Processes and theory of Motoo, Journal of Zhengzhou Unversty (Natural Scence Edton), (24), [6] X. J. Zhao, G. J. Yan, Paths of Brth-Death Processes, Chnese Journal of Appled Probablty and Statstcs, (23), [7] C. W. Xu, G. J Yan, Martn Entrance Boundary and ay-knght Compactfcaton of Mnmal Q-Processes, Chnese Journal of Appled Probablty and Statstcs, 6 (2), [8] S. S. Hao, C. W. Xu, Case Study of ay-knght Compactfcaton and Martn Entrance Boundary of Mnmal Q-Processes, Journal of Tonghua Teachers College, 6 (24), [9] ogers, L. C. G. and Wllams, D., Dffusons, Marov Processes and Martngales (Volume Ito Calculus), Cambrdge Mathematcal Lbrary, Cambrdge Unversty Press, 2. [] Z. K. Wang, X. Q. Yang, brth and death processes and Marov chans, Scence Press, Beng, 25. eceved: October 5, 24; Publshed: December 8, 24