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On the Root solution to the Skorokhod embedding problem given full marginals

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❖♥ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ t♦ t❤❡ ❙❦♦r♦❦❤♦❞ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠

❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s

∗ ❆❧❡①❛♥❞r❡ ❘✐❝❤❛r❞ † ❳✐❛♦❧✉ ❚❛♥◆✐③❛r ❚♦✉③✐➓ ❖❝t♦❜❡r ✷✸✱ ✷✵✶✽ ❆❜str❛❝t ❚❤✐s ♣❛♣❡r ❡①❛♠✐♥❡s t❤❡ ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❦♦r♦❤♦❞ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠ ❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s ♦♥ s♦♠❡ ❝♦♠♣❛❝t t✐♠❡ ✐♥t❡r✈❛❧✳ ❖✉r r❡s✉❧ts ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❧✐♠✐t✐♥❣ ❛r❣✉♠❡♥ts ❜❛s❡❞ ♦♥ ✜♥✐t❡❧②✲♠❛♥② ♠❛r❣✐♥❛❧s ❘♦♦t s♦❧✉t✐♦♥ ♦❢ ❈♦①✱ ❖❜➟ó❥✱ ❛♥❞ ❚♦✉③✐ ❬✾❪✳ ❖✉r ♠❛✐♥ r❡s✉❧t ♣r♦✈✐❞❡s ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❜② ♠❡❛♥s ♦❢ ❛ ❝♦♥✈❡♥✐❡♥t ♣❛r❛❜♦❧✐❝ P❉❊✳

✶ ■♥tr♦❞✉❝t✐♦♥

❚❤❡ ❙❦♦r♦❦❤♦❞ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠✱ ✐♥✐t✐❛❧❧② s✉❣❣❡st❡❞ ❜② ❙❦♦r♦❦❤♦❞ ❬✸✵❪✱ ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ ❛ st♦♣♣✐♥❣ t✐♠❡ τ t♦❣❡t❤❡r ✇✐t❤ ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ B s✉❝❤ t❤❛t Bτ ∼ µ ❢♦r ❛ ❣✐✈❡♥ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥ µ ♦♥ R✳ ❚❤❡ ❡①✐st✐♥❣ ❧✐t❡r❛t✉r❡ ❝♦♥t❛✐♥s ✈❛r✐♦✉s s♦❧✉t✐♦♥s s✉❣❣❡st❡❞ ✐♥ ❞✐✛❡r❡♥t ❝♦♥t❡①ts✳ ❙♦♠❡ ♦❢ t❤❡♠ s❛t✐s❢② ❛♥ ♦♣t✐♠❛❧✐t② ♣r♦♣❡rt② ❛♠♦♥❣ ❛❧❧ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s✱ ❡✳❣✳ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ ❬✷✽❪✱ t❤❡ ❘♦st s♦❧✉t✐♦♥ ❬✷✾❪✱ t❤❡ ❆③é♠❛✲❨♦r s♦❧✉t✐♦♥ ❬✶❪✱ t❤❡ ❱❛❧❧♦✐s s♦❧✉t✐♦♥ ❬✸✶❪✱ t❤❡ P❡r❦✐♥s s♦❧✉t✐♦♥ ❬✷✼❪✱ ❡t❝✳ ❚❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ ❡①t❡♥s✐✈❡❧② r❡✈✐✈❡❞ ✐♥ t❤❡ r❡❝❡♥t ❧✐t❡r❛t✉r❡ ❞✉❡ t♦ t❤❡ ✐♠♣♦rt❛♥t ❝♦♥♥❡①✐♦♥ ✇✐t❤ t❤❡ ♣r♦❜❧❡♠ ♦❢ r♦❜✉st ❤❡❞❣✐♥❣ ✐♥ ✜♥❛♥❝✐❛❧ ♠❛t❤❡♠❛t✐❝s✳ ❲❡ r❡❢❡r t♦ ❖❜➟ó❥ ❬✷✺❪ ❛♥❞ ❍♦❜s♦♥ ❬✶✾❪ ❢♦r ❛ s✉r✈❡② ♦♥ ❞✐✛❡r❡♥t s♦❧✉t✐♦♥s ❛♥❞ t❤❡ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ✜♥❛♥❝❡✳ ❖✉r ✐♥t❡r❡st ✐♥ t❤✐s ♣❛♣❡r ✐s ♦♥ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❦♦r♦❤♦❞ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❛s ❛ ❤✐tt✐♥❣ t✐♠❡ ♦❢ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ B ♦❢ s♦♠❡ t✐♠❡✲s♣❛❝❡ ❞♦♠❛✐♥ R ✉♥❧✐♠✐t❡❞ t♦ t❤❡ r✐❣❤t✱ t❤❛t ✐s✱ τR:= inf{t ≥ 0 : (t, Bt) ∈ R}✳ ❚❤✐s s♦❧✉t✐♦♥ ✇❛s s❤♦✇♥ ❜② ❘♦st ❬✷✾❪ t♦ ❤❛✈❡ t❤❡ ♠✐♥✐♠❛❧ ✈❛r✐❛♥❝❡ ❛♠♦♥❣ ❛❧❧ s♦❧✉t✐♦♥s t♦ t❤❡ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠✳ ❆s ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ✜♥❛♥❝❡✱ ✐t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡❞✉❝❡ r♦❜✉st ♥♦✲❛r❜✐tr❛❣❡ ♣r✐❝❡ ❜♦✉♥❞s ❢♦r ❛ ❝❧❛ss ♦❢ ✈❛r✐❛♥❝❡ ♦♣t✐♦♥s ✭s❡❡ ❡✳❣✳ ❍♦❜s♦♥ ❬✶✾❪✮✳ ❚♦ ✜♥❞ t❤❡ ❜❛rr✐❡r R ✐♥ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❘♦♦t s♦❧✉t✐♦♥✱ ❈♦① ❛♥❞ ❲❛♥❣ ❬✽❪ ♣r♦✈✐❞❡❞ ❛ ❝♦♥str✉❝t✐♦♥ ❜② s♦❧✈✐♥❣ ❛ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t②✳ ❚❤✐s ∗❚❤✐s ✇♦r❦ ✇❛s st❛rt❡❞ ✇❤✐❧❡ t❤❡ ❛✉t❤♦rs ❤❛❞ t❤❡ ♦♣♣♦rt✉♥✐t② t♦ ❜❡♥❡✜t ❢r♦♠ t❤❡ ❊❘❈ ❣r❛♥t ✸✶✶✶✶✶ ❘♦❋✐❘▼ ✷✵✶✷✲✷✵✶✽✳ †▼■❈❙ ❛♥❞ ❋é❞ér❛t✐♦♥ ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❈◆❘❙ ❋❘✲✸✹✽✼✱ ❈❡♥tr❛❧❡❙✉♣é❧❡❝✱ ❛❧❡①❛♥✲ ❞r❡✳r✐❝❤❛r❞❅❝❡♥tr❛❧❡s✉♣❡❧❡❝✳❢r✳ ❚❤✐s ❛✉t❤♦r ✐s ❣r❛t❡❢✉❧ t♦ t❤❡ ❈▼❆P ❢♦r ✐ts ❤♦s♣✐t❛❧✐t②✱ ✇❤❡r❡ ♠♦st ♦❢ t❤✐s ✇♦r❦ ✇❛s ❝❛rr✐❡❞ ♦✉t✳ ‡❈❡r❡♠❛❞❡✱ ❯♥✐✈❡rs✐t② ♦❢ P❛r✐s ❉❛✉♣❤✐♥❡✱ P❙▲ ❯♥✐✈❡rs✐t②✱ t❛♥❅❝❡r❡♠❛❞❡✳❞❛✉♣❤✐♥❡✳❢r✳ ❚❤✐s ❛✉t❤♦r ❛❝✲ ❦♥♦✇❧❡❞❣❡s t❤❡ ✜♥❛♥❝✐❛❧ s✉♣♣♦rt ♦❢ t❤❡ ■♥✐t✐❛t✐✈❡ ❞❡ ❘❡❝❤❡r❝❤❡ ✏▼ét❤♦❞❡s ♥♦♥✲❧✐♥é❛✐r❡s ♣♦✉r ❧❛ ❣❡st✐♦♥ ❞❡s r✐sq✉❡s ✜♥❛♥❝✐❡rs✑ s♣♦♥s♦r❡❞ ❜② ❆❳❆ ❘❡s❡❛r❝❤ ❋✉♥❞✳ ➓❈▼❆P✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ ♥✐③❛r✳t♦✉③✐❅♣♦❧②t❡❝❤♥✐q✉❡✳❡❞✉✳ ❚❤✐s ❛✉t❤♦r ❛❝❦♥♦✇❧❡❞❣❡s t❤❡ ✜♥❛♥❝✐❛❧ s✉♣✲ ♣♦rt ♦❢ t❤❡ ❈❤❛✐r❡s ❋✐♥❛♥❝✐❛❧ ❘✐s❦s✱ ❛♥❞ ❋✐♥❛♥❝❡ ❛♥❞ ❙✉st❛✐♥❛❜❧❡ ❉❡✈❡❧♦♣♠❡♥t✱ ❤♦st❡❞ ❜② t❤❡ ▲♦✉✐s ❇❛❝❤❡❧✐❡r ■♥st✐t✉t❡✳ ✶

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❛♣♣r♦❛❝❤ ✐s t❤❡♥ ❡①♣❧♦r❡❞ ✐♥ ●❛ss✐❛t✱ ❖❜❡r❤❛✉s❡r✱ ❛♥❞ ❞♦s ❘❡✐s ❬✶✸❪ ❛♥❞ ●❛ss✐❛t✱ ▼✐❥❛t♦✈✐➣✱ ❛♥❞ ❖❜❡r❤❛✉s❡r ❬✶✷❪ t♦ ❝♦♥str✉❝t R ✉♥❞❡r ♠♦r❡ ❣❡♥❡r❛❧ ❝♦♥❞✐t✐♦♥s✳ ❲❡ ❛❧s♦ r❡❢❡r t♦ t❤❡ r❡♠❛r❦❛❜❧❡ ✇♦r❦ ♦❢ ❇❡✐❣❧❜ö❝❦✱ ❈♦①✱ ❛♥❞ ❍✉❡s♠❛♥♥ ❬✹❪ ✇❤✐❝❤ ❞❡r✐✈❡s t❤❡ ❘♦♦t ❡♠❜❡❞❞✐♥❣✱ ❛♠♦♥❣ ♦t❤❡r s♦❧✉t✐♦♥s✱ ❛s ❛ ♥❛t✉r❛❧ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ♠♦♥♦t♦♥✐❝✐t② ♣r✐♥❝✐♣❧❡ ✐♥ ♦♣t✐♠❛❧ tr❛♥s♣♦rt✳ ■t ✐s ❛❧s♦ ♥❛t✉r❛❧ t♦ ❡①t❡♥❞ t❤❡ ❙❦♦r♦❤♦❞ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠ t♦ t❤❡ ♠✉❧t✐♣❧❡ ♠❛r❣✐♥❛❧s ❝❛s❡✳ ▲❡t (µk)0≤k≤n ❜❡ ❛ ❢❛♠✐❧② ♦❢ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s✱ ♥♦♥❞❡❝r❡❛s✐♥❣ ✐♥ t❤❡ ❝♦♥✈❡① ♦r❞❡r✱ ✐✳❡✳ µk−1(φ) ≤ µk(φ)✱ k = 1, . . . , n✱ ❢♦r ❛❧❧ ❝♦♥✈❡① ❢✉♥❝t✐♦♥s φ : R → R✳ ❚❤❡ ♠✉❧t✐♣❧❡✲♠❛r❣✐♥❛❧s ❙❦♦r♦❤♦❞ ❡♠❜❞❞✐♥❣ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ B✱ t♦❣❡t❤❡r ✇✐t❤ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ st♦♣♣✐♥❣ t✐♠❡s (τk)1≤k≤n✱ s✉❝❤ t❤❛t Bτk ∼ µk ❢♦r ❡❛❝❤ k = 1, · · · , n✳ ▼❛❞❛♥ ❛♥❞ ❨♦r ❬✷✸❪ ♣r♦✈✐❞❡❞ ❛ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ♠❛r❣✐♥❛❧s✱ ✉♥❞❡r ✇❤✐❝❤ t❤❡ ❆③é♠❛✲❨♦r ❡♠❜❡❞❞✐♥❣ st♦♣♣✐♥❣ t✐♠❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡❛❝❤ ♠❛r❣✐♥❛❧ ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ♦r❞❡r❡❞✱ s♦ t❤❛t t❤❡ ✐t❡r❛t✐♦♥ ♦❢ ❆③é♠❛✲❨♦r s♦❧✉t✐♦♥s ♣r♦✈✐❞❡s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ♠✉❧t✐♣❧❡ ♠❛r❣✐♥❛❧s ❙❦♦r♦❦❤♦❞ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠✳ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ❆③é♠❛✲❨♦r ❡♠❜❡❞❞✐♥❣ st♦♣♣✐♥❣ t✐♠❡s ♠❛② ♥♦t ❜❡ ♦r❞❡r❡❞✳ ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❆③é♠❛✲❨♦r ❡♠❜❡❞❞✐♥❣ ✇❛s ♦❜t❛✐♥❡❞ ❜② ❇r♦✇♥✱ ❍♦❜s♦♥✱ ❛♥❞ ❘♦❣❡rs ❬✻❪ ✐♥ t❤❡ t✇♦✲♠❛r❣✐♥❛❧s ❝❛s❡✱ ❛♥❞ ❧❛t❡r ❜② ❖❜➟ó❥ ❛♥❞ ❙♣♦✐❞❛ ❬✷✻❪ ❢♦r ❛♥ ❛r❜✐tr❛r② ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♠❛r❣✐♥❛❧s✳ ▼♦r❡♦✈❡r✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡♠❜❡❞❞✐♥❣s ❡♥❥♦②s t❤❡ s✐♠✐❧❛r ♦♣t✐♠❛❧✐t② ♣r♦♣❡rt② ❛s ✐♥ t❤❡ ♦♥❡ ♠❛r❣✐♥❛❧ ❝❛s❡✳ ■♥ ❈❧❛✐ss❡✱ ●✉♦✱ ❛♥❞ ❍❡♥r②✲▲❛❜♦r❞èr❡ ❬✼❪✱ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❱❛❧❧♦✐s s♦❧✉t✐♦♥ t♦ t❤❡ t✇♦✲♠❛r❣✐♥❛❧s ❝❛s❡ ✐s ♦❜t❛✐♥❡❞ ❢♦r ❛ s♣❡❝✐✜❝ ❝❧❛ss ♦❢ ♠❛r❣✐♥❛❧s✳ ❲❡ ❛❧s♦ r❡❢❡r t♦ ❇❡✐❣❧❜ö❝❦✱ ❈♦①✱ ❛♥❞ ❍✉❡s♠❛♥♥ ❬✺❪ ❢♦r ❛ ❣❡♦♠❡tr✐❝ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♦♣t✐♠❛❧ ❙❦♦r♦❦❤♦❞ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s ❣✐✈❡♥ ♠✉❧t✐♣❧❡ ♠❛r❣✐♥❛❧s✳ ❚❤❡ ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❦♦r♦❤♦❞ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠ ✇❛s r❡❝❡♥t❧② ❡①t❡♥❞❡❞ ❜② ❈♦①✱ ❖❜➟ó❥✱ ❛♥❞ ❚♦✉③✐ ❬✾❪ t♦ t❤❡ ♠✉❧t✐♣❧❡ ♠❛r❣✐♥❛❧s ❝❛s❡✳ ❖✉r ♦❜❥❡❝t✐✈❡ ✐♥ t❤✐s ♣❛♣❡r ✐s t♦ ❝❤❛r✲ ❛❝t❡r✐③❡ t❤❡ ❧✐♠✐t ❝❛s❡ ✇✐t❤ ❛ ❢❛♠✐❧② ♦❢ ❢✉❧❧ ♠❛r❣✐♥❛❧s µ = (µt)t∈[0,1]✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t ❡❛❝❤ µt❤❛s ✜♥✐t❡ ✜rst ♠♦♠❡♥t ❛♥❞ t 7→ µt✐s r✐❣❤t ❝♦♥t✐♥✉♦✉s ❛♥❞ ✐♥❝r❡❛s✐♥❣ ✐♥ ❝♦♥✈❡① ♦r❞❡r✳ ❙✉❝❤ ❛ ❢❛♠✐❧② ✐s ❝❛❧❧❡❞ ❛ ♣❡❛❝♦❝❦ ✭♦r P❈❖❈ ✏Pr♦❝❡ss✉s ❈r♦✐ss❛♥t ♣♦✉r ❧✬❖r❞r❡ ❈♦♥✈❡①❡✑ ✐♥ ❋r❡♥❝❤✮ ❜② ❍✐rs❝❤✱ Pr♦❢❡t❛✱ ❘♦②♥❡tt❡✱ ❛♥❞ ❨♦r ❬✶✽❪✳ ❚❤❡♥ ❑❡❧❧❡r❡r✬s ❚❤❡♦r❡♠ ❬✷✷❪ ❡♥s✉r❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ r✐❣❤t✲❝♦♥t✐♥✉♦✉s ♠❛rt✐♥❣❛❧❡ M = (Mt)0≤t≤1 s✉❝❤ t❤❛t Mt ∼ µt ❢♦r ❡❛❝❤ t ∈ [0, 1]✳ ❋✉rt❤❡r✱ ❜② ▼♦♥r♦❡✬s r❡s✉❧t ❬✷✹❪✱ ♦♥❡ ❝❛♥ ✜♥❞ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ st♦♣♣✐♥❣ t✐♠❡s (τt)0≤t≤1 t♦❣❡t❤❡r ✇✐t❤ ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ B = (Bs)s≥0 s✉❝❤ t❤❛t Bτt ∼ µt ❢♦r ❡❛❝❤ t ∈ [0, 1]✳ ❚❤✐s ❝♦♥s✐sts ✐♥ ❛♥ ❡♠❜❡❞❞✐♥❣ ❢♦r t❤❡ ❢✉❧❧ ♠❛r❣✐♥❛❧s µ✳ ❲❡ r❡❢❡r t♦ ❬✶✽❪ ❢♦r ❞✐✛❡r❡♥t ❡①♣❧✐❝✐t ❝♦♥str✉❝t✐♦♥s ♦❢ t❤❡ ♠❛rt✐♥❣❛❧❡s ♦r ❡♠❜❡❞❞✐♥❣s ✜tt✐♥❣ t❤❡ ♣❡❛❝♦❝❦ ♠❛r❣✐♥❛❧s✳ ❆♠♦♥❣ ❛❧❧ ♠❛rt✐♥❣❛❧❡s ♦r µ✲❡♠❜❡❞❞✐♥❣s✱ ✐t ✐s ✐♥t❡r❡st✐♥❣ t♦ ✜♥❞ s♦❧✉t✐♦♥s ❡♥❥♦②✐♥❣ s♦♠❡ ♦♣t✐♠❛❧✐t② ♣r♦♣❡rt✐❡s✳ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ▼❛❞❛♥ ❛♥❞ ❨♦r ❬✷✸❪✱ t❤❡ ❆③é♠❛✲❨♦r ❡♠❜❡❞❞✐♥❣ τAY t ♦❢ t❤❡ ♦♥❡ ♠❛r❣✐♥❛❧ ♣r♦❜❧❡♠ ✇✐t❤ µt ✐s ♦r❞❡r❡❞ ✇✳r✳t✳ t✱ ❛♥❞ t❤✉s (τtAY)0≤t≤1 ✐s t❤❡ ❡♠❜❡❞❞✐♥❣ ♠❛①✲ ✐♠✐③✐♥❣ t❤❡ ❡①♣❡❝t❡❞ ♠❛①✐♠✉♠ ❛♠♦♥❣ ❛❧❧ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥s✳ ❚❤✐s ♦♣t✐♠❛❧✐t② ✐s ❢✉rt❤❡r ❡①t❡♥❞❡❞ ❜② ❑ä❧❧❜❧❛❞✱ ❚❛♥✱ ❛♥❞ ❚♦✉③✐ ❬✷✶❪ ❛❧❧♦✇✐♥❣ ❢♦r ♥♦♥✲♦r❞❡r❡❞ ❜❛rr✐❡rs✳ ❍♦❜s♦♥ ❬✷✵❪ ❣❛✈❡ ❛ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ♠❛rt✐♥❣❛❧❡ ✇✐t❤ ♠✐♥✐♠❛❧ ❡①♣❡❝t❡❞ t♦t❛❧ ✈❛r✐❛t✐♦♥ ❛♠♦♥❣ ❛❧❧ ♠❛rt✐♥❣❛❧❡s ✜tt✐♥❣ t❤❡ ♠❛r❣✐♥❛❧s✳ ❍❡♥r②✲▲❛❜♦r❞èr❡✱ ❚❛♥✱ ❛♥❞ ❚♦✉③✐ ❬✶✼❪ ♣r♦✈✐❞❡❞ ❛ ❧♦❝❛❧ ▲é✈② ♠❛rt✐♥❣❛❧❡✱ ❛s ❧✐♠✐t ♦❢ t❤❡ ❧❡❢t✲♠♦♥♦t♦♥❡ ♠❛rt✐♥❣❛❧❡s ✐♥tr♦❞✉❝❡❞ ❜② ❇❡✐❣❧❜ö❝❦ ❛♥❞ ❏✉✐❧❧❡t ❬✸❪ ✭s❡❡ ❛❧s♦ ❍❡♥r②✲▲❛❜♦r❞èr❡ ❛♥❞ ❚♦✉③✐ ❬✶✻❪✮✱ ✇❤✐❝❤ ✐♥❤❡r✐ts ✐ts ♦♣t✐♠❛❧✐t② ♣r♦♣❡rt②✳ ❋♦r ❣❡♥❡r❛❧ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❛♥❞ t❤❡ ❛ss♦❝✐❛t❡❞ ❞✉❛❧✐t② r❡s✉❧t✱ ♦♥❡ ♥❡❡❞s ❛ t✐❣❤t♥❡ss ❛r❣✉♠❡♥t✱ ✇❤✐❝❤ ✐s st✉❞✐❡❞ ✐♥ ●✉♦✱ ❚❛♥✱ ❛♥❞ ❚♦✉③✐ ❬✶✺❪ ❜② ✉s✐♥❣ t❤❡ ❙✲t♦♣♦❧♦❣② ♦♥ t❤❡ ❙❦♦r♦❦❤♦❞ s♣❛❝❡✱ ❛♥❞ ✐♥ ❑ä❧❧❜❧❛❞✱ ❚❛♥✱ ❛♥❞ ❚♦✉③✐ ❬✷✶❪ ❜② ✉s✐♥❣ t❤❡ ❙❦♦r♦❦❤♦❞ ❡♠❜❡❞❞✐♥❣ ❛♣♣r♦❛❝❤✳ ❚❤❡ ❛✐♠ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ st✉❞② t❤❡ ❢✉❧❧ ♠❛r❣✐♥❛❧s ❧✐♠✐t ♦❢ t❤❡ ♠✉❧t✐♣❧❡✲♠❛r❣✐♥❛❧s ❘♦♦t ❡♠❜❡❞❞✐♥❣ ❛s ❞❡r✐✈❡❞ ✐♥ ❬✾❪✳ ❚❤✐s ❧❡❛❞s t♦ ❛ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ ❢♦r t❤❡ ✷

(3)

❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠ ❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s✳ ❯s✐♥❣ t❤❡ t✐❣❤t♥❡ss r❡s✉❧t ✐♥ ❬✷✶❪✱ ✇❡ ❝❛♥ ❡❛s✐❧② ♦❜t❛✐♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ❧✐♠✐t ❛s ✇❡❧❧ ❛s ✐ts ♦♣t✐♠❛❧✐t②✳ ❲❡ t❤❡♥ ♣r♦✈✐❞❡ s♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t ❘♦♦t s♦❧✉t✐♦♥ ❛s ✇❡❧❧ ❛s t❤❛t ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ✐s ✉s❡❞ ✐♥ t❤❡ ✜♥✐t❡❧② ♠❛♥② ♠❛r❣✐♥❛❧s ❝❛s❡ t♦ ❞❡s❝r✐❜❡ t❤❡ ❜❛rr✐❡rs✳ ■♥ t❤❡ r❡st ♦❢ t❤❡ ♣❛♣❡r✱ ✇❡ ✇✐❧❧ ✜rst ❢♦r♠✉❧❛t❡ ♦✉r ♠❛✐♥ r❡s✉❧ts ✐♥ ❙❡❝t✐♦♥ ✷✳ ❚❤❡♥ ✐♥ ❙❡❝t✐♦♥ ✸✱ ✇❡ r❡❝❛❧❧ s♦♠❡ ❞❡t❛✐❧s ♦♥ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ ❣✐✈❡♥ ✜♥✐t❡❧② ♠❛♥② ♠❛r❣✐♥❛❧s ✐♥ ❬✾❪ ❛♥❞ t❤❡ ❧✐♠✐t ❛r❣✉♠❡♥t ♦❢ ❬✷✶❪✱ ✇❤✐❝❤ ✐♥❞✉❝❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❧✐♠✐t ❘♦♦t s♦❧✉t✐♦♥ ❢♦r t❤❡ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠ ❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s✳ ❲❡ t❤❡♥ ♣r♦✈✐❞❡ t❤❡ ♣r♦♦❢s ♦❢ ♦✉r ♠❛✐♥ r❡s✉❧ts ♦♥ s♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❧✐♠✐t ❘♦♦t s♦❧✉t✐♦♥ ✐♥ ❙❡❝t✐♦♥ ✹✳ ❙♦♠❡ ❢✉rt❤❡r ❞✐s❝✉ss✐♦♥s ❛r❡ ✜♥❛❧❧② ♣r♦✈✐❞❡❞ ✐♥ ❙❡❝t✐♦♥✺✳

✷ Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥ ❛♥❞ ♠❛✐♥ r❡s✉❧ts

❲❡ ❛r❡ ❣✐✈❡♥ ❛ ❢❛♠✐❧② ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s µ = (µs)s∈[0,1]♦♥ R✱ s✉❝❤ t❤❛t µs✐s ❝❡♥tr❡❞ ✇✐t❤ ✜♥✐t❡ ✜rst ♠♦♠❡♥t ❢♦r ❛❧❧ s ∈ [0, 1]✱ s 7→ µs ✐s ❝à❞❧à❣ ✉♥❞❡r t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ t♦♣♦❧♦❣②✱ ❛♥❞ t❤❡ ❢❛♠✐❧② µ ✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣ ✐♥ ❝♦♥✈❡① ♦r❞❡r✱ ✐✳❡✳ ❢♦r ❛♥② ❝♦♥✈❡① ❢✉♥❝t✐♦♥ φ : R → R✱ Z R φ(x)µs(dx) ≤ Z R φ(x)µt(dx) ❢♦r ❛❧❧ s ≤ t. ❉❡✜♥✐t✐♦♥ ✷✳✶✳ (i) ❆ st♦♣♣✐♥❣ r✉❧❡ ✐s ❛ t❡r♠ α = (Ωα, Fα, Fα, Pα, Bα, (τsα)s∈[0,1]), s✉❝❤ t❤❛t (Ωα, Fα, Fα, Pα) ✐s ❛ ✜❧t❡r❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ Bα ❛♥❞ ❛ ❢❛♠✐❧② ♦❢ st♦♣♣✐♥❣ t✐♠❡s (τα s)s∈[0,1] s✉❝❤ t❤❛t s 7→ τsα ✐s ❝à❞❧à❣ ❛♥❞ ♥♦♥✲ ❞❡❝r❡❛s✐♥❣✳ ❲❡ ❞❡♥♦t❡ A :=❆❧❧ st♦♣♣✐♥❣ r✉❧❡s , ❛♥❞ At:=  α ∈ A : τ1α≤ t , ❢♦r ❛❧❧ t ≥ 0. (ii) ❆ st♦♣♣✐♥❣ r✉❧❡ α ∈ A ✐s ❝❛❧❧❡❞ ❛ µ✲❡♠❜❡❞❞✐♥❣ ✐❢ (Bα t∧τα 1)t≥0 ✐s ✉♥✐❢♦r♠❧② ✐♥t❡❣r❛❜❧❡ ❛♥❞ Bταα s ∼ µs ❢♦r ❛❧❧ s ∈ [0, 1]✳ ❲❡ ❞❡♥♦t❡ ❜② A(µ) t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ µ✲❡♠❜❡❞❞✐♥❣s✳ (iii) ▲❡t πn = {0 = s0 < s1 < · · · < sn = 1} ❜❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ [0, 1]✳ ❆ st♦♣♣✐♥❣ r✉❧❡ α ∈ A ✐s ❝❛❧❧❡❞ ❛ (µ, πn)✲❡♠❜❡❞❞✐♥❣ ✐❢ (Bt∧τα α 1)t≥0 ✐s ✉♥✐❢♦r♠❧② ✐♥t❡❣r❛❜❧❡ ❛♥❞ B α τα sk ∼ µsk ❢♦r ❛❧❧ k = 1, · · · , n✳ ❲❡ ❞❡♥♦t❡ ❜② A(µ, πn) t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ (µ, πn)✲❡♠❜❡❞❞✐♥❣s✳ ❖✉r ❛✐♠ ✐s t♦ st✉❞② t❤❡ ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❦♦r♦❤♦❞ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠ ✭❙❊P✱ ❤❡r❡❛❢t❡r✮ ❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s (µs)s∈[0,1]✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ✜rst r❡❝❛❧❧ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❊P ❣✐✈❡♥ ✜♥✐t❡❧② ♠❛♥② ♠❛r❣✐♥❛❧s✱ ❝♦♥str✉❝t❡❞ ✐♥ ❬✾❪✳ ▲❡t (πn)n≥1 ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ [0, 1]✱ ✇❤❡r❡ πn = {0 = sn0 < sn1 < · · · < snn = 1} ❛♥❞ |πn| := maxnk=1|snk − snk−1| → 0 ❛s n → ∞✳ ❚❤❡♥ ❢♦r ❡✈❡r② ✜①❡❞ n✱ ♦♥❡ ♦❜t❛✐♥s n ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s (µsn k)1≤k≤n❛♥❞ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❘♦♦t s♦❧✉t✐♦♥ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❙❊P✳ ❚❤❡♦r❡♠ ✭❈♦①✱ ❖❜➟ó❥ ❛♥❞ ❚♦✉③✐✱ ✷✵✶✽✮✳ ❋♦r ❛♥② n ≥ 1✱ t❤❡r❡ ❡①✐sts ❛ (µ, πn)✲❡♠❜❡❞❞✐♥❣ α∗n ❝❛❧❧❡❞ ❘♦♦t ❡♠❜❡❞❞✐♥❣✱ ✇❤❡r❡ σn k := τ α∗ n sn k ✐s ❞❡✜♥❡❞ ❜② σ0n:= 0 ❛♥❞ σkn:= inf{t ≥ σk−1n : (t, Bα∗n t ) ∈ Rnk}, ✸

(4)

❢♦r s♦♠❡ ❢❛♠✐❧② ♦❢ ❜❛rr✐❡rs (Rn k)1≤k≤n ✐♥ R+× R✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛♥② ♥♦♥✲❞❡❝r❡❛s✐♥❣ ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥ f : R+→ R+✱ ♦♥❡ ❤❛s EPα∗n h Z τα∗n 1 0 f (t)dti = inf α∈A(µ,πn) EPα h Z τα 1 0 f (t)dti. ❚❤❡ ❜❛rr✐❡rs (Rn k)1≤k≤n❛r❡ ❣✐✈❡♥ ❡①♣❧✐❝✐t❧② ✐♥ ❬✾❪ ❜② s♦❧✈✐♥❣ ❛♥ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠✱ s❡❡ ❙❡❝t✐♦♥✸✳✶❜❡❧♦✇✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② A([0, 1], R+) t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❝à❞❧à❣ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s a : [0, 1] → R+✱ ✇❤✐❝❤ ✐s ❛ P♦❧✐s❤ s♣❛❝❡ ✉♥❞❡r t❤❡ ▲é✈② ♠❡tr✐❝✳ ◆♦t✐❝❡ ❛❧s♦ t❤❛t t❤❡ ▲é✈② ♠❡tr✐❝ ♠❡tr✐❝✐③❡s t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ t♦♣♦❧♦❣② ♦♥ A([0, 1], R+)s❡❡♥ ❛s ❛ s♣❛❝❡ ♦❢ ✜♥✐t❡ ♠❡❛s✉r❡s✳ ❉❡♥♦t❡ ❛❧s♦ ❜② C(R+, R)t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❝♦♥t✐♥✉♦✉s ♣❛t❤s ω : R+→ R ✇✐t❤ ω0 = 0✱ ✇❤✐❝❤ ✐s ❛ P♦❧✐s❤ s♣❛❝❡ ✉♥❞❡r t❤❡ ❝♦♠♣❛❝t ❝♦♥✈❡r❣❡♥❝❡ t♦♣♦❧♦❣②✳ ❚❤❡♥ ❢♦r ❛ ❣✐✈❡♥ ❡♠❜❡❞❞✐♥❣ α✱ ♦♥❡ ❝❛♥ s❡❡ (Bα · , τ·α) ❛s ❛ r❛♥❞♦♠ ❡❧❡♠❡♥t t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ C(R+, R) × A([0, 1], R+)✱ ✇❤✐❝❤ ❛❧❧♦✇s t♦ ❞❡✜♥❡ t❤❡✐r ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡✳ ❖✉r ✜rst ♠❛✐♥ r❡s✉❧t ❡♥s✉r❡s t❤❛t t❤❡ (µ, πn)✲❘♦♦t ❡♠❜❡❞❞✐♥❣ ❤❛s ❛ ❧✐♠✐t ✐♥ s❡♥s❡ ♦❢ t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡✱ ✇❤✐❝❤ ❡♥❥♦②s t❤❡ s❛♠❡ ♦♣t✐♠❛❧✐t② ♣r♦♣❡rt②✱ ❛♥❞ t❤✉s ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ❢✉❧❧ ♠❛r❣✐♥❛❧s ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❊P✳ ❖✉r ♣r♦♦❢ r❡q✉✐r❡s t❤❡ ❢♦❧❧♦✇✐♥❣ t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥✳ ❆ss✉♠♣t✐♦♥ ✷✳✷✳ ▲❡t U : [0, 1] × R → R ❜❡ t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ µ ❞❡✜♥❡❞ ❜② U (s, x) := − Z R|x − y|µ s(dy). ✭✷✳✶✮ ❆ss✉♠❡ t❤❛t U ✐s C1 ✐♥ s✱ ✇✐t❤ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ∂ sU s✉❝❤ t❤❛t x 7→ sups∈[0,1]∂sU (s, ·) ❤❛s ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤✳ ❚❤❡♦r❡♠ ✶✳ (i) ▲❡t (πn)n≥1❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ [0, 1] s✉❝❤ t❤❛t |πn| → 0 ❛s n → ∞✳ ❉❡♥♦t❡ ❜② α∗ nt❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ (µ, πn)✲❘♦♦t ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts α∗∈ A(µ) s✉❝❤ t❤❛t t❤❡ s❡q✉❡♥❝❡ (Bα∗ n · , τα ∗ n · )n≥1 ✇❡❛❦❧② ❝♦♥✈❡r❣❡s t♦ (Bα ∗ · , τα ∗ · )✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛❧❧ ♥♦♥✲ ❞❡❝r❡❛s✐♥❣ ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥s f : R+→ R+✱ ♦♥❡ ❤❛s EPα∗h Z τα∗ 1 0 f (t)dti = inf α∈A(µ) EPαh Z τα 1 0 f (t)dti. (ii) ❯♥❞❡r ❆ss✉♠♣t✐♦♥ ✷✳✷✱ ❢♦r ❛❧❧ ✜①❡❞ (s, t) ∈ [0, 1] × R+✱ t❤❡ ❧❛✇ ♦❢ Bα ∗ τα∗ s ∧t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐t✐♦♥s (πn)n≥1 ❛♥❞ ♦❢ t❤❡ ❧✐♠✐t α∗✳ ❲❡ ♥❡①t ♣r♦✈✐❞❡ s♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❢✉❧❧ ♠❛r❣✐♥❛❧s ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❊P α∗ ❣✐✈❡♥ ✐♥ ❚❤❡♦r❡♠✶✳ ▲❡t u(s, t, x) := −EPα∗|Bt∧τα∗α∗ s − x|  , (s, t, x) ∈ Z := [0, 1] × R+× R. ✭✷✳✷✮ ❖✉r ♥❡①t ♠❛✐♥ r❡s✉❧t✱ ❚❤❡♦r❡♠ ✷ ❜❡❧♦✇✱ ♣r♦✈✐❞❡s ❛ ✉♥✐q✉❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ u ✇❤✐❝❤ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ❧✐♠✐t α∗✱ t❤✉s ❥✉st✐❢②✐♥❣ ❈❧❛✐♠ ✭✐✐✮ ♦❢ ❚❤❡♦r❡♠ ✳ ▼♦r❡♦✈❡r✱ ✐t ❢♦❧❧♦✇s ❜② ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ t❤❛t ♦♥❡ ❤❛s

−|x| −√t E|B1| ≤ UN(0,t)(x) ≤ u(1, t, x) ≤ u(s, t, x) ≤ U(0, x),

(5)

❢♦r ❛❧❧ (s, t, x) ∈ [0, 1] × R+× R✱ ✇❤❡r❡ ✇❡ ❞❡♥♦t❡❞ ❜② UN(0,t) t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ N(0, t) ❞✐str✐❜✉t✐♦♥ ✭s❡❡ ✭✷✳✶✮ ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥✮✳ ■♥ t❤❡ ✜♥✐t❡❧② ♠❛♥② ♠❛r❣✐♥❛❧s ❝❛s❡ ✐♥ ❬✾❪✱ t❤❡ ❢✉♥❝t✐♦♥ u ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ ❛♥ ♦♣t✐♠❛❧ st♦♣✲ ♣✐♥❣ ♣r♦❜❧❡♠ ❛♥❞ ✐s t❤❡♥ ✉s❡❞ t♦ ❞❡✜♥❡ t❤❡ ❜❛rr✐❡rs ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❘♦♦t s♦❧✉t✐♦♥✳ ❙✐♠✐❧❛r t♦ ❡q✉❛t✐♦♥s ✭✷✳✶✵✮ ❛♥❞ ✭✸✳✶✮ ✐♥ ❬✾❪✱ ✇❡ ❝❛♥ ❝❤❛r❛❝t❡r✐③❡ u ❛s ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ♦❢ ❛♥ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠✱ ❛♥❞ t❤❡♥ ❛s ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t②✿ ( min∂tu − 12∂xx2 u, ∂s(u − U) = 0, ♦♥ ✐♥t(Z), u t=0 = U ❛♥❞ u s=0= U (0, .). ✭✷✳✸✮ ▲❡t Du ❛♥❞ D2u ❞❡♥♦t❡ t❤❡ ❣r❛❞✐❡♥t ❛♥❞ ❍❡ss✐❛♥ ♦❢ u ✇✳r✳t✳ z = (s, t, x)✱ ❛♥❞ s❡t✿ F (Du, D2u) := min∂tu − 1 2∂ 2 xxu, ∂s(u − U) . ✭✷✳✹✮ ❉❡✜♥✐t✐♦♥ ✷✳✸✳ (i) ❆♥ ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ v : Z → R ✐s ❛ ✈✐s❝♦s✐t② s✉❜s♦❧✉t✐♦♥ ♦❢ ✭✷✳✸✮ ✐❢ v|s=0 ≤ U(0, ·)✱ v|t=0 ≤ U ❛♥❞ F (Dϕ, D2ϕ)(z0) ≤ 0 ❢♦r ❛❧❧ (z0, ϕ) ∈ ✐♥t(Z) × C2(Z) s❛t✐s❢②✐♥❣ (v − ϕ)(z0) = maxz∈Z(v − ϕ)(z)✳ (ii) ❆ ❧♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ w : [0, 1] × R+× R → R ✐s ❛ ✈✐s❝♦s✐t② s✉♣❡rs♦❧✉t✐♦♥ ♦❢ ✭✷✳✸✮ ✐❢ w|s=0≥ U(0, ·)✱ w|t=0 ≥ U ❛♥❞ F (Dϕ, D2ϕ)(z0) ≥ 0 ❢♦r ❛❧❧ (z0, ϕ) ∈ ✐♥t(Z) × C2(Z) s❛t✐s❢②✐♥❣ (w − ϕ)(z0) = minz∈Z(w − ϕ)(z)✳ (iii) ❆ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ v ✐s ❛ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ✭✷✳✸✮ ✐❢ ✐t ✐s ❜♦t❤ ✈✐s❝♦s✐t② s✉❜s♦❧✉t✐♦♥ ❛♥❞ s✉♣❡rs♦❧✉t✐♦♥✳ ❚❤❡♦r❡♠ ✷✳ ▲❡t ❆ss✉♠♣t✐♦♥✷✳✷ ❤♦❧❞ tr✉❡✳ (i) ❚❤❡ ❢✉♥❝t✐♦♥ u ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ♦❢ ❛♥ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠✱ u(s, t, x) = sup α∈At EPαhU (0, x + Bα τα s) + Z s 0 ∂sU (s − k, x + Bατα k)1{τ α k<t}dk i . ✭✷✳✺✮ (ii) ❚❤❡ ❢✉♥❝t✐♦♥ u(s, t, x) ✐s ❞❡❝r❡❛s✐♥❣ ❛♥❞ ❧♦❝❛❧❧② ▲✐♣s❝❤✐t③ ✐♥ s✱ ✉♥✐❢♦r♠❧② ▲✐♣s❝❤✐t③ ✐♥ x ❛♥❞ ✉♥✐❢♦r♠❧② 1 2✲❍ö❧❞❡r ✐♥ t✳ ▼♦r❡♦✈❡r✱ u ✐s ❛ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ❡q✉❛t✐♦♥ ✭✷✳✸✮✳ (iii) ▼♦r❡♦✈❡r✱ u ✐s t❤❡ ✉♥✐q✉❡ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥ ♦❢ ✭✷✳✸✮ s❛t✐s❢②✐♥❣ |u(s, t, x)| ≤ C(1 + t + |x|), (s, t, x) ∈ Z, ❢♦r s♦♠❡ ❝♦♥st❛♥t C > 0.

✸ ▼✉❧t✐♣❧❡ ♠❛r❣✐♥❛❧s ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❊P ❛♥❞ ✐ts ❧✐♠✐t

❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t♦ r❡❝❛❧❧ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ t♦ t❤❡ ❙❊P ❣✐✈❡♥ ♠✉❧t✐♣❧❡ ♠❛r❣✐♥❛❧s ❢r♦♠ ❬✾❪✳ ❆s ❛♥ ❡①t❡♥s✐♦♥ t♦ t❤❡ ♦♥❡ ♠❛r❣✐♥❛❧ ❘♦♦t s♦❧✉t✐♦♥ st✉❞✐❡❞ ✐♥ ❬✽❪ ❛♥❞ ❬✶✷❪✱ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♠✉❧t✐♣❧❡ ♠❛r❣✐♥❛❧s✬ ❝❛s❡ ❡♥❥♦②s s♦♠❡ ♦♣t✐♠❛❧✐t② ♣r♦♣❡rt② ❛♠♦♥❣ ❛❧❧ ❡♠❜❡❞❞✐♥❣s✳ ❲❡ t❤❡♥ ❛❧s♦ r❡❝❛❧❧ t❤❡ ❧✐♠✐t ❛r❣✉♠❡♥t ✐♥ ❬✷✶❪ t♦ s❤♦✇ ❤♦✇ t❤❡ ♦♣t✐♠❛❧✐t② ♣r♦♣❡rt② ✐s ♣r❡s❡r✈❡❞ ✐♥ t❤❡ ❧✐♠✐t ❝❛s❡✳ ✺

(6)

✸✳✶ ❚❤❡ ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❊P ❣✐✈❡♥ ♠✉❧t✐♣❧❡ ♠❛r❣✐♥❛❧s ▲❡t n ∈ N ❛♥❞ πn ❜❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ [0, 1]✱ ✇✐t❤ πn = {0 = sn0 < sn1 < · · · < snn = 1}✱ ✇❡ t❤❡♥ ♦❜t❛✐♥ n ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s µn := {µsn j}j=1,··· ,n ❛♥❞ r❡❝❛❧❧ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ t♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡♠❜❡❞❞✐♥❣ ♣r♦❜❧❡♠✳ ▲❡t Ω = C(R+, R) ❞❡♥♦t❡ t❤❡ ❝❛♥♦♥✐❝❛❧ s♣❛❝❡ ♦❢ ❛❧❧ ❝♦♥t✐♥✉♦✉s ♣❛t❤s ω : R+ −→ R ✇✐t❤ ω0 = 0✱ B ❜❡ t❤❡ ❝❛♥♦♥✐❝❛❧ ♣r♦❝❡ss✱ Bx := x + B✱ F = (Ft)t≥0 ❜❡ t❤❡ ❝❛♥♦♥✐❝❛❧ ✜❧tr❛t✐♦♥✱ F := F∞✱ ❛♥❞ P0 t❤❡ ❲✐❡♥❡r ♠❡❛s✉r❡ ✉♥❞❡r ✇❤✐❝❤ B ✐s ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❋♦r ❡❛❝❤ t ≥ 0✱ ❧❡t T0,t ❞❡♥♦t❡ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ F✲st♦♣♣✐♥❣ t✐♠❡s t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ [0, t]✳ ❉❡♥♦t❡ δnU (snj, x) := U (snj, x) − U(snj−1, x), x ∈ R, ✇❤✐❝❤ ✐s ♥♦♥✲♣♦s✐t✐✈❡ s✐♥❝❡ {µs}s∈[0,1] ✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣ ✐♥ ❝♦♥✈❡① ♦r❞❡r✐♥❣✳ ❲❡ t❤❡♥ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ un(·) ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠s✿ un s=0:= U (sn0, .), ❛♥❞ un(snj, t, x) := sup θ∈T0,t Eun(snj−1, t − θ, Bθx) + δnU (snj, Bθx)1{θ<t}. ✭✸✳✶✮ ❉❡♥♦t✐♥❣ s✐♠✐❧❛r❧② δnu(sn j, t, x) = un(snj, t, x) − un(snj−1, t, x)✱ ✇❡ ❞❡✜♥❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ st♦♣♣✐♥❣ r❡❣✐♦♥s Rnj :=  (t, x) ∈ [0, ∞] × [−∞, ∞] : δnu(snj, t, x) = δnU (snj, x) , j = 1, . . . , n. ✭✸✳✷✮ ●✐✈❡♥ t❤❡ ❛❜♦✈❡✱ t❤❡ ❘♦♦t s♦❧✉t✐♦♥ ♦♥ t❤❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ B ✐♥ t❤❡ s♣❛❝❡ (Ω, F, P0)✱ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢❛♠✐❧② σn= (σn 1, . . . , σnn) ♦❢ st♦♣♣✐♥❣ t✐♠❡s σn0 := 0, ❛♥❞ σjn:= inft ≥ σj−1n : (t, Bt) ∈ Rnj , ∀j ∈ {1, . . . , n}. ✭✸✳✸✮ ❚❤❡ st♦♣♣✐♥❣ t✐♠❡s σn✐♥❞✉❝❡ ❛ st♦♣♣✐♥❣ r✉❧❡ α∗ n✐♥ t❤❡ s❡♥s❡ ♦❢ ❉❡✜♥✐t✐♦♥✷✳✶✿ αn∗ = Ωα∗n, Fα∗n, Fαn∗, Pα∗n, Bα∗n, τα∗n := Ω, F, F, P 0, B, τα ∗ n, ✭✸✳✹✮ ✇✐t❤ τα∗ n s := σnj ❢♦r s ∈ [snj, snj+1)✳ ❚❤❡♦r❡♠ ✭❈♦①✱ ❖❜➟ó❥✱ ❛♥❞ ❚♦✉③✐ ❬✾❪✮✳ ❚❤❡ st♦♣♣✐♥❣ r✉❧❡ α∗ n ✐s ❛ (µ, πn)✲❡♠❜❡❞❞✐♥❣✱ ✇✐t❤ un(snj, t, x) = −E|Bt∧σn j − x|. ✭✸✳✺✮ ▼♦r❡♦✈❡r✱ ❢♦r ❛❧❧ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈❡ f : R+→ R+✱ ✇❡ ❤❛✈❡ EP0 h Z σn n 0 f (t)dti = EP0 h Z τα∗n 1 0 f (t)dti = inf α∈A(µ,πn) EPα h Z τα 1 0 f (t)dti. ❯s✐♥❣ ❛ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❛r❣✉♠❡♥t✱ ♦♥❡ ❝❛♥ ❛❧s♦ r❡❢♦r♠✉❧❛t❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ un ✐♥ ✭✸✳✶✮ ❜② ✐♥❞✉❝t✐♦♥ ❛s ❛ ❣❧♦❜❛❧ ♠✉❧t✐♣❧❡ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② Tn 0,t t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ t❡r♠s (τ1, · · · , τn)✱ ✇❤❡r❡ ❡❛❝❤ τj✱ j = 1, · · · , n✱ ✐s ❛ F✲st♦♣♣✐♥❣ t✐♠❡ ♦♥ (Ω, F, P0) s❛t✐s❢②✐♥❣ 0 ≤ τ1 ≤ . . . ≤ τn≤ t✳ Pr♦♣♦s✐t✐♦♥ ✸✳✶✳ ❋♦r ❛❧❧ j = 1, · · · , n✱ ✇❡ ❤❛✈❡ un(snj, t, x) = sup (τ1,...,τn)∈T0,tn EhU (0, x + Bτ j) + j X k=1 δnU (snk, x + Bτj−k+1)1{τj−k+1<t} i . ✭✸✳✻✮ ✻

(7)

Pr♦♦❢✳ ❲❡ ✇✐❧❧ ✉s❡ ❛ ❜❛❝❦✇❛r❞ ✐♥❞✉❝t✐♦♥ ❛r❣✉♠❡♥t✳ ❋✐rst✱ ❧❡t ✉s ❞❡♥♦t❡ ❜② Tr,t t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ F✲st♦♣♣✐♥❣ t✐♠❡s t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ [r, t]✱ ❛♥❞ ❜② Br,x s := x + Bs− Br ❢♦r ❛❧❧ s ≥ r✳ ❚❤❡♥ ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❡①♣r❡ss✐♦♥ ✭✸✳✶✮ t❤❛t un(snj−1, t − r, x) = sup τ∈Tr,t Eun(sn j−2, t − τ, Bτr,x) + δnU (snj−1, Bτr,x)1{τ <t}  . ❯s✐♥❣ t❤❡ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ♣r✐♥❝✐♣❧❡✱ ♦♥❡ ❤❛s un(snj, t, x) = sup τ1∈T0,t E  un(snj−1, t − τ1, Bτ0,x1 ) + δ nU (sn j, B0,xτ1 )1{τ1<t}  = sup τ1∈T0,t E  ess sup τ2∈Tτ1,t E h un(snj−2, t − τ2, B τ1,Bτ10,x τ2 ) + δ nU (sn j−1, B τ1,B0,xτ1 τ2 )1{τ1<t} Fτ1 i + δnU (snj, Bτ0,x1 )1{τ1<t}  = sup (τ1,τ2)∈T0,t2 E  un(snj−2, t − τ2, Bτ0,x2 ) + j X k=j−1 δnU (snk, Bτ0,xj−k+1)1{τj−k+1<t}  . ❚♦ ❝♦♥❝❧✉❞❡✱ ✐t ✐s ❡♥♦✉❣❤ t♦ ❛♣♣❧② t❤❡ s❛♠❡ ❛r❣✉♠❡♥t t♦ ✐t❡r❛t❡ ❛♥❞ t♦ ✉s❡ t❤❡ ❢❛❝t t❤❛t un(0, t, x) = U (0, x) ❢♦r ❛♥② t ❛♥❞ x✳ ❘❡♠❛r❦ ✸✳✷✳ ❋♦r ❧❛t❡r ✉s❡s✱ ✇❡ ❛❧s♦ ♦❜s❡r✈❡ t❤❛t ✐t ✐s ♥♦t ♥❡❝❡ss❛r② t♦ r❡str✐❝t t❤❡ st♦♣♣✐♥❣ t✐♠❡s ✇✳r✳t✳ t❤❡ ❇r♦✇♥✐❛♥ ✜❧tr❛t✐♦♥✱ ✐♥ t❤❡ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠ ✭✸✳✻✮✳ ■♥ ❢❛❝t✱ ♦♥❡ ❝❛♥ ❝♦♥s✐❞❡r ❛ ❧❛r❣❡r ✜❧tr❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ✇❤✐❝❤ B ✐s st✐❧❧ ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❧❡t An t ❞❡♥♦t❡ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ st♦♣♣✐♥❣ r✉❧❡s α = (Ωα, Fα, Fα, Pα, Bα, τjα, j = 1, · · · n) s✉❝❤ t❤❛t (Ωα, Fα, Fα, Pα) ✐s ❛ ✜❧t❡r❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ Bα ❛♥❞ (τα j )j=1,··· ,n ✐s ❛ s❡q✉❡♥❝❡ ♦❢ st♦♣♣✐♥❣ t✐♠❡s s❛t✐s❢②✐♥❣ 0 ≤ τ1α≤ . . . ≤ τnα ≤ t✳ ❚❤❡♥ ♦♥❡ ❤❛s un(snj, t, x) = sup α∈An t EPαhU (0, x + Bταα j) + j X k=1 δnU (snk, x + Bατα j−k+1)1{τ α j−k+1<t} i . ❚❤✐s ❡q✉✐✈❛❧❡♥❝❡ ✐s st❛♥❞❛r❞ ❛♥❞ ✈❡r② ✇❡❧❧ ❦♥♦✇♥ ✐♥ ❝❛s❡ n = 1✱ s❡❡ ❛❧s♦ ▲❡♠♠❛ ✹✳✾ ✵❢ ❬✶✹❪ ❢♦r t❤❡ ♠✉❧t✐♣❧❡ st♦♣♣✐♥❣ ♣r♦❜❧❡♠ ✇❤❡r❡ n ≥ 1✳ ✸✳✷ ❚❤❡ ❘♦♦t s♦❧✉t✐♦♥ ❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s ✭❚❤❡♦r❡♠ ✶✳(i)✮ ■♥ ❑ä❧❧❜❧❛❞✱ ❚❛♥✱ ❛♥❞ ❚♦✉③✐ ❬✷✶❪✱ ✐t ✐s s❤♦✇♥ t❤❛t t❤❡ s❡q✉❡♥❝❡ ♦❢ ❘♦♦t st♦♣♣✐♥❣ t✐♠❡s (σn 1, · · · , σnn)n≥1 ✐s t✐❣❤t ✐♥ s♦♠❡ s❡♥s❡ ❛♥❞ ❛♥② ❧✐♠✐t ♣r♦✈✐❞❡s ❛♥ ❡♠❜❡❞❞✐♥❣ s♦❧✉t✐♦♥ ❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❧❡t (σn k)k=1,··· ,n ❜❡ t❤❡ ❘♦♦t ❡♠❜❡❞❞✐♥❣ ❣✐✈❡♥ n✲♠❛r❣✐♥❛❧s (µsnk)k=1,··· ,n ❞❡✜♥❡❞ ✐♥ ✭✸✳✸✮✱ ✇❡ ❞❡✜♥❡ α∗ n❜② ✭✸✳✹✮ ❛s ❛ (µ, πn)✲❡♠❜❡❞❞✐♥❣ ✐♥ s❡♥s❡ ♦❢ ❉❡✜♥✐t✐♦♥✷✳✶✳ ◆♦t✐❝❡ t❤❛t Pn:= Pα∗ n◦ (Bα ∗ n · , τα ∗ n · )−1 ✐s ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥ C(R+, R) × A([0, 1], R+)✱ ✇❤✐❝❤ ✐s ❛ P♦❧✐s❤ s♣❛❝❡ ✐❢ C(R+, R)✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❝♦♠♣❛❝t ❝♦♥✈❡r❣❡♥❝❡ t♦♣♦❧♦❣② ❛♥❞ A([0, 1], R+) ✼

(8)

✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ▲é✈② ♠❡tr✐❝✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ❝♦♥s✐❞❡r t❤❡ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s❡q✉❡♥❝❡ (α∗ n)n≥1✳ ❚❤❡♦r❡♠ ✶ ✐s t❤❡♥ ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❣❛t❤❡r❡❞ ❢r♦♠ s❡✈❡r❛❧ r❡s✉❧ts ✐♥ ❬✷✶❪✳ ❘❡❝❛❧❧ ❛❧s♦ t❤❛t A(µ) ❛♥❞ A(µ, πn) ❛r❡ ❞❡✜♥❡❞ ✐♥ ❉❡✜♥✐t✐♦♥✷✳✶✳ Pr♦♣♦s✐t✐♦♥ ✸✳✸✳ ▲❡t (πn)n≥1 ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ ♣❛rt✐t✐♦♥s ♦❢ [0, 1] ✇✐t❤ ♠❡s❤ |πn| → 0✱ ❛♥❞ ❧❡t α∗ n ❜❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠✉❧t✐♣❧❡✲♠❛r❣✐♥❛❧s ❘♦♦t ❡♠❜❡❞❞✐♥❣ ✭✸✳✹✮✳ ❚❤❡♥✱ t❤❡ s❡q✉❡♥❝❡ Bα∗n · , τα ∗ n ·  ✐s t✐❣❤t✱ ❛♥❞ ❛♥② ❧✐♠✐t α∗ ✐s ❛ ❢✉❧❧ ♠❛r❣✐♥❛❧s ❡♠❜❡❞❞✐♥❣✱ ✐✳❡✳ α∈ A(µ)✱ ✇✐t❤ Pα∗nk ◦ (Bα ∗ nk τsα∗nk )−1−→ Pα∗◦ (Bταα∗∗ s ) −1, ❢♦r ❛❧❧ s ∈ [0, 1] \ T, ❢♦r s♦♠❡ ❝♦✉♥t❛❜❧❡ s❡t T ⊂ [0, 1)✱ ❛♥❞ s♦♠❡ s✉❜s❡q✉❡♥❝❡ (nk)k≥1✱ ❛♥❞ EPα∗h Z τα∗ 1 0 f (t)dti = inf α∈A(µ) EPαh Z τα 1 0 f (t)dti, ❢♦r ❛♥② ♥♦♥✲❞❡❝r❡❛s✐♥❣ ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈❡ ❢✉♥❝t✐♦♥ f : R+→ R+✳ Pr♦♦❢✳ (i) ❚❤❡ ✜rst ✐t❡♠ ✐s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ ▲❡♠♠❛ ✹✳✺ ♦❢ ❬✷✶❪✳ (ii) ❋♦r t❤❡ s❡❝♦♥❞ ✐t❡♠✱ ✇❡ ♥♦t✐❝❡ t❤❛t Φ(ω·, θ·) := − Rθ1 0 f (t)dt ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ C(R+, R) × A([0, 1], R+) ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❛❜♦✈❡✳ ❚❤❡♥ ✐t ✐s ❡♥♦✉❣❤ t♦ ❛♣♣❧② ❚❤❡♦r❡♠ ✷✳✸ ❛♥❞ Pr♦♣♦s✐t✐♦♥ ✸✳✻ ♦❢ ❬✷✶❪ t♦ ♦❜t❛✐♥ t❤❡ ♦♣t✐♠❛❧✐t② ♦❢ α∗

✹ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠s

❛♥❞

✳(ii)

❘❡❝❛❧❧ t❤❛t u(s, t, ·) ✐s ❞❡✜♥❡❞ ✐♥ ✭✷✳✷✮ ❛s t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ Bα∗ t∧τα∗ s ❢♦r ❛♥ ❛r❜✐tr❛r② ❘♦♦t s♦❧✉t✐♦♥ α∗ ❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s✳ ❲❡ ♣r♦✈✐❞❡ ❛♥ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❛s ✇❡❧❧ ❛s ❛ P❉❊ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r t❤❡ ❢✉♥❝t✐♦♥ u ✉♥❞❡r ❆ss✉♠♣t✐♦♥ ✷✳✷✱ ✇❤✐❝❤ ❝♦♥s✐sts ✐♥ ❛ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳ ❋✉rt❤❡r✱ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ P❉❊ ✐♥❞✉❝❡s t❤❡ ✉♥✐q✉❡♥❡ss r❡s✉❧t ✐♥ ♣❛rt (ii) ♦❢ ❚❤❡♦r❡♠✶✳ ✹✳✶ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ u ❜② ❛♥ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠ ✭❚❤❡♦r❡♠ ✷✳(i)✮ ❇② ❛ s❧✐❣❤t ❛❜✉s❡ ♦❢ ♥♦t❛t✐♦♥✱ ✇❡ ❝❛♥ ❡①t❡♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ un❣✐✈❡♥ ✐♥ ✭✸✳✶✮ t♦ [0, 1]×R +×R ❜② s❡tt✐♥❣ un(s, t, x) := un(snj, t, x) ✇❤❡♥❡✈❡r s ∈ (snj−1, snj]. ❲✐t❤ At✐♥ ❉❡✜♥✐t✐♦♥ ✷✳✶✱ ✇❡ ❛❧s♦ ❞❡✜♥❡ eu ❛s ❛ ♠❛♣♣✐♥❣ ❢r♦♠ [0, 1] × R+× R t♦ R ❜② e u(s, t, x) := sup α∈At EPα  U (0, x + Bταα s) + Z s 0 ∂sU (s − k, x + Bατα k)1{τ α k<t}dk  . ✭✹✳✶✮ ❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t♦ ♣r♦✈✐❞❡ s♦♠❡ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ t❤✐s ❧✐♠✐t ❧❛✇ ❛s ✇❡❧❧ ❛s t❤❡ ❧✐♠✐t ♣r♦❜❧❡♠ ♦❢ un✸✳✶✮ ✉s❡❞ ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❘♦♦t s♦❧✉t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✹✳✶✳ ❋♦r ❛❧❧ (s, t, x)✱ ♦♥❡ ❤❛s un(s, t, x) → eu(s, t, x)❛s n → ∞✳

(9)

Pr♦♦❢✳ ❲❡ st❛rt ❜② r❡✇r✐t✐♥❣ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛ ♦❢ un(k, t, x)✐♥ ❘❡♠❛r❦✸✳✷❛s un(snj, t, x) = sup α∈An t EPα " U (0, x + Bταα j) + j X k=1 δnU (snj−k+1, x + Bταα k)1{τ α k<t} # . (i) ❋✐rst✱ ❢♦r ❛ ✜①❡❞ n ∈ N✱ ♦♥❡ ❝❛♥ s❡❡ An t ❛s ❛ s✉❜s❡t ♦❢ At ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✳ ●✐✈❡♥ α ∈ An t✱ ❛♥❞ ❛ss✉♠❡ t❤❛t s ∈ (snj−1, snj]✳ ▲❡t ✉s s❡t ˆτkα:= τiα✇❤❡♥❡✈❡r k ∈ [s−snj−i+1, s−snj−i)✳ ◆♦t✐❝❡ t❤❛t U(sn j, x + Bταα 1) − U(s, x + B α τα 1) ≤ 0✱ t❤❡♥ ✐t ❢♦❧❧♦✇s ❜② ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ t❤❛t j X k=1 δnU (snj−k+1, x + Bταα k)1{τ α k<t} ≤ U(s, x + Bταα 1) − U(s n j−1, x + Bταα 1) + j X k=2 δnU (snj−k+1, x + Bταα k)1{τ α k<t} = Z s 0 ∂sU (s − k, x + Bταˆα k)1{ˆτ α k<t}dk ❛♥❞ ✐t ❢♦❧❧♦✇s ❜② ✭✹✳✶✮ t❤❛t un(s, t, x) = un(sn j, t, x) ≤ eu(s, t, x)✳ (ii)▲❡t α ∈ At✱ ❛♥❞ ❞❡✜♥❡ αn∈ Ant ❜② ταn j := τ α sn j, ❢♦r j = 1, · · · , n. ▲❡t jn ❜❡ s✉❝❤ t❤❛t snjn ❝♦♥✈❡r❣❡s t♦ s ❛s n → ∞✱ t❤❡♥ ✐t ❢♦❧❧♦✇s t❤❛t Xn := U (0, x + Bτααn jn ) + jn X k=1 δnU (snjn−k+1, x + Bτααn k )1{τ αn k <t} −→ n→∞U (0, x + B α τα s) + Z s 0 ∂sU (s − k, x + Bταα k)1{τ α k<t}dk, ❛✳s✳ ❘❡❝❛❧❧ t❤❛t ❜② ❆ss✉♠♣t✐♦♥✷✳✷✱ t❤❡r❡ ❡①✐sts s♦♠❡ C > 0 ❛♥❞ p > 0 s✉❝❤ t❤❛t |U(0, x)| ≤ C +|x| ❛♥❞ sups∈[0,1]|∂sU (s, x)| ≤ C(1 + |x|p) ❢♦r ❛♥② x ∈ R✱ t❤❡♥ (Xn)n≥1 ✐s ✐♥ ❢❛❝t ✉♥✐❢♦r♠❧② ✐♥t❡✲ ❣r❛❜❧❡✳ ❍❡♥❝❡ ❢♦r ε > 0 s✉❝❤ t❤❛t α ✐s ε✲♦♣t✐♠❛❧ ✐♥ ✭✹✳✶✮✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s r❡♠❛r❦ ❛♥❞ ❢r♦♠ ❋❛t♦✉✬s ❧❡♠♠❛ t❤❛t lim infn→∞E[Xn] ≥ eu(s, t, x) − ε✳ ❚❤✉s✱ limn→∞un(s, t, x) ≥

e

u(s, t, x).❍❡♥❝❡ t❤✐s ♣r♦✈❡s t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ ❤♦❧❞s✿ limn→∞un(s, t, x) = eu(s, t, x)✳

▲❡♠♠❛ ✹✳✷✳ ❚❤❡ ❢✉♥❝t✐♦♥ eu(s, t, x) ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❛♥❞ ❧♦❝❛❧❧② ▲✐♣s❝❤✐t③ ✐♥ s✱ ❛♥❞ ✐s ✉♥✐✲ ❢♦r♠❧② ▲✐♣s❝❤✐t③ ✐♥ x ❛♥❞ ✉♥✐❢♦r♠❧② 1 2✲❍ö❧❞❡r ✐♥ t✳ Pr♦♦❢✳ ❋✐rst✱ ✉s✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛ ♦❢ un✐♥ ✭✸✳✺✮ ❛♥❞ ♥♦t✐❝✐♥❣ t❤❛t y 7→ |y−x| ✐s ❝♦♥✈❡①✱ ✇❡ s❡❡ t❤❛t s 7→ un(s, t, x)✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣✳ ❋✉rt❤❡r✱ ✉s✐♥❣ ✭✸✳✶✮✱ ✐t ❢♦❧❧♦✇s ✐♠♠❡❞✐❛t❡❧② t❤❛t un(snj, t, x) − un(snj−1, t, x) ≥ U(snj, x) − U(snj−1, x). ❚❤❡♥ ✉♥❞❡r ❆ss✉♠♣t✐♦♥ ✷✳✷✱ ♦♥❡ ❤❛s 0 ≥ ∂sun(s, t, x) ≥ −C(1 + |x|p) ❢♦r s♦♠❡ ❝♦♥st❛♥t C > 0 ❛♥❞ p > 0 ✐♥❞❡♣❡♥❞❡♥t ♦❢ n✳ ❇② t❤❡ ❧✐♠✐t r❡s✉❧t un → eu✱ ✐t ❢♦❧❧♦✇s t❤❛t eu(s, t, x) ✐s ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❛♥❞ ❧♦❝❛❧❧② ▲✐♣s❝❤✐t③ ✐♥ s✳ ❋✐♥❛❧❧②✱ ✉s✐♥❣ ❛❣❛✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠✉❧❛ ♦❢ un ✐♥ ✭✸✳✺✮✱ ✐t ✐s ❡❛s② t♦ ❞❡❞✉❝❡ t❤❛t un(k, t, x) ✐s ✉♥✐❢♦r♠❧② ▲✐♣s❝❤✐t③ ✐♥ x ❛♥❞ 1/2✲❍ö❧❞❡r ✐♥ t✱ ✉♥✐❢♦r♠❧② ✐♥ n✳ ❆s ❧✐♠✐t ♦❢ un✱ ✐t ❢♦❧❧♦✇s t❤❛t eu ✐s ❛❧s♦ ✉♥✐❢♦r♠❧② ▲✐♣s❝❤✐t③ ✐♥ x ❛♥❞ 1/2✲❍ö❧❞❡r ✐♥ t✳ ✾

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❲❡ ♥❡①t s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ u ❞❡✜♥❡❞ ❜② ✭✷✳✷✮ ✐s ❛❧s♦ t❤❡ ❧✐♠✐t ♦❢ un✱ ✇❤✐❝❤ ❧❡❛❞s t♦ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ u ❛♥❞ eu✱ ❛♥❞ t❤❡♥ ❚❤❡♦r❡♠✷✳(i) r❡❛❞✐❧② ❢♦❧❧♦✇s✳ Pr♦♣♦s✐t✐♦♥ ✹✳✸✳ ❋♦r ❛❧❧ (s, t, x) ∈ [0, 1] × R+× R✱ ♦♥❡ ❤❛s u(s, t, x) = eu(s, t, x)✳ Pr♦♦❢✳ ❇② ❚❤❡♦r❡♠ ✸✳✶ ♦❢ ❬✾❪ t♦❣❡t❤❡r ✇✐t❤ ♦✉r ❡①t❡♥❞❡❞ ❞❡✜♥✐t✐♦♥ ✐♥ ✭✸✳✹✮✱ un(s, t, x) = −EPα∗nh Bα∗n t∧τα∗n s −x i .▼♦r❡♦✈❡r✱ ❜② Pr♦♣♦s✐t✐♦♥✸✳✸✱ t❤❡r❡ ❡①✐sts ❛ ❝♦✉♥t❛❜❧❡ s❡t T ⊂ [0, 1) ❛♥❞ ❛ s✉❜s❡q✉❡♥❝❡ (nk)k≥1s✉❝❤ t❤❛t Pα ∗ nk◦(Bα ∗ nk τsα∗nk )−1 → Pα∗ ◦(Bα∗ τα∗ s ) −1✳ ❆s E[max 0≤r≤t|Br−x|  < ∞ ❢♦r ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ✐t ❢♦❧❧♦✇s t❤❛t EPα∗nk Bα∗nk t∧τsα∗nk − x  −→ EPα∗ Bα∗ t∧τα∗ s − x ✱ ❢♦r ❛❧❧ s ∈ [0, 1] \ T✳ ❍❡♥❝❡✱ un(s, t, x) −→ u(s, t, x), ❢♦r ❛❧❧ s ∈ [0, 1] \ T. ❋✉rt❤❡r✱ ❜② t❤❡ r✐❣❤t✲❝♦♥t✐♥✉✐t② ♦❢ s 7→ τα∗ s ✱ ✐t ✐s ❡❛s② t♦ ❞❡❞✉❝❡ t❤❛t s 7→ u(s, t, x) := −EPα∗ |Bα∗ t∧τα∗ s − x| ✐s r✐❣❤t✲❝♦♥t✐♥✉♦✉s✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✇❡ ❦♥♦✇ ❢r♦♠ Pr♦♣♦s✐t✐♦♥ ✹✳✶ t❤❛t un(s, t, x) → eu(s, t, x)✱ ❛♥❞ ❢r♦♠ ▲❡♠♠❛✹✳✷t❤❛t eu ✐s ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✐♥ ❛❧❧ ❛r❣✉♠❡♥ts✱ ✐t ❢♦❧❧♦✇s t❤❛t u(s, t, x) = eu(s, t, x) ❤♦❧❞s ❢♦r ❛❧❧ (s, t, x) ∈ [0, 1] × R+× R✳ ❘❡♠❛r❦ ✹✳✹✳ ❋♦r♠❛❧❧②✱ ✇❡ ❝❛♥ ✉♥❞❡rst❛♥❞ t❤❡ ❛❜♦✈❡ r❡s✉❧t ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉✐✈❛❧❡♥t ✇❛②✳ ❚❤❡ n✲♠❛r❣✐♥❛❧s ❘♦♦t s♦❧✉t✐♦♥ (σn, B)❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ ❛ ❢✉❧❧ ♠❛r❣✐♥❛❧ ❘♦♦t s♦❧✉t✐♦♥ (σ, B) ✇❤✐❝❤ t❤❡♥ s❛t✐s✜❡s✿ u(s, t, x) = −E Bt∧σs− x . ✹✳✷ P❉❊ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ u ✭❚❤❡♦r❡♠ ✷✳(ii)✮ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳(ii)✳ ❙t❡♣ ✶✳ ❲❡ ✜rst ♥♦t✐❝❡ t❤❛t t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ u(s, t, x) ✐♥ (s, t, x) ❢♦❧❧♦✇s ❞✐r❡❝t❧② ❜② ▲❡♠♠❛✹✳✷❛♥❞ Pr♦♣♦s✐t✐♦♥✹✳✸✳ ❙t❡♣ ✷✳ ■♥ ❛ ✜❧t❡r❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, F, F, P) ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ W ✱ ✇❡ ❞❡♥♦t❡ ❜② Ut t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ F✲♣r❡❞✐❝t❛❜❧❡ ♣r♦❝❡ss❡s γ = (γr)r≥0 s✉❝❤ t❤❛t R01γr2dr ≤ t✳ ●✐✈❡♥ ❛ ❝♦♥tr♦❧ ♣r♦❝❡ss γ✱ ✇❡ ❞❡✜♥❡ t✇♦ ❝♦♥tr♦❧❧❡❞ ♣r♦❝❡ss❡s Xγ ❛♥❞ Yγ ❜② Xsγ := x + Z s 0 γrdWr, Ysγ := Z s 0 γr2dr. ❇② ❛ t✐♠❡ ❝❤❛♥❣❡ ❛r❣✉♠❡♥t✱ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t u(s, t, x) = sup γ∈Ut EhU (0, Xsγ) + Z s 0 ∂sU (s − k, Xkγ)1{Ykγ<t}dk i . ✭✹✳✷✮ ■♥❞❡❡❞✱ ❣✐✈❡♥ γ ∈ Ut✱ ♦♥❡ ♦❜t❛✐♥s ❛ sq✉❛r❡ ✐♥t❡❣r❛❜❧❡ ♠❛rt✐♥❣❛❧❡ Xγ ✇❤✐❝❤ ❤❛s t❤❡ r❡♣r❡s❡♥t❛✲ t✐♦♥ Xγ s = WYsγ✱ ✇❤❡r❡ W ✐s ❛ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❛♥❞ Y γ s ❛r❡ ❛❧❧ st♦♣♣✐♥❣ t✐♠❡s✱ ❛♥❞ ✐t ✐♥❞✉❝❡s ❛ st♦♣♣✐♥❣ r✉❧❡ ✐♥ At✳ ❇② ✭✹✳✶✮ ❛♥❞ Pr♦♣♦s✐t✐♦♥✹✳✸✱ ✐t ❢♦❧❧♦✇s t❤❛t ✐♥ ✭✹✳✷✮✱ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ✐s ❧❛r❣❡r t❤❛♥ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❣✐✈❡♥ ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ st♦♣♣✐♥❣ t✐♠❡s (τ1, · · · , τn) ∈ T0,tn✱ ✇❡ ❞❡✜♥❡ Γs := τj∨ s−sn j sn j+1−s∧ τj+1 ❢♦r ❛❧❧ s ∈ [s n j, snj+1)✳ ◆♦t✐❝❡ t❤❛t ✶✵

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s 7→ Γs ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ❛♥❞ Γsn j = τj ❢♦r ❛❧❧ j = 1, · · · , n✳ ❚❤❡♥ ♦♥❡ ❝❛♥ ❝♦♥str✉❝t ❛ ♣r❡❞✐❝t❛❜❧❡ ♣r♦❝❡ss γ s✉❝❤ t❤❛t P0 Z · 0 γrdBr, Z · 0 γr2dt −1 = P0◦  BΓ·, Γ· −1 . ❯s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ un✐♥ ✭✸✳✻✮ ❛♥❞ ✐ts ❝♦♥✈❡r❣❡♥❝❡ ✐♥ Pr♦♣♦s✐t✐♦♥ ✹✳✶✱ ♦♥❡ ♦❜t❛✐♥s t❤❛t ✐♥ ✭✹✳✷✮✱ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✐s ❧❛r❣❡r t❤❛♥ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✳ ❚❤❡ ❛❜♦✈❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ s❛t✐s✜❡s t❤❡ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ♣r✐♥❝✐♣❧❡ ✭s❡❡ ❡✳❣✳ ❬✶✶❪✮✿ ❢♦r ❛ ❢❛♠✐❧② ♦❢ st♦♣♣✐♥❣ t✐♠❡s (τγ) γ∈Ut ❞♦♠✐♥❛t❡❞ ❜② s✱ ♦♥❡ ❤❛s u(s, t, x) = sup γ∈Ut Ehu s − τγ, t − Yτγγ, Xτγγ  + Z τγ 0 ∂sU (s − k, Xτγγ  1{Ykγ<t}dk i . ✭✹✳✸✮ ❙t❡♣ ✸ ✭s✉♣❡rs♦❧✉t✐♦♥✮✳ ▲❡t z = (s, t, x) ∈ int(Z) ❜❡ ✜①❡❞✱ ❛♥❞ ϕ ∈ C2(Z) ❜❡ s✉❝❤ t❤❛t 0 = (u − ϕ)(z) = minz′∈Z(u − ϕ)✳ ❇② ❛ s❧✐❣❤t ❛❜✉s❡ ♦❢ ♥♦t❛t✐♦♥✱ ❞❡♥♦t❡ ❜② ∂sU (z′) t❤❡ q✉❛♥t✐t② ∂sU (s′, x′) ❢♦r ❛♥② z′ = (s′, t′, x′)✳ ❚❤❡♥ ❜② ✭✹✳✸✮✱ ❢♦r ❛♥② ❢❛♠✐❧② ♦❢ st♦♣♣✐♥❣ t✐♠❡s (τγ) γ∈Ut ❞♦♠✐♥❛t❡❞ ❜② s✱ ♦♥❡ ❤❛s✱ sup γ∈Ut E h Z τγ 0 − ∂ sϕ(Zk) + ∂sU (Zk)1{Ykγ<t}  dk + Z τγ 0 γk2 − ∂tϕ + 1 2∂ 2 xxϕ)(Zk)dk i ≤ 0, ✇❤❡r❡ Zk := (s − k, t − Ykγ, Xkγ) ❛♥❞ X0γ= x✳ ❈❤♦♦s✐♥❣ γ·≡ 0 ❛♥❞ τγ ≡ h ❢♦r h s♠❛❧❧ ❡♥♦✉❣❤✱ ✇❡ ❣❡t − ∂sϕ + ∂sU(z) ≤ 0. ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❝❤♦♦s✐♥❣ γ· ≡ γ0 ❢♦r s♦♠❡ ❝♦♥st❛♥t γ0 ❛♥❞ τγ := inf{k ≥ 0 : |Xkγ− x| + |Ykγ| ≥ h}✱ t❤❡♥ ❜② ❧❡tt✐♥❣ γ0 ❜❡ ❧❛r❣❡ ❡♥♦✉❣❤ ❛♥❞ h ❜❡ s♠❛❧❧ ❡♥♦✉❣❤✱ ♦♥❡ ❝❛♥ ❞❡❞✉❝❡ t❤❛t − ∂tϕ + 1 2∂ 2 xxϕ  (s, t, x) ≤ 0. ❙t❡♣ ✹ ✭s✉❜s♦❧✉t✐♦♥✮✳ ❆ss✉♠❡ t❤❛t u ✐s ♥♦t ❛ ✈✐s❝♦s✐t② s✉❜✲s♦❧✉t✐♦♥✱ t❤❡♥ t❤❡r❡ ❡①✐sts z = (s, t, x) ∈ ✐♥t(Z) ❛♥❞ ϕ ∈ C2(Z)✱ s✉❝❤ t❤❛t 0 = (u − ϕ)(z) = maxz′∈Z(u − ϕ)(z′)✱ ❛♥❞ min∂tϕ − 1 2∂ 2 xxϕ, ∂s(ϕ − U) (s, t, x) > 0. ❇② ❝♦♥t✐♥✉✐t② ♦❢ u ❛♥❞ ϕ✱ ✇❡ ♠❛② ✜♥❞ R > 0 s✉❝❤ t❤❛t min∂tϕ − 1 2∂ 2 xxϕ, ∂s(ϕ − U) ≥ 0, ♦♥ BR(z), ✭✹✳✹✮ ✇❤❡r❡ BR(z) ✐s t❤❡ ♦♣❡♥ ❜❛❧❧ ✇✐t❤ r❛❞✐✉s R ❛♥❞ ❝❡♥t❡r z✳ ▲❡t τγ := inf{k : Zkγ ∈/ BR(z) ♦r Ykγ ≥ t}✱ ❛♥❞ ♥♦t✐❝❡ t❤❛t max∂BR(s,t,x)(u − ϕ) = −η < 0✱ ❜② t❤❡ str✐❝t ♠❛①✐♠❛❧✐t② ♣r♦♣❡rt②✳ ❚❤❡♥ ✐t ❢♦❧❧♦✇s ❢r♦♠ ✭✹✳✸✮ t❤❛t 0 = sup γ E h u s − τγ, t − Yτγγ, Xτγγ  − u(s, t, x) + Z τγ 0 ∂sU (s − k, Xτγγ  1{Ykγ<t}dk i ≤ −η + sup γ Eh Z τγ 0  − ∂s(ϕ − U1{Yγ k<t}) − (∂tϕ − 1 2∂ 2 xxϕ)γk2  (Zk)dk i ≤ −η, ✇❤❡r❡ t❤❡ ❧❛st ✐♥❡q✉❛❧✐t② ❢♦❧❧♦✇s ❜② ✭✹✳✹✮✳ ❚❤✐s ✐s t❤❡ r❡q✉✐r❡❞ ❝♦♥tr❛❞✐❝t✐♦♥✳ ✶✶

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✹✳✸ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ♣r✐♥❝✐♣❧❡ ♦❢ t❤❡ P❉❊ ✭❚❤❡♦r❡♠s ✷✳(iii) ❛♥❞ ✶✳(ii)✮ ❘❡❝❛❧❧ t❤❛t t❤❡ ♦♣❡r❛t♦r F ✐s ❞❡✜♥❡❞ ✐♥ ✭✷✳✹✮ ❛♥❞ ✇❡ ✇✐❧❧ st✉❞② t❤❡ P❉❊ ✭✷✳✸✮✳ ❋♦r ❛♥② η ≥ 0✱ ❛ ❧♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ w : Z → R ✐s ❝❛❧❧❡❞ ❛♥ η✲str✐❝t ✈✐s❝♦s✐t② s✉♣❡rs♦❧✉t✐♦♥ ♦❢ ✭✷✳✸✮ ✐❢ w|s=0≥ η + U(0, ·)✱ w|t=0 ≥ η + U ❛♥❞ F (Dϕ, D2ϕ)(z0) ≥ η ❢♦r ❛❧❧ (z0, ϕ) ∈ ✐♥t(Z) × C2(Z) s❛t✐s❢②✐♥❣ (w − ϕ)(z0) = minz∈Z(w − ϕ)(z)✳ Pr♦♣♦s✐t✐♦♥ ✹✳✺ ✭❈♦♠♣❛r✐s♦♥✮✳ ▲❡t v ✭r❡s♣✳ w✮ ❜❡ ❛♥ ✉♣♣❡r ✭r❡s♣✳ ❧♦✇❡r✮ s❡♠✐❝♦♥t✐♥✉♦✉s ✈✐s❝♦s✐t② s✉❜s♦❧✉t✐♦♥ ✭r❡s♣✳ s✉♣❡rs♦❧✉t✐♦♥✮ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✷✳✸✮ s❛t✐s❢②✐♥❣ v(z) ≤ C(1 + t + |x|) ❛♥❞ w(z) ≥ −C(1 + t + |x|), z ∈ Z, ❢♦r s♦♠❡ ❝♦♥st❛♥t C > 0. ❚❤❡♥ v ≤ w ♦♥ Z✳ Pr♦♦❢✳ ❲❡ ♣r♦❝❡❡❞ ✐♥ t❤r❡❡ st❡♣s✳ (i) ■♥ t❤✐s st❡♣✱ ✇❡ ♣r♦✈❡ t❤❡ r❡s✉❧t ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❝♦♠♣❛r✐s♦♥ r❡s✉❧t ❤♦❧❞s tr✉❡ ✐❢ t❤❡ s✉♣❡rs♦❧✉t✐♦♥ ✐s η−str✐❝t ❢♦r s♦♠❡ η > 0✳ ❋✐rst✱ ❞✐r❡❝t ✈❡r✐✜❝❛t✐♦♥ r❡✈❡❛❧s t❤❛t t❤❡ ❢✉♥❝t✐♦♥✿ w1(s, t, x) := U (s, x) + η(1 + s + t), (s, t, x) ∈ Z, ✐s ❛♥ η−str✐❝t s✉♣❡rs♦❧✉t✐♦♥✳ ❋♦r ❛❧❧ µ ∈ (0, 1)✱ ✇❡ ❝❧❛✐♠ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ wµ:= (1−µ)w+µw1 ✐s ❛ µη−str✐❝t ✈✐s❝♦s✐t② s✉♣❡rs♦❧✉t✐♦♥✳ ■♥❞❡❡❞✱ t❤✐s ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ❆✳✸ ✭♣✳✺✷✮ ♦❢ ❇❛r❧❡s ❛♥❞ ❏❛❦♦❜s❡♥ ❬✷❪✱ ✇❤✐❝❤ s❤♦✇s t❤❛t wµ ✐s ❛ ✈✐s❝♦s✐t② s✉♣❡rs♦❧✉t✐♦♥ ♦❢ ❜♦t❤ ❧✐♥❡❛r ❡q✉❛t✐♦♥s✿ ∂twµ− 1 2∂ 2 xxwµ≥ µη ❛♥❞ ∂swµ− ∂sU ≥ µη. ❆ss✉♠❡ t❤❛t t❤❡ ❝♦♠♣❛r✐s♦♥ ♣r✐♥❝✐♣❧❡ ❤♦❧❞s tr✉❡ ✐❢ t❤❡ s✉♣❡rs♦❧✉t✐♦♥ ✐s str✐❝t✱ t❤❡♥ ✐t ❢♦❧❧♦✇s t❤❛t v ≤ wµ ♦♥ Z✳ ▲❡t µ ց 0✱ ✇❡ ♦❜t❛✐♥ v ≤ w ♦♥ Z✳ (ii) ■♥ ✈✐❡✇ ♦❢ t❤❡ ♣r❡✈✐♦✉s st❡♣✱ ✇❡ ♠❛② ❛ss✉♠❡ ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t② t❤❛t w ✐s ❛♥ η−str✐❝t s✉♣❡rs♦❧✉t✐♦♥✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ ❝♦♠♣❛r✐s♦♥ r❡s✉❧t ✐♥ t❤✐s s❡tt✐♥❣✱ ✇❡ ❛ss✉♠❡ t♦ t❤❡ ❝♦♥tr❛r② t❤❛t δ := (v − w)(ˆz) > 0, ❢♦r s♦♠❡ ˆz ∈ Z, ✭✹✳✺✮ ❛♥❞ ✇❡ ✇♦r❦ t♦✇❛r❞ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❋♦❧❧♦✇✐♥❣ t❤❡ st❛♥❞❛r❞ ❞♦✉❜❧✐♥❣ ✈❛r✐❛❜❧❡s t❡❝❤♥✐q✉❡✱ ✇❡ ✐♥tr♦❞✉❝❡ ❢♦r ❛r❜✐tr❛r② α, ε > 0✿ Φα,ε(z, z′) := α 2 z − z′ 2 + ε ϕ(z) + ϕ(z′), ✇✐t❤ ϕ(z) := ln (1 − s) + 1 2 t 2+ x2, z, z∈ Z, ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❛①✐♠✉♠ Mα,ε:= sup (z,z′)∈Z×Z  v(z) − w(z′) − Φα,ε(z, z′) ≥ δ − 2εϕ(ˆz) > 0, ✭✹✳✻✮ ❜② ✭✹✳✺✮✱ ❢♦r s✉✣❝✐❡♥t❧② s♠❛❧❧ ε > 0✳ ❆❧s♦✱ r❡❝❛❧❧✐♥❣ t❤❛t ❜♦t❤ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥s U ❛♥❞ UN(0, 1) ❤❛✈❡ ❧✐♥❡❛r ❣r♦✇t❤ ✐♥ x✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❜♦✉♥❞s ♦♥ v ❛♥❞ w t❤❛t t❤❡ ❛❜♦✈❡ ✶✷

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s✉♣r❡♠✉♠ ♠❛② ❜❡ ❝♦♥✜♥❡❞ t♦ ❛ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ Z × Z✳ ❚❤❡♥ t❤❡ ✉♣♣❡r s❡♠✐❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐♠♣❧✐❡s t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠✐♥✐♠✐③❡r (zα,ε, z′α,ε) ∈ Z × Z✱ ✐✳❡✳ Mα,ε = v zα,ε− w z′α,ε α 2 zα,ε − z′α,ε 2− ε ϕ(zα,ε) + ϕ(z′α,ε), ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥✈❡r❣✐♥❣ s✉❜s❡q✉❡♥❝❡ zε n, z′εn  := zεαn, z ′ε αn  −→ (zε, z′ε) ∈ Z × Z✱ ❢♦r s♦♠❡ (αn)n ❝♦♥✈❡r❣✐♥❣ t♦ ∞✳ ▼♦r❡♦✈❡r✱ ❞❡♥♦t✐♥❣ ❜② z∗ ❛♥② ♠✐♥✐♠✐③❡r ♦❢ v − w − 2εϕ✱ ✇❡ ♦❜t❛✐♥ ❢r♦♠ t❤❡ ✐♥❡q✉❛❧✐t② (v − w − 2εϕ)(z∗) ≤ Mαn,ε t❤❛t ℓ := lim sup n→∞ α 2 zε n− z′εn 2 ≤ lim sup n→∞ v(z ε n) − w(z′εn) − ε ϕ(znε) − ϕ(z′εn)  − (v − w − 2εϕ)(z∗) ≤ v(zε) − w(z′ε) − ε ϕ(zε) − ϕ(z′ε)− (v − w − 2εϕ)(z) < ∞. ❚❤❡♥ zε= z′ε✱ ❛♥❞ 0 ≤ ℓ ≤ (v − w − 2εϕ)(zε) − (v − w − 2εϕ)(z) ≤ 0 ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ z ❈♦♥s❡q✉❡♥t❧②✿ zε = z′ε, αn zε n− z′εn 2 −→ 0, ❛♥❞ Mαn −→ sup Z (u − v) − 2εϕ, ❛s n → ∞. ✭✹✳✼✮ ❋✐♥❛❧❧②✱ ♦✉r ❞❡✜♥✐t✐♦♥ ♦❢ ϕ ✐♠♣❧✐❡s t❤❛t sε< 1✳ ▼♦r❡♦✈❡r✱ ❛s v ✐s ❛ s✉❜s♦❧✉t✐♦♥ ❛♥❞ w ❛ s✉♣❡r✲ s♦❧✉t✐♦♥✱ ✇❡ s❡❡ t❤❛t ✐❢ ˆz ❧✐❡s ✐♥ t❤❡ r❡♠❛✐♥✐♥❣ ♣❛rt ♦❢ ∂Z✱ ✇❡ ✇♦✉❧❞ ❤❛✈❡ lim supn→∞Mαn ≤ −2εϕ zε≤ 0✱ ✇❤✐❝❤ ✐s ✐♥ ❝♦♥tr❛❞✐❝t✐♦♥ ✇✐t❤ t❤❡ ♣♦s✐t✐✈❡ ❧♦✇❡r ❜♦✉♥❞ ✐♥ ✭✹✳✻✮✳ ❈♦♥s❡q✉❡♥t❧② zε✐s ❛♥ ✐♥t❡r✐♦r ♣♦✐♥t ♦❢ Z✱ ❛♥❞ t❤❡r❡❢♦r❡ ❜♦t❤ zε n❛♥❞ z′εn❛r❡ ✐♥t❡r✐♦r ♣♦✐♥ts ♦❢ Z ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ n✳ (iii)❲❡ ♥♦✇ ✉s❡ t❤❡ ✈✐s❝♦s✐t② ♣r♦♣❡rt✐❡s ♦❢ v ❛♥❞ w ❛t t❤❡ ✐♥t❡r✐♦r ♣♦✐♥ts znε ❛♥❞ z′εn✱ ❢♦r ❧❛r❣❡ n✳ ❇② t❤❡ ❈r❛♥❞❛❧❧✲■s❤✐✐ ▲❡♠♠❛✱ s❡❡ ❡✳❣✳ ❈r❛♥❞❛❧❧✱ ■s❤✐✐✱ ❛♥❞ ▲✐♦♥s ❬✶✵❪✱ ✇❡ ♠❛② ✜♥❞ ❢♦r ❡❛❝❤ s✉❝❤ n t✇♦ ♣❛✐rs (pε n, Aεn) ❛♥❞ (qnε, Bnε) ✐♥ R3× S3✱ s✉❝❤ t❤❛t pεn+ DΦα,ε(znε), Aεn+ D2Φα,ε(znε) ∈ Jw(znε), qεn− DΦα,ε(z′εn), Bnε− D2Φα,ε(z′εn) ∈ Jv(z′εn), pεn= qnε = αn(zεn− z′εn) ❛♥❞ Aεn≤ Bnε, ✇❤❡r❡ J ❛♥❞ J ❞❡♥♦t❡ t❤❡ s❡❝♦♥❞ ♦r❞❡r s✉♣❡r ❛♥❞ s✉❜❥❡ts✱ s❡❡ ❬✶✵❪✳ ❚❤❡♥✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ s✉❜s♦❧✉t✐♦♥ ♣r♦♣❡rt② ♦❢ v ❛♥❞ t❤❡ η−str✐❝t s✉♣❡rs♦❧✉t✐♦♥ ♦❢ w t❤❛t minnαn(tεn− t′εn) + εtεn− 1 2(A ε 3,3,n+ ε), αn(sεn− s′εn) + ε 1 − sε n − ∂ sU (sεn, xεn) o ≤ 0 ≤ −η + minnαn(tεn− t′εn) − εt′εn− 1 2(B ε 3,3,n− ε), αn(sεn− s′εn) − ε 1 − s′ε n− ∂ sU (s′εn, x′εn) o ≤ −η + minnαn(tεn− t′εn) − εt′εn− 1 2(A ε 3,3,n− ε), αn(sεn− s′εn) − ε 1 − s′ε n − ∂ sU (s′εn, x′εn) o , ❜② t❤❡ ✐♥❡q✉❛❧✐t② Aε n≤ Bεn✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t 0 ≤ −η − ε(tεn+ t′εn) + 2ε − ε 1 − sε n − ε 1 − s′ε n + ∂sU (s′εn, x′εn) − ∂sU (s′εn, x′εn) ≤ −η + 2ε + ∂sU (sεn, xnε) − ∂sU (s′εn, x′εn) −→ −η + 2ε, ❛s n → ∞, ✇❤✐❝❤ ♣r♦✈✐❞❡s t❤❡ r❡q✉✐r❡❞ ❝♦♥tr❛❞✐❝t✐♦♥ ❢♦r s✉✣❝✐❡♥t❧② s♠❛❧❧ ε > 0✳ ✶✸

(14)

❘❡♠❛r❦ ✹✳✻✳ ❚♦ ❝♦♥❝❧✉❞❡ t❤❡ ♣r♦♦❢s ♦❢ ❚❤❡♦r❡♠ ✷ (iii) ❛s ✇❡❧❧ ❛s ❚❤❡♦r❡♠ ✶ (ii)✱ ✇❡ ♥♦t✐❝❡ t❤❛t t❤❡ ❝♦♠♣❛r✐s♦♥ r❡s✉❧t ✐♥ Pr♦♣♦s✐t✐♦♥ ✹✳✺✐♥❞✉❝❡s ✐♠♠❡❞✐❛t❡❧② t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ P❉❊ ✭✷✳✸✮ ✐♥ ♣❛rt (iii) ♦❢ ❚❤❡♦r❡♠ ✷✳ ❋✉rt❤❡r✱ t❤✐s ✐♠♣❧✐❡s ❛❧s♦ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ♦❢ Bα∗ τα∗ s ∧t ❢♦r ❛❧❧ s ∈ [0, 1] ❛♥❞ t ≥ 0✱ ❛♥❞ ❤❡♥❝❡ t❤❡ ✉♥✐q✉❡♥❡ss ♦❢ ❧❛✇ ♦❢ B α∗ τα∗ s ∧t ✐♥ ♣❛rt (ii) ♦❢ ❚❤❡♦r❡♠ ✶✳

✺ ▼♦r❡ ❞✐s❝✉ss✐♦♥s

❘❡❝❛❧❧ t❤❛t ✐♥ t❤❡ ❝❛s❡ ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② ♠❛r❣✐♥❛❧s (µsn j)j=1,··· ,n✱ t❤❡ ❘♦♦t st♦♣♣✐♥❣ t✐♠❡s {σn j}j=1..n ❛r❡ ❞❡✜♥❡❞ s✉❝❝❡ss✐✈❡❧② ❛s ❤✐tt✐♥❣ t✐♠❡s ♦❢ ❜❛rr✐❡rs✱ t❤❛t ✐s σjn:= inft ≥ σj−1n : (t, Bt) ∈ Rnj , ✇✐t❤ ❜❛rr✐❡rs Rn j ❞❡✜♥❡❞ ❜② Rnj = {(t, x) : δnun(snj, t, x) = δnU (snj, x)}. ■♥ ❝♦♥s❡q✉❡♥❝❡✱ ♦♥❡ ❤❛s ❢♦r ❛♥② n ≥ 1✱ δnun(snj, σjn, Bσn j) = δ nU (sn j, Bσn j), ∀j = 1, . . . , n. ❉❡♥♦t❡ ❢♦r s✐♠♣❧✐❝✐t② ❛ ❘♦♦t s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❊P ❣✐✈❡♥ ❢✉❧❧ ♠❛r❣✐♥❛❧s ❜② (σ∞ s )s∈[0,1]✳ ❆s✲ s✉♠❡ t❤❛t t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ∂su(s, t, x)❡①✐sts ❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s✱ t❤❡♥ ♦♥❡ ♠❛② ♥❛t✉r❛❧❧② ❡①♣❡❝t t♦ ❤❛✈❡ Z t 0 ∂su(s, σ∞s , Bσ∞ s )ds = Z t 0 ∂sU (s, Bσ∞ s )ds, ❢♦r ❛❧❧ t ∈ [0, 1]. ◆❡✈❡rt❤❡❧❡ss✱ ✐t ✐s ♥♦t ❡❛s② t♦ ❢♦r♠✉❧❛t❡ ❛ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ♦♥ U t♦ ❡♥s✉r❡ t❤❛t ∂su(s, t, x) ✐s ✇❡❧❧✲❞❡✜♥❡❞✱ ❛s u ✐s ♦♥❧② t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ♦❢ ❛♥ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ ♣r♦❜❧❡♠✳ ❲❡ ❝♦✉❧❞ ❛❧s♦ ❡①♣❡❝t t♦ ❞❡✜♥❡ t❤❡ ❧✐♠✐t ❘♦♦t s♦❧✉t✐♦♥ σ∞ s ❛s ❛ ❤✐tt✐♥❣ t✐♠❡ s✉❝❤ t❤❛t σs= inf {t ≥ σs−: (t, Wt) ∈ Rs} , ✭✺✳✶✮ ❢♦r ❜❛rr✐❡rs R = {Rs}s∈[0,1] ❞❡✜♥❡❞ ❜② Rs := {(t, x) : ∂su(s, t, x) = ∂sU (s, x)} . ❇✉t ❛❣❛✐♥ ❤❡r❡✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ∂su(s, t, x) ✐s ♥♦t ❝❧❡❛r✳ ▼♦r❡♦✈❡r✱ ❛s t❤❡ ♥✉♠❜❡r ♦❢ ♠❛r❣✐♥❛❧s ✐s ♥♦t ❝♦✉♥t❛❜❧❡ ✐♥ t❤❡ ❢✉❧❧ ♠❛r❣✐♥❛❧s ❝❛s❡✱ t❤❡ ❡q✉❛t✐♦♥ ✭✺✳✶✮ ❝❛♥♥♦t ♣r♦✈✐❞❡ ❛ ❞❡✜♥✐t✐♦♥ ❢♦r ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ❢❛♠✐❧② ♦❢ st♦♣♣✐♥❣ t✐♠❡s✳ ❋✐♥❛❧❧②✱ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ❞✉❛❧ ♣r♦❜❧❡♠ ♦❢ t❤❡ ♦♣t✐♠❛❧ ❙❊P ❤❛s ❜❡❡♥ ♣r♦✈✐❞❡❞ ✐♥ ❬✾❪✳ ■t ✐s ❛❧s♦ ✐♥t❡r❡st✐♥❣ t♦ ❧♦♦❦ ❛t t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❞✉❛❧ s♦❧✉t✐♦♥s✳ ◆❡✈❡rt❤❡❧❡ss✱ ❛s t❤❡ ❞✉❛❧ s♦❧✉t✐♦♥ ❛r❡ ♦♥❧② ❞❡✜♥❡❞ ✐♥ ❛♥ ✐♥❞✉❝t✐✈❡ ✇❛② ✉s✐♥❣ t❤❡ ❜❛rr✐❡rs (Rn k)k=1,··· ,n✱ ✐t ✐s ♥♦t ❝❧❡❛r ❤♦✇ t♦ ✜❣✉r❡ ♦✉t t❤❡ ❧✐♠✐t ❜❛rr✐❡rs ❛♥❞ t❤❡ ❧✐♠✐t ❞✉❛❧ s♦❧✉t✐♦♥s✳ ✶✹

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