А.В. Булинский, А.Н. Ширяев - Теория случайных процессов (1134115), страница 19
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+(Tt)t>0 ), M = 1 (8.37), .. kTtf k 6 kf k f 2 B t > 0.E. 8.12. + P (x t B ) { (Xt t > 0), . (7.19). C, (Ttf )(x) =ZXf (y)P (x t dy)(8.39) ) B (X Q R), . 6.15.', (8.39) (Ttf )(x) = Exf (Xt)(8.40) Ex , L(X0) = x. C , (Xt t > 0) t = 0 x 2 X .+ (Tt)t>0 , C0-, f 2 Bs ; tlimT f = f!0+ t(8.41) s ; lim B , .. kTtf ; f k ! 0 t ! 0+.E. 8.13. C, x 2 Xlim P (x t B ) = x(B ) B 2 Bt!0+(8.42) x { C, (8.39) C0- B (X R). (8.39) , B (X R) C0- (Tt)t>0, Q(x t B ) := (Tt1 B )(x) x 2 X t > 0 B 2 B(8.43)) , , ..
1') { 4') . ??.156E. 8.14. + (Tt)t>0 (8.39) - (7.18) m- . +, " C0 B (Rm R). , P (x t V"(x)) = 1 ; o(t) t ! 0+ V" (x) { ( ) " x 2 Rm.E. 8.15. + (Tt)t>0 { ) B B0 f , Ttf , t = 0. +, Ttf O0 1) f 2 B0. +, B0 ( ) TtB0 B0 t > 0.E. 8.16.
(. O?, . ??]) + Q { Rn, .. . C, (Ttf )(x) =ZRnf (e;t x + (1 ; e;2t)1=2y)Q(dy) x 2 Rn t > 0(8.44) Lp(Rn B(Rn) Q) p > 1 ) C0-, *. { J.K A ( ) (Tt )t>0 Tt f ; fAf = s ; tlim(8.45)!0+t " f , ). 3 , DA A ( f1 f2 2 DA , f1 + f2 2 DA 2 R). * ,Af = ddt (Ttf )jt=0 f 2 DA :+(8.46) 98.17. + A { , 8.14.= Cbu(Rm) , Rm . +@ 2f 2 C (Rm) k j = 1 : : : m:(8.47)@xk@xj buC, f (, . 8.14 , " f )(8.48)Af = 21 ^f ^ { 3.2 = k k Rm.
C t > 0 f "#1 (T f (x) ; f (x)) = 11 Z f (y)e;ky;xk2=(2t)dy ; f (x) =t tt (2t)m=2 Rm157Zp1= t(2)m=2 (f (x + z t) ; f (x))e;kzk2 =2dz:Rm+ @m pm2f (x + z pt)XX@f(x)1@f (x + z t) = f (x) + zk t @x + 2zk zj t @x @x (8.49)kkjk=1kj =1pR = (x z t) 2 R, jj 6 1. 9, g(u)du = 0, g { pR , jt;1(Ttf (x) ; f (x)) ; (1=2)^f (x)j 6p @ 2f (x) Zm 2X1@f(x+z6 m (2)m=2 @x @x t) ; @x @x e;kzk2=2jzkjjzj jdz =: I (x t): (8.50)k jk jRkj =1* (8.47) , I (x t) (8.50) ) x 2 Rm, t > 0.
M I (x t) Ir(x t), ) Br = fkzk 6 rg Jr (x t) { fkzk > rg. f " r(") >p 0 ,p Jr (x t) < "=2 x 2 Rm t > 0. 9 (8.47) , kz tk 6 r t z 2 Br , Ir(x t) < "=2 x 2 Rm 0 < t < t0("). 2G -% % & % & %%. D g t 2 R - B , t ( ))s ; hlim(g(t + h) ; g(t))=h:!0C g : Oa b] ! B , Oa b] ( { , Oa b], )R b g(),a t)dt, , M , B .@ . * Z b Z g(t)dt 6 b kg(t)kdtaa(8.51)( , , ). ' , g Oa + h b + h], Zbag(t + h)dt =Z b+ha+hg(t)dt:(8.52)D dg=dt { Oa b] ( { ), Z b dgdt = g(b) ; g(a):(8.53)a dt2 kg(u) ; g(v)k 6 supt2vu] kdg=dtk(u ; v) a 6 v < u 6 b (, ). D g { u 2 R, )Z u+h1s ; hlimg(t)dt = g(u):(8.54)!0+ h u158E.
8.18. C, g : Oa b] ! B Oa b], L { B , Z bLa Zbg(t)dt =aLg(t)dt:(8.55)E. 8.19. + (Tt)t>0 { ) C0- A. C, f 2 DA, Ttf t > 0( t = 0 ), Tt f 2 DA (8.56)dTtf = AT f = T Afttdt(8.57)Ttf ; f =(8.58)Zt0TsAfds:E. 8.20. C, A C0- (Tt)t>0 .. $ ( f(f Af ) f 2 DA g) B B . C , fn 2 DA fn ! f , Afn ! g n ! 1 ( B ), f 2 DA Af = g.E.
8.21. C, B0 = ODA ], B0 8.14, A { ) (Tt)t>0, O] B . , DA B ) C0- (Tt)t>0.E. 8.22. C, m > 1 3 M f : Rm ! R, ) (8.47). +" . 8.20 C8.17 , DA " , M.E. 8.23. ( DA , . 8.14 m = 1.* (., ., O?]), L, DL = B , L . ', C8.17 A , ) (8.47), .' ) .E.
8.24. 9 <& % L & %% B %% C0-, < L %.K , , Tt = etL t > 0(8.59)P (tL)n =n! (" etL := 1n=0P ktLkn=n! < 1) % < , t > 0, 1n=0 kTtk 6 ejtj t > 0(8.60) = kLk.' , . 8.19 8.24 ) .159E. 8.25. D A { ( ) (Tt)t>0, f 2 DA Ttf z(t) dz(t) = Az(t)dt) kz(t)k 6 cet c > 0 t > 0, kz(t) ; f k ! 0 t ! 0+.J A { % % DA B (, , DA 6= B ), % , % % % C0-( , C0- ) %, '. @, '<, % % ,%.+ (Tt)t>0 { ) .
0Rg :=Z10e;tTtgdt g 2 B > 0(8.61) . * (8.61) ( ) 0 u u ! 1. * , Rg { " 3 () Ttg. * (8.61) , kRk 6 1= > 0:E. 8.26. C, g 2 Bs ; limR g = g:!1 (8.62)(8.63)A 98.27. (Tt)t>0 { #) A. - > 0 R : B ! DA (8.64)(I ; A) : DA ! B(8.65)I { , R = (I ; A);1:(8.66)2 (8.64). + (8.55), (8.52) , , g 2 B > 01 (T ; I )R g = 1 (T ; I ) Z 1 e;sT gds = 1 Z 1 e;s(T g ; T g)ds =st+sst tt tt 00 Z1Z11t;u;se Tugdu ; e Tsgds ==t et0ZZ1t11t;ut= ;tee Tugdu + t (e ; 1) e;sTsgds =00Zt= ; 1t et e;uTu gdu + 1t (et ; 1)Rg:0160+ t ! 0+, (8.54), ARg = ;g + Rg:(8.67)@ , (8.64) %.
0 (8.67) , g 2 B > 0 f ; Af = g(8.68)% = f = Rg. 9%, % (8.68) % = f1 f2. @v = f1 ; f2 2 DA v ; Av = 0. 0 (8.57)dTtv = T Av = T v:ttdt+", , d (e;tT v) = 0:(8.69)tdt( (7), (8.69) , e;tTtv = x 2 B t > 0: t ! 0+, , x = v. 9, (Tt)t>0 { ) ( (8.37) (8.60) > 0, C8.27 > ), , 0 6 kvk = e;tkTtvk 6 e;tkvk t ! 1:0, v = 0 , , (8.68) f = Rg > 0 g 2 B . @ , I ; A %, %&' DA B (8.66). 2% 98.28.
(Tt)t>0 (St)t>0 { C0- #) B . " # , Tt = St t > 0.2 * C8.27 , (Tt)t>0 (St)t>0 (8.66). F 2 B . @, (8.55), g 2 B > 0Z1e;tF (Ttg ; Stg)dt = 0:0+" Ttg = Stg t > 0, g 2 B . 4 , & %, - %&', , O0 1) %%, (., .,O?]). 22 , % L '%&%, % %, (8.59).A 98.29 (K { 3). B { A { # DA B . * A #) C0- B , #) .161ODA] = B (DA B ).2. 4 # g 2 B # > 0 f ; Af = g . f 2 DA .3. 4 . f kf k 6 kg k=.1.162 9.
$ * 0.1 20 . F # Q P (t), t > 0. " # %. QT . $ &. 4# !, 2 . % # !.0 % , , % %<% %% , O0 1), % & ' -% % ,%%.= 4 fXt t > 0g ( P (t) = (pij (t))) , P (t) ! I t ! 0+, . . i j 2 Xlim p (t) = ij :t!0+ ij(9.1) 9.1. 4 % fXt t > 0g i j 2 X # t h > 0 jpij (t + h) ; pij (t)j 6 1 ; pii (h):: , % , i j 2 X $% pij (t) O0 1).2 *pij (t + h) ; pij (t) =Xkpik (h)pkj (t) ; pij (t) = pij (t)(pii(h) ; 1) ++Xk6=ipik (h)pkj (t) 6Xk6=ipik (h) = 1 ; pii(h)- , pij (t + h) ; pij (t) > pij (t)(pii (h) ; 1) > pii (h) ; 1. 2A 9.2. " P (t), t > 0, | )d+ P (t) = Qdt t=0,(9.2).
. )# pij (t):+qij = d pdtij (t) :t=0(9.3) 0 6 qij < 1i 6= jqi = ;qii 2 O0 1]:(9.4)1634 Q $.2 1 , i 2 X . (9.1) > 0 , pii(h) > 0 h 2 O0 ]. C t > 0, (8.4), , pii (t) > (pii(t=n))n ,n 2 N. n = n(t ) , t=n < , , pii (t) > 0 t > 0.+" O0 1) H (t) = ; log pii (t) ( ). ', H (s + t) 6 H (s) + H (t) s t 2 O0 1)(9.5) pii(s + t) > pii (s)pii(t) (8.19) s t > 0. +q = sup H (t)=t:t>0(9.6)=, q 2 O0 1]. M% %, .1. %, q < 1. @ " > 0 t0 = t0(") > 0, , H (t0)=t0 > q ; ". C h 2 (0 t0) t0 = nh + ^, n = Ot0=h], 0 6 ^ < h, O] { . + (9.5), H (^) :q ; " 6 H (t0)=t0 6 (nH (h) + H (^))=t0 = Hh(h) nh+(9.7)t0t0 (9.1) H (^) ! 0 ^ ! 0 (^ ! 0, h ! 0+). +"q ; " 6 liminf H (h)=h:h!0+(9.8)* (9.6) , lim suph!0+ H (h)=h 6 q.
0, ) limh!0+ H (h)=h = q.@ , H (h) = lim ; log(1 ; (1 ; pii (h))) = lim 1 ; pii (h) :q = hlim(9.9)!0+ hh!0+h!0+hh2. %, q = 1. @ M > 0 t0 = t0(M ) > 0 , H (t0)=t0 > M . C , (9.8), , liminf H (h)=h > M:h!0+@ , ) limh!0+ H (h)=h = 1. = (9.9).M% %, pij (t) i 6= j . = fij(k)(h) i i k h ( ), , " j mh, m = 1 : : : k. C n > 2 pij (nh) >n;1Xk=1fij(k)(h)pij ((n ; k)h):(9.10)C, 8.2, pij (nh) = P (Xnh = j jX0 = i) =164Xj1 :::jn;1 2XP (Xnh = j X(n;1)h = jn;1 : : : Xh = j1jX0 = i) >>n;1 XXXk=1 Jk jk+1 :::jn;1 2XP (Xnh = j X(n;1)h = jn;1 : : : X(k+1)h = jk+1jXkh = i) P (Xkh = i X(k;1)h = jk;1 : : : Xh = j1jX0 = i) Jk = f(j1 : : : jk;1) : jm 2= fi j g m = 1 : : : k ; 1g. 4 , (8.2)1 P (X = j X(n;1)h = jn;1 : : : Xkh = i : : : Xh = j1 X0 = i) =P (X0 = i) nh1 P (X = j X=(n;1)h = jn;1 : : : X(k+1)h = jk+1 jXkh = i : : : X0 = i)P (X0 = i) nh P (Xkh = i : : : X0 = i) == P (Xnh = j X(n;1)h = jn;1 : : : X(k+1)h = jk+1 jXkh = i)P (Xkh = i : : : Xh = j1jX0 = i):= , Xjk+1 :::jn;1P (Xnh = j X(n;1)h = jn;1 : : : X(k+1)h = jk+1 jXkh = i) = pij ((n ; k)h)XP (Xkh = i : : : Xh = j1jX0 = i) = fij(k)(h):Jk(k)pij (h) = i j k P h ( ).
@, (9.10) k > 1 ( = 0)pii(kh) = fij(k)(h) +=,k;1Xm=1p(ijm)(h)pji ((k ; m)h):(9.11)Pk;1 p(m)(h) 6 1, "m=1 ijfij(k) (h) > pii (kh) ; 16maxp ((k ; m)h)m6k;1 ji( ). (9.1) " > 0 t0 = t0(" i j ) > 0 , pji (t) 6 " pii (t) > 1 ; " pjj (t) > 1 ; " t 2 O0 t0]:(9.12)@fij(k)(h) > 1 ; 2" nh 6 t0 k = 1 : : : n ; 1:(9.13)', pij ((n ; k)h) > pij (h)pjj ((n ; k ; 1)h) k = 1 : : : n ; 1 n > 2, (9.10) { (9.13) pij (nh) > (1 ; 2")pij (h)n;1Xk=1pjj ((n ; k ; 1)h) > (1 ; 2")(1 ; ")(n ; 2)pij (h): n = Ot0=h], pij (t) ( 9.1), 1 > pijt(t0) > (1 ; 2")(1 ; ") lim sup pijh(h) :0h!0+165@, t0 , , liminf pijt(t) > (1 ; 2")(1 ; ") lim sup pijh(h) :t!0+h!0++ " > 0 , , ) limt!0+ pij (t)=t.
2PC N 2 N t > 0, , pij (t) = 1, jX pij (t)1 ; pii (t) :6ttj 6Nj 6=i (9.3) P q 6 q , ,ijij 6=ij 6NXj 6=iqij 6 qiQ(9.14) qi = 1 (9.14) .z , " Q " iXj 6=iqij = qi:(9.15)@, (8.8) 0 ; BB ; Q=BB...@0...1CCCA0C (9.15) .A 9.3 ( ). % ; . t>0P 0(t) = QP (t). . #-(9.16)i j 2 X t > 0 )# p0ij (t), p0ij (t) =Xkqik pkj (t):(9.17) (9.2) pij (0) = ij , (9.17) t = 0, .2 C t > 0, h > 0 i j 2 X XLij (h t) = h1 pik (h)pkj (t):k6=i166@pij (t + h) ; pij (t) = pii (h) ; 1 p (t) + L (h t):ijijhh+ N 2 N (9.3)X pik (h)Xlim Lij (h t) > limpqikpkj (t):kj (t) =h!0+h!0+ k6=i hk6=ik6N(9.18)k 6N0,lim Lij (h t) >h!0+Xk6=iqik pkj (t):(9.19)P+ N > i, , pkj (t) 6 1 pik (h) = 1, i j 2 X t > 0,kh > 0 XXLij (h t) 6 pikh(h) pkj (t) + h1 pik (h) =k6=ik>NX pik (h)Xk6N1=h pkj (t) + h 1 ; pii (h) ; pik (h) :k6=ik6N+"lim L (h t) 6h!0+ ij0,lim L (h t) 6h!0+ ijXk6=iXk6=ik 6Nk6Nk6=iqik pkj (t) ; qii ;qik pkj (t) + qi ;Xk6=iXk6Nk6=iqik =qik :Xk6=iqik pkj (t)(9.20) (9.15).
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