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❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❡❛♥ ❋✐❡❧❞ ❚②♣❡ ❈♦♥tr♦❧ ❚❤❡♦r②

❆❧❛✐♥ ❇❡♥s♦✉ss❛♥✶✱ P✳ ❏❛♠❡s♦♥ ●r❛❜❡r✷✱ ❙✳ ❈✳ P❤✐❧❧✐♣ ❨❛♠✸

✶■♥t❡r♥❛t✐♦♥❛❧ ❈❡♥t❡r ❢♦r ❉❡❝✐s✐♦♥ ❛♥❞ ❘✐s❦ ❆♥❛❧②s✐s

❏✐♥❞❛❧ ❙❝❤♦♦❧ ♦❢ ▼❛♥❛❣❡♠❡♥t✱ ❯♥✐✈❡rs✐t② ♦❢ ❚❡①❛s ❛t ❉❛❧❧❛s

✷❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s

❇❛②❧♦r ❯♥✐✈❡rs✐t②

✸❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s

❚❤❡ ❈❤✐♥❡s❡ ❯♥✐✈❡rs✐t② ♦❢ ❍♦♥❣ ❑♦♥❣

❖♥❧✐♥❡ ❲♦r❦s❤♦♣ ♦♥ ▼❡❛♥ ❋✐❡❧❞ ●❛♠❡s✱ ❏✉♥❡ ✶✾

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶ ✴ ✹✵

❆❝❦♥♦✇❧❡❞❣♠❡♥t

❚❤✐s ✇♦r❦ ✇❛s ♣❛rt✐❛❧❧② s✉♣♣♦rt❡❞ ❜② t❤❡ ◆❙❋ t❤r♦✉❣❤ r❡s❡❛r❝❤ ❣r❛♥ts ❉▼❙✲✶✻✶✷✽✽✵ ❛♥❞ ❉▼❙✲✶✾✵✺✹✹✾✱ ❛♥❞ ❜② t❤❡ ❍♦♥❣ ❑♦♥❣ ❘❡s❡❛r❝❤ ●r❛♥ts ❈♦✉♥❝✐❧ t❤r♦✉❣❤ ♣r♦❥❡❝t ❍❑●❘❋✲✶✹✸✵✵✼✶✼✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

✷

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷ ❍✐❧❜❡rt s♣❛❝❡ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

✸

▼❛✐♥ r❡s✉❧ts

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

✷

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷ ❍✐❧❜❡rt s♣❛❝❡ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

✸

▼❛✐♥ r❡s✉❧ts

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✹ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥

❚❤❡ ▼❛st❡r ❊q✉❛t✐♦♥

❖✉r ♣r♦❥❡❝t ✐s t♦ st✉❞② − ∂U ∂t (x, m, t) + AxU(x, m, t) +

• Rn Aξ

d dm U(ξ, m, t)(x) dm(ξ) + ✶ ✷λ|DxU(x, m, t)|✷ + ✶ λ

• Rn DξU(ξ, m, t) · Dξ

d dm U(ξ, m, t)(x) dm(ξ) = d dm F(m)(x), U(x, m, T) = d dm FT(m)(x) ✇❤❡r❡ x ∈ Rn, m ∈ P(Rn), ❛♥❞ t ∈ [✵, T]✳ ❍❡r❡ Ax ✐s s❡❝♦♥❞ ♦r❞❡r ❡❧❧✐♣t✐❝ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ✇✐t❤ ✉s✉❛❧ ❛ss✉♠♣t✐♦♥s✿ Axϕ(x) = −✶ ✷tr

• σσ∗D✷ϕ(x)
• .

❙❡❡ P✳ ▲✳ ▲✐♦♥s ❧❡❝t✉r❡s ✐♥ ❈♦❧❧è❣❡ ❞❡ ❋r❛♥❝❡ ❇♦♦❦ ❜② ❈❛r❞❛❧✐❛❣✉❡t✱ ❉❡❧❛r✉❡✱ ▲❛sr②✱ ▲✐♦♥s ✭❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s ❙t✉❞✐❡s✮ ❚✇♦✲✈♦❧✉♠❡ s❡t ❜② ❈❛r♠♦♥❛✱ ❉❡❧❛r✉❡

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✺ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥

■♥t❡r♣r❡t✐♥❣ t❤❡ ▼❛st❡r ❊q✉❛t✐♦♥

❋✐rst ✐♥t❡r♣r❡t❛t✐♦♥✿ ♠❡❛♥ ✜❡❧❞ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ❡q✉❛t✐♦♥ d dm F(m) ❛♥❞ d dm FT(m) ❣✐✈❡ t❤❡ ♠❡❛♥ ✜❡❧❞✬s ❝♦♥tr✐❜✉t✐♦♥ t♦ ✐♥❞✐✈✐❞✉❛❧ ❝♦st ✭r✉♥♥✐♥❣ ❛♥❞ t❡r♠✐♥❛❧✱ r❡s♣✳✮

• ❛♠❡ ✐s ♣♦t❡♥t✐❛❧

❙❡❝♦♥❞ ✐♥t❡r♣r❡t❛t✐♦♥✿ U(x, m, t) ✐s t❤❡ ❞❡❝♦✉♣❧✐♥❣ ✜❡❧❞ ❢♦r ❛ ♠❡❛♥ ✜❡❧❞ t②♣❡ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ✐s ❢♦r♠❛❧❧② inf

• Jm,t(v) := λ

✷ T

t

• Rn |v(ξ, s)|✷ dms(ξ) ds +

T

t

F(ms) ds + FT(mT)

• ∂sms + Axms + ∇ · (vms) = ✵, mt = m
• .

❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ❢♦r♠❛❧❧② ❣✐✈❡ ❍❏✲❋P s②st❡♠ t②♣✐❝❛❧ ♦❢ ▼❋●

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✻ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥

❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥

❋♦r t❤❡ ♠❡❛♥ ✜❡❧❞ t②♣❡ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ s❛t✐s✜❡s − ∂V ∂t (m, t) +

• Rn Ax

d dm V (m, t)(x) dm(x) + ✶ ✷λ

• Rn
• D d

dm V (m, t)(x)

• ✷

dm(x) = F(m), V (m, T) = FT(m). ❲❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ✇✐t❤ r❡s♣❡❝t t♦ m t♦ ❣❡t t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥✳ ❙❡❡ ❇❡♥s♦✉ss❛♥✱ ❋r❡❤s❡✱ ❨❛♠ ✏■♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ▼❛st❡r ❊q✉❛t✐♦♥✳✳✳✧ ❙❡❡ ❛❧s♦ P❤❛♠✱ ❲❡✐ ✷✵✶✽ ❢♦r ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s t❤❡♦r② ♦❢ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥s ♦♥ ❲❛ss❡rst❡✐♥ s♣❛❝❡ ❖✉r ❣♦❛❧ Pr♦✈❡ ❡①✐st❡♥❝❡ ♦❢ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥s t♦ ❇❡❧❧♠❛♥ ❛♥❞ ▼❛st❡r ❡q✉❛t✐♦♥s ✉s✐♥❣ ♦♥❧② ♦♣t✐♠❛❧ ❝♦♥tr♦❧ t❡❝❤♥✐q✉❡s✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✼ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

✷

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷ ❍✐❧❜❡rt s♣❛❝❡ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

✸

▼❛✐♥ r❡s✉❧ts

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✽ ✴ ✹✵

■♥tr♦❞✉❝t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

▲✐❢t✐♥❣ t❤❡ ♣r♦❜❧❡♠

▼❛✐♥ ✐❞❡❛ ✏▲✐❢t✑ t❤❡ ♠❡❛♥ ✜❡❧❞ t②♣❡ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❢r♦♠ t❤❡ s♣❛❝❡ ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s t♦ ❛ ❍✐❧❜❡rt s♣❛❝❡✳ ▼♦t✐✈❛t✐♦♥✿ ▲✐♦♥s ♣r♦♣♦s❡❞ ❧✐❢t✐♥❣ ❢r♦♠ ♠❡❛s✉r❡s t♦ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛s ❛ ✇❛② ♦❢ ❞❡✜♥✐♥❣ ❞❡r✐✈❛t✐✈❡s ♦♥ ❲❛ss❡rst❡✐♥ s♣❛❝❡ L✷ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❢♦r♠ ❛ ❍✐❧❜❡rt s♣❛❝❡ ❈❧❛ss✐❝❛❧ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ✇♦r❦s ✇❡❧❧ ♦♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ ▲✐❢t✐♥❣ t❤❡ ♣r♦❜❧❡♠ ❣✐✈❡s r❡❛s♦♥❛❜❧❡ r❡s✉❧ts ✐♥ t❤❡ ✜rst✲♦r❞❡r ❝❛s❡✿ ❇❡♥s♦✉ss❛♥ ❨❛♠ ✷✵✶✽ ❆ ♣r❡✈✐♦✉s ♣r❡♣r✐♥t ✉s❡s ❡①❛❝t❧② t❤✐s ✐❞❡❛ ◆❡✇ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤❡ ♣r❡s❡♥t ✇♦r❦ ✭✐✮ ❖✉r ✈❡rs✐♦♥ ♦❢ ✏❧✐❢t✐♥❣✑ ✐s r❛❞✐❝❛❧❧② ❞✐✛❡r❡♥t ❢r♦♠ t❤❛t ♦❢ ▲✐♦♥s✳ ✭✐✐✮ ❚❤❡ r❡s✉❧ts ❛r❡ ♥♦t ♥❡✇❀ ❤♦✇❡✈❡r✱ t❤❡ ♠❡t❤♦❞ ♦❢ ♣r♦♦❢ ✐s ❝♦♠♣❧❡t❡❧② ❞✐✛❡r❡♥t✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✾ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

✷

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷ ❍✐❧❜❡rt s♣❛❝❡ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

✸

▼❛✐♥ r❡s✉❧ts

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✵ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

✷

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷ ❍✐❧❜❡rt s♣❛❝❡ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

✸

▼❛✐♥ r❡s✉❧ts

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✶ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷

❲❛ss❡rst❡✐♥ s♣❛❝❡

❲❡ ❞❡♥♦t❡ ❜② P✷(Rn) t❤❡ ❲❛ss❡rst❡✐♥ s♣❛❝❡ ♦❢ ❇♦r❡❧ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s m ♦♥ Rn s✉❝❤ t❤❛t

• Rn|x|✷ dm(x) < ∞✱ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ♠❡tr✐❝

W✷(µ, ν) =

• inf
• |x − y|✷ dπ(x, y) : π ∈ Π(µ, ν)
• .

✭✶✮ ❈♦♥s✐❞❡r ❛♥ ❛t♦♠❧❡ss ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ (Ω, A, P), ❛♥❞ ♦♥ ✐t t❤❡ s♣❛❝❡ L✷(Ω, A, P; Rn) ♦❢ sq✉❛r❡ ✐♥t❡❣r❛❜❧❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ ✈❛❧✉❡s ✐♥ Rn. ❋♦r X ∈ L✷(Ω, A, P; Rn) ✇❡ ❞❡♥♦t❡ ❜② LX t❤❡ ❧❛✇ ♦❢ X✱ ❣✐✈❡♥ ❜② LX(A) = P(X ∈ A)✳ ❚♦ ❛♥② m ✐♥ P✷(Rn), ♦♥❡ ❝❛♥ ✜♥❞ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ Xm ✐♥ L✷(Ω, A, P; Rn) s✉❝❤ t❤❛t LXm = m✳ ❲❡ t❤❡♥ ❤❛✈❡ W ✷

✷ (m, m′) =

inf

LXm =m, LXm′ =m′ E[|Xm − Xm′|✷],

✭✷✮ ✇❤❡r❡ t❤❡ ✐♥✜♠✉♠ ✐s ❛tt❛✐♥❡❞✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✷ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷

❋✐rst ❞❡r✐✈❛t✐✈❡

❲❡ s❛② F ✐s ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ dF dm : P✷ × Rn → R s✉❝❤ t❤❛t✱ ❢♦r s♦♠❡ c : P✷(Rn) → [✵, ∞) t❤❛t ✐s ❜♦✉♥❞❡❞ ♦♥ ❜♦✉♥❞❡❞ s✉❜s❡ts✱ ✇❡ ❤❛✈❡

• dF

dm (m, x)

• ≤ c(m)
• ✶ +|x|✷

✭✸✮ ❛♥❞ lim

ǫ→✵

F(m + ǫ(m′ − m)) − F(m) ǫ = dF dm (m, x) d (m′ − m)(x) ✭✹✮ ❢♦r ❛♥② m′ ∈ P✷✳ ❙✐♥❝❡ dF dm ✐s ✉♥✐q✉❡ ♦♥❧② ✉♣ t♦ ❛ ❝♦♥st❛♥t✱ ✇❡ r❡q✉✐r❡ t❤❡ ♥♦r♠❛❧✐③❛t✐♦♥ ❝♦♥❞✐t✐♦♥ dF dm (m, x) dm(x) = ✵, ✭✺✮ ✇❤✐❝❤ ✐♥ ♣❛rt✐❝✉❧❛r ❡♥s✉r❡s t❤❡ ❢✉♥❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♥st❛♥t ✐s ✵. ❲❡ ✇✐❧❧ ♦❢t❡♥ ❞❡♥♦t❡ dF dm (m)(x) := dF dm (m, x)✳ ❚❤❡♥ dF dm (m) ∈ L✷

m(Rn)✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✸ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷

❙❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ■

❲❡ s❛② F ✐s t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ♣r♦✈✐❞❡❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ d✷F dm✷ : P✷ × Rn × Rn → R s✉❝❤ t❤❛t✱ ❢♦r s♦♠❡ c : P✷(Rn) → [✵, ∞) t❤❛t ✐s ❜♦✉♥❞❡❞ ♦♥ ❜♦✉♥❞❡❞ s✉❜s❡ts✱

• d✷F

dm✷ (m, x, ˜ x)

• ≤ c(m)
• ✶ +|x|✷ +|˜

x|✷ ✭✻✮ ❛♥❞ lim

ǫ→✵

✶ ǫ dF dm (m + ǫ( ˜ m′ − m), x) − dF dm (m, x)

• d

(m′ − m)(x) = d✷F dm✷ (m, x, ˜ x) d (m′ − m)(x) d ( ˜ m′ − m)(˜ x) ✭✼✮

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✹ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷

❙❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ■■

❢♦r ❛♥② m′, ˜ m′ ∈ P✷✳ ❚♦ ❡♥s✉r❡ d✷F dm✷ (m, x, ˜ x) ✐s ✉♥✐q✉❡❧② ❞❡✜♥❡❞✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡ ♥♦r♠❛❧✐③❛t✐♦♥ ❝♦♥✈❡♥t✐♦♥ d✷F dm✷ (m, x, ˜ x) dm(˜ x) = ✵ ∀x, d✷F dm✷ (m, x, ˜ x) dm(x) = ✵ ∀˜ x. ✭✽✮ ❆❣❛✐♥✱ ✇❡ ✇✐❧❧ ✇r✐t❡ d✷F dm✷ (m, x, ˜ x) = d✷F dm✷ (m)(x, ˜ x)✱ ✇❤❡r❡ ✇❡ ♥♦t❡ t❤❛t d✷F dm✷ (m) ∈ L✷

m×m✳

❙t❛♥❞❛r❞ ❛r❣✉♠❡♥ts s❤♦✇ t❤❛t d✷F dm✷ (m) ✐s s②♠♠❡tr✐❝✱ ✐✳❡✳ d✷F dm✷ (m)(x, ˜ x) = d✷F dm✷ (m)(˜ x, x).

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✺ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

✷

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷ ❍✐❧❜❡rt s♣❛❝❡ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

✸

▼❛✐♥ r❡s✉❧ts

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✻ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

▲❡t (Ω, A, P) ❜❡ ❛♥ ❛t♦♠❧❡ss ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✳ ❋♦r m ∈ P✷✱ ❧❡t Hm := L✷(Ω, A, P; L✷

m(Rn; Rn))✳ ❖♥ Hm ✇❡ ❞❡✜♥❡ t❤❡ ✐♥♥❡r ♣r♦❞✉❝t

X, Y Hm = E

• X(x)·Y (x) dm(x) =
• Ω
• Rn X(ω, x)·Y (ω, x) dm(x) dP(ω). ✭✾✮

❲❤❡♥ ✐t ✐s s✉✣❝✐❡♥t❧② ❝❧❡❛r ✇❤✐❝❤ ✐♥♥❡r ♣r♦❞✉❝t ✇❡ ♠❡❛♥✱ ✇❡ ✇✐❧❧ ♦❢t❡♥ ❞r♦♣ t❤❡ s✉❜s❝r✐♣t Hm✳ ◆♦t❡ Hm ∼ = L✷(Ω × Rn, A ⊗ B, P × m; Rn)✱ ✇❤❡r❡ B ✐s t❤❡ ❇♦r❡❧ σ✲❛❧❣❡❜r❛ ♦♥ Rn✳ ❉❡✜♥✐t✐♦♥ ▲❡t m ∈ P✷, X ∈ Hm✳ ❲❡ ❞❡✜♥❡ X ⊗ m ∈ P✷ ❜② ❞✉❛❧✐t②✿ ❢♦r ❛❧❧ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s φ : Rn → R s✉❝❤ t❤❛t x → |φ(x)|

✶+ |x|✷ ✐s ❜♦✉♥❞❡❞✱ ✇❡ ❤❛✈❡

• φ(x) d

(X ⊗m)(x) = E

• φ
• X(x)
• dm(x)
• =
• Ω
• Rn φ
• X(ω, x)
• dm(x) dP(ω).

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✼ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

▲❡♠♠❛ ❚❤❡ ♠❛♣ X → X ⊗ m ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ❢r♦♠ Hm t♦ P✷✱ ✐✳❡✳ W✷(X ⊗ m, Y ⊗ m) ≤X − Y Hm✳ ■❢ X(ω, x) = X(x) ✐s ❞❡t❡r♠✐♥✐st✐❝✱ t❤❡♥ X ⊗ m = X♯m ✇❤❡r❡ X♯m(E) := m(X −✶(E))✳ ▲❡♠♠❛ ▲❡t X, Y ∈ Hm✱ ❛♥❞ s✉♣♣♦s❡ X ◦ Y ∈ Hm✳ ❚❤❡♥ (X ◦ Y ) ⊗ m = X ⊗ (Y ⊗ m)✳ ❊①❛♠♣❧❡s ✭✐✮ ■❢ X(x) = x ✐s t❤❡ ✐❞❡♥t✐t② ♠❛♣✱ t❤❡♥ X ⊗ m = m✳ ✭✐✐✮ ■❢ X(x) = a ✐s ❛ ❝♦♥st❛♥t ♠❛♣✱ t❤❡♥ X ⊗ m = δa✱ t❤❡ ❉✐r❛❝ ❞❡❧t❛ ♠❛ss ❝♦♥❝❡♥tr❛t❡❞ ❛t a✳ ■❢ X(ω, x) = X(ω) ✐s ❥✉st ❛♥ L✷ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✐♥ Rn✱ t❤❡♥ X ⊗ m = LX✳ Pr♦♦❢✿

• φ(x) d

(X ⊗ m)(x) = E

• φ (X) dm(x)
• = E
• φ (X)
• ✭✶✵✮

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✽ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

▲✐❢t✐♥❣ ❢✉♥❝t✐♦♥❛❧s ♦♥ P✷ ■

▲❡t F : P✷ → R✳ ❋♦r ❡✈❡r② m ∈ P✷✱ t❤❡ ♠❛♣ X → F(X ⊗ m) ✐s ❛ ❢✉♥❝t✐♦♥❛❧ ♦♥ Hm✳ ❲❡ ❞❡✜♥❡ t❤❡ ✏♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡✑ ♦❢ F ✇✐t❤ r❡s♣❡❝t t♦ X ∈ Hm ❛s t❤❡ ✉♥✐q✉❡ ❡❧❡♠❡♥t DXF(X ⊗ m) ♦❢ Hm✱ ✐❢ ✐t ❡①✐sts✱ s✉❝❤ t❤❛t lim

ǫ→✵

F

• (X + ǫY ) ⊗ m
• − F(X ⊗ m)

ǫ = DXF(X ⊗ m), Y ∀Y ∈ Hm. ✭✶✶✮

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✶✾ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

▲✐❢t✐♥❣ ❢✉♥❝t✐♦♥❛❧s ♦♥ P✷ ■■

❚❤❡♦r❡♠ ✭❇❡♥s♦✉ss❛♥✱ P❏●✱ ❨❛♠✮ ▲❡t F : P✷ → R ❜❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ❛ss✉♠❡ x → dF

dm (m, x) ✐s

❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ✐♥ Rn✳ ❆ss✉♠❡ t❤❛t ✐ts ❞❡r✐✈❛t✐✈❡ D dF

dm (m)(x) ✐s

❝♦♥t✐♥✉♦✉s ✐♥ ❜♦t❤ m ❛♥❞ x ✇✐t❤

• D dF

dm (m)(x)

• ≤ c(m)
• ✶ +|x|
• ✭✶✷✮

❢♦r s♦♠❡ ❝♦♥st❛♥t c(m) ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ m✳ ❚❤❡♥ DXF(X ⊗ m) = D dF dm (X ⊗ m)(X(·)). ✭✶✸✮ ■❢ X(x) = x✱ t❤❡♥ X ⊗ m = m✱ ❛♥❞ t❤✉s ✭✶✸✮ ❣✐✈❡s t❤❡ ▲✲❞❡r✐✈❛t✐✈❡ DXF(m) = D dF dm (m)(·). ✭✶✹✮

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✵ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

▲✐❢t✐♥❣ ❢✉♥❝t✐♦♥❛❧s ♦♥ P✷ ■■■

Pr♦♦❢✳ ◆♦t❡ t❤❛t✱ ❜② ✭✶✷✮✱ D dF

dm (X ⊗ m, X(·)) ∈ Hm ❢♦r ❛♥② X ∈ Hm✳ ▲❡t Y ∈ Hm ❜❡

❛r❜✐tr❛r②✳ ❋♦r ǫ = ✵✱ ❧❡t µ = (X + ǫY ) ⊗ m, ν = X ⊗ m✱ ❛♥❞ ❢♦r t ∈ [✵, ✶] s❡t νt = ν + t(µ − ν)✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ✶ ǫ

• F
• (X + ǫY ) ⊗ m
• − F(X ⊗ m)
• = ✶

ǫ ✶

✵

• Rn

dF dm (νt, x) d (µ − ν)(x) dt = ✶ ǫ E ✶

✵

• Rn

dF dm

• νt, X(x) + ǫY (x)
• − dF

dm

• νt, X(x)
• dm(x) dt

→ E

• Rn D dF

dm

• X ⊗ m, X(x)
• · Y (x) dm(x)

= D dF dm

• X ⊗ m, X(·)
• , Y Hm

✉s✐♥❣ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ D dF

dm ✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✶ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

P❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ✇rt m ■

▲❡t F : P✷ → R ❛♥❞ ❧❡t X ∈ ∩m∈P✷Hm✳ ❲❡ ❞❡✜♥❡ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ F(X ⊗ m) ✇✐t❤ r❡s♣❡❝t t♦ m✱ ❞❡♥♦t❡❞ ∂F

∂m (X ⊗ m)(x)✱ t♦ ❜❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢

m → F(X ⊗ m) ✐♥ t❤❡ s❡♥s❡ ❣✐✈❡♥ ❜❡❢♦r❡✳ ▲❡♠♠❛ ▲❡t X : Ω × Rn → Rn ❜❡ ❛ (A ⊗ B, B) ♠❡❛s✉r❛❜❧❡ ✈❡❝t♦r ✜❡❧❞ ✭✇❤❡r❡ B ✐s t❤❡ ❇♦r❡❧ σ✲❛❧❣❡❜r❛ ♦♥ Rn✮ s✉❝❤ t❤❛t E

• X(x)
• ✷ ≤ c(X)
• ✶ +|x|✷

∀x ∈ Rn, ✭✶✺✮ ✇❤❡r❡ c(X) ✐s ❛ ❝♦♥st❛♥t ❞❡♣❡♥❞✐♥❣ ♦♥❧② ♦♥ X✳ ❚❤❡♥ X ∈ ∩m∈P✷Hm✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✷ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

P❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ✇rt m ■■

Pr♦♣♦s✐t✐♦♥ ▲❡t F : P✷ → R ❜❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ❧❡t X ∈ ∩m∈P✷Hm✳ ❚❤❡♥ ∂F ∂m (X ⊗ m)(x) = E dF dm (X ⊗ m)(X(x)). ✭✶✻✮

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✸ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

P❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ✇rt m ■■■

Pr♦♦❢✳ ❋♦r ǫ = ✵ ❧❡t µ = X ⊗

• m + ǫ(m′ − m)
• , ν = X ⊗ m✱ ❛♥❞ ❢♦r t ∈ [✵, ✶] s❡t

νt = ν + t(µ − ν)✳ ❲❡ ❤❛✈❡✱ ❛s ǫ → ✵✱ ✶ ǫ

• F
• X ⊗
• m + ǫ(m′ − m)
• − F (X ⊗ m)
• = ✶

ǫ ✶

✵

• Rn

dF dm (νt, x) d (µ − ν)(x) = E ✶

✵

• Rn

dF dm

• νt, X(x)
• d

(m′ − m)(x) → E

• Rn

dF dm

• X ⊗ m, X(x)
• d

(m′ − m)(x), ✉s✐♥❣ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ dF

dm ✳ ❚❤❡ ❝❧❛✐♠ ❢♦❧❧♦✇s✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✹ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❍✐❧❜❡rt s♣❛❝❡

❆ ❢♦r♠✉❧❛ ❢♦r s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s

▲❡t F : P✷ → R ❜❡ t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ❧❡t X, Z ∈ Hm✳ ❚❤❡♥ D✷

XF(X ⊗ m) ❡①✐sts ✐♥ ❛ ●ât❡❛✉① s❡♥s❡ ❛♥❞

D✷

XF(X ⊗ m)(Z)(x) = D✷ dF

dm (X ⊗ m)(X(x))Z(x) + ˜ E

• Rn D✶D✷

d✷F dm✷ (X ⊗ m)( ˜ X(˜ x), X(x)) ˜ Z(˜ x) dm(˜ x) ✭✶✼✮ ✐♥ ✇❤✐❝❤ ˜ X(˜ x), ˜ Z(˜ x) ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❝♦♣✐❡s ♦❢ X(x), Z(x)✱ ❛♥❞ ✐♥ ✇❤✐❝❤ t❤❡ ❡①♣❡❝t❛t✐♦♥ ˜ E ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ X(x).

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✺ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

✷

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷ ❍✐❧❜❡rt s♣❛❝❡ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

✸

▼❛✐♥ r❡s✉❧ts

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✻ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

❘❡❝❛❧❧ t❤❛t (Ω, A, P) ✐s ❛♥ ❛t♦♠❧❡ss ♣r♦❜❛❜✐❧✐t② s♣❛❝❡✱ ❛♥❞ ❢♦r m ∈ P✷✱ Hm := L✷(Ω, A, P; L✷

m(Rn; Rn))✳ ◆♦✇ ❛ss✉♠❡

(Ω, A, P) ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ t♦ ❝♦♥t❛✐♥ ❛ st❛♥❞❛r❞ ❲✐❡♥❡r ♣r♦❝❡ss ✐♥ Rn✱ ❞❡♥♦t❡❞ w(t)✱ ✇✐t❤ ✜❧tr❛t✐♦♥ Wt = {Ws

t }s≥t ✇❤❡r❡

Ws

t = σ

• (w(τ) − w(t)) : t ≤ τ ≤ s
• ✳

(Ω, A, P) ✐s r✐❝❤ ❡♥♦✉❣❤ t♦ s✉♣♣♦rt r❛♥❞♦♠ ✈❛r✐❛❜❧❡s t❤❛t ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❡♥t✐r❡ ❲✐❡♥❡r ♣r♦❝❡ss✳ ❉❡♥♦t❡ ❜② Hm,t t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ X = Xt ∈ Hm s✉❝❤ t❤❛t X ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ Wt✳ ❋♦r X ∈ Hm,t ✇❡ ❞❡✜♥❡ σ✲❛❧❣❡❜r❛s Ws

Xt = σ(X) ∨ Ws t ✱ ❛♥❞ t❤❡ ✜❧tr❛t✐♦♥

❣❡♥❡r❛t❡❞ ❜② t❤❡s❡ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ WXt✳ L✷

WXt(t, T; Hm) ✇✐❧❧ ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ♣r♦❝❡ss❡s ✐♥ L✷(t, T; Hm) t❤❛t ❛r❡ ❛❞❛♣t❡❞ t♦

WXt✱ L✷

Wt(t, T; HX⊗m) t❤❡ s❡t ♦❢ ❛❧❧ ♣r♦❝❡ss❡s ✐♥ L✷(t, T; HX⊗m) t❤❛t ❛r❡

❛❞❛♣t❡❞ t♦ Wt✳ ▲❡♠♠❛ ❚❤❡r❡ ✐s ❛ ❧✐♥❡❛r ✐s♦♠❡tr② L✷

Wt(t, T; HX⊗m) → L✷ WXt(t, T; Hm) ❣✐✈❡♥ ❜②

v(s)(x) → v(s)(X(x))✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✼ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

❈♦♥tr♦❧ ♣r♦❜❧❡♠✿ ✜rst ❢♦r♠✉❧❛t✐♦♥

❉❡✜♥❡ t❤❡ ❝♦st ❢✉♥❝t✐♦♥❛❧ JX⊗m,t : L✷

Wt(t, T; HX⊗m) → R ❜②

JX⊗m,t(v·t(·)) = λ ✷ T

t

• Rn E|vξt(s)|✷ d

(X ⊗ m)(ξ) ds + T

t

F(X·t(s; v·t(·)) ⊗ (X·t ⊗ m)) ds + FT(X·t(T; v·t(·)) ⊗ (X·t ⊗ m)) ✭✶✽✮ ✇❤❡r❡ X·t(s; v·t(·)) ✐s ❞❡✜♥❡❞ ❜② Xxt(s; v·t(·)) = x + s

t

vxt(τ) dτ + σ(w(s) − w(t)). ✭✶✾✮ ❚❤✐s ✐s ❥✉st ❛ ❝❧❛ss✐❝❛❧ ❙❉❊ ❢♦r ❡❛❝❤ x ∈ Rn✳ ❚❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s V (X ⊗ m, t) := inf

v JX⊗m,t(v·t(·)).

✭✷✵✮

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✽ ✴ ✹✵

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

❈♦♥tr♦❧ ♣r♦❜❧❡♠✿ s❡❝♦♥❞ ❢♦r♠✉❧❛t✐♦♥

❇② t❤❡ ✐s♦♠❡tr② L✷

Wt(t, T; HX⊗m) → L✷ WXt(t, T; Hm) ❣✐✈❡♥ ❜② vxt(s) → vXt(s)✱

t❤❡ ✜rst ❢♦r♠✉❧❛t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ s❡❝♦♥❞✿ ▲❡t JXt : L✷

WXt(t, T; Hm) → R ❜❡ ❣✐✈❡♥ ❜②

JXt(vXt(·)) = λ ✷ T

t

• vXt(s)
• ✷

Hm ds

+ T

t

F(XXt(s; vXt(·)) ⊗ m) ds + FT(XXt(T; vXt(·)) ⊗ m) ✭✷✶✮ ✇❤❡r❡ XXt(s; vXt(·)) = X + s

t

vXt(τ) dτ + σ(w(s) − w(t)). ✭✷✷✮ ❚❤✐s ✐s ❛♥ ❙❉❊ ♦♥ t❤❡ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡ Hm✳ ❘❡♠❛r❦ ✭✐✮ XXt(s; vXt(·)) ∈ Hm,s ❢♦r ❛❧❧ s ≥ t✳ ✭✐✐✮ XXt(s; vXt(·)) ⊗ m ✐s ❛♥ ❛❜✉s❡ ♦❢ ♥♦t❛t✐♦♥✳ ■t ❛❝t✉❛❧❧② ♠❡❛♥s X·t(s; v·(·)) ⊗ (X·t ⊗ m)✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✷✾ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

✶

■♥tr♦❞✉❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❖✉r ❛♣♣r♦❛❝❤

✷

◆❡✇ ♠❡t❤♦❞ ♦❢ ❧✐❢t✐♥❣ ❉❡r✐✈❛t✐✈❡s ✐♥ t❤❡ s♣❛❝❡ P✷ ❍✐❧❜❡rt s♣❛❝❡ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ r❡❢♦r♠✉❧❛t❡❞

✸

▼❛✐♥ r❡s✉❧ts

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✵ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❆ss✉♠♣t✐♦♥s ■

F ❛♥❞ FT ❛r❡ t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❊ss❡♥t✐❛❧ ❡st✐♠❛t❡s✿

• D✷ dF

dm (m)(x)

• ≤ c,
• DD✷

d✷F dm✷ (m)(x, ˜ x)

• ≤ c,
• D dFT

dm (m)(x)

• ≤ cT,
• D✷D✶

d✷FT dm✷ (m)(x, ˜ x)

• ≤ cT, ∀x, ˜

x ∈ Rn. ✭✷✸✮ ❙❡♠✐✲❝♦♥✈❡①✐t② ❝♦♥❞✐t✐♦♥s✿ D✷ dF dm (m)(x)ξ · ξ + D✷D✶ d✷F dm✷ (m)(x, ˜ x)ξ · ˜ ξ ≥ −c′|ξ|(|ξ| + |˜ ξ|), D✷ dF dm (m)(x)ξ · ξ + D✷D✶ d✷F dm✷ (m)(x, ˜ x)ξ · ˜ ξ ≥ −c′|ξ|(|ξ| + |˜ ξ|), ∀x, ˜ x, ξ, ˜ ξ. ✭✷✹✮

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✶ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❆ss✉♠♣t✐♦♥s ■■

❈♦♥t✐♥✉✐t② ❝♦♥❞✐t✐♦♥✿ (m, x) → D✷ dF dm (m)(x), (m, x, ˜ x) → D✷D✶ d✷F dm✷ (m)(x, ˜ x) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢r♦♠ P✷(Rn) × Rn❛♥❞ P✷(Rn) × Rn × Rn → L(Rn; Rn), r❡s♣❡❝t✐✈❡❧②✱ ✭✷✺✮ ❛♥❞ ❧✐❦❡✇✐s❡ ❢♦r FT. ❈♦♥❞✐t✐♦♥ t♦ ❣✉❛r❛♥t❡❡ ♦❜❥❡❝t✐✈❡ ✐s ❝♦♥✈❡①✿ λ − T(c′

T + c′T

✷ ) > ✵. ✭✷✻✮ ❙♠❛❧❧♥❡ss ❝♦♥❞✐t✐♦♥✿ λ − T(cT + c T ✷ ) > ✵ ✭✷✼✮

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✷ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

▲✐❢t❡❞ ❇❡❧❧♠❛♥ ❊q✉❛t✐♦♥

❚❤❡ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥ ❢♦r V (X ⊗ m, t) ✐s ∂V ∂t (X ⊗ m, t) + ✶ ✷D✷

XV (X ⊗ m, t)(σN), σN

− ✶ ✷λ||DXV (X ⊗ m, t)||✷ + F(X ⊗ m) = ✵, V (X ⊗ m, T) = FT(X ⊗ m). ✭✷✽✮ ❖✉r ❞❡✜♥✐t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ ✐♥❝❧✉❞❡s t❤❡s❡ ❡ss❡♥t✐❛❧ ❢❡❛t✉r❡s✿ V , DX, D✷

XV ❛r❡ ❝♦♥t✐♥✉♦✉s ✭V ❛♥❞ DXV ❛r❡ ❍ö❧❞❡r ✐♥ t✐♠❡✮

V ✐s r✐❣❤t✲❞✐✛❡r❡♥t✐❛❜❧❡ ✐♥ t✐♠❡ ❚❤❡♦r❡♠ ✭❇❡♥s♦✉ss❛♥✱ P❏●✱ ❨❛♠✮ ❯♥❞❡r t❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s✱ V (X ⊗ m, t) ✐s t❤❡ ✉♥✐q✉❡ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✸ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❇❛❝❦ ❞♦✇♥ t♦ ♦r✐❣✐♥❛❧ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥

❆ s✐♠♣❧❡ ❝♦r♦❧❧❛r② ♦❢ ♦✉r t❤❡♦r❡♠ ✐s t❤❛t V (m, t) s♦❧✈❡s t❤❡ ♦r✐❣✐♥❛❧ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥✿ − ∂V ∂t (m, t) +

• Rn Ax

d dm V (m, t)(x) dm(x) + ✶ ✷λ

• Rn
• D d

dm V (m, t)(x)

• ✷

dm(x) = F(m), V (m, T) = FT(m). ❚♦ ❞❡r✐✈❡ t❤✐s✱ ❥✉st ♣❧✉❣ ✐♥ X(x) = x✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✹ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❙♦❧✈✐♥❣ t❤❡ ▼❛st❡r ❊q✉❛t✐♦♥ ■

▲❡t U(x, m, t) = d dm V (m, t)(x)✳ ❚❤❡♥ ❢♦r♠❛❧❧② U s❛t✐s✜❡s t❤❡ ▼❛st❡r ❊q✉❛t✐♦♥✿ − ∂U ∂t (x, m, t) + AxU(x, m, t) +

• Rn Aξ

d dm U(ξ, m, t)(x) dm(ξ) + ✶ ✷λ|DxU(x, m, t)|✷ + ✶ λ

• Rn DξU(ξ, m, t) · Dξ

d dm U(ξ, m, t)(x) dm(ξ) = d dm F(m)(x), U(x, m, T) = d dm FT(m)(x).

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✺ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❙♦❧✈✐♥❣ t❤❡ ▼❛st❡r ❊q✉❛t✐♦♥ ■■

❚♦ ❥✉st✐❢② t❤✐s✱ ✇❡ ♥❡❡❞ ♠♦r❡ r❡❣✉❧❛r✐t②✿

• D d

dm F(m)(x)

• ≤ c(✶ + |x|),
• D✷ d

dm F(m)(x)

• ≤ c,
• D✸ d

dm F(m)(x)

• ≤ c,
• D✶

d✷ dm✷ F(m)(x, x)

• ≤ c(✶ + |

x|),

• D✷D✶

d✷ dm✷ F(m)(x, x)

• ≤ c,
• D✷

✶

d✷ dm✷ F(m)(x, x)

• ≤ c(✶ + |

x|),

• D✷

✶D✷

d✷ dm✷ F(m)(x, x)

• ≤ c.

✭✷✾✮ ❚❤❡♦r❡♠ ✭❇❡♥s♦✉ss❛♥✱ P❏●✱ ❨❛♠✮ ■♥ ❛❞❞✐t✐♦♥ t♦ ❛❧❧ t❤❡ ♣r❡✈✐♦✉s ❛ss✉♠♣t✐♦♥s✱ t❛❦❡ λ ≥ λT ❢♦r λT s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❞❡♣❡♥❞✐♥❣ ♦♥ c, cT✱ ❛♥❞ T✳ ❚❤❡♥ U s❛t✐s✜❡s t❤❡ ▼❛st❡r ❊q✉❛t✐♦♥ ✐♥ ❛ ♣♦✐♥t✇✐s❡ s❡♥s❡✳ ❲❡ ❞♦ ♥♦t ❢✉❧❧② tr❡❛t ✉♥✐q✉❡♥❡ss❀ s❡❡ ♣r❡✈✐♦✉s r❡❢❡r❡♥❝❡s✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✻ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

▼❛✐♥ ✐❞❡❛ ❜❡❤✐♥❞ ❛❧❧ t❤❡ ♣r♦♦❢s

❚❤❡ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r② ❢♦r t❤❡ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ❞❡r✐✈❡❞ ❢r♦♠ ❛ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ s②st❡♠ ♦❢ ❙❉❊s✿ YXt(s) = X − ✶ λ s

t

ZXt(τ) dτ + σ(w(s) − w(t)), ✭✸✵✮ ZXt(s) = E   T

s

DXF(YXt(τ) ⊗ m) dτ + DXFT(YXt(T) ⊗ m)

• Ws

Xt

  . ✭✸✶✮ ❲❡ ❧♦♦❦ ❢♦r ❛ ♣r✐♦r✐ ❡st✐♠❛t❡s ♦♥ t❤❡ ♣❛✐r (YXt(s), ZXt(s))✱ ✐♥❝❧✉❞✐♥❣ ❛ s❡♥s✐t✐✈✐t② ❛♥❛❧②s✐s ✇rt m✳ ❆❧❧ ♦❢ ♦✉r ❡st✐♠❛t❡s ♦♥ V ❛r✐s❡ ❛s ❛ ❝♦r♦❧❧❛r②✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✼ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❘❡❢❡r❡♥❝❡s ■

❆✳ ❇❡♥s♦✉ss❛♥✱ ❏✳ ❋r❡❤s❡✱ ❛♥❞ ❙✳ ❈✳ P✳ ❨❛♠✱ ❚❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ✐♥ ♠❡❛♥ ✜❡❧❞ t❤❡♦r②✱ ❏♦✉r♥❛❧ ❞❡ ▼❛t❤é♠❛t✐q✉❡s P✉r❡s ❡t ❆♣♣❧✐q✉é❡s✱ ✶✵✸ ✭✷✵✶✺✮✱ ♣♣✳ ✶✹✹✶✕✶✹✼✹✳ ❆✳ ❇❡♥s♦✉ss❛♥✱ P✳ ❏✳ ●r❛❜❡r✱ ❛♥❞ ❙✳ ❨❛♠✱ ❙t♦❝❤❛st✐❝ ❝♦♥tr♦❧ ♦♥ s♣❛❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✾✵✸✳✶✷✻✵✷✱ ✭✷✵✶✾✮✳ ❆✳ ❇❡♥s♦✉ss❛♥✱ P✳ ❏✳ ●r❛❜❡r✱ ❛♥❞ ❙✳ ❈✳ P✳ ❨❛♠✱ ❈♦♥tr♦❧ ♦♥ ❤✐❧❜❡rt s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ t♦ ♠❡❛♥ ✜❡❧❞ t②♣❡ ❝♦♥tr♦❧ t❤❡♦r②✱ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✷✵✵✺✳✶✵✼✼✵✱ ✭✷✵✷✵✮✳ ❆✳ ❇❡♥s♦✉ss❛♥ ❛♥❞ P✳ ❨❛♠✱ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ ♦♥ s♣❛❝❡ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ ♠❛st❡r ❡q✉❛t✐♦♥✱ ❊❙❆■▼✿ ❈❖❈❱✱ ✭✷✵✶✽✮✳ ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✵✺✶✴❝♦❝✈✴✷✵✶✽✵✸✹✳ P✳ ❈❛r❞❛❧✐❛❣✉❡t✱ ▼✳ ❈✐r❛♥t✱ ❛♥❞ ❆✳ P♦rr❡tt❛✱ ❙♣❧✐tt✐♥❣ ♠❡t❤♦❞s ❛♥❞ s❤♦rt t✐♠❡ ❡①✐st❡♥❝❡ ❢♦r t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥s ✐♥ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡s✱ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✷✵✵✶✳✶✵✹✵✻✱ ✭✷✵✷✵✮✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✽ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❘❡❢❡r❡♥❝❡s ■■

P✳ ❈❛r❞❛❧✐❛❣✉❡t✱ ❋✳ ❉❡❧❛r✉❡✱ ❏✳ ▲❛sr②✱ ❛♥❞ P✳ ▲✐♦♥s✱ ❚❤❡ ▼❛st❡r ❊q✉❛t✐♦♥ ❛♥❞ t❤❡ ❈♦♥✈❡r❣❡♥❝❡ Pr♦❜❧❡♠ ✐♥ ▼❡❛♥ ❋✐❡❧❞ ●❛♠❡s✿ ✭❆▼❙✲✷✵✶✮✱ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s ❙t✉❞✐❡s✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷✵✶✾✳ ❘✳ ❈❛r♠♦♥❛ ❛♥❞ ❋✳ ❉❡❧❛r✉❡✱ Pr♦❜❛❜✐❧✐st✐❝ ❚❤❡♦r② ♦❢ ▼❡❛♥ ❋✐❡❧❞ ●❛♠❡s✿ ✈♦❧✳ ■✱ ▼❡❛♥ ❋✐❡❧❞ ❋❇❙❉❊s✱ ❈♦♥tr♦❧✱ ❛♥❞ ●❛♠❡s✱ ❙t♦❝❤❛st✐❝ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ✭✷✵✶✼✮✳ ✱ Pr♦❜❛❜✐❧✐st✐❝ ❚❤❡♦r② ♦❢ ▼❡❛♥ ❋✐❡❧❞ ●❛♠❡s✿ ✈♦❧✳ ■■✱ ▼❡❛♥ ❋✐❡❧❞ ●❛♠❡s ✇✐t❤ ❈♦♠♠♦♥ ◆♦✐s❡ ❛♥❞ ▼❛st❡r ❊q✉❛t✐♦♥s✱ ❙t♦❝❤❛st✐❝ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✳ ❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ✭✷✵✶✼✮✳ ❲✳ ●❛♥❣❜♦ ❛♥❞ ❆✳ ❘✳ ▼és③ár♦s✱ ●❧♦❜❛❧ ✇❡❧❧✲♣♦s❡❞♥❡ss ♦❢ ♠❛st❡r ❡q✉❛t✐♦♥s ❢♦r ❞❡t❡r♠✐♥✐st✐❝ ❞✐s♣❧❛❝❡♠❡♥t ❝♦♥✈❡① ♣♦t❡♥t✐❛❧ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡s✱ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✷✵✵✹✳✵✶✻✻✵✱ ✭✷✵✷✵✮✳ ❲✳ ●❛♥❣❜♦ ❛♥❞ ❆✳ ➅✇✐➛❝❤✱ ❊①✐st❡♥❝❡ ♦❢ ❛ s♦❧✉t✐♦♥ t♦ ❛♥ ❡q✉❛t✐♦♥ ❛r✐s✐♥❣ ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡s✱ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✷✺✾ ✭✷✵✶✺✮✱ ♣♣✳ ✻✺✼✸✕✻✻✹✸✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✸✾ ✴ ✹✵

▼❛✐♥ r❡s✉❧ts

❘❡❢❡r❡♥❝❡s ■■■

❙✳ ▼❛②♦r❣❛✱ ❙❤♦rt t✐♠❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♠❛st❡r ❡q✉❛t✐♦♥ ♦❢ ❛ ✜rst ♦r❞❡r ♠❡❛♥ ✜❡❧❞ ❣❛♠❡✱ ❏♦✉r♥❛❧ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✭✷✵✶✾✮✳ ❈✳ ▼♦✉ ❛♥❞ ❏✳ ❩❤❛♥❣✱ ❲❡❛❦ s♦❧✉t✐♦♥s ♦❢ ♠❡❛♥ ✜❡❧❞ ❣❛♠❡ ♠❛st❡r ❡q✉❛t✐♦♥s✱ ❛r❳✐✈ ♣r❡♣r✐♥t ❛r❳✐✈✿✶✾✵✸✳✵✾✾✵✼✱ ✭✷✵✶✾✮✳ ❍✳ P❤❛♠ ❛♥❞ ❲✳ ❳✐❛♦❧✐✱ ❇❡❧❧♠❛♥ ❡q✉❛t✐♦♥ ❛♥❞ ✈✐s❝♦s✐t② s♦❧✉t✐♦♥s ❢♦r ♠❡❛♥✲✜❡❧❞ st♦❝❤❛st✐❝ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ ❊❙❆■▼✿ ❈♦♥tr♦❧✱ ❖♣t✐♠✐s❛t✐♦♥ ❛♥❞ ❈❛❧❝✉❧✉s ♦❢ ❱❛r✐❛t✐♦♥s✱ ✭✷✵✶✺✮✳ ❈✳ ❲✉✱ ❏✳ ❩❤❛♥❣✱ ❡t ❛❧✳✱ ❱✐s❝♦s✐t② s♦❧✉t✐♦♥s t♦ ♣❛r❛❜♦❧✐❝ ♠❛st❡r ❡q✉❛t✐♦♥s ❛♥❞ ♠❝❦❡❛♥✕✈❧❛s♦✈ s❞❡s ✇✐t❤ ❝❧♦s❡❞✲❧♦♦♣ ❝♦♥tr♦❧s✱ ❆♥♥❛❧s ♦❢ ❆♣♣❧✐❡❞ Pr♦❜❛❜✐❧✐t②✱ ✸✵ ✭✷✵✷✵✮✱ ♣♣✳ ✾✸✻✕✾✽✻✳

P✳ ❏✳ ●r❛❜❡r ❡t ❛❧✳ ❈♦♥tr♦❧ ♦♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❖♥❧✐♥❡ ▼❋● ❏✉♥❡ ✶✾ ✹✵ ✴ ✹✵