Tích phân ngẫu nhiên trên thang thời gian

❇é ●✐➳♦ ❉ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤ ◆❣➠ ➜×♥❤ ❙➡♥ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ❱✐♥❤ ✲ ✷✵✶✻ ❇é ●✐➳♦ ❉ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❍ä❝ ❱✐♥❤ ◆❣➠ ➜×♥❤ ❙➡♥ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ❈❤✉②➟♥ ♥❣➭♥❤✿ ▲ý t❤✉②Õt ①➳❝ s✉✃t ✈➭ ❚❤è♥❣ ❦➟ t♦➳♥ ❤ä❝ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✻ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ t♦➳♥ ❤ä❝ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ ❱✐♥❤ ✲ ✷✵✶✻ ❚❙✳ ◆❣✉②Ơ♥ ❚❤❛♥❤ ❉✐Ư✉ ▼ơ❝ ❧ơ❝ ▼ơ❝ ❧ơ❝ ✐ ▼ë ➤➬✉ ✶ ✶ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✸ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ị ❣✐➯✐ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ◗✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✶✽ ✷✳✶ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤ ✳ ✳ ✳ ✶✽ ✷✳✷ ❚Ý❝❤ t rt ị ì tí ✳ ✳ ✷✸ ✷✳✸ ❈➠♥❣ t❤ø❝ ■t➠ ✈➭ ø♥❣ ❞ô♥❣ ❑Õt ❧✉❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✐ ▼ë ➤➬✉ ◆➝♠ ✶✾✷✸✱ ◆♦r❜❡rt ❲✐❡♥❡r ➤➲ sư ❞ơ♥❣ ❧ý t❤✉②Õt ➤é ➤♦ ➤Ĩ ①➞② ❞ù♥❣ q✉➳ tr×♥❤ ❝❤✉②Ĩ♥ ➤é♥❣ ❇r♦✇♥ ✈➭ ❝❤ø♥❣ ♠✐♥❤ sù tå♥ t➵✐ ❞✉② ♥❤✃t ❝ñ❛ ♥ã✳ ❚r♦♥❣ ❝➠♥❣ tr×♥❤ ❝đ❛ ♠×♥❤✱ ❲✐❡♥❡r ➤➲ ❝❤Ø r❛ r➺♥❣ q✉ü ➤➵♦ ❝đ❛ q✉➳ tr×♥❤ ❝❤✉②Ĩ♥ ➤é♥❣ ❇r♦✇♥ ❝ã ❜✐Õ♥ ♣❤➞♥ ❦❤➠♥❣ ❣✐í✐ ♥é✐✳ ❉♦ ➤ã✱ tÝ❝❤ ♣❤➞♥ t❤❡♦ q✉➳ tr×♥❤ ❲✐❡♥❡r ❦❤➠♥❣ t❤Ĩ ➤Þ♥❤ ♥❣❤Ü❛ t❤❡♦ ❝➳❝❤ t❤➠♥❣ t❤➢ê♥❣ ♥❤➢ ❧➭ tÝ❝❤ ♣❤➞♥ ▲❡❜❡s❣✉❡✲❙t✐❡❧❥❡s✳ ➜✐Ị✉ ♥➭② ➤➲ ➤➢ỵ❝ ❦❤➽❝ ♣❤ơ❝ ❜ë✐ ❑✐②♦s❤✐ ■t➠✱ ♥❤➭ ❚♦➳♥ ❤ä❝ ♥❣➢ê✐ ◆❤❐t ❜➯♥✳ ↕♥❣ ➤➲ ①➞② ❞ù♥❣ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ t❤❡♦ q✉➳ tr×♥❤ ❲✐❡♥❡r ✈➭♦ ♥➝♠ ✶✾✹✹ ✭❬✻❪✮✳ ❙❛✉ ➤ã✱ ❏♦s❡♣❤ ▲❡♦ ❉♦♦❜ ➤➲ ♠ë ré♥❣ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ t❤❡♦ q✉➳ tr×♥❤ ❝ã ❣✐❛ sè trù❝ ❣✐❛♦ ✈➭♦ ♥➝♠ ✶✾✺✸ ✭❬✷❪✮✳ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ t✐Õ♣ tơ❝ ➤➢ỵ❝ ♠ë ré♥❣ ➤è✐ ✈í✐ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤ ❜ë✐ P❛✉❧✲❆♥❞rÐ ▼❡②❡r ♥➝♠ ✶✾✻✷ ✭❬✶✸❪✮ ✈➭ ❜ë✐ ❍✐r♦s❤✐ ❑✉♥✐t❛ ✈➭ ❙❤✐♥③♦ ❲❛t❛♥❛❜❡ ♥➝♠ ✶✾✻✼ ✭❬✶✵❪✮✳ ◆➝♠ ✶✾✼✵✱ P❛✉❧✲❆♥❞rÐ ▼❛②❡r ✈➭ ❈❛t❤❡r✐♥❡ ❉♦❧Ð❛♥s✲ ❉❛❞❡ ➤➲ tí t rt ị ì tí ✭❬✶✹❪✮✳ ①➞② ❞ù♥❣ ❈ò♥❣ tr♦♥❣ ♥➝♠ ➤ã✱ ❈✳ ❉❡❧❧❛❝❤❡r✐❡ ✈➭ ❑❧❛✉s ❇✐❝❤t❡❧❡r ➤➲ ①➞② ❞ù♥❣ tÝ❝❤ ♣❤➞♥ ❝ñ❛ ♠ét ❤➭♠ ♥❣➱✉ ♥❤✐➟♥ ❦❤➯ ➤♦➳♥ t❤❡♦ ♠ét s❡♠✐♠❛rt✐♥❣❛❧❡✳ ◆❣➭② ♥❛②✱ ❝❤ó♥❣ t❛ ❣ä✐ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ➤➢ỵ❝ ❝❤Ø r❛ ë tr➟♥ ❧➭ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ■t➠✳ ➜è✐ ✈í✐ ❝➳❝ tÝ♥❤ t♦➳♥ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ t❤ê✐ ❣✐❛♥ rê✐ r➵❝ t❤× ❝➳❝ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ ♠❛rt✐♥❣❛❧❡ ❝ã t❤Ĩ ①❡♠ ♥❤➢ ❧➭ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ ■t➠✳ ▲ý t❤✉②Õt t❤❛♥❣ t❤ê✐ ❣✐❛♥ ➤➢ỵ❝ ❙t❡❢❛♥ ❍✐❧❣❡r ♥❤➭ ❚♦➳♥ ❤ä❝ ♥❣➢ê✐ ➜ø❝ ➤➢❛ r❛ ❧➬♥ ➤➬✉ t✐➟♥ ♥➝♠ ✶✾✽✽ tr♦♥❣ ❧✉❐♥ ➳♥ t✐Õ♥ sü ❝đ❛ ➠♥❣ ✭❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ❇❡r♥❞ ❆✉❧❜❛❝❤✮✭❬✺❪✮✱ ♥❤➺♠ ♠ơ❝ ➤Ý❝❤ ❤ỵ♣ ♥❤✃t ♠ét sè ✈✃♥ ➤Ị ✈Ị ❣✐➯✐ tÝ❝❤ rê✐ r➵❝ ✈➭ ❧✐➟♥ tơ❝✳ ❚õ ➤ã✱ ❝❤đ ➤Ị t❤❛♥❣ t❤ê✐ ❣✐❛♥ t❤✉ ❤ót ➤➢ỵ❝ sù q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ tr➟♥ t❤Õ ❣✐í✐✳ ❚✉② ♥❤✐➟♥✱ ❝➳❝ ❦Õt q✉➯ ➤➢ỵ❝ ❦Ĩ r❛ ë tr➟♥ ❝❤Ø t❐♣ tr✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ë ❣✐➯✐ tÝ❝❤ t✃t ➤Þ♥❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ➜✐ t❤❡♦ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ❣✐➯✐ tÝ❝❤ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✱ ❣➬♥ ➤➞② ♠ét sè ♥❤ã♠ t➳❝ ❣✐➯ tr♦♥❣ ❬✸✱ ✹✱ ✶✺❪ ➤➲ ❜➢í❝ ➤➬✉ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ✈✃♥ ➤Ò ✈Ò tÝ♥❤ t♦➳♥ ♥❣➱✉ ♥❤✐➟♥ ✶ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ❚r➟♥ ❝➡ së t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✸✱ ✹❪ ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❝❤♦ ❧✉❐♥ ✈➝♥ ❧➭✿ ✧❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✧ ✱ ♥❤➺♠ ♠ơ❝ ➤Ý❝❤ t×♠ ❤✐Ĩ✉ ❝➳❝❤ ①➞② ❞ù♥❣ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ♥ã✳ ❱í✐ ♠ơ❝ ➤Ý❝❤ ➤ã✱ ♥é✐ ❞✉♥❣ ❝đ❛ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ t❤➭♥❤ ✷ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶✳ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳ rì ữ ề t ề tÝ❝❤ t✃t ➤Þ♥❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✈➭ ♠ét sè q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ ✈í✐ t❐♣ ❝❤Ø sè ❧➭ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ❈❤➢➡♥❣ ✷✳ ❚Ý❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ❈❤➢➡♥❣ ✷ ❧➭ ♥é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝ñ❛ ▲✉❐♥ ✈➝♥✱ tr×♥❤ ❜➭② ✈Ị tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤❀ tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ị ì tí r tr ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ✈Ị ❝➠♥❣ t❤ø❝ ■t➠ ✈➭ ❝➳❝ ø♥❣ ❞ơ♥❣✳ ❉♦ ♥➝♥❣ ❧ù❝ ❝ß♥ ♥❤✐Ị✉ ❤➵♥ ❝❤Õ✱ ♥➟♥ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✱ ♥❣➢ê✐ ết rt ợ ữ ó ý ❝đ❛ ❝➳❝ ♥❤➭ ❦❤♦❛ ❤ä❝ ✈➭ ➤å♥❣ ♥❣❤✐Ư♣ ➤Ĩ ❧✉❐♥ ✈➝♥ ❝ã t❤Ĩ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ◆❣❤Ư ❆♥✱ ♥❣➭②✳✳✳✳✳ t❤➳♥❣ ✳✳✳✳ ♥➝♠ ✷✵✶✻ ❍ä❝ ✈✐➟♥ ◆❣➠ ➜×♥❤ ❙➡♥ ✷ ❈❤➢➡♥❣ ✶ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✈Ị ❣✐➯✐ tÝ❝❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ❚r♦♥❣ ♠ơ❝ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ✈✃♥ ➤Ị ❝➡ ❜➯♥ ❝đ❛ ❣✐➯✐ tÝ❝❤ t✃t ➤Þ♥❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✱ ❝➳❝ ❦Õt q✉➯ tr×♥❤ ❜➭② tr♦♥❣ ụ ợ t từ t ệ ị ♥❣❤Ü❛ ✶✳✶✳✶ ❣ä✐ ❧➭ ♠ét ✳ ✭❬✶❪✮ ▼ét t❐♣ ❝♦♥ ó rỗ ủ t số tự t tờ ✭t✐♠❡ s❝❛❧❡✮✳ ❑ý ❤✐Ư✉ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ❧➭ T ❉Ơ t❤✃② r➺♥❣ ❝➳❝ t❐♣ ❤ỵ♣ R, Z, N, N0 , [0, 1] ∪ [2, 3], [0, 1] ∪ N, ✈➭ t❐♣ ❈❛♥t♦r , ❧➭ ❝➳❝ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ❚r♦♥❣ ❦❤✐ ➤ã ❝➳❝ t❐♣ ❤ỵ♣ Q, R \ Q, (0, 1), ❦❤➠♥❣ ♣❤➯✐ ❧➭ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✈× ❝❤ó♥❣ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❝➳❝ t➞♣ ❤ỵ♣ ➤ã♥❣✳ ✸ R ➤➢ỵ❝ ❚r♦♥❣ ❧✉❐♥ ✈➝♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ ❧✉➠♥ ①Ðt t➠♣➠ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ❧➭ t➠♣➠ ❝➯♠ s✐♥❤ ❝ñ❛ t➠♣➠ t❤➠♥❣ t❤➢ê♥❣ tr➟♥ t ợ số tự ị ĩ T ●✐➯ sö ➳♥❤ ❧➭ ♠ét t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ①➵ σ:T→T ①➳❝ ➤Þ♥❤ ❜ë✐ σ(t) = inf{s ∈ T : s > t}, ➤➢ỵ❝ ❣ä✐ t♦➳♥ tư ❜➢í❝ ♥❤➯② t✐Õ♥ ❧➭ ρ:T→T T ➳ ❣✐❛♥ ♥❤ ①➵ ✭ ❢♦r✇❛r❞ ❥✉♠♣ ♦♣❡r❛t♦r✮ tr➟♥ t❤❛♥❣ t❤ê✐ tr➟♥ t❤❛♥❣ t❤ê✐ ①➳❝ ➤Þ♥❤ ❜ë✐ ρ(t) = sup{s ∈ T : s < t}, ➤➢ỵ❝ ❣✐❛♥ ❣ä✐ t♦➳♥ tư ❜➢í❝ ♥❤➯② ❧ï✐ ➤➞② ❝❤ó♥❣ t❛ q✉② ➢í❝ T ❣✐❛♥ ❝ã t❤❛♥❣ t❤ê✐ ❣✐❛♥ ♠❛①✐♠✉♠ T ❜❛❝❦✇❛r❞ ❥✉♠♣ ♦♣❡r❛t♦r✮ trï ♠❐t ♣❤➯✐ σ(t) > t ρ(t) < t sử T ỗ rts ý ❤✐Ö✉ sup ∅ = inf T ✈➭ ✱ ❝➳❝ t❐♣ σ(M ) = M ✭♥❣❤Ü❛ ρ(m) = m ❦ý σ(t) = t ♥Õ✉ ♥Õ✉ ❤✐Ư✉ ❤ỵ♣ ♥Õ✉ ✮✳ ✱ ❝➠ ❧❐♣ ♣❤➯✐ ρ(t) = t ✱ ♥Õ✉ [a, b] t ❧➭ ♥Õ✉ ❝➠ ❧❐♣ tr➳✐ ✭❧❡❢t✲s❝❛tt❡r❡❞✮ ♥Õ✉ ✈õ❛ ❝➠ ❧❐♣ tr➳✐ ✈õ❛ ❝➠ ❧❐♣ ♣❤➯✐✳ t❐♣ ❤ỵ♣ {t ∈ T : a t➢➡♥❣ b}; {t ∈ T : a < t < b}; {t ∈ T : a Ta = {t ∈ T : t a} ø♥❣ ✈➭ ♥Õ✉  T \ [m, σ(m)) ✹ ➤➢ỵ❝ ❣ä✐ r✐❣❤t✲s❝❛tt❡r❡❞✮ (a, b]; (a, b); [a, b)   T t∈T ✭ {t ∈ T : a < t kT = ❧➭ ♥Õ✉ t❤❛♥❣ m ❝➠ ❧❐♣ ✭✐s♦❧❛t❡❞✮ a, b ∈ T ✭♥❣❤Ü❛ ❧➭ ❧➭ ♠ét t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ▼ét ➤✐Ó♠ ✭ ✈➭ ❧➭ ➤✐Ĩ♠ tù✱ M ❧➭ trï ♠❐t tr➳✐ ✭❧❡❢t✲❞❡♥s❡✮ ❱í✐ t➢➡♥❣ inf ∅ = sup T ❝ã ♠✐♥✐♠✉♠ ❧➭ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✸✳ ❧➭ ✭ T ë t❤ê✐ ❧➭ ♥Õ✉ T = −∞ T = m, ❧➭ t ❝➳❝ t < b} ✳ t❐♣ b} ✱ ❤ỵ♣ ❑ý ❤✐Ư✉ k T =   T  T \ (ρ(M ), M ] ♥Õ✉ max T = +∞ ♥Õ✉ max T = M ❑ý ❤✐Ö✉ I1 = {t : t ▼Ư♥❤ ➤Ị ✶✳✶✳✹ ❝➠ ❧❐♣ tr➳✐ ✳ ✭❬✶❪✮ ❚❐♣ ❤ỵ♣ ♣❤➯✐ ❝đ❛ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✺✳ }, I2 = {t : t I ❝➠ ❧❐♣ ♣❤➯✐ }, I = I1 ∪ I2 ✭✶✳✶✮ ❣å♠ t✃t ❝➯ ❝➳❝ ➤✐Ó♠ ❝➠ ❧❐♣ tr➳✐ ❤♦➷❝ ❝➠ ❧❐♣ T ❧➭ ❦❤➠♥❣ q✉➳ ➤Õ♠ ➤➢ỵ❝✳ ●✐➯ sư T ❧➭ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ : T k R+ ị à(t) = (t) t, ợ ọ ①➵ ❤➭♠ ❤➵t t✐Õ♥ ✭❢♦r✇❛r❞ ❣r❛✐♥✐♥❡ss ❢✉♥❝t✐♦♥✮ ν : T → R+ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ T tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ T ①➳❝ ➤Þ♥❤ ❜ë✐ ν(t) = t − ρ(t), ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➭♠ ❤➵t ❧ï✐ ✭❜❛❝❦✇❛r❞ ❣r❛✐♥✐♥❡ss ❢✉♥❝t✐♦♥✮ ❱Ý ❞ơ ✶✳✶✳✻✳ ✰✮ ❱í✐ ✰✮ ❱í✐ T=Z ✰✮ ❱í✐ h T=R tì tì (t) = t = (t), à(t) = ρ(t) = 0; ρ(t) = t − 1, σ(t) = t + 1, µ(t) = ν(t) = ❧➭ sè t❤ù❝ ❞➢➡♥❣✳ ❈❤ó♥❣ t❛ ➤Þ♥❤ ♥❣❤Ü❛ t❤❛♥❣ t❤ê✐ ❣✐❛♥ T = hZ ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉✿ hZ = {kh : k ∈ Z} = {· · · − 3h, −2h, −h, 0, h, 2h, 3h, · · · }, ❦❤✐ ➤ã ρ(t) = t − h, σ(t) = t + h, µ(t) = ν(t) = h ✰✮ ❱í✐ a, b ❧➭ ❝➳❝ sè t❤ù❝ ❞➢➡♥❣✳ ❳Ðt t❤❛♥❣ t❤ê✐ ❣✐❛♥ ∞ Pa,b = [k(a + b), k(a + b) + b] k=1 ✺ T = Pa,b ♥❤➢ s❛✉ ❑❤✐ ➤ã✱ σ(t) = ∞    t ♥Õ✉ k=1 ∞ ♥Õ✉ t∈ ♥Õ✉ t∈ ∞ t∈ ♥Õ✉ n ∈ N0 , [k(a + b), k(a + b) + b) k=1 ∞ ♥Õ✉ t∈ {k(a + b) + b} k=1 ✈➭ ∞    0   a {k(a + b)}, k=1    0   a ✰✮ ❱í✐ (k(a + b), k(a + b) + b] k=1 ∞   t − a ν(t) = {k(a + b) + b}, t∈ ♥Õ✉    t µ(t) = [k(a + b), k(a + b) + b) k=1 ∞   t + a ρ(t) = t∈ t∈ ♥Õ✉ (k(a + b), k(a + b) + b] k=1 ∞ ♥Õ✉ t∈ {k(a + b)} k=1 ①Ðt ❞➲② sè ➤✐Ò✉ ❤ß❛ n H0 = 0, Hn = k=1 , n k ❳➳❝ ➤Þ♥❤ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ♥❤➢ s❛✉ H = {Hn : n ∈ N} ❑❤✐ ➤ã✱ n+1 σ(Hn ) = k=1  n−1    k , ρ(Hn ) = k=1  k  0 ✈➭ µ(Hn ) =   1 n , ν(Hn ) =  n+1 0 ✻ ♥Õ✉ n ♥Õ✉ n = 0, 1, ♥Õ✉ n ♥Õ✉ n = ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✼✳ f ❈❤♦ ❤➭♠ sè f T ①➳❝ ➤Þ♥❤ tr➟♥ ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ R ✳ ❍➭♠ sè ➤➢ỵ❝ ❣ä✐ ❧➭ ✐✮ ❝❤Ý♥❤ q✉② ✭r❡❣✉❧❛t❡❞✮ f ♥Õ✉ ❝ã ❣✐í✐ ❤➵♥ tr➳✐ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t tr➳✐ ✈➭ ❝ã ❣✐í✐ ❤➵♥ ♣❤➯✐ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t ♣❤➯✐✳ ✐✐✮ rd✲❧✐➟♥ tơ❝ rd✲❝♦♥t✐♥✉♦✉s ✭ ❣✐í✐ rd ✐✐✐✮ ❤➵♥ ✮ ♥Õ✉ tr➳✐ t➵✐ ♥❤÷♥❣ ✲❧✐➟♥ tơ❝ ❦ý ❤✐Ư✉ ❧➭ ✭ ♣❤➯✐ ♥❤÷♥❣ ld ✲❧✐➟♥ tơ❝ ❦ý ❤✐Ư✉ ❧➭ ●✐➯ sư fρ : T → R k T✳ r➺♥❣ ♥Õ✉ f :T→R ➤✐Ó♠ Cld f ♠❐t tr➳✐✳ ♠❐t ♣❤➯✐✳ lim f (s) ❜ë✐ f ρ = f◦ ρ f (t− ) ❧➭ ➤✐Ĩ♠ ❝➠ ❧❐♣ tr➳✐ t❤× ❤ỵ♣ ❝➳❝ ❤➭♠ ❚❐♣ ❤ỵ♣ ❝➳❝ ❤➭♠ Cld (T, R) ❧➭ ♠ét ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr➟♥ σ(s)↑t ❚❐♣ ❧✐➟♥ tơ❝ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t tr➳✐✱ ❝ã ❣✐í✐ trï ❧➭ ❤➭♠ sè ①➳❝ ➤Þ♥❤ ❜ë✐ ❑ý ❤✐Ư✉ t ❤♦➷❝ trï Crd (T, R) ❤♦➷❝ ✮ ♥Õ✉ t➵✐ ❧✐➟♥ tô❝ t➵✐ ♥❤÷♥❣ ➤✐Ĩ♠ trï ♠❐t ♣❤➯✐ ✈➭ ❝ã ➤✐Ĩ♠ ld✲❧✐➟♥ tô❝ ld✲❝♦♥t✐♥✉♦✉s ❤➵♥ t∈ Crd f ❤♦➷❝ ft− = ftρ ft− T ✳ ✱ ♥❣❤Ü❛ ❧➭ ❑❤✐ ➤ã✱ ❝❤ó♥❣ t❛ ✈✐Õt ftρ = f (ρ(t)) ✈í✐ ♠ä✐ ♥Õ✉ tå♥ t➵✐ ❣✐í✐ ❤➵♥ tr➳✐✳ ❚❛ t❤✃② ✳ ➜Þ♥❤ ❧ý ✶✳✶✳✽✳ ●✐➯ sö f : T → R ❧➭ ♠ét ❤➭♠ sè ①➳❝ ➤Þ♥❤ tr➟♥ T✳ ❑❤✐ ➤ã✱ ❝❤ó♥❣ t❛ ❝ã✿ ✐✮ ◆Õ✉ f ❧➭ ❤➭♠ sè ❧✐➟♥ tơ❝ t❤× ✐✐✮ ◆Õ✉ f ❧➭ ❤➭♠ sè ✐✈✮ ❚♦➳♥ tư ❜➢í❝ ♥❤➯② ❧ï✐ f ❧➭ ❤➭♠ sè ❧➭ ❤➭♠ sè rd✲❧✐➟♥ tơ❝ t❤× f ✐✐✐✮ ❚♦➳♥ tư ❜➢í❝ ♥❤➯② t✐Õ♥ ✈✮ ◆Õ✉ f σ rd✲❧✐➟♥ tô❝ ✈➭ ld✲❧✐➟♥ tô❝✳ ❧➭ ❤➭♠ sè ❝❤Ý♥❤ q✉②✳ ❧➭ ❤➭♠ sè rd✲❧✐➟♥ tô❝✳ ρ ❧➭ ❤➭♠ sè ld✲❧✐➟♥ tơ❝✳ ld✲❧✐➟♥ tơ❝ t❤× f ρ ❝ị♥❣ ❧➭ ❤➭♠ sè ✼ ld✲❧✐➟♥ tơ❝✳ ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✶✵✮✱ ❤Ư t❤ø❝ tr➟♥ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ t G(Mτ − )∇[M ]τ = a kn kn G(Mti−1 )(Mt2i = lim n→∞ − Mt2i−1 ) −2 G(Mti−1 )Mti−1 (Mti − Mti−1 ) i=1 i=1 kn G(Mti−1 )(Mti − Mti−1 )2 = lim n→∞ i=1 ❙✉② r❛ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❑ý ❤✐Ư✉ s❛♦ ❝❤♦ ∇ C 1,2 (Ta ×Rd ; R) ❧➭ ❤ä t✃t ❝➯ ❝➳❝ ❤➭♠ ✲❦❤➯ ✈✐ ❧✐➟♥ tô❝ t❤❡♦ ❜✐Õ♥ ➜Þ♥❤ ❧ý ✷✳✸✳✺ ✭❈➠♥❣ t❤ø❝ ■t➠✮ V ∈ C 1,2 (Ta × Rd ; R)✳ t V (t, x) ①➳❝ ➤Þ♥❤ tr➟♥ ✈➭ ❦❤➯ ✈✐ ❧✐➟♥ tơ❝ ✷ ❧➬♥ t❤❡♦ ❜✐Õ♥ Ta ×Rd x ✳ ✳ ●✐➯ sư X = (X1 , · · · , Xd ) ❧➭ ❜é d✲s❡♠✐♠❛rt✐♥❣❛❧❡✱ ❑❤✐ ➤ã✱ V (t, X) ❧➭ ♠ét s❡♠✐♠❛rt✐♥❣❛❧❡ ✈➭ ❝➠♥❣ t❤ø❝ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ t V (t, X(t)) = V (a, X(a)) + a d t + a i=1 ∂ ∇V (τ, X(τ− ))∇τ ∂ ∇τ ∂V (τ, X(τ− ))∇Xi (τ ) + ∂xi t i,j a ∂ 2V (τ, X(τ− ))∇[Xi , Xj ]τ ∂xi xj d V (s, X(s)) − V (s, X(s− )) − + s∈(a,t] i=1 s∈(a,t] − tr♦♥❣ ➤ã s∈(a,t] i,j ∂V (s, X(s− ))∇∗ Xi (s) ∂xi ∂ 2V (s, X(s− ))(∇∗ Xi (s))(∇∗ Xj (s)), ∂xi xj ∇∗ Xi (s) = Xi (s) − Xi (s− ) ✈➭ ∂∇V ∂∇τ (t, x) ❧➭ ∇✲➤➵♦ ✭✷✳✶✽✮ ❤➭♠ r✐➟♥❣ ❝ñ❛ V (t, x) t❤❡♦ ❜✐Õ♥ t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚õ ❝➠♥❣ t❤ø❝ ❦❤❛✐ tr✐Ĩ♥ ❚❛②❧♦r ➤è✐ ✈í✐ ❤➭♠ ♥❤✐Ị✉ ❜✐Õ♥✱ t❛ ❝ã V (t, y) − V (t, x) d = i=1 ∂V (t, x) (yi − xi ) + ∂xi i,j ∂ V (t, x) (yi − xi )(yj − xj ) + R(x, y), ∂xi ∂xj ✸✵ tr♦♥❣ ➤ã ❝❤♦ |R(x, y)| lim r(u) = 0, ➤ó♥❣ ✈í✐ u↓0 ❝❤➷♥ ❜ë✐ X f ∈ C2 τm = inf{t : X(t) ➤✐Ĩ♠ ❞õ♥❣ m ✈í✐ r : R+ → R+ ❧➭ ❤➭♠ t➝♥❣ s❛♦ ①➳❝ ➤Þ♥❤ tr➟♥ t❐♣ ❝♦♠♣❛❝t✳ m} ❳➳❝ ➤Þ♥❤ t❤ê✐ X(t ∧ τm ) ❑❤✐ ➤ã✱ s❡♠✐♠❛rt✐♥❣❛❧❡ ✈➭ ♥Õ✉ ❝➠♥❣ t❤ø❝ ■t➠ ➤ó♥❣ ✈í✐ ✳ ❉♦ ➤ã✱ ❝❤ó♥❣ t❛ ❧✉➠♥ ❣✐➯ t❤✐Õt r➺♥❣ ▲✃② r( y − x ) y − x X X(t m ) ỗ ị m tì ú ị >0 ì ể ❣✐➳♥ ➤♦➵♥ ❝đ❛ ♠❛rt✐♥❣❛❧❡ X ❧➭ ❦❤➠♥❣ q✉➳ ➤Õ♠ ➤➢ỵ❝ ✈➭ X(s) − X(s− ) < ∞, s∈(a,t] ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ♣❤➞♥ t❐♣ ❝➳❝ ➤✐Ĩ♠ ❣✐➳♥ ➤♦➵♥ ❝đ❛ C1 ❧➭ t❐♣ ❤÷✉ ❤➵♥ ✈➭ C2 X tr➟♥ (a, t] t❤➭♥❤ ❤❛✐ ❧í♣✿ ❧➭ t❐♣ ❝➳❝ ➤✐Ĩ♠ ❣✐➳♥ ➤♦➵♥ s❛♦ ❝❤♦ X(s) − X(s− ) ε2 s∈C2 ❳Ðt ♣❤➞♥ ❤♦➵❝❤ π (n) ❝đ❛ [a, t] ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐ ✭✷✳✶✶✮ ✭✷✳✶✷✮✱ t❛ ❝ã V (t, X(t)) − V (a, X(a)) = V (tk , X(tk )) − V (tk−1 , X(tk−1 )) k V (tk , X(tk−1 )) − V (tk−1 , X(tk−1 )) = k V (tk , X(tk )) − V (tk , X(tk−1 )) + k C1 ì ữ X V (tk , X(tk )) − V (tk , X(tk−1 )) lim n→∞ ❧➭ ❝❛❞❧❛❣✱ ♥➟♥ C1 ∩(tk−1 ,tk ]=∅ V (s, Xs ) − V (s, Xs− ) = ✭✷✳✶✾✮ s∈C1 ➜Ĩ ➤➡♥ ❣✐➯♥ tr♦♥❣ tr×♥❤ ❜➭② ❝❤ó♥❣ t❛ ❦ý ❤✐Ö✉ ❜ë✐ C1 ∩(ti−1 ,ti ]=∅ ✈➭ (1) C1 ∩(ti−1 ,ti ]=∅ ✳ ❇➺♥❣ ❝➳❝❤ ♣❤➞♥ tÝ❝❤ t❤➭♥❤ tæ♥❣ ❝➳❝ sè ❤➵♥❣ tr➟♥ ❝➳❝ ❦❤♦➯♥❣ rê✐ ♥❤❛✉ ❜ë✐ (2) ✸✶ ✈➭ ❞ï♥❣ ❝➠♥❣ t❤ø❝ ❚❛②❧♦r✱ t❛ ❝ã V (tk , X(tk )) − V (tk , X(tk−1 )) (2) d ∂V (tk , X(tk−1 ))(Xi (tk ) − Xi (tk−1 )) ∂xi = + (2) 2 (2) i,j i=1 ∂ V (tk , X(tk−1 ))(Xi (tk ) − Xi (tk−1 ))(Xj (tk ) − Xj (tk−1 )) ∂xi ∂xj R(X(tk−1 ), X(tk )) + (2) kn d k=1 i=1 ∂V (tk , X(tk−1 ))(Xi (tk ) − Xi (tk−1 )) ∂xi = + kn k=1 i,j ∂ 2V (tk , X(tk−1 ))(Xi (tk ) − Xi (tk−1 ))(Xj (tk ) − Xj (tk−1 )) ∂xi ∂xj d − (1) − (1) i,j i=1 ∂V (tk , X(tk−1 ))(Xi (tk ) − Xi (tk−1 )) ∂xi ∂ 2V (tk , X(tk−1 ))(Xi (tk ) − Xi (tk−1 ))(Xj (tk ) − Xj (tk−1 )) ∂xi ∂xj + R(X(tk1 ), X(tk )) (2) ì C1 ữ sè ❤➵♥❣ tr♦♥❣ ❤➭♥❣ t❤ø ✺ ❝đ❛ ❤Ư t❤ø❝ tr➟♥ ❤é✐ tơ ✈Ị d s∈C1 i=1 ∂V (s, X(s− ))(Xi (s) − Xi (s− )) , ∂xi ✭✷✳✷✵✮ ✈➭ ❤➭♥❣ t❤ø ✻ ❤é✐ tơ ✈Ị s∈C1 i,j ∂ 2V (s, X(s− ))(Xi (s) − Xi (s− ))(Xj (s) − Xj (s− )) ∂xi ∂xj ✸✷ ✭✷✳✷✶✮ ❈❤ó♥❣ t❛ ❝ã t❤Ĩ ➢í❝ ❧➢ỵ♥❣ sè ❤➵♥❣ ❝✉è✐ ❝ï♥❣ ♥❤➢ s❛✉ R(X(tk−1 ), X(tk )) (2) r( max C1 ∩(tk−1 ,tk ]=∅ X(tk ) − X(tk−1 ) X(tk ) − X(tk−1 ) ) (2) ❉♦ ➤ã✱ R(X(tk−1 ), X(tk )) lim sup n→∞ (2) d X(tk ) − X(tk−1 ) r(ε+) lim sup n→∞ ❈❤♦ r(ε+) [Xi ]t i=1 tk ∈π (n) ε↓0 ✱ t❛ ❝ã R(X(tk−1 ), X(tk )) → 0, lim sup n→∞ ✭✷✳✶✾✮ (2) V (s, X(s)) − V (s, X(s− )), ❤é✐ tơ ✈Ị s∈(a,t] d ✭✷✳✷✵✮ ❤é✐ tơ ✈Ị s∈(a,t] i=1 ✭✷✳✷✶✮ ❤é✐ tơ ✈Ị s∈(a,t] i,j ∂V (s, X(s− ))(∇∗ Xi (s)), ∂xi ∂ 2V (s, X(s− ))∇∗ Xi (s)∇∗ Xj (s) ∂xi ∂xj ❚õ ✭✷✳✶✵✮✱ s✉② r❛ kn d k=1 i=1 lim n→∞ ∂V (tk , X(tk−1 ))(Xi (tk ) − Xi (tk−1 )) ∂xi d t = i=1 ✸✸ a ∂V (τ, X(τ− ))∇Xi (τ ) ∂xi ✭✷✳✷✷✮ ❙ư ❞ơ♥❣ ❇ỉ ➤Ị ✷✳✸✳✹✱ t❛ ❝ã kn d lim n→∞ k=1 i=1 ∂ V (tk , X(tk−1 )) (Xi (tk ) − Xi (tk−1 ))(Xj (tk ) − Xj (tk−1 )) ∂xi ∂xj t ∂ 2V (τ, Xτ− )∇[Xi , Xj ]τ ∂xi ∂xj = a i,j ✭✷✳✷✸✮ ❍➡♥ ♥÷❛✱ (V (tk ,X(tk−1 )) − V (tk−1 , X(tk−1 ))) lim n→∞ k ( = lim n→∞ t = a k ∇ V (tk , X(tk−1 )) − V (tk−1 , X(tk−1 )) )(tk − tk−1 ) tk − tk−1 ∂ V (τ, X(τ− ))∇τ ∂ ∇τ ▼➷t ❦❤➳❝✱ V (tk , X(tk−1 )) − V (tk−1 , X(tk−1 )) V (t, X(t)) − V (a, X(a)) = k V (tk , X(tk )) − V (tk , X(tk−1 )) + k V (tk , X(tk−1 )) − V (tk−1 , X(tk−1 )) = k V (tk , X(tk )) − V (tk , X(tk−1 )) + (1) V (tk , X(tk )) − V (tk , X(tk−1 )) + (2) ❱Õ ♣❤➯✐ ❤é✐ tô ➤Õ♥ t a ∂ ∇V (τ, X(τ− ))∇τ + ∂ ∇τ d t a i=1 + ∂V (τ, X(τ− ))∇Xτ ∂xi t i,j ✸✹ a ∂ 2V (τ, X(τ− ))∇[Xi , Xj ]τ ∂xi xj d V (s, X(s)) − V (s, X(s− )) − + s∈(a,t] i=1 s∈(a,t] − s∈(a,t] i,j ∂V (s, X(s− ))∇∗ Xi (s) ∂xi ∂ 2V (s, X(s− ))(∇∗ Xi (s))(∇∗ Xj (s)) ∂xi xj ❙✉② r❛ ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❍Ö q✉➯ s❛✉ ❧➭ ❝➠♥❣ t❤ø❝ ■t➠ rê✐ r➵❝ ➤➢ỵ❝ ①➞② ❞ù♥❣ ❜ë✐ ❉✳ ❑❛♥♥❛♥ ✈➭ ❇✳ ❩❤❛♥ ♥➝♠ ✷✵✵✷ ✭❬✽❪✮✳ ❍Ö q✉➯ ✷✳✸✳✻ ✳ ✭❬✽❪✮ ▲✃② T = N, a = ✈➭ Xn ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥➭♦ ➤ã✳ ❑❤✐ ➤ã✱ ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✸✳✺ t❤× ❝➠♥❣ t❤ø❝ ✭✷✳✶✽✮ ❝ã ❞➵♥❣✳ n (V (k, Xk−1 ) − V (k − 1, Xk−1 )) V (n, Xn ) = V (0, X0 ) + k=1 n Vx (k, Xk−1 )∇ Xk + n ∗ + k=1 n + k=1 (Vx (k, Xk ) − Vx (k, Xk−1 )∇∗ Xk k=1 V (k, Xk ) − V (k, Xk−1 ) − n (Vx (k, Xk ) + Vx (k, Xk−1 ))∇∗ Xk , k=1 ✭✷✳✷✹✮ tr♦♥❣ ➤ã Vx (k, ·) = ❚r➢ê♥❣ ❤ỵ♣ ❑ý Ft ❤✐Ư✉ ∂V (k,·) ✈➭ ∂x ∇∗ Xk = Xk − Xk−1 T = R, a = Lloc (Ta , R) ❧➭ t❛ ❝ã ➤➢ỵ❝ ❝➠♥❣ t❤ø❝ ■t➠ ✈í✐ t❤ê✐ ❣✐❛♥ ❧✐➟♥ tơ❝✳ ❤ä q trì {f (t)}tTa trị tự ù ❤ỵ♣✱ t❤á❛ ♠➲♥ T |f (τ )|∇τ < +∞ ❤✳❝✳❝ , ✈í✐ ♠ä✐ T ∈ Ta ✭✷✳✷✺✮ a ▲✃② fi ∈ Lloc (Ta , R) ✈➭ M ∈ M2 gi ∈ L2 (Ta ; M ) ❀ t i = 1, ✳ ❳Ðt ✷ q✉➳ tr×♥❤ t fi (τ ) ∇τ + Xi (t) = Xi (a) + ✈í✐ a gi (τ )∇Mτ ∀ i = 1, a ✸✺ ✭✷✳✷✻✮ ❇ỉ ➤Ị ✷✳✸✳✼✳ ●✐➯ sư X1 , X2 srt ợ ị ✭✷✳✷✻✮ ❑❤✐ ➤ã✱ t [X1 , X2 ]t = f1 (τ )f2 (τ )ν(τ )∇τ a t ∗ + (f1 (s)g2 (s) + f2 (s)g1 (s))ν(s)∇ Ms + g1 (τ )g2 (τ )∇[M ]τ , a a0 t❤× ln t ln q Et (M ) = [1 + (Mqτ − Mqτ −1 )] a τ = ln ln q +1 ▼Ư♥❤ ➤Ị ✷✳✸✳✶✸✳ ●✐➯ sư X, Y t✐➟✉ t➵✐ ❧➭ ❤❛✐ s❡♠✐♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tÝ❝❤✱ tr✐Ưt a t❤× Et (X)Et (Y ) = Et (X + Y + [X, Y ]) ✹✵ ✭✷✳✸✺✮ ❈❤ø♥❣ ♠✐♥❤✳ ❚õ ➤➻♥❣ t❤ø❝ ✭✷✳✶✸✮ t❛ ❝ã t Et (X)Et (Y ) = t Eτ− (X)∇Eτ (Y ) + a Eτ− (Y )∇Eτ (X) + [E(X), E(Y )]t a t t Eτ− (X)Eτ− (Y )∇(Xτ + Yτ ) + = a Eτ− (Y )Eτ− (X)∇[X, Y ]τ a t Eτ− (X)Eτ− (Y )∇(Xτ + Yτ + [X, Y ]τ ) = a ❚õ ➤ã✱ s✉② r❛ Et (X)Et (Y ) ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ t Zτ− ∇(Xτ + Yτ + [X, Y ]τ ) Z(t) = + a ◆❣❤Ü❛ ❧➭✱ Et (X)Et (Y ) = Et (X + Y + [X, Y ]) ❙✉② r❛ ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✹✶ ❦Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ t❤✉ ợ ết q í s rì ột sè ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ị ❣✐➯✐ tÝ❝❤ t✃t ➤Þ♥❤ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ ✈➭ q✉➳ tr×♥❤ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥✳ ✷✮ ❚×♠ ❤✐Ĩ✉ ✈Ị tÝ❝❤ ♣❤➞♥ ♥❣➱✉ ♥❤✐➟♥ tr➟♥ t❤❛♥❣ t❤ê✐ ❣✐❛♥ t❤❡♦ ♠❛rt✐♥❣❛❧❡ ❜×♥❤ ♣❤➢➡♥❣ ❦❤➯ tí rt ị ì tí ré♥❣ tÝ❝❤ ♣❤➞♥ t❤❡♦ s❡♠✐♠❛rt✐♥❣❛❧❡ ➤å♥❣ t❤ê✐ ❝❤Ø r❛ ❝➳❝ tÝ♥❤ ❝❤✃t q✉❡♥ t❤✉é❝ ❝đ❛ ❝❤ó♥❣✳ ✸✮ ❚r×♥❤ ❜➭② ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝➠♥❣ t❤ø❝ ■t➠✱ ❝❤Ø r❛ ❝➳❝ ❤Ö q✉➯ ➤è✐ ✈í✐ ❝➠♥❣ t❤ø❝ ■t➠✱ ➤å♥❣ t❤ê✐ sư ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ■t➠ ❣✐➯✐ ♣❤➢➡♥❣ tr×♥❤ t✉②Õ♥ tÝ♥❤ ❉♦❧Ð❛♥s ✲ ❉❛❞❡✳ ✹✷ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ▼✳ ❇♦❤♥❡r ❛♥❞ ❚✐♠❡ ❙❝❛❧❡s✱ ❆✳ P❡t❡rs♦♥✳ ❆❞✈❛♥❝❡s ✐♥ ❉②♥❛♠✐❝ ❊q✉❛t✐♦♥s ♦♥ ✭✷✵✵✸✮✱ ❇✐r❦❤❛✉s❡r ❇♦st♦♥✱ ❇❛s❡❧✱ ❇❡r❧✐♥✳ ❬✷❪ ❏✳ ▲✳ ❉♦♦❜✳ ✭✶✾✺✸✮✱ ❙t♦❝❤❛st✐❝ Pr♦❝❡ss❡s✱ ❏♦❤♥ ❲✐❧❡② ❛♥❞ ❙♦♥s✱ ◆❡✇ ❨♦r❦✳ ❬✸❪ ◆✳ ❍✳ ❉✉ ❛♥❞ ◆✳ ❚✳ ❉✐❡✉✳ ✭✷✵✶✶✮✱ ❚❤❡ ❢✐rst ❛tt❡♠♣t ♦♥ t❤❡ st♦❝❤❛st✐❝ ❝❛❧❝✉❧✉s ♦♥ t✐♠❡ s❝❛❧❡✱ ❏♦✉r♥❛❧ ♦❢ ❙t♦❝❤❛st✐❝ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥✳ ✷✾ ✱ ✶✵✺✼ ✲ ✶✵✽✵✳ ❬✹❪ ◆✳ ❍✳ ❉✉ ❛♥❞ ◆✳ ❚✳ ❉✐❡✉✳ ✭✷✵✶✶✮✱ ❙t♦❝❤❛st✐❝ ❞②♥❛♠✐❝ ❡q✉❛t✐♦♥ ♦♥ t✐♠❡ s t t t tta ă t trs r tt a ă P t❤❡s✐s✱ ❯♥✐✈❡rs✐t❛❛t ❲ ❬✻❪ ❑✳ ■t➠✳ ✭✶✾✹✹✮✱ ❙t♦❝❤❛st✐❝ ■♥t❡❣r❛❧✱ ❬✼❪ ❑✳ ■t➠✳ ✭✶✾✺✶✮✱ ▼❛t❤✳ ❏✳ ❖♥ ❛ ❢♦r♠✉❧❛ ✉r③❜✉r❣✳ Pr♦❝✳ ■♠♣✳ ❆❝❛❞✳ ❚♦❦②♦✳ ❝♦♥❝❡r♥✐♥❣ st♦❝❤❛st✐❝ ✷✵ ✱ ✺✶✾ ✲ ✺✷✹✳ ❞✐❢❢❡r❡♥t✐❛❧s✱ ✸ ◆❛❣♦②❛ ✱ ✺✺ ✲ ✻✺✳ ❬✽❪ ❉✳ ❑❛♥♥❛♥ ✈➭ ❇✳ ❩❤❛♥✳ ✭✷✵✵✷✮✱ ❆ ❞✐s❝r❡t❡ ✲ t✐♠❡ ■t➠✬s ❢♦r♠✉❧❛✱ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✳ ✷✵ ❙t♦❝❤❛st✐❝ ✱ ✶✶✸✸ ✲ ✶✶✹✵✳ ❬✾❪ ◆✳ ❑❛③❛♠❛❦✐✳ ✭✶✾✼✷✮✱ ❖♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ ♠❛rt✐♥❣❛❧❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✱ ❚➠❤♦❦✉ ▼❛t❤✳ ❏♦✉r♥✳ ✷✹ ✱ ✹✻✸ ✲ ✹✻✽✳ ❖♥ sq✉❛r❡ ✐♥t❡❣r❛❜❧❡ ♠❛rt✐♥❣❛❧❡s✱ ❬✶✵❪ ❍✳ ❑✉♥✐t❛ ❛♥❞ ❙✳ ❲❛♥t❛♥❛❜❡✳ ✭✶✾✻✼✮✱ ◆❛❣♦②❛ ▼❛t❤✳ ✸✵ ✱ ✷✵✾ ✲ ✷✹✺✳ ❬✶✶❪ ❍✳ P✳ ▼❝❑❡❛♥✳ ❏r✳ ✭✶✾✻✾✮✱ ❬✶✷❪ P✳ ❆✳ ▼❡②❡r✳ ✭✶✾✻✼✮✱ ♠❛t✐❝s✱ ❏✳ ❙t♦❝❤❛st✐❝ ■♥t❡❣r❛❧s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ◆❡✇ ❨♦r❦✳ ■♥tÐ❣r❛❧❡s st♦❝❤❛st✐q✉❡s ■✱ ■■✱ ▲❡❝t✉r❡ ♥♦t❡s ✐♥ ♠❛t❤❡✲ ❙♣r✐♥❣❡r ✲ ❱❡r❧❡❣✱ ❇❡r❧✐♥✳ ✸✾ ✱ ✼✷ ✲ ✶✶✼✳ ✹✸ ❆ ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❡♦r❡♠ ❢♦r s✉♣❡r♠❛rt✐♥❣❛❧❡s✱ ❬✶✸❪ P✳ ❆✳ ▼❡②❡r✳ ✭✶✾✻✷✮✱ ▼❛t❤✳ ✻ ✱ ✶✾✸ ✲ ✷✵✺✳ ❬✶✹❪ P✳ ❆✳ ▼❡②❡r ❛♥❞ ❈✳ ❉♦❧Ð❛♥s✲❉❛❞❡✳ ✭✶✾✼✵✮✱ ♣♦rt ❛✉① ♠❛rt✐♥❣❛❧❡s ❧♦❝❛❧❡s✱ ✐♥ ▼❛t❤❡♠❛t✐❝s✳ ❬✶✺❪ ❙✳ ■❧❧✳ ❏✳ ❙❛♥②❛❧✳ ■♥tÐ❣r❛❧❡s st♦❝❤❛st✐q✉❡s ♣❛r r❛♣✲ ❙Ð♠✐♥❛✐r❡ ❞❡ Pr♦❜❛❜✐❧✐tÐs ■❱✱ ▲❡❝t✉r❡ ◆♦t❡s ✶✷✹ ✭✷✵✵✽✮✱ ✱ ✼✼ ✲ ✶✵✼✳ ❙t♦❝❤❛st✐❝ ❞②♥❛♠✐❝ ❡q✉❛t✐♦♥s✱ P❤✳❉✳ ❉✐ss❡rt❛t✐♦♥✱ ♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ▼✐ss♦✉r✐ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✳ ✹✹ ❆♣✲