Диссертация (1155084), страница 6

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Ñîîòâåòñòâóþùèå ðåçóëüòàòû ïðèâåäåíû â [71].3.2.1Ââåäåíèå è îñíîâíûå ïîíÿòèÿÇäåñü X = X(t), t ≥ 0 - ÏÐà ñ èíòåíñèâíîñòÿìè ðîæäåíèÿ è ãèáåëè λn (t),µn (t) ñîîòâåòñòâåííî.Âåðîÿòíîñòíàÿ äèíàìèêà ïðîöåññà îïèñûâàåòñÿ ñèñòåìîé Êîëìîãîðîâà:dp0dtdpkdt= −λ0 (t)p0 + µ1 (t)p1 ,= λk−1 (t)pk−1 − (λk (t) + µk (t)) pk ++µk+1 (t)pk+1 , k ≥ 1.(3.2.35)47Ïîëàãàåìλn (t) ≤ Λn ≤ L < ∞,(3.2.36)µn (t) ≤ ∆n ≤ L < ∞,ïðè ïî÷òè âñåõ t ≥ 0.Òàêèì îáðàçîì, ìîæíî ðàññìîòðåòü ñèñòåìó (3.2.35) â êà÷åñòâå äèôôåðåíöèàëüíîãî óðàâíåíèÿ:dp= A (t) p,dtp = p(t),(3.2.37)t ≥ 0,â ïðîñòðàíñòâå l1 ñ îãðàíè÷åííûì îïåðàòîðîì A(t).Ìû ïðèìåíÿåì îáùèé ìåòîä ëîãàðèôìè÷åñêîé íîðìû ìàòðèöû è èññëåäóåì ïðîáëåìó óñòîé÷èâîñòè ñèñòåìû Êîëìîãîðîâà äëÿ íåîäíîðîäíûõ ìàðêîâñêèõöåïåé. Ìåòîä îñíîâàí íà ëîãàðèôìè÷åñêîé íîðìå ëèíåéíîãî îïåðàòîðà è ïðåîáðàçîâàíèÿõ ìàòðèöû èíòåíñèâíîñòåé ìàðêîâñêîé öåïè (ñì.[77, 51, 83, 84, 92]).3.2.2Äîïîëíèòåëüíûå ñâåäåíèÿ è ðåçóëüòàòûÒàê êàê p(t) ∈ Ω äëÿ ëþáîãî t ≥ 0, ïîëîæèì pi (t) = 1 −j6=i pj (t),Päëÿ ïðîèçâîëüíîãî ôèêñèðîâàííîãî i.

Òàêèì îáðàçîì, ìû ïîëó÷àåì ñëåäóþùóþñèñòåìó èç (3.2.37)dz(t)= B(t)z(t) + f (t),dt(3.2.38)ãäå z (t) - ýòî p (t) áåç êîîðäèíàòû pi , à èìåííî, z (t) = (p0 , p1 , . . . , pi−1 , pi+1 , . . . ).Ñîîòâåòñòâåííî èìååì f (t) = (0, 0, . . . , µi , λi , 0, . . . ), è B (t) - ìàòðèöà âèäà:i−2i−1i+1i+2···0000·········2···i − 1···i + 1 · · ·i + 2······00000000λi−2 − µi−µi−1 − λi−1 − µi−µi − µi−µi−λi−λi−µi+1 − λi+1 − λiµi+2 − λi00λi+1−µi+2 − λi+2··· .··· ··· ··· 0148Ïóñòü D∗ - ìàòðèöà âèäà:i−2 i−1 i+1 i+2 i+30−1···i − 2 −1i − 1 −1i + 1 0i + 2 0i + 3 0···0···00000···,························−10000······−1−1000······00111······00011······00001························òîãäàD∗ BD∗−1=−µ1 − λ0µ1000···· · ·· · ·· · ·.· · ·· · ·λ1−µ2 − λ1µ2000λ2−µ3 − λ2µ3000λ3−µ4 − λ3µ4000λ4−µ5 − λ40000λ5 · · ·00000···Çàìåòèì, ÷òî D∗ BD∗−1 íå çàâèñèò îò i.49Ïóñòü òåïåðü {dk } - ïîñëåäîâàòåëüíîñòü ïîëîæèòåëüíûõ ÷èñåë, è D∗∗ =diag (d0 , d1 , .

. . , di−1 , di+1 , di+2 , . . . ).−d00 −d−d11 ...D = −di−1 −di−1 00 00···Ïîëîæèì D = D∗∗ D∗ ,000···000.···di+1 · · ·di+2 · · ·· · · −di−10···0di+1···00···ÒîãäàDBDÏóñòül1D−1-=−µ1 − λ0d0 µ1d100···d1 λ1d0−µ2 − λ1d1 µ2d200d2 λ2d1−µ3 − λ2d2 µ3d300d3 λ3d2−µ4 − λ3000d4 λ4d3· · ·· · ·.· · ·· · ····ïðîñòðàíñòâîïîñëåäîâàòåëüíîñòåé:l1D={z=(p0 , p1 , ..., pi−1 , pi+1 , ...)> : kzk1D ≡ kDzk < ∞}. Êðîìå òîãî, ââåäåì äîïîëíèòåëüíîå ïðîñòðàíñòâî l1E :l1E = {z = (p0 , p1 , ..., pi−1 , pi+1 , ...)> : kzk1E ≡Xk|pk | < ∞}.k6=iÐàññìîòðèì âûðàæåíèÿ:αk (t) =λk+1 (t) − ddk−1µk (t) , k < i − 1,λk (t) + µk+1 (t) − ddk+1kkλi−1 (t) + µi (t) − di+1 λi (t) − di−2 µi−1 (t) , k = i − 1,di−1di−1λi (t) + µi+1 (t) − ddi+2λi+1 (t) − ddi−1µi (t) , k = i,i+1i+1λk (t) + µk+1 (t) − dk+2 λk+1 (t) − dk µk (t) , k > idk+1dk+1(3.2.39)50è(3.2.40)α (t) = inf αk (t) .k≥0Ðàññìîòðèì (3.2.38) êàê äèôôåðåíöèàëüíîå óðàâíåíèå â ïðîñòðàíñòâå l1D .Òîãäà åãî ðåøåíèåì ÿâëÿåòñÿtZz(t) = V (t, 0)z(0) +V (t, τ )f (τ ) dτ,(3.2.41)0ãäå V (t, z) - îïåðàòîð Êîøè äëÿ (3.2.38), ñì.

[83].Èìååì kf (t)k1D = di−1 µi (t) + di+1 λi (t) ≤ di−1 ∆i + di+1 Λi ïðè ïî÷òè âñåõt ≥ 0. Ñ äðóãîé ñòîðîíû, åñëè ïîëîæèìβk (t) =λk (t) + µk+1 (t) + ddk+1λk+1 (t) + ddk−1µk (t) , k < i − 1,kkλi−1 (t) + µi (t) + di+1 λi (t) + di−2 µi−1 (t) , k = i − 1,di−1di−1λi+1 (t) + ddi−1µi (t) , k = i,λi (t) + µi+1 (t) + ddi+2i+1i+1λk (t) + µk+1 (t) + dk+2 λk+1 (t) + dk µk (t) , k > i,dk+1dk+1(3.2.42)òî ìîæíî ïîëó÷èòükB(t)k1D = sup βk (t) ≤ 4L − α(t),k≥0ïðè ïî÷òè âñåõ t ≥ 0.Òîãäà f (t) è B(t) îãðàíè÷åííûå è ëîêàëüíî èíòåãðèðóåìûå íà ïðîìåæóòêå[0, ∞) âåêòîð-ôóíêöèÿ è ôóíêöèÿ îïåðàòîðà â ïðîñòðàíñòâå l1D ñîîòâåòñòâåííî.Òåïåðü ïîëó÷àåì ñëåäóþùóþ îöåíêó äëÿ ëîãàðèôìè÷åñêîé íîðìûγ (B(t)) â l1D :γ (B)1D = γ DB(t)D−11= − inf (αk (t)) = −α(t),k≥0(3.2.43)â ñîîòâåòñòâèè ñ (3.2.40), ïîäðîáíåå ñì.

[77, 51, 83, 84, 92].ÈìååìkV (t, s)k1D ≤ eRt− α(τ ) dτs.(3.2.44)51Ïóñòü ñóùåñòâóåò ïîëîæèòåëüíîå M è α òàêèå, ÷òî−eRtα(τ ) dτs≤ M e−α(t−s) ,(3.2.45)äëÿ ëþáîãî 0 ≤ s ≤ t. Òîãäà ïðîöåññ X(t) ñëàáî ýêñïîíåíöèàëüíî ýðãîäè÷åí ïîíîðìå 1D.Ïîëîæèì òåïåðü z (0) = 0 (òî åñòü, p (0) = ei ). Òîãäàkz (t) k1D ≤ kV (t, 0) k1D kz (0) k1D +Rt+ 0 kV (t, s) k1D kf (s) k1D ds ≤Rt≤ 0 M e−α(t−s) kf (s) k1D ds ≤≤ α1 M (di−1 ∆i + di+1 Λi ) .(3.2.46)Ñ äðóãîé ñòîðîíû,kzk1D = (d0 + ···+ di−1 ) p0 + (d1 + ···+ di−1 ) p1 +···+ di−1 pi−1 + di+1 pi+1 + (di+1 + di+2 ) pi+2 + ···.Îáîçíà÷èìgk =i−1Xdj ,kXGk =dj .j=i+1j=kÇàòåìkz (t) k1D = kD z (t) k =Xpk (t) gk +k<i(+Xpk (t) Gk ≥k>i(Òàêèì îáðàçîì, pk (t) ≤óòâåðæäåíèå.pk (t) gk , k < i,pk (t) Gk , k > i.kz(t)k1D,gkkz(t)k1DGk ,k<ik>i(3.2.47), è ìû ïîëó÷àåì ñëåäóþùååÐàññìîòðèì ïðîöåññ ðîæäåíèÿ è ãèáåëè X(t) ñ èíòåíñèâíîñòÿìè λk (t) è µk (t).

Ïóñòü ñóùåñòâóåò ïîñëåäîâàòåëüíîñòü {dk } òàêàÿ, ÷òîÒåîðåìà 13.52(3.2.45) âûïîëíÿåòñÿ. Òîãäà X(t) ñëàáî ýêñïîíåíöèàëüíî ýðãîäè÷åí ïî íîðìå1D, è ñïðàâåäëèâû ñëåäóþùèå îöåíêè(pk (t) ≤M (di−1 ∆i +di+1 Λi ),α gkM (di−1 ∆i +di+1 Λi ),α Gkk<ik>i,(3.2.48)äëÿ ëþáîãî k.3.2.3ÀïïðîêñèìàöèÿÐàññìîòðèì óñå÷åííûé ïðîöåññ ðîæäåíèÿ è ãèáåëè ñ ïðîñòðàíñòâîì ñîñòîÿíèé N1 , N1 + 1, . .

. , N2 è èíòåíñèâíîñòÿìè λ∗k (t) = λk (t) ïðè N1 ≤ k < N2 ,è µ∗k (t) = µk (t) ïðè N1 < k ≤ N2 , ïîëàãàÿ îñòàëüíûå èíòåíñèâíîñòè ðîæäåíèÿ è ãèáåëè ðàâíûìè íóëþ. Îáîçíà÷èì ÷åðåç A∗ (t), p∗ (t) ñîîòâåòñòâóþùèåõàðàêòåðèñòèêè ðàññìàòðèâàåìîãî ïðîöåññà.Äëÿ óñå÷åííîãî ïðîöåññà èìååì ñëåäóþùåå ðàâåíñòâî:dp∗= A∗ (t) p∗ (t) ,dt(3.2.49)âìåñòî(3.2.37). Òåïåðü, ñâîéñòâî p∗ (t) ∈ Ω äëÿ ëþáîãî t ≥ 0 ïîçâîëÿåò ïîëîæèòüp∗i (t) = 1 −P∗j6=i pj (t),ïîëó÷àåìäëÿ ïðîèçâîëüíîãî ôèêñèðîâàííîãî i.

Òîãäà èç (3.2.49)dz∗= B ∗ (t) z∗ (t) + f ∗ (t) ,dt(3.2.50)âìåñòî(3.2.38).Ïåðåïèøåì (3.2.50) â âèäå:dz∗= B (t) z∗ (t) + (B ∗ (t) − B (t)) z∗ (t) + f ∗ (t) .dt(3.2.51)Òàêèì îáðàçîì, ìû ïîëó÷àåì ñëåäóþùåå ðàâåíñòâî äëÿ (3.2.38)è (3.2.51):z (t) − z∗ (t) = V (t, 0) (z (0) − z∗ (0)) +Z tV (t, s) (B (s) − B ∗ (s)) z∗ (s) ds+0Z tV (t, s) (f (s) − f ∗ (s)) ds.0(3.2.52)53Áóäåì ïîëàãàòü, ÷òî z (0) = z (0) = 0 (òî åñòü, p (0) = p∗ (0) = ei èëè∗X(0) = X ∗ (0) = i), ãäå N1 < i < N2 . Òîãäà f (s) = f ∗ (s), äëÿ ëþáîãî s.Ïîëó÷àåìZ∗z (t) − z (t) =tV (t, s) (B (s) − B ∗ (s)) z∗ (s) ds(3.2.53)0è((B (s) − B ∗ (s)) z∗ (s)) =0, · · · , 0, µN1 p∗N1 , −µN1 p∗N1 , 0, · · · ,>0, −λN2 p∗N2 , λN2 p∗N2 , 0, · · · .(3.2.54)Ïóñòü {d∗k } - ïîñëåäîâàòåëüíîñòü ïîëîæèòåëüíûõ ÷èñåë òàêàÿ, ÷òîeRt− α∗ (τ ) dτs≤ M ∗ e−α∗(t−s)(3.2.55),äëÿ ïîëîæèòåëüíûõ M ∗ , α∗ è ëþáîãî 0 ≤ s ≤ t, âìåñòî (3.2.45), ãäåα∗ (t) = min αk∗ (t)(3.2.56)èαk∗ (t) =λ∗k+1 (t) − ddk−1µ∗k (t) , k < i − 1,λ∗k (t) + µ∗k+1 (t) − ddk+1kkλ∗ (t) + µ∗ (t) − di+1 λ∗ (t) − di−2 µ∗ (t) , k = i − 1,i−1idi−1 idi−1 i−1i+2 ∗λ∗i (t) + µ∗i+1 (t) − ddi+1λi+1 (t) − ddi−1µ∗ (t) , k = i,i+1 iλ∗ (t) + µ∗ (t) − dk+2 λ∗ (t) − dk µ∗ (t) , k > i.kk+1dk+1 k+1dk+1 k(3.2.57)54Òàêèì îáðàçîì, ïîëó÷àåìk (B (s) − B ∗ (s)) z∗ (s) k1D ≤≤ |gN1 −1 + gN1 | µN1 (s)p∗N1 (s)+(3.2.58)+ |GN2 +1 + GN2 | λN2 (s)p∗N2 (s) ≤≤ 2gN1 −1 ∆N1 p∗N1 (s) + 2GN2 +1 ΛN2 p∗N2 (s).Ïîëîæèì gk∗ =i−1Pd∗j è G∗k =j=kkPd∗j .j=i+1Âìåñòî (3.2.48) èìååìp∗k (t) ≤M ∗ (∆i d∗i−1 +Λi d∗i+1 ),α∗ gk∗M ∗ (∆i d∗i−1 +Λi d∗i+1 ),α∗ G∗kk < i,(3.2.59)k > i.Ïîýòîìó èç (3.2.53), (3.2.58) è (3.2.59) âûòåêàåò îöåíêà:kz (t) − z∗ (t) k1D ≤2M M ∗ ∆i d∗i−1 + Λi d∗i+1≤·∗ααgN1 −1 ∆N1 GN2 +1 ΛN2·+.∗∗gNGN21Ïóñòüd = min (di−1 , di+1 ) ,W = infkgk d Gk, ,k i k(3.2.60).(3.2.61)Èìååì ñëåäóþùèå íåðàâåíñòâà:|pi − p∗i | ≤ |p0 − p∗0 | + |pi−1 − p∗i−1 | +1+|pi+1 − p∗i+1 | + · · · ≤ kz (t) − z∗ (t) k1D ,d(3.2.62)2kp (t) − p∗ (t) k ≤ kz (t) − z∗ (t) k1D ,d(3.2.63)55d1 + · · · + di−12kzk1D ≥ 1p1 + · · · +1di−1ddi+1(i − 1)pi−1 + i pi + (i + 1)pi+1 +i−1ii+1di+1 + di+2(i + 2)pi+2 + · · · ≥ W kpk1E .i+2(3.2.64)Ðàññìîòðèì ïðîöåññû ðîæäåíèÿ è ãèáåëè X(t), X ∗(t) òàêèå,÷òî âûïîëíÿþòñÿ (3.2.45) è (3.2.55).

Ïóñòü p (0) = p∗ (0) = ei (òî åñòüX(0) = X ∗ (0) = i). Òîãäà:Òåîðåìà 14.èkp (t) − p∗ (t) k ≤4M M ∗ ∆i d∗i−1 + Λi d∗i+1≤dα α∗gN1 −1 ∆N1 GN2 +1 ΛN2·+∗gNG∗N21(3.2.65)kp (t) − p∗ (t) k1E ≤4M M ∗ ∆i d∗i−1 + Λi d∗i+1≤W α α∗gN1 −1 ∆N1 GN2 +1 ΛN2+.·∗∗gNGN21(3.2.66)Ïóñòü âûïîëíÿþòñÿ óñëîâèÿ Òåîðåìû 14 è, êðîìå òîãî, N2 =∞. Òàêèì îáðàçîì, ñïðàâåäëèâî ñëåäóþùåå:Ñëåäñòâèå 4.≤≤äëÿ ëþáîãî i > N1.4M M ∗kp (t) − p∗ (t) k ≤∆i d∗i−1 + Λi d∗i+1 gN1 −1 ∆N1,∗dα α∗ gN1(3.2.67)4M M ∗kp (t) − p∗ (t) k1E ≤∆i d∗i−1 + Λi d∗i+1 gN1 −1 ∆N1,∗W α α∗ gN1(3.2.68)56Ïóñòü â óñëîâèÿõ Òåîðåìû 14 N1 = 0. Òîãäà ñïðàâåäëèâû ñëåäóþùèå îöåíêè:Ñëåäñòâèå 5.4M M ∗kp (t) − p∗ (t) k ≤∆i d∗i−1 + Λi d∗i+1 GN2 +1 ΛN2,dα α∗ G∗N2(3.2.69)4M M ∗kp (t) − p∗ (t) k1E ≤∆i d∗i−1 + Λi d∗i+1 GN2 +1 ΛN2,W α α∗ G∗N2(3.2.70)äëÿ ëþáîãî i < N2.3.2.4ÏðèìåðûÐàññìîòðèì ïðîöåññ ðîæäåíèÿ è ãèáåëè ñ ïåðèîäè÷åñêèìè èíòåíñèâíîñòÿìè ðîæäåíèÿ è ãèáåëè:Ïðèìåð 1.λk (t) = λ (t) = 10 + cos t,µk (t) = 1 + sin t, 0 < k < 1000,µk (t) = 24 + sin t, k ≥ 1000.Âîçüìåì i = 1000.Äëÿ íà÷àëà ïîëîæèì.

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