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❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❯♥✐✈❡rs✐t② ♦❢ P❛r✐s ❱■✱ ▲❛❜♦r❛t♦✐r❡ ❏✳✲▲✳ ▲✐♦♥s ❖❧✐✈✐❡r P✐r♦♥♥❡❛✉ ✇✐t❤ ❨✈❡s ❆❝❤❞♦✉✱ ◆✳ ▲❛♥t♦s✱ ❛♥❞ ✐♥♣✉t ❢r♦♠ ❈✳ ◆❣✉②❡♥✱ ❊✳❲✳ ❙❛❝❤s ❛♥❞ ▼✳ ❙❝❤✉ ✫ ❆✳ ❈♦♥③❡ ✭◆❛t✐①✐s✮ ✰ ❛♥❞ ❩❡❧✐❛❞❡ ✇✇✇✳❛♥♥✳❥✉ss✐❡✉✳❢r✴♣✐r♦♥♥❡❛✉ ❘■❈❆▼ s♣❡❝✐❛❧ ❙❡♠❡st❡r ✷✵✵✽ ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶ ✴ ✺✶

❖✉t❧✐♥❡ ✳ ✶ ■♥tr♦❞✉❝t✐♦♥ ✳ ✷ ❈❛❧✐❜r❛t✐♦♥ ✳ ✸ ❉✉♣✐r❡✬s ❡q✉❛t✐♦♥ ❘❡❞✉❝❡❞ ❇❛s✐s ❢♦r ❙♦❧✈❡rs ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷ ✴ ✺✶

▼♦♥t❡✲❈❛r❧♦ ❙✐♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❇❧❛❝❦ ✫ ❙❝❤♦❧❡s ▼♦❞❡❧ ❆ ✜♥❛♥❝✐❛❧ ❛ss❡t ✇✐t❤ t❡♥❞❡♥❝② µ ❛♥❞ ✈♦❧❛t✐❧✐t② σ ❞ ❙ t = ❙ t ( µ ❞ t + σ ❞ ❲ t ) , ❙ ✵ ❦♥♦✇♥ • µ = r ( t ) t❤❡ ✐♥t❡r❡st r❛t❡ ✉♥❞❡r t❤❡ r✐s❦✲♥❡✉tr❛❧ ♣r♦❜❛❜✐❧✐t② ❧❛✇✳ ( ❲ t ) ✐s ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❊✉r♦♣❡❛♥ ❝❛❧❧s ❛♥❞ ♣✉ts ♦♥ ❙ ❛r❡ ✈❛❧✉❡❞ ❛t t ❜② t❤❡ ❡①♣❡❝t❡❞ ♣r♦✜t✱ ❞✐s❝♦✉♥t❡❞ ❛t t ✿ ✸ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ✶✳ ❚r❡❡ ♠❡t❤♦❞s ✷✳ ▼♦♥t❡❝❛r❧♦✿ ❈ t = ❡ − r ( ❚ − t ) ❊ ( ❙ ❚ − ❑ ) + � � √ ❞ ❙ t = ❙ t ( µ ❞ t + σ ❞ ❲ t ) , ❙ ( ✵ ) = ❙ ✵ . ❞ ❲ t ≈ ❞ t N ( ✵ , ✶ ) ✸✳ ■t♦ ❈❛❧❝✉❧✉s ✿ ∂ t ❈ + σ ✷ ① ✷ ∂ ✷ ❈ ( ❚ , ① ) = ( ① − ❑ ) + ①① ❈ + r① ∂ ① ❈ − r❈ = ✵ ✷ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸ ✴ ✺✶

▼♦♥t❡✲❈❛r❧♦✿ ●♦♦❞s ❛♥❞ ❜❛❞s ◆❡❛r t♦ t❤❡ ♠♦❞❡❧✐s❛t✐♦♥ ●✐✈❡s ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡ ❡rr♦r √ ◆ ⇒ ◗✉❛s✐✲▼♦♥t❡ ❈❛r❧♦ ❈♦♥✈❡r❣❡s ✐♥ ✶ / √ ❈❛♥ ❝♦♠♣✉t❡ s❡✈❡r❛❧ ❞❡r✐✈❛t✐✈❡s ✇✐t❤ ❖ ( ◆ ) ♦♣❡r❛t✐♦♥s ❊❛s② t♦ ♣❛r❛❧❧❡❧✐③❡✳ ❈♦♠♣❧❡①✐t② ❣r♦✇s ✐♥ ❖ ( ❞ ) ✇✐t❤ t❤❡ ❞✐♠❡♥s✐♦♥ ❞ ✳ ●r❡❡❦s ❜② ▼❛❧❧✐❛✈✐♥ ❝❛❧❝✉❧✉s✳ ❈❛❧✐❜r❛t✐♦♥ ✐s ✈❡r② ❤❛r❞✳ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹ ✴ ✺✶

P❉❊s✿ ●♦♦❞s ❛♥❞ ❜❛❞s ◆❡❛r❡r t♦ t❤❡ ❛♥❛❧②t✐❝❛❧ ❢♦r♠✉❧❛s ❯♣♣❡r ❛♥❞ ❧♦✇❡r ❡rr♦r ❜♦✉♥❞s ✇✐t❤ ❛ ♣♦st❡r✐♦r✐ ❡st✐♠❛t❡s ❈♦♥✈❡r❣❡s ✐♥ ❖ ( ◆ − ♣ ) ✇✐t❤ ♦r❞❡r ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ♣ ❈♦♠♣✉t❡ s❡✈❡r❛❧ ❞❡r✐✈❛t✐✈❡s ✐♥ ❖ ( ◆ ❧♦❣ ◆ ) ♦♣❡r❛t✐♦♥s ✭❉✉♣✐r❡✮✳ ❈❛♥ ❜❡ ♣❛r❛❧❧❡❧✐③❡❞✳ ❈✉rs❡❞ ❜② ❞✐♠❡♥s✐♦♥ ❞ ✐♥ ❖ ( ◆ ❞ ) ❡①❝❡♣t ✇✐t❤ s♣❛rs❡ ❣r✐❞ ✳ ●r❡❡❦s ❛r❡ ✈❡r② ❡❛s② ❈❛❧✐❜r❛t✐♦♥ ✐s r❡❛s♦♥❛❜❧② ❡❛s② ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✺ ✴ ✺✶

❊①✐st❡♥❝❡✱ ❯♥✐q✉❡♥❡ss✱ ❘❡❣✉❧❛r✐t② ✭ ❆❝❤❞♦✉✱ ❈r❡♣❡②✮ ∂ t P − σ ✷ ① ✷ ∂ ✷ ①① P − r① ∂ ① P + rP = ✵ ✷ P ( ✵ , ① ) = ( ❑ − ① ) + , ① →∞ P ( ① , t ) = ✵ ❧✐♠ ❚❤❡♦r❡♠ ❆ss✉♠❡ ✵ < σ ♠ ≤ σ ( ① , t ) < σ ▼ , ① ∂ ① σ ∈ ▲ ∞ ❛♥❞ r , σ, ① ∂ ① σ ▲✐♣s❝❤✐t③ ✐♥ t✱ t❤❡♥ P ❡①✐sts✱ ✐s ❝♦♥t✐♥✉♦✉s ✐♥ t✐♠❡ ❛♥❞ ① ✷ ∂ ①① P ∈ ▲ ✷ ( R + ) ✳ ❋✉rt❤❡r♠♦r❡ ✐❢ ① ✷ ∂ ①① σ ∈ ▲ ∞ t❤❡♥ P ✐s ❝♦♥✈❡① ❛♥❞ ♣♦s✐t✐✈❡✳ ❯s❡ ❛ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠ ✐♥ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ ❱ ✐♥ ✇❤✐❝❤ ❛ t ( ., . ) ✐s ●❛r❞✐♥❣✲❝♦❡r❝✐✈❡✿ ❱ = { ✉ ∈ ▲ ✷ ( R + ) : ① ∂ ① ✉ ∈ ▲ ✷ ( R + ) } P ( ✵ ) = ( ❑ − ① ) + , P ∈ ▲ ✷ ( ✵ , ❚ , ❱ ) ( ∂ t P , ✇ ) + ❛ t ( P , ✇ ) = ✵ , ∀ ✇ ∈ ❱ , ❘ + ∂ ① ( ✇ σ ✷ ① ✷ � ❛ t ( P , ✇ ) = ) ∂ ① P − r①✇ ∂ ① P + rP✇ ] ✷ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✻ ✴ ✺✶

❚❤❡ ❋✐♥✐t❡ ❊❧❡♠❡♥t ▼❡t❤♦❞ ✐♥ ① ∂ t − ① ✷ σ ✷ ∂ ✷ ✉ ∂ ✉ ∂ ① ✷ − r① ∂ ✉ ✉ ( ① , ✵ ) = ( ❑ − ① ) + , ∂ ① + r✉ = ✵ , ✉ ( ▲ , t ) = ✵ ✷ ∂ ① ( ① ✷ σ ✷ ∂ ① ( ① ✷ σ ✷ ∂ ✉ ∂ t − ∂ ∂ ✉ ∂ ① ) − ① ν ∂ ✉ ∂ ① + r✉ = ✵ ✇✐t❤ ν = r − ∂ ) ✷ ✷ ✶ ■♠♣❧✐❝✐t ✐♥ t✐♠❡✱ ❝❡♥t❡r❡❞ ✐♥ s♣❛❝❡ ✭✉♣✇✐♥❞ ✉s✉❛❧❧② ♥♦t ♥❡❝❡ss❛r②✮✿ ✉ ♥ + ✶ − ✉ ♥ ∂ ① ( ① ✷ σ ✷ ♥ + ✶ ♥ + ✶ − ∂ ∂ ✉ ) − ① ν ∂ ✉ + r✉ ♥ + ✶ = ✵ δ t ♥ ✷ ∂ ① ∂ ① ✷ ❱❛r✐❛t✐♦♥❛❧ ❢♦r♠ ✐♥ ❱ = { ✈ ∈ ▲ ✷ ( R + ) : ① ∂ ① ✈ ∈ ▲ ✷ ( R + ) } ✸ ❋✐♥✐t❡ ❡❧❡♠❡♥t ❜❛s✐s ❱ ❤ ≈ ❱ ✹ ( ❇ + δ t❆ ) ❯ ♥ + ✶ = ❈❯ ♥ s♦❧✈❡❞ ❜② ●❛✉ss ▲❯ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✼ ✴ ✺✶

❆ ♣♦st❡r✐♦r✐ ❡st✐♠❛t❡s ✭❨✳❆❝❤❞♦✉✮ Pr♦♣♦s✐t✐♦♥ [[ ✉ − ✉ ❤ ,δ t ]]( t ♥ ) ≤ ❝ ( ✉ ✵ ) δ t ✶   ✷ ♥ ♠ − ✶ + µ ✷ + δ t ♠ � � � ✷ η ♠ ❣ ( ρ δ t ) ( ✶ − ✷ λδ t ✐ ) η ♠ ,ω   σ ✷ σ ✷ ♠✐♥ ♠✐♥ ♠ = ✶ ✐ = ✶ ω ∈T ♠❤ ✇❤❡r❡ ▲ , µ ❛r❡ t❤❡ t✐♠❡✲❝♦♥t✐♥✉✐t② ❝♦♥st❛♥ts ♦❢ σ ✷ , r , ① σ∂ ① σ ✐♥ ▲ ∞ ✱ ❝ ( ✉ ✵ ) = ( � ✉ ✵ � ✷ + δ t �∇ ✉ ✵ � ✷ ) ✶ / ✷ ✱ ❣ ( ρ δ t ) = ( ✶ + ρ δ t ) ✷ ♠❛① ( ✷ , ✶ + ρ δ t ) ♠ = δ t ♠ ❡ − ✷ λ t ♠ − ✶ σ ✷ η ✷ ❤ − ✉ ♠ − ✶ | ✷ ♠✐♥ ✷ | ✉ ♠ ❱ , ❤ ❤ − ✉ ♠ − ✶ ① ♠❛① ( ω ) � ✉ ♠ − r① ∂ ✉ ♠ ❤ ω ❤ ∂ ① + r✉ ♠ ❤ η ♠ ,ω = ❤ � ▲ ✷ ( ω ) δ t ♠ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✽ ✴ ✺✶