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❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ s✐❣♥ ❢❛st ❞✐✛✉s✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②✳

❇❡♥❥❛♠✐♥ ●❡ss

❋❛❦✉❧tät ❢ür ▼❛t❤❡♠❛t✐❦ ❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞

❙✐①t❤ ❲♦r❦s❤♦♣ ♦♥ ❘❛♥❞♦♠ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s✱ ❇✐❡❧❡❢❡❧❞✱ ❖❝t♦❜❡r ✷✵✶✸

♣r❡♣r✐♥t✿❬❛r❳✐✈✿✶✸✶✵✳✻✾✼✶❪✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶ ✴ ✸✵

❖✉t❧✐♥❡

✶

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

✷

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

✸

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

✹

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✸ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

▼❛♥② ✭❝♦♠♣❧❡①✮ s②st❡♠s ✐♥ ♥❛t✉r❡ ❡①❤✐❜✐t ♣♦✇❡r ❧❛✇ s❝❛❧✐♥❣✿ ❚❤❡ ♥✉♠❜❡r ♦❢ ❛♥ ❡✈❡♥t ◆(s) s❝❛❧❡s ✇✐t❤ t❤❡ ❡✈❡♥t s✐③❡ s ❛s ◆(s) ∼ s−α ❋♦r ❡①❛♠♣❧❡✿ ❡❛rt❤q✉❛❦❡s ✺✵ ❧❛r❣❡st ❝✐t✐❡s ✐♥ t❤❡ ❯❙❆

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✹ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

▼❛♥② ✭❝♦♠♣❧❡①✮ s②st❡♠s ✐♥ ♥❛t✉r❡ ❡①❤✐❜✐t ♣♦✇❡r ❧❛✇ s❝❛❧✐♥❣✿ ❚❤❡ ♥✉♠❜❡r ♦❢ ❛♥ ❡✈❡♥t ◆(s) s❝❛❧❡s ✇✐t❤ t❤❡ ❡✈❡♥t s✐③❡ s ❛s ◆(s) ∼ s−α ❋♦r ❡①❛♠♣❧❡✿ ❡❛rt❤q✉❛❦❡s ✺✵ ❧❛r❣❡st ❝✐t✐❡s ✐♥ t❤❡ ❯❙❆

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✹ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

P❤❛s❡✲tr❛♥s✐t✐♦♥s✿ ❚❤❡ ■s✐♥❣ ♠♦❞❡❧✱ ❢❡rr♦♠❛❣♥❡t✐s♠ ❈r✐t✐❝❛❧ t❡♠♣❡r❛t✉r❡ ❚ = ❚❝✿

str♦♥❣❧② ❝♦rr❡❧❛t❡❞✿ s♠❛❧❧ ♣❡rt✉r❜❛t✐♦♥s ❝❛♥ ❤❛✈❡ ❣❧♦❜❛❧ ❡✛❡❝ts ♥♦ s♣❡❝✐✜❝ ❧❡♥❣t❤ s❝❛❧❡ ✭❝♦♠♣❧❡① s②st❡♠✱ ❝r✐t✐❝❛❧✐t②✮

❖❜s❡r✈❡✿ ❋♦r ❚ = ❚❝✱ ♣♦✇❡r✲❧❛✇ s❝❛❧✐♥❣ ❢♦r ◆(s) ❜❡✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ +✶ ❝❧✉st❡rs ♦❢ s✐③❡ s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✺ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

P❤❛s❡✲tr❛♥s✐t✐♦♥s✿ ❚❤❡ ■s✐♥❣ ♠♦❞❡❧✱ ❢❡rr♦♠❛❣♥❡t✐s♠ ❈r✐t✐❝❛❧ t❡♠♣❡r❛t✉r❡ ❚ = ❚❝✿

str♦♥❣❧② ❝♦rr❡❧❛t❡❞✿ s♠❛❧❧ ♣❡rt✉r❜❛t✐♦♥s ❝❛♥ ❤❛✈❡ ❣❧♦❜❛❧ ❡✛❡❝ts ♥♦ s♣❡❝✐✜❝ ❧❡♥❣t❤ s❝❛❧❡ ✭❝♦♠♣❧❡① s②st❡♠✱ ❝r✐t✐❝❛❧✐t②✮

❖❜s❡r✈❡✿ ❋♦r ❚ = ❚❝✱ ♣♦✇❡r✲❧❛✇ s❝❛❧✐♥❣ ❢♦r ◆(s) ❜❡✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ +✶ ❝❧✉st❡rs ♦❢ s✐③❡ s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✺ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

P❤❛s❡✲tr❛♥s✐t✐♦♥s✿ ❚❤❡ ■s✐♥❣ ♠♦❞❡❧✱ ❢❡rr♦♠❛❣♥❡t✐s♠ ❈r✐t✐❝❛❧ t❡♠♣❡r❛t✉r❡ ❚ = ❚❝✿

str♦♥❣❧② ❝♦rr❡❧❛t❡❞✿ s♠❛❧❧ ♣❡rt✉r❜❛t✐♦♥s ❝❛♥ ❤❛✈❡ ❣❧♦❜❛❧ ❡✛❡❝ts ♥♦ s♣❡❝✐✜❝ ❧❡♥❣t❤ s❝❛❧❡ ✭❝♦♠♣❧❡① s②st❡♠✱ ❝r✐t✐❝❛❧✐t②✮

❖❜s❡r✈❡✿ ❋♦r ❚ = ❚❝✱ ♣♦✇❡r✲❧❛✇ s❝❛❧✐♥❣ ❢♦r ◆(s) ❜❡✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ +✶ ❝❧✉st❡rs ♦❢ s✐③❡ s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✺ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

■s✐♥❣ ♠♦❞❡❧ ♥❡❡❞s ♣r❡❝✐s❡ t✉♥✐♥❣ ❚ = ❚❝ t♦ ❞✐s♣❧❛② ♣♦✇❡r ❧❛✇ s❝❛❧✐♥❣ ❍♦✇ ❝❛♥ t❤✐s ♦❝❝✉r ✐♥ ♥❛t✉r❡❄ ■❞❡❛ ♦❢ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②✿ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪ ✏❈r✐t✐❝❛❧✐t②✑ r❡❢❡rs t♦ t❤❡ ♣♦✇❡r✲❧❛✇ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♣❛t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❞✐str✐❜✉t✐♦♥s✱ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛✳ ✏❙❡❧❢✲♦r❣❛♥✐③❡❞✑ r❡❢❡rs t♦ t❤❡ ❢❛❝t t❤❛t t❤❡s❡ s②st❡♠s ♥❛t✉r❛❧❧② ❡✈♦❧✈❡ ✐♥t♦ ❛ ❝r✐t✐❝❛❧ st❛t❡ ✇✐t❤♦✉t ❛♥② t✉♥✐♥❣ ♦❢ t❤❡ ❡①t❡r♥❛❧ ♣❛r❛♠❡t❡rs✱ ✐✳❡✳ t❤❡ ❝r✐t✐❝❛❧ st❛t❡ ✐s ❛♥ ❛ttr❛❝t♦r ♦❢ t❤❡ ❞②♥❛♠✐❝s✳ ❇❛❦✱ ❚❛♥❣✱ ❲✐❡s❡♥❢❡❧❞✿ ❙❛♥❞♣✐❧❡ ❛s ❛ t♦② ♠♦❞❡❧ ♦❢ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✻ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

■s✐♥❣ ♠♦❞❡❧ ♥❡❡❞s ♣r❡❝✐s❡ t✉♥✐♥❣ ❚ = ❚❝ t♦ ❞✐s♣❧❛② ♣♦✇❡r ❧❛✇ s❝❛❧✐♥❣ ❍♦✇ ❝❛♥ t❤✐s ♦❝❝✉r ✐♥ ♥❛t✉r❡❄ ■❞❡❛ ♦❢ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②✿ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪ ✏❈r✐t✐❝❛❧✐t②✑ r❡❢❡rs t♦ t❤❡ ♣♦✇❡r✲❧❛✇ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♣❛t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❞✐str✐❜✉t✐♦♥s✱ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛✳ ✏❙❡❧❢✲♦r❣❛♥✐③❡❞✑ r❡❢❡rs t♦ t❤❡ ❢❛❝t t❤❛t t❤❡s❡ s②st❡♠s ♥❛t✉r❛❧❧② ❡✈♦❧✈❡ ✐♥t♦ ❛ ❝r✐t✐❝❛❧ st❛t❡ ✇✐t❤♦✉t ❛♥② t✉♥✐♥❣ ♦❢ t❤❡ ❡①t❡r♥❛❧ ♣❛r❛♠❡t❡rs✱ ✐✳❡✳ t❤❡ ❝r✐t✐❝❛❧ st❛t❡ ✐s ❛♥ ❛ttr❛❝t♦r ♦❢ t❤❡ ❞②♥❛♠✐❝s✳ ❇❛❦✱ ❚❛♥❣✱ ❲✐❡s❡♥❢❡❧❞✿ ❙❛♥❞♣✐❧❡ ❛s ❛ t♦② ♠♦❞❡❧ ♦❢ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✻ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

■s✐♥❣ ♠♦❞❡❧ ♥❡❡❞s ♣r❡❝✐s❡ t✉♥✐♥❣ ❚ = ❚❝ t♦ ❞✐s♣❧❛② ♣♦✇❡r ❧❛✇ s❝❛❧✐♥❣ ❍♦✇ ❝❛♥ t❤✐s ♦❝❝✉r ✐♥ ♥❛t✉r❡❄ ■❞❡❛ ♦❢ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②✿ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪ ✏❈r✐t✐❝❛❧✐t②✑ r❡❢❡rs t♦ t❤❡ ♣♦✇❡r✲❧❛✇ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♣❛t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❞✐str✐❜✉t✐♦♥s✱ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛✳ ✏❙❡❧❢✲♦r❣❛♥✐③❡❞✑ r❡❢❡rs t♦ t❤❡ ❢❛❝t t❤❛t t❤❡s❡ s②st❡♠s ♥❛t✉r❛❧❧② ❡✈♦❧✈❡ ✐♥t♦ ❛ ❝r✐t✐❝❛❧ st❛t❡ ✇✐t❤♦✉t ❛♥② t✉♥✐♥❣ ♦❢ t❤❡ ❡①t❡r♥❛❧ ♣❛r❛♠❡t❡rs✱ ✐✳❡✳ t❤❡ ❝r✐t✐❝❛❧ st❛t❡ ✐s ❛♥ ❛ttr❛❝t♦r ♦❢ t❤❡ ❞②♥❛♠✐❝s✳ ❇❛❦✱ ❚❛♥❣✱ ❲✐❡s❡♥❢❡❧❞✿ ❙❛♥❞♣✐❧❡ ❛s ❛ t♦② ♠♦❞❡❧ ♦❢ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✻ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

■s✐♥❣ ♠♦❞❡❧ ♥❡❡❞s ♣r❡❝✐s❡ t✉♥✐♥❣ ❚ = ❚❝ t♦ ❞✐s♣❧❛② ♣♦✇❡r ❧❛✇ s❝❛❧✐♥❣ ❍♦✇ ❝❛♥ t❤✐s ♦❝❝✉r ✐♥ ♥❛t✉r❡❄ ■❞❡❛ ♦❢ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②✿ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪ ✏❈r✐t✐❝❛❧✐t②✑ r❡❢❡rs t♦ t❤❡ ♣♦✇❡r✲❧❛✇ ❜❡❤❛✈✐♦r ♦❢ t❤❡ s♣❛t✐❛❧ ❛♥❞ t❡♠♣♦r❛❧ ❞✐str✐❜✉t✐♦♥s✱ ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ ❝r✐t✐❝❛❧ ♣❤❡♥♦♠❡♥❛✳ ✏❙❡❧❢✲♦r❣❛♥✐③❡❞✑ r❡❢❡rs t♦ t❤❡ ❢❛❝t t❤❛t t❤❡s❡ s②st❡♠s ♥❛t✉r❛❧❧② ❡✈♦❧✈❡ ✐♥t♦ ❛ ❝r✐t✐❝❛❧ st❛t❡ ✇✐t❤♦✉t ❛♥② t✉♥✐♥❣ ♦❢ t❤❡ ❡①t❡r♥❛❧ ♣❛r❛♠❡t❡rs✱ ✐✳❡✳ t❤❡ ❝r✐t✐❝❛❧ st❛t❡ ✐s ❛♥ ❛ttr❛❝t♦r ♦❢ t❤❡ ❞②♥❛♠✐❝s✳ ❇❛❦✱ ❚❛♥❣✱ ❲✐❡s❡♥❢❡❧❞✿ ❙❛♥❞♣✐❧❡ ❛s ❛ t♦② ♠♦❞❡❧ ♦❢ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✻ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❛♥❞♣✐❧❡s

❚✇♦ s❝❛❧❡s✿ ❙❧♦✇ ❡♥❡r❣② ✐♥❥❡❝t✐♦♥ ✭❛❞❞✐♥❣ s❛♥❞✮✱ ❢❛st ❡♥❡r❣② ❞✐✛✉s✐♦♥ ✭❛✈❛❧❛♥❝❤❡s✮ ❈r✐t✐❝❛❧✐t②✿ ◆♦ t②♣✐❝❛❧ ❛✈❛❧❛♥❝❤❡ s✐③❡✱ ❧♦❝❛❧ ♣❡rt✉r❜❛t✐♦♥ ♠❛② ❤❛✈❡ ❣❧♦❜❛❧ ❡✛❡❝ts P♦✇❡r ❧❛✇ s❝❛❧✐♥❣✿ ◆(s) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✈❛❧❛♥❝❡s ♦❢ s✐③❡ s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✼ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❛♥❞♣✐❧❡s

❚✇♦ s❝❛❧❡s✿ ❙❧♦✇ ❡♥❡r❣② ✐♥❥❡❝t✐♦♥ ✭❛❞❞✐♥❣ s❛♥❞✮✱ ❢❛st ❡♥❡r❣② ❞✐✛✉s✐♦♥ ✭❛✈❛❧❛♥❝❤❡s✮ ❈r✐t✐❝❛❧✐t②✿ ◆♦ t②♣✐❝❛❧ ❛✈❛❧❛♥❝❤❡ s✐③❡✱ ❧♦❝❛❧ ♣❡rt✉r❜❛t✐♦♥ ♠❛② ❤❛✈❡ ❣❧♦❜❛❧ ❡✛❡❝ts P♦✇❡r ❧❛✇ s❝❛❧✐♥❣✿ ◆(s) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✈❛❧❛♥❝❡s ♦❢ s✐③❡ s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✼ ✴ ✸✵

❙❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❙❛♥❞♣✐❧❡s

❚✇♦ s❝❛❧❡s✿ ❙❧♦✇ ❡♥❡r❣② ✐♥❥❡❝t✐♦♥ ✭❛❞❞✐♥❣ s❛♥❞✮✱ ❢❛st ❡♥❡r❣② ❞✐✛✉s✐♦♥ ✭❛✈❛❧❛♥❝❤❡s✮ ❈r✐t✐❝❛❧✐t②✿ ◆♦ t②♣✐❝❛❧ ❛✈❛❧❛♥❝❤❡ s✐③❡✱ ❧♦❝❛❧ ♣❡rt✉r❜❛t✐♦♥ ♠❛② ❤❛✈❡ ❣❧♦❜❛❧ ❡✛❡❝ts P♦✇❡r ❧❛✇ s❝❛❧✐♥❣✿ ◆(s) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✈❛❧❛♥❝❡s ♦❢ s✐③❡ s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✼ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✽ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪✳ ❆✐♠✿ ❉❡✜♥❡ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥ ❞✐s♣❧❛②✐♥❣ ❙❖❈✳ ❈♦♥s✐❞❡r ❛♥ ◆ ×◆ sq✉❛r❡ ❧❛tt✐❝❡✱ r❡♣r❡s❡♥t✐♥❣ ❛ ❞✐s❝r❡t❡ r❡❣✐♦♥ O = {(✐,❥)}◆

✐,❥=✶✳

❆t ❡❛❝❤ s✐t❡ (✐,❥) t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ s❛♥❞♣✐❧❡ ❛t t✐♠❡ t ✐s ❤t

✐❥✳

❚❤❡ s②st❡♠ ✐s ♣❡rt✉r❜❡❞ ❡①t❡r♥❛❧❧② ✉♥t✐❧ t❤❡ ❤❡✐❣❤t ❤ ❡①❝❡❡❞s ❛ t❤r❡s❤♦❧❞ ✭❝r✐t✲ ✐❝❛❧✮ ✈❛❧✉❡ ❤❝✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✾ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪✳ ❆✐♠✿ ❉❡✜♥❡ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥ ❞✐s♣❧❛②✐♥❣ ❙❖❈✳ ❈♦♥s✐❞❡r ❛♥ ◆ ×◆ sq✉❛r❡ ❧❛tt✐❝❡✱ r❡♣r❡s❡♥t✐♥❣ ❛ ❞✐s❝r❡t❡ r❡❣✐♦♥ O = {(✐,❥)}◆

✐,❥=✶✳

❆t ❡❛❝❤ s✐t❡ (✐,❥) t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ s❛♥❞♣✐❧❡ ❛t t✐♠❡ t ✐s ❤t

✐❥✳

❚❤❡ s②st❡♠ ✐s ♣❡rt✉r❜❡❞ ❡①t❡r♥❛❧❧② ✉♥t✐❧ t❤❡ ❤❡✐❣❤t ❤ ❡①❝❡❡❞s ❛ t❤r❡s❤♦❧❞ ✭❝r✐t✲ ✐❝❛❧✮ ✈❛❧✉❡ ❤❝✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✾ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪✳ ❆✐♠✿ ❉❡✜♥❡ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥ ❞✐s♣❧❛②✐♥❣ ❙❖❈✳ ❈♦♥s✐❞❡r ❛♥ ◆ ×◆ sq✉❛r❡ ❧❛tt✐❝❡✱ r❡♣r❡s❡♥t✐♥❣ ❛ ❞✐s❝r❡t❡ r❡❣✐♦♥ O = {(✐,❥)}◆

✐,❥=✶✳

❆t ❡❛❝❤ s✐t❡ (✐,❥) t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ s❛♥❞♣✐❧❡ ❛t t✐♠❡ t ✐s ❤t

✐❥✳

❚❤❡ s②st❡♠ ✐s ♣❡rt✉r❜❡❞ ❡①t❡r♥❛❧❧② ✉♥t✐❧ t❤❡ ❤❡✐❣❤t ❤ ❡①❝❡❡❞s ❛ t❤r❡s❤♦❧❞ ✭❝r✐t✲ ✐❝❛❧✮ ✈❛❧✉❡ ❤❝✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✾ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪✳ ❆✐♠✿ ❉❡✜♥❡ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥ ❞✐s♣❧❛②✐♥❣ ❙❖❈✳ ❈♦♥s✐❞❡r ❛♥ ◆ ×◆ sq✉❛r❡ ❧❛tt✐❝❡✱ r❡♣r❡s❡♥t✐♥❣ ❛ ❞✐s❝r❡t❡ r❡❣✐♦♥ O = {(✐,❥)}◆

✐,❥=✶✳

❆t ❡❛❝❤ s✐t❡ (✐,❥) t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ s❛♥❞♣✐❧❡ ❛t t✐♠❡ t ✐s ❤t

✐❥✳

❚❤❡ s②st❡♠ ✐s ♣❡rt✉r❜❡❞ ❡①t❡r♥❛❧❧② ✉♥t✐❧ t❤❡ ❤❡✐❣❤t ❤ ❡①❝❡❡❞s ❛ t❤r❡s❤♦❧❞ ✭❝r✐t✲ ✐❝❛❧✮ ✈❛❧✉❡ ❤❝✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✾ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪✳ ❆✐♠✿ ❉❡✜♥❡ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥ ❞✐s♣❧❛②✐♥❣ ❙❖❈✳ ❈♦♥s✐❞❡r ❛♥ ◆ ×◆ sq✉❛r❡ ❧❛tt✐❝❡✱ r❡♣r❡s❡♥t✐♥❣ ❛ ❞✐s❝r❡t❡ r❡❣✐♦♥ O = {(✐,❥)}◆

✐,❥=✶✳

❆t ❡❛❝❤ s✐t❡ (✐,❥) t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ s❛♥❞♣✐❧❡ ❛t t✐♠❡ t ✐s ❤t

✐❥✳

❚❤❡ s②st❡♠ ✐s ♣❡rt✉r❜❡❞ ❡①t❡r♥❛❧❧② ✉♥t✐❧ t❤❡ ❤❡✐❣❤t ❤ ❡①❝❡❡❞s ❛ t❤r❡s❤♦❧❞ ✭❝r✐t✲ ✐❝❛❧✮ ✈❛❧✉❡ ❤❝✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✾ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛♥t❛②✱ ■❛♥♦s✐❀ P❤②s✐❝❛ ❆✱ ✶✾✾✷❪✳ ❆✐♠✿ ❉❡✜♥❡ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥ ❞✐s♣❧❛②✐♥❣ ❙❖❈✳ ❈♦♥s✐❞❡r ❛♥ ◆ ×◆ sq✉❛r❡ ❧❛tt✐❝❡✱ r❡♣r❡s❡♥t✐♥❣ ❛ ❞✐s❝r❡t❡ r❡❣✐♦♥ O = {(✐,❥)}◆

✐,❥=✶✳

❆t ❡❛❝❤ s✐t❡ (✐,❥) t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ s❛♥❞♣✐❧❡ ❛t t✐♠❡ t ✐s ❤t

✐❥✳

❚❤❡ s②st❡♠ ✐s ♣❡rt✉r❜❡❞ ❡①t❡r♥❛❧❧② ✉♥t✐❧ t❤❡ ❤❡✐❣❤t ❤ ❡①❝❡❡❞s ❛ t❤r❡s❤♦❧❞ ✭❝r✐t✲ ✐❝❛❧✮ ✈❛❧✉❡ ❤❝✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✾ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡♥✱ ❛ t♦♣♣❧✐♥❣ ✭❛✈❛❧❛♥❝❤❡✮ ❡✈❡♥t ♦❝❝✉rs✿ ❚❤❡ t♦♣♣❧✐♥❣ ❛t ❛♥② ❵❛❝t✐✈❛t❡❞✬ s✐t❡ (❦,❧) ✐s ❞❡s❝r✐❜❡❞ ❜②✿ ❤t+✶

✐❥

→ ❤t

✐❥ −▼❦❧ ✐❥ ,

∀(✐,❥) ∈ O, ✇❤❡r❡ ▼❦❧

✐❥ =

     ✹ (❦,❧) = (✐,❥) −✶ (❦,❧) ∼ (✐,❥) ✵ ♦t❤❡r✇✐s❡✳ ❘❡✇r✐t❡ ❛s✿ ❤t+✶

✐❥

−❤t

✐❥ = −▼❦❧ ✐❥ ❍(❤t ✐❥ −❤❝ ✐❥),

∀(✐,❥) ∈ O, ✇❤❡r❡ ❍ ✐s t❤❡ ❍❡❛✈✐s✐❞❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❛✈❛❧❛♥❝❤❡s ❛r❡ ❝♦♥t✐♥✉❡❞ ✉♥t✐❧ ♥♦ s✐t❡ ❡①❝❡❡❞s t❤❡ t❤r❡s❤♦❧❞ ✭✇❤✐❝❤ ♦❜✈✐✲ ♦✉s❧② ❤❛♣♣❡♥s ❛❢t❡r ✜♥✐t❡❧② ♠❛♥② st❡♣s✮✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✵ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡♥✱ ❛ t♦♣♣❧✐♥❣ ✭❛✈❛❧❛♥❝❤❡✮ ❡✈❡♥t ♦❝❝✉rs✿ ❚❤❡ t♦♣♣❧✐♥❣ ❛t ❛♥② ❵❛❝t✐✈❛t❡❞✬ s✐t❡ (❦,❧) ✐s ❞❡s❝r✐❜❡❞ ❜②✿ ❤t+✶

✐❥

→ ❤t

✐❥ −▼❦❧ ✐❥ ,

∀(✐,❥) ∈ O, ✇❤❡r❡ ▼❦❧

✐❥ =

     ✹ (❦,❧) = (✐,❥) −✶ (❦,❧) ∼ (✐,❥) ✵ ♦t❤❡r✇✐s❡✳ ❘❡✇r✐t❡ ❛s✿ ❤t+✶

✐❥

−❤t

✐❥ = −▼❦❧ ✐❥ ❍(❤t ✐❥ −❤❝ ✐❥),

∀(✐,❥) ∈ O, ✇❤❡r❡ ❍ ✐s t❤❡ ❍❡❛✈✐s✐❞❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❛✈❛❧❛♥❝❤❡s ❛r❡ ❝♦♥t✐♥✉❡❞ ✉♥t✐❧ ♥♦ s✐t❡ ❡①❝❡❡❞s t❤❡ t❤r❡s❤♦❧❞ ✭✇❤✐❝❤ ♦❜✈✐✲ ♦✉s❧② ❤❛♣♣❡♥s ❛❢t❡r ✜♥✐t❡❧② ♠❛♥② st❡♣s✮✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✵ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❚❤❡♥✱ ❛ t♦♣♣❧✐♥❣ ✭❛✈❛❧❛♥❝❤❡✮ ❡✈❡♥t ♦❝❝✉rs✿ ❚❤❡ t♦♣♣❧✐♥❣ ❛t ❛♥② ❵❛❝t✐✈❛t❡❞✬ s✐t❡ (❦,❧) ✐s ❞❡s❝r✐❜❡❞ ❜②✿ ❤t+✶

✐❥

→ ❤t

✐❥ −▼❦❧ ✐❥ ,

∀(✐,❥) ∈ O, ✇❤❡r❡ ▼❦❧

✐❥ =

     ✹ (❦,❧) = (✐,❥) −✶ (❦,❧) ∼ (✐,❥) ✵ ♦t❤❡r✇✐s❡✳ ❘❡✇r✐t❡ ❛s✿ ❤t+✶

✐❥

−❤t

✐❥ = −▼❦❧ ✐❥ ❍(❤t ✐❥ −❤❝ ✐❥),

∀(✐,❥) ∈ O, ✇❤❡r❡ ❍ ✐s t❤❡ ❍❡❛✈✐s✐❞❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❛✈❛❧❛♥❝❤❡s ❛r❡ ❝♦♥t✐♥✉❡❞ ✉♥t✐❧ ♥♦ s✐t❡ ❡①❝❡❡❞s t❤❡ t❤r❡s❤♦❧❞ ✭✇❤✐❝❤ ♦❜✈✐✲ ♦✉s❧② ❤❛♣♣❡♥s ❛❢t❡r ✜♥✐t❡❧② ♠❛♥② st❡♣s✮✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✵ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈❡❧❧✉❧❛r ❛✉t♦♠❛t❛ ♠♦❞❡❧

❆s ❛♥ ❡①❛♠♣❧❡✿

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✶ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈♦♥t✐♥✉✉♠ ❧✐♠✐t

P❛ss✐♥❣ t♦ ❛ ❝♦♥t✐♥✉✉♠ ❧✐♠✐t ✐♥ ❤t+✶

✐❥

−❤t

✐❥ = −▼❦❧ ✐❥ ❍(❤t ✐❥ −❤❝ ✐❥),

∀(✐,❥) ∈ O, ❣✐✈❡s ✭✐♥❢♦r♠❛❧❧②✮ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ✇❤❡r❡ ❳ ✐s t❤❡ ❝♦♥t✐♥✉♦✉s ❤❡✐❣❤t✲❞❡♥s✐t② ❢✉♥❝t✐♦♥✳ ■♥ ❛❞❞✐t✐♦♥ ✇❡ ✐♠♣♦s❡ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ ❍(❳(t,ξ)−❳ ❝(ξ)) = ✵, ♦♥ ∂O. ◆♦t❡✿ ❖♥❧② t❤❡ r❡❧❛①❛t✐♦♥✴❞✐✛✉s✐♦♥ ♣❛rt ♠♦❞❡❧❡❞ ❤❡r❡✳ ❋♦r ❢✉❧❧ ❙❖❈✲♠♦❞❡❧ ✇❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ✐♥❝❧✉❞❡ t❤❡ ❡①t❡r♥❛❧✱ r❛♥❞♦♠ ❡♥❡r❣② ✐♥♣✉t✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✷ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈♦♥t✐♥✉✉♠ ❧✐♠✐t

P❛ss✐♥❣ t♦ ❛ ❝♦♥t✐♥✉✉♠ ❧✐♠✐t ✐♥ ❤t+✶

✐❥

−❤t

✐❥ = −▼❦❧ ✐❥ ❍(❤t ✐❥ −❤❝ ✐❥),

∀(✐,❥) ∈ O, ❣✐✈❡s ✭✐♥❢♦r♠❛❧❧②✮ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ✇❤❡r❡ ❳ ✐s t❤❡ ❝♦♥t✐♥✉♦✉s ❤❡✐❣❤t✲❞❡♥s✐t② ❢✉♥❝t✐♦♥✳ ■♥ ❛❞❞✐t✐♦♥ ✇❡ ✐♠♣♦s❡ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ ❍(❳(t,ξ)−❳ ❝(ξ)) = ✵, ♦♥ ∂O. ◆♦t❡✿ ❖♥❧② t❤❡ r❡❧❛①❛t✐♦♥✴❞✐✛✉s✐♦♥ ♣❛rt ♠♦❞❡❧❡❞ ❤❡r❡✳ ❋♦r ❢✉❧❧ ❙❖❈✲♠♦❞❡❧ ✇❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ✐♥❝❧✉❞❡ t❤❡ ❡①t❡r♥❛❧✱ r❛♥❞♦♠ ❡♥❡r❣② ✐♥♣✉t✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✷ ✴ ✸✵

❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❇❚❲ ♠♦❞❡❧ ❢r♦♠ ❛ ❝❡❧❧✉❧❛r ❛✉t♦♠❛t♦♥

❈♦♥t✐♥✉✉♠ ❧✐♠✐t

P❛ss✐♥❣ t♦ ❛ ❝♦♥t✐♥✉✉♠ ❧✐♠✐t ✐♥ ❤t+✶

✐❥

−❤t

✐❥ = −▼❦❧ ✐❥ ❍(❤t ✐❥ −❤❝ ✐❥),

∀(✐,❥) ∈ O, ❣✐✈❡s ✭✐♥❢♦r♠❛❧❧②✮ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ✇❤❡r❡ ❳ ✐s t❤❡ ❝♦♥t✐♥✉♦✉s ❤❡✐❣❤t✲❞❡♥s✐t② ❢✉♥❝t✐♦♥✳ ■♥ ❛❞❞✐t✐♦♥ ✇❡ ✐♠♣♦s❡ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ ❍(❳(t,ξ)−❳ ❝(ξ)) = ✵, ♦♥ ∂O. ◆♦t❡✿ ❖♥❧② t❤❡ r❡❧❛①❛t✐♦♥✴❞✐✛✉s✐♦♥ ♣❛rt ♠♦❞❡❧❡❞ ❤❡r❡✳ ❋♦r ❢✉❧❧ ❙❖❈✲♠♦❞❡❧ ✇❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ✐♥❝❧✉❞❡ t❤❡ ❡①t❡r♥❛❧✱ r❛♥❞♦♠ ❡♥❡r❣② ✐♥♣✉t✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✷ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✸ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄ ❘❡❝❛❧❧✿ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ❲❡ ✇✐❧❧ r❡str✐❝t t♦ t❤❡ s✉♣❡r❝r✐t✐❝❛❧ ❝❛s❡✱ ✐✳❡✳ s✉♣♣♦s✐♥❣ ①✵ ≥ ❳ ❝✳ ❙✉❜st✐t✉t✐♥❣ ❳ → ❳ −❳ ❝ ❛♥❞ ✉s✐♥❣ ❳ ≥ ✵ ②✐❡❧❞s ∂ ∂t ❳(t,ξ) = ∆s❣♥(❳(t,ξ)), ❳(✵,ξ) = ①✵(ξ) ✇✐t❤ ①✵ ≥ ✵ ❛♥❞ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ s❣♥(❳(t,ξ)) = ✵, ♦♥ ∂O. ■♥❢♦r♠❛❧❧②✿ ∆s❣♥(❳) = δ✵(❳)∆❳ +s❣♥′′(❳)|∇❳|✷. ❆✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡ = ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✹ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄ ❘❡❝❛❧❧✿ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ❲❡ ✇✐❧❧ r❡str✐❝t t♦ t❤❡ s✉♣❡r❝r✐t✐❝❛❧ ❝❛s❡✱ ✐✳❡✳ s✉♣♣♦s✐♥❣ ①✵ ≥ ❳ ❝✳ ❙✉❜st✐t✉t✐♥❣ ❳ → ❳ −❳ ❝ ❛♥❞ ✉s✐♥❣ ❳ ≥ ✵ ②✐❡❧❞s ∂ ∂t ❳(t,ξ) = ∆s❣♥(❳(t,ξ)), ❳(✵,ξ) = ①✵(ξ) ✇✐t❤ ①✵ ≥ ✵ ❛♥❞ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ s❣♥(❳(t,ξ)) = ✵, ♦♥ ∂O. ■♥❢♦r♠❛❧❧②✿ ∆s❣♥(❳) = δ✵(❳)∆❳ +s❣♥′′(❳)|∇❳|✷. ❆✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡ = ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✹ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄ ❘❡❝❛❧❧✿ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ❲❡ ✇✐❧❧ r❡str✐❝t t♦ t❤❡ s✉♣❡r❝r✐t✐❝❛❧ ❝❛s❡✱ ✐✳❡✳ s✉♣♣♦s✐♥❣ ①✵ ≥ ❳ ❝✳ ❙✉❜st✐t✉t✐♥❣ ❳ → ❳ −❳ ❝ ❛♥❞ ✉s✐♥❣ ❳ ≥ ✵ ②✐❡❧❞s ∂ ∂t ❳(t,ξ) = ∆s❣♥(❳(t,ξ)), ❳(✵,ξ) = ①✵(ξ) ✇✐t❤ ①✵ ≥ ✵ ❛♥❞ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ s❣♥(❳(t,ξ)) = ✵, ♦♥ ∂O. ■♥❢♦r♠❛❧❧②✿ ∆s❣♥(❳) = δ✵(❳)∆❳ +s❣♥′′(❳)|∇❳|✷. ❆✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡ = ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✹ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄ ❘❡❝❛❧❧✿ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ❲❡ ✇✐❧❧ r❡str✐❝t t♦ t❤❡ s✉♣❡r❝r✐t✐❝❛❧ ❝❛s❡✱ ✐✳❡✳ s✉♣♣♦s✐♥❣ ①✵ ≥ ❳ ❝✳ ❙✉❜st✐t✉t✐♥❣ ❳ → ❳ −❳ ❝ ❛♥❞ ✉s✐♥❣ ❳ ≥ ✵ ②✐❡❧❞s ∂ ∂t ❳(t,ξ) = ∆s❣♥(❳(t,ξ)), ❳(✵,ξ) = ①✵(ξ) ✇✐t❤ ①✵ ≥ ✵ ❛♥❞ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ s❣♥(❳(t,ξ)) = ✵, ♦♥ ∂O. ■♥❢♦r♠❛❧❧②✿ ∆s❣♥(❳) = δ✵(❳)∆❳ +s❣♥′′(❳)|∇❳|✷. ❆✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡ = ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✹ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄ ❘❡❝❛❧❧✿ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ❲❡ ✇✐❧❧ r❡str✐❝t t♦ t❤❡ s✉♣❡r❝r✐t✐❝❛❧ ❝❛s❡✱ ✐✳❡✳ s✉♣♣♦s✐♥❣ ①✵ ≥ ❳ ❝✳ ❙✉❜st✐t✉t✐♥❣ ❳ → ❳ −❳ ❝ ❛♥❞ ✉s✐♥❣ ❳ ≥ ✵ ②✐❡❧❞s ∂ ∂t ❳(t,ξ) = ∆s❣♥(❳(t,ξ)), ❳(✵,ξ) = ①✵(ξ) ✇✐t❤ ①✵ ≥ ✵ ❛♥❞ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ s❣♥(❳(t,ξ)) = ✵, ♦♥ ∂O. ■♥❢♦r♠❛❧❧②✿ ∆s❣♥(❳) = δ✵(❳)∆❳ +s❣♥′′(❳)|∇❳|✷. ❆✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡ = ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✹ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄ ❘❡❝❛❧❧✿ ∂ ∂t ❳(t,ξ) = ∆❍(❳(t,ξ)−❳ ❝(ξ)), ❲❡ ✇✐❧❧ r❡str✐❝t t♦ t❤❡ s✉♣❡r❝r✐t✐❝❛❧ ❝❛s❡✱ ✐✳❡✳ s✉♣♣♦s✐♥❣ ①✵ ≥ ❳ ❝✳ ❙✉❜st✐t✉t✐♥❣ ❳ → ❳ −❳ ❝ ❛♥❞ ✉s✐♥❣ ❳ ≥ ✵ ②✐❡❧❞s ∂ ∂t ❳(t,ξ) = ∆s❣♥(❳(t,ξ)), ❳(✵,ξ) = ①✵(ξ) ✇✐t❤ ①✵ ≥ ✵ ❛♥❞ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ s❣♥(❳(t,ξ)) = ✵, ♦♥ ∂O. ■♥❢♦r♠❛❧❧②✿ ∆s❣♥(❳) = δ✵(❳)∆❳ +s❣♥′′(❳)|∇❳|✷. ❆✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡ = ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✹ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r ❞❡t❡r♠✐♥✐st✐❝ P❉❊

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r ❞❡t❡r♠✐♥✐st✐❝ P❉❊

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✺ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r s✐♥❣✉❧❛r ❖❉❊

❈♦♥s✐❞❡r t❤❡ s✐♥❣✉❧❛r ❖❉❊ ˙ ❢ = −❝❢ α, α ∈ (✵,✶), ❝ > ✵. ❚❤❡♥✿ (❢ ✶−α)

′ = −(✶−α).

❲❡ ♦❜t❛✐♥ ❢ ✶−α(t) = ❢ ✶−α(✵)−(✶−α)❝t ✇❤✐❝❤ ✐♠♣❧✐❡s ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✻ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r s✐♥❣✉❧❛r ❖❉❊

❈♦♥s✐❞❡r t❤❡ s✐♥❣✉❧❛r ❖❉❊ ˙ ❢ = −❝❢ α, α ∈ (✵,✶), ❝ > ✵. ❚❤❡♥✿ (❢ ✶−α)

′ = −(✶−α).

❲❡ ♦❜t❛✐♥ ❢ ✶−α(t) = ❢ ✶−α(✵)−(✶−α)❝t ✇❤✐❝❤ ✐♠♣❧✐❡s ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✻ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r s✐♥❣✉❧❛r ❖❉❊

❈♦♥s✐❞❡r t❤❡ s✐♥❣✉❧❛r ❖❉❊ ˙ ❢ = −❝❢ α, α ∈ (✵,✶), ❝ > ✵. ❚❤❡♥✿ (❢ ✶−α)

′ = −(✶−α).

❲❡ ♦❜t❛✐♥ ❢ ✶−α(t) = ❢ ✶−α(✵)−(✶−α)❝t ✇❤✐❝❤ ✐♠♣❧✐❡s ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✻ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

❬❉✐❛③✱ ❉✐❛③❀ ❈P❉❊✱ ✶✾✼✾❪ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ✭❋❚❊✮ ✇❛s ✜rst ♣r♦✈❡♥ ❢♦r ∂ ∂t ❳(t,ξ) = ∆s❣♥(❳(t,ξ)). ■♥ ❬❇❛r❜✉❀ ▼▼❆❙✱ ✷✵✶✷❪ ❛♥♦t❤❡r ✭♠♦r❡ r♦❜✉st✮ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ ❡♥❡r❣② ♠❡t❤♦❞s ✇❛s ✐♥tr♦❞✉❝❡❞✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✼ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

■♥❢♦r♠❛❧❧② t❤❡ ♣r♦♦❢ ❜♦✐❧s ❞♦✇♥ t♦ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛♥ ▲✶ ❛♥❞ ❛♥ ▲∞ ❡st✐♠❛t❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✿ ■♥❢♦r♠❛❧ ▲∞ ❡st✐♠❛t❡✿ ❳(t)∞ ≤ ①✵∞, ∀t ≥ ✵. ■♥❢♦r♠❛❧ ▲✶✲❡st✐♠❛t❡✿ ∂t

• O |❳(t,ξ)|❞ξ =
• O s❣♥(❳(t,ξ))∆s❣♥(❳(t,ξ))❞ξ

= −

• O |∇s❣♥(❳(t,ξ))|✷❞ξ

≤ −

• O |s❣♥(❳(t,ξ))|♣❞ξ

✷

♣

≤ −(|{ξ|❳(t,ξ) = ✵}|)

✷ ♣ ,

❢♦r s♦♠❡ ✭❞✐♠❡♥s✐♦♥ ❞❡♣❡♥❞❡♥t✮ ♣ > ✷✳ ◆♦t❡✿

✷ ♣ < ✶✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✽ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

■♥❢♦r♠❛❧❧② t❤❡ ♣r♦♦❢ ❜♦✐❧s ❞♦✇♥ t♦ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛♥ ▲✶ ❛♥❞ ❛♥ ▲∞ ❡st✐♠❛t❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✿ ■♥❢♦r♠❛❧ ▲∞ ❡st✐♠❛t❡✿ ❳(t)∞ ≤ ①✵∞, ∀t ≥ ✵. ■♥❢♦r♠❛❧ ▲✶✲❡st✐♠❛t❡✿ ∂t

• O |❳(t,ξ)|❞ξ =
• O s❣♥(❳(t,ξ))∆s❣♥(❳(t,ξ))❞ξ

= −

• O |∇s❣♥(❳(t,ξ))|✷❞ξ

≤ −

• O |s❣♥(❳(t,ξ))|♣❞ξ

✷

♣

≤ −(|{ξ|❳(t,ξ) = ✵}|)

✷ ♣ ,

❢♦r s♦♠❡ ✭❞✐♠❡♥s✐♦♥ ❞❡♣❡♥❞❡♥t✮ ♣ > ✷✳ ◆♦t❡✿

✷ ♣ < ✶✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✽ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

■♥❢♦r♠❛❧❧② t❤❡ ♣r♦♦❢ ❜♦✐❧s ❞♦✇♥ t♦ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛♥ ▲✶ ❛♥❞ ❛♥ ▲∞ ❡st✐♠❛t❡ ♦❢ t❤❡ s♦❧✉t✐♦♥✿ ■♥❢♦r♠❛❧ ▲∞ ❡st✐♠❛t❡✿ ❳(t)∞ ≤ ①✵∞, ∀t ≥ ✵. ■♥❢♦r♠❛❧ ▲✶✲❡st✐♠❛t❡✿ ∂t

• O |❳(t,ξ)|❞ξ =
• O s❣♥(❳(t,ξ))∆s❣♥(❳(t,ξ))❞ξ

= −

• O |∇s❣♥(❳(t,ξ))|✷❞ξ

≤ −

• O |s❣♥(❳(t,ξ))|♣❞ξ

✷

♣

≤ −(|{ξ|❳(t,ξ) = ✵}|)

✷ ♣ ,

❢♦r s♦♠❡ ✭❞✐♠❡♥s✐♦♥ ❞❡♣❡♥❞❡♥t✮ ♣ > ✷✳ ◆♦t❡✿

✷ ♣ < ✶✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✽ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

❖❜s❡r✈❡

• O |❳(t,ξ)|❞ξ ≤ ❳(t)∞|{ξ|❳(t,ξ) = ✵}|.

≤ ①✵∞|{ξ|❳(t,ξ) = ✵}|. ❯s✐♥❣ t❤✐s ❛❜♦✈❡ ❣✐✈❡s ∂t

• O |❳(t,ξ)|❞ξ ≤ −

✶ ①✵

✷ ♣

∞

• O |❳(t,ξ)|❞ξ

✷

♣

. ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ t❤❡ s✐♥❣✉❧❛r ❖❉❊ ˙ ❢ = −❝❢ α, α ∈ (✵,✶), ❝ > ✵ ❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❤♦❧❞s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✾ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

❖❜s❡r✈❡

• O |❳(t,ξ)|❞ξ ≤ ❳(t)∞|{ξ|❳(t,ξ) = ✵}|.

≤ ①✵∞|{ξ|❳(t,ξ) = ✵}|. ❯s✐♥❣ t❤✐s ❛❜♦✈❡ ❣✐✈❡s ∂t

• O |❳(t,ξ)|❞ξ ≤ −

✶ ①✵

✷ ♣

∞

• O |❳(t,ξ)|❞ξ

✷

♣

. ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ t❤❡ s✐♥❣✉❧❛r ❖❉❊ ˙ ❢ = −❝❢ α, α ∈ (✵,✶), ❝ > ✵ ❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❤♦❧❞s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✾ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ s❡❧❢✲♦r❣❛♥✐③❡❞ ❝r✐t✐❝❛❧✐t②

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈

❖❜s❡r✈❡

• O |❳(t,ξ)|❞ξ ≤ ❳(t)∞|{ξ|❳(t,ξ) = ✵}|.

≤ ①✵∞|{ξ|❳(t,ξ) = ✵}|. ❯s✐♥❣ t❤✐s ❛❜♦✈❡ ❣✐✈❡s ∂t

• O |❳(t,ξ)|❞ξ ≤ −

✶ ①✵

✷ ♣

∞

• O |❳(t,ξ)|❞ξ

✷

♣

. ❲❡ ❛r❡ ❧❡❢t ✇✐t❤ t❤❡ s✐♥❣✉❧❛r ❖❉❊ ˙ ❢ = −❝❢ α, α ∈ (✵,✶), ❝ > ✵ ❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡ s❡❡♥ t❤❛t ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❤♦❧❞s✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✶✾ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✵ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

■♥ ❬❉í❛③✲●✉✐❧❡r❛❀ ❊P▲ ✭❊✉r♦♣❤②s✐❝s ▲❡tt❡rs✮✱ ✶✾✾✹❪✱ ❬●✐❛❝♦♠❡tt✐✱ ❉✐❛③✲●✉✐❧❡r❛❀ P❤②s✳ ❘❡✈✳ ❊✱ ✶✾✾✽❪✱ ❬❉í❛③✲●✉✐❧❡r❛❀ P❤②s✳ ❘❡✈✳ ❆✱ ✶✾✾✷❪ ✐t ✇❛s ♣♦✐♥t❡❞ ♦✉t t❤❛t ✐t ✐s ♠♦r❡ r❡❛❧✐st✐❝ t♦ ✐♥❝❧✉❞❡ st♦❝❤❛st✐❝ ♣❡rt✉r❜❛t✐♦♥s✳ ❚❤✐s ❧❡❛❞s t♦ ❙P❉❊ ♦❢ t❤❡ ❢♦r♠ ❞❳t = ∆❍(❳t −❳ ❝)+❇(❳t −❳ ❝)❞❲t, ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥ts ❇✳ ❲❡ st✉❞② ❧✐♥❡❛r ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✱ ✐✳❡✳ ❞❳t = ∆❍(❳t −❳ ❝)+

◆

∑

❦=✶

❢❦(❳t −❳ ❝)❞β ❦

t .

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✶ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

■♥ ❬❉í❛③✲●✉✐❧❡r❛❀ ❊P▲ ✭❊✉r♦♣❤②s✐❝s ▲❡tt❡rs✮✱ ✶✾✾✹❪✱ ❬●✐❛❝♦♠❡tt✐✱ ❉✐❛③✲●✉✐❧❡r❛❀ P❤②s✳ ❘❡✈✳ ❊✱ ✶✾✾✽❪✱ ❬❉í❛③✲●✉✐❧❡r❛❀ P❤②s✳ ❘❡✈✳ ❆✱ ✶✾✾✷❪ ✐t ✇❛s ♣♦✐♥t❡❞ ♦✉t t❤❛t ✐t ✐s ♠♦r❡ r❡❛❧✐st✐❝ t♦ ✐♥❝❧✉❞❡ st♦❝❤❛st✐❝ ♣❡rt✉r❜❛t✐♦♥s✳ ❚❤✐s ❧❡❛❞s t♦ ❙P❉❊ ♦❢ t❤❡ ❢♦r♠ ❞❳t = ∆❍(❳t −❳ ❝)+❇(❳t −❳ ❝)❞❲t, ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥ts ❇✳ ❲❡ st✉❞② ❧✐♥❡❛r ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✱ ✐✳❡✳ ❞❳t = ∆❍(❳t −❳ ❝)+

◆

∑

❦=✶

❢❦(❳t −❳ ❝)❞β ❦

t .

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✶ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

■♥ ❬❉í❛③✲●✉✐❧❡r❛❀ ❊P▲ ✭❊✉r♦♣❤②s✐❝s ▲❡tt❡rs✮✱ ✶✾✾✹❪✱ ❬●✐❛❝♦♠❡tt✐✱ ❉✐❛③✲●✉✐❧❡r❛❀ P❤②s✳ ❘❡✈✳ ❊✱ ✶✾✾✽❪✱ ❬❉í❛③✲●✉✐❧❡r❛❀ P❤②s✳ ❘❡✈✳ ❆✱ ✶✾✾✷❪ ✐t ✇❛s ♣♦✐♥t❡❞ ♦✉t t❤❛t ✐t ✐s ♠♦r❡ r❡❛❧✐st✐❝ t♦ ✐♥❝❧✉❞❡ st♦❝❤❛st✐❝ ♣❡rt✉r❜❛t✐♦♥s✳ ❚❤✐s ❧❡❛❞s t♦ ❙P❉❊ ♦❢ t❤❡ ❢♦r♠ ❞❳t = ∆❍(❳t −❳ ❝)+❇(❳t −❳ ❝)❞❲t, ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥ts ❇✳ ❲❡ st✉❞② ❧✐♥❡❛r ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✱ ✐✳❡✳ ❞❳t = ∆❍(❳t −❳ ❝)+

◆

∑

❦=✶

❢❦(❳t −❳ ❝)❞β ❦

t .

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✶ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

■♥ ❬❉í❛③✲●✉✐❧❡r❛❀ ❊P▲ ✭❊✉r♦♣❤②s✐❝s ▲❡tt❡rs✮✱ ✶✾✾✹❪✱ ❬●✐❛❝♦♠❡tt✐✱ ❉✐❛③✲●✉✐❧❡r❛❀ P❤②s✳ ❘❡✈✳ ❊✱ ✶✾✾✽❪✱ ❬❉í❛③✲●✉✐❧❡r❛❀ P❤②s✳ ❘❡✈✳ ❆✱ ✶✾✾✷❪ ✐t ✇❛s ♣♦✐♥t❡❞ ♦✉t t❤❛t ✐t ✐s ♠♦r❡ r❡❛❧✐st✐❝ t♦ ✐♥❝❧✉❞❡ st♦❝❤❛st✐❝ ♣❡rt✉r❜❛t✐♦♥s✳ ❚❤✐s ❧❡❛❞s t♦ ❙P❉❊ ♦❢ t❤❡ ❢♦r♠ ❞❳t = ∆❍(❳t −❳ ❝)+❇(❳t −❳ ❝)❞❲t, ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❞✐✛✉s✐♦♥ ❝♦❡✣❝✐❡♥ts ❇✳ ❲❡ st✉❞② ❧✐♥❡❛r ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡✱ ✐✳❡✳ ❞❳t = ∆❍(❳t −❳ ❝)+

◆

∑

❦=✶

❢❦(❳t −❳ ❝)❞β ❦

t .

◗✉❡st✐♦♥✿ ❉♦ ❛✈❛❧❛♥❝❤❡s ❡♥❞ ✐♥ ✜♥✐t❡ t✐♠❡❄

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✶ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

❘❡❝❛❧❧✿ ❞❳t = ∆s❣♥(❳t)+

◆

∑

❦=✶

❢❦❳t❞β ❦

t ,

✇✐t❤ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❡①t✐♥❝t✐♦♥ t✐♠❡ τ✵(ω) := ✐♥❢{t≥✵|❳t(ω) = ✵, ❛✳❡✳ ✐♥ O}. ❲❡ ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣ts✿ ✭❋✶✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t② ❢♦r s♠❛❧❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ P[τ✵ < ∞] > ✵✱ ❢♦r s♠❛❧❧ ❳✵ = ①✵✳ ✭❋✷✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✿ P[τ✵ < ∞] > ✵✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳ ✭❋✸✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✿ P[τ✵ < ∞] = ✶✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✷ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

❘❡❝❛❧❧✿ ❞❳t = ∆s❣♥(❳t)+

◆

∑

❦=✶

❢❦❳t❞β ❦

t ,

✇✐t❤ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❡①t✐♥❝t✐♦♥ t✐♠❡ τ✵(ω) := ✐♥❢{t≥✵|❳t(ω) = ✵, ❛✳❡✳ ✐♥ O}. ❲❡ ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣ts✿ ✭❋✶✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t② ❢♦r s♠❛❧❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ P[τ✵ < ∞] > ✵✱ ❢♦r s♠❛❧❧ ❳✵ = ①✵✳ ✭❋✷✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✿ P[τ✵ < ∞] > ✵✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳ ✭❋✸✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✿ P[τ✵ < ∞] = ✶✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✷ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

❘❡❝❛❧❧✿ ❞❳t = ∆s❣♥(❳t)+

◆

∑

❦=✶

❢❦❳t❞β ❦

t ,

✇✐t❤ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❡①t✐♥❝t✐♦♥ t✐♠❡ τ✵(ω) := ✐♥❢{t≥✵|❳t(ω) = ✵, ❛✳❡✳ ✐♥ O}. ❲❡ ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣ts✿ ✭❋✶✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t② ❢♦r s♠❛❧❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ P[τ✵ < ∞] > ✵✱ ❢♦r s♠❛❧❧ ❳✵ = ①✵✳ ✭❋✷✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✿ P[τ✵ < ∞] > ✵✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳ ✭❋✸✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✿ P[τ✵ < ∞] = ✶✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✷ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

❘❡❝❛❧❧✿ ❞❳t = ∆s❣♥(❳t)+

◆

∑

❦=✶

❢❦❳t❞β ❦

t ,

✇✐t❤ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❡①t✐♥❝t✐♦♥ t✐♠❡ τ✵(ω) := ✐♥❢{t≥✵|❳t(ω) = ✵, ❛✳❡✳ ✐♥ O}. ❲❡ ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣ts✿ ✭❋✶✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t② ❢♦r s♠❛❧❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ P[τ✵ < ∞] > ✵✱ ❢♦r s♠❛❧❧ ❳✵ = ①✵✳ ✭❋✷✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✿ P[τ✵ < ∞] > ✵✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳ ✭❋✸✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✿ P[τ✵ < ∞] = ✶✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✷ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❡ st♦❝❤❛st✐❝ ❇❚❲ ♠♦❞❡❧

❘❡❝❛❧❧✿ ❞❳t = ∆s❣♥(❳t)+

◆

∑

❦=✶

❢❦❳t❞β ❦

t ,

✇✐t❤ ③❡r♦ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❡①t✐♥❝t✐♦♥ t✐♠❡ τ✵(ω) := ✐♥❢{t≥✵|❳t(ω) = ✵, ❛✳❡✳ ✐♥ O}. ❲❡ ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❝❡♣ts✿ ✭❋✶✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t② ❢♦r s♠❛❧❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ P[τ✵ < ∞] > ✵✱ ❢♦r s♠❛❧❧ ❳✵ = ①✵✳ ✭❋✷✮ ❊①t✐♥❝t✐♦♥ ✇✐t❤ ♣♦s✐t✐✈❡ ♣r♦❜❛❜✐❧✐t②✿ P[τ✵ < ∞] > ✵✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳ ✭❋✸✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥✿ P[τ✵ < ∞] = ✶✱ ❢♦r ❛❧❧ ❳✵ = ①✵✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✷ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙♦♠❡ ❦♥♦✇♥ r❡s✉❧ts

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s t♦ ❞❳t ∈ ∆φ(❳t)❞t +

◆

∑

❦=✶

❢❦❳t❞β ❦

t

✇✐t❤ φ ❜❡✐♥❣ ♣♦ss✐❜❧② ♠✉❧t✐✲✈❛❧✉❡❞ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❈▼P✱ ✷✵✵✾❪✳ ■♥ t❤❡ s❛♠❡ ♣❛♣❡r ✭❋✶✮ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ✐s s❤♦✇♥ ❢♦r ❞ = ✶✳ ■♥ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❏▼❆❆✱ ✷✵✶✷❪ t❤✐s ✇❛s ❡①t❡♥❞❡❞ t♦ ♣r♦✈❡ ✭❋✶✮ ❢♦r t❤❡ ❇❚❲ ♠♦❞❡❧ ❢♦r ❞ = ✶✳ ■♥ t❤❡ r❡❝❡♥t ✇♦r❦ ❬❘ö❝❦♥❡r✱ ❲❛♥❣❀ ❏▲▼❙✱ ✷✵✶✸❪ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ❤❛s ❜❡❡♥ s♦❧✈❡❞✳ ■♥ ❝❛s❡ ♦❢ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ❞❳t ∈ ∆s❣♥(❳t)❞t +❞❲t, ❡r❣♦❞✐❝✐t② ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r ❞ = ✶ ✐♥ ❬●❡ss✱ ❚ö❧❧❡❀ ❏▼P❆✱ t♦ ❛♣♣❡❛r❪✳ ■♥ ❬❇❛r❜✉✱ ❘ö❝❦♥❡r❀ ❆❘▼❆✱ ✷✵✶✸❪ ✭❋✶✮ ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r t❤❡ r❡❧❛t❡❞ st♦❝❤❛s✲ t✐❝ t♦t❛❧ ✈❛r✐❛t✐♦♥ ✢♦✇ ❢♦r ❞ ≤ ✸✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✸ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙♦♠❡ ❦♥♦✇♥ r❡s✉❧ts

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s t♦ ❞❳t ∈ ∆φ(❳t)❞t +

◆

∑

❦=✶

❢❦❳t❞β ❦

t

✇✐t❤ φ ❜❡✐♥❣ ♣♦ss✐❜❧② ♠✉❧t✐✲✈❛❧✉❡❞ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❈▼P✱ ✷✵✵✾❪✳ ■♥ t❤❡ s❛♠❡ ♣❛♣❡r ✭❋✶✮ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ✐s s❤♦✇♥ ❢♦r ❞ = ✶✳ ■♥ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❏▼❆❆✱ ✷✵✶✷❪ t❤✐s ✇❛s ❡①t❡♥❞❡❞ t♦ ♣r♦✈❡ ✭❋✶✮ ❢♦r t❤❡ ❇❚❲ ♠♦❞❡❧ ❢♦r ❞ = ✶✳ ■♥ t❤❡ r❡❝❡♥t ✇♦r❦ ❬❘ö❝❦♥❡r✱ ❲❛♥❣❀ ❏▲▼❙✱ ✷✵✶✸❪ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ❤❛s ❜❡❡♥ s♦❧✈❡❞✳ ■♥ ❝❛s❡ ♦❢ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ❞❳t ∈ ∆s❣♥(❳t)❞t +❞❲t, ❡r❣♦❞✐❝✐t② ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r ❞ = ✶ ✐♥ ❬●❡ss✱ ❚ö❧❧❡❀ ❏▼P❆✱ t♦ ❛♣♣❡❛r❪✳ ■♥ ❬❇❛r❜✉✱ ❘ö❝❦♥❡r❀ ❆❘▼❆✱ ✷✵✶✸❪ ✭❋✶✮ ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r t❤❡ r❡❧❛t❡❞ st♦❝❤❛s✲ t✐❝ t♦t❛❧ ✈❛r✐❛t✐♦♥ ✢♦✇ ❢♦r ❞ ≤ ✸✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✸ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙♦♠❡ ❦♥♦✇♥ r❡s✉❧ts

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s t♦ ❞❳t ∈ ∆φ(❳t)❞t +

◆

∑

❦=✶

❢❦❳t❞β ❦

t

✇✐t❤ φ ❜❡✐♥❣ ♣♦ss✐❜❧② ♠✉❧t✐✲✈❛❧✉❡❞ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❈▼P✱ ✷✵✵✾❪✳ ■♥ t❤❡ s❛♠❡ ♣❛♣❡r ✭❋✶✮ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ✐s s❤♦✇♥ ❢♦r ❞ = ✶✳ ■♥ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❏▼❆❆✱ ✷✵✶✷❪ t❤✐s ✇❛s ❡①t❡♥❞❡❞ t♦ ♣r♦✈❡ ✭❋✶✮ ❢♦r t❤❡ ❇❚❲ ♠♦❞❡❧ ❢♦r ❞ = ✶✳ ■♥ t❤❡ r❡❝❡♥t ✇♦r❦ ❬❘ö❝❦♥❡r✱ ❲❛♥❣❀ ❏▲▼❙✱ ✷✵✶✸❪ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ❤❛s ❜❡❡♥ s♦❧✈❡❞✳ ■♥ ❝❛s❡ ♦❢ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ❞❳t ∈ ∆s❣♥(❳t)❞t +❞❲t, ❡r❣♦❞✐❝✐t② ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r ❞ = ✶ ✐♥ ❬●❡ss✱ ❚ö❧❧❡❀ ❏▼P❆✱ t♦ ❛♣♣❡❛r❪✳ ■♥ ❬❇❛r❜✉✱ ❘ö❝❦♥❡r❀ ❆❘▼❆✱ ✷✵✶✸❪ ✭❋✶✮ ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r t❤❡ r❡❧❛t❡❞ st♦❝❤❛s✲ t✐❝ t♦t❛❧ ✈❛r✐❛t✐♦♥ ✢♦✇ ❢♦r ❞ ≤ ✸✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✸ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙♦♠❡ ❦♥♦✇♥ r❡s✉❧ts

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s t♦ ❞❳t ∈ ∆φ(❳t)❞t +

◆

∑

❦=✶

❢❦❳t❞β ❦

t

✇✐t❤ φ ❜❡✐♥❣ ♣♦ss✐❜❧② ♠✉❧t✐✲✈❛❧✉❡❞ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❈▼P✱ ✷✵✵✾❪✳ ■♥ t❤❡ s❛♠❡ ♣❛♣❡r ✭❋✶✮ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ✐s s❤♦✇♥ ❢♦r ❞ = ✶✳ ■♥ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❏▼❆❆✱ ✷✵✶✷❪ t❤✐s ✇❛s ❡①t❡♥❞❡❞ t♦ ♣r♦✈❡ ✭❋✶✮ ❢♦r t❤❡ ❇❚❲ ♠♦❞❡❧ ❢♦r ❞ = ✶✳ ■♥ t❤❡ r❡❝❡♥t ✇♦r❦ ❬❘ö❝❦♥❡r✱ ❲❛♥❣❀ ❏▲▼❙✱ ✷✵✶✸❪ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ❤❛s ❜❡❡♥ s♦❧✈❡❞✳ ■♥ ❝❛s❡ ♦❢ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ❞❳t ∈ ∆s❣♥(❳t)❞t +❞❲t, ❡r❣♦❞✐❝✐t② ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r ❞ = ✶ ✐♥ ❬●❡ss✱ ❚ö❧❧❡❀ ❏▼P❆✱ t♦ ❛♣♣❡❛r❪✳ ■♥ ❬❇❛r❜✉✱ ❘ö❝❦♥❡r❀ ❆❘▼❆✱ ✷✵✶✸❪ ✭❋✶✮ ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r t❤❡ r❡❧❛t❡❞ st♦❝❤❛s✲ t✐❝ t♦t❛❧ ✈❛r✐❛t✐♦♥ ✢♦✇ ❢♦r ❞ ≤ ✸✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✸ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙♦♠❡ ❦♥♦✇♥ r❡s✉❧ts

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s t♦ ❞❳t ∈ ∆φ(❳t)❞t +

◆

∑

❦=✶

❢❦❳t❞β ❦

t

✇✐t❤ φ ❜❡✐♥❣ ♣♦ss✐❜❧② ♠✉❧t✐✲✈❛❧✉❡❞ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❈▼P✱ ✷✵✵✾❪✳ ■♥ t❤❡ s❛♠❡ ♣❛♣❡r ✭❋✶✮ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ✐s s❤♦✇♥ ❢♦r ❞ = ✶✳ ■♥ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❏▼❆❆✱ ✷✵✶✷❪ t❤✐s ✇❛s ❡①t❡♥❞❡❞ t♦ ♣r♦✈❡ ✭❋✶✮ ❢♦r t❤❡ ❇❚❲ ♠♦❞❡❧ ❢♦r ❞ = ✶✳ ■♥ t❤❡ r❡❝❡♥t ✇♦r❦ ❬❘ö❝❦♥❡r✱ ❲❛♥❣❀ ❏▲▼❙✱ ✷✵✶✸❪ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ❤❛s ❜❡❡♥ s♦❧✈❡❞✳ ■♥ ❝❛s❡ ♦❢ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ❞❳t ∈ ∆s❣♥(❳t)❞t +❞❲t, ❡r❣♦❞✐❝✐t② ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r ❞ = ✶ ✐♥ ❬●❡ss✱ ❚ö❧❧❡❀ ❏▼P❆✱ t♦ ❛♣♣❡❛r❪✳ ■♥ ❬❇❛r❜✉✱ ❘ö❝❦♥❡r❀ ❆❘▼❆✱ ✷✵✶✸❪ ✭❋✶✮ ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r t❤❡ r❡❧❛t❡❞ st♦❝❤❛s✲ t✐❝ t♦t❛❧ ✈❛r✐❛t✐♦♥ ✢♦✇ ❢♦r ❞ ≤ ✸✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✸ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙♦♠❡ ❦♥♦✇♥ r❡s✉❧ts

❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ s♦❧✉t✐♦♥s t♦ ❞❳t ∈ ∆φ(❳t)❞t +

◆

∑

❦=✶

❢❦❳t❞β ❦

t

✇✐t❤ φ ❜❡✐♥❣ ♣♦ss✐❜❧② ♠✉❧t✐✲✈❛❧✉❡❞ ❣♦❡s ❜❛❝❦ t♦ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❈▼P✱ ✷✵✵✾❪✳ ■♥ t❤❡ s❛♠❡ ♣❛♣❡r ✭❋✶✮ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ✐s s❤♦✇♥ ❢♦r ❞ = ✶✳ ■♥ ❬❇❛r❜✉✱ ❉❛ Pr❛t♦✱ ❘ö❝❦♥❡r❀ ❏▼❆❆✱ ✷✵✶✷❪ t❤✐s ✇❛s ❡①t❡♥❞❡❞ t♦ ♣r♦✈❡ ✭❋✶✮ ❢♦r t❤❡ ❇❚❲ ♠♦❞❡❧ ❢♦r ❞ = ✶✳ ■♥ t❤❡ r❡❝❡♥t ✇♦r❦ ❬❘ö❝❦♥❡r✱ ❲❛♥❣❀ ❏▲▼❙✱ ✷✵✶✸❪ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r t❤❡ ❩❤❛♥❣ ♠♦❞❡❧ ❤❛s ❜❡❡♥ s♦❧✈❡❞✳ ■♥ ❝❛s❡ ♦❢ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ❞❳t ∈ ∆s❣♥(❳t)❞t +❞❲t, ❡r❣♦❞✐❝✐t② ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r ❞ = ✶ ✐♥ ❬●❡ss✱ ❚ö❧❧❡❀ ❏▼P❆✱ t♦ ❛♣♣❡❛r❪✳ ■♥ ❬❇❛r❜✉✱ ❘ö❝❦♥❡r❀ ❆❘▼❆✱ ✷✵✶✸❪ ✭❋✶✮ ❤❛s ❜❡❡♥ s❤♦✇♥ ❢♦r t❤❡ r❡❧❛t❡❞ st♦❝❤❛s✲ t✐❝ t♦t❛❧ ✈❛r✐❛t✐♦♥ ✢♦✇ ❢♦r ❞ ≤ ✸✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✸ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ✭▼❛✐♥ r❡s✉❧t✮ ▲❡t ①✵ ∈ ▲∞(O)✱ ❳ ❜❡ t❤❡ ✉♥✐q✉❡ ✈❛r✐❛t✐♦♥❛❧ s♦❧✉t✐♦♥ t♦ ❇❚❲ ❛♥❞ ❧❡t τ✵(ω) := ✐♥❢{t ≥ ✵|❳t(ω) = ✵, ❢♦r ❛✳❡✳ ξ ∈ O}. ❚❤❡♥ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❤♦❧❞s✱ ✐✳❡✳ P[τ✵ < ∞] = ✶. ❋♦r ❡✈❡r② ♣ > ❞

✷ ∨ ✶✱ t❤❡ ❡①t✐♥❝t✐♦♥ t✐♠❡ τ✵(ω) ♠❛② ❜❡ ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❢♦r ①✵

❜♦✉♥❞❡❞ ✐♥ ▲♣(O)✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✹ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ✭▼❛✐♥ r❡s✉❧t✮ ▲❡t ①✵ ∈ ▲∞(O)✱ ❳ ❜❡ t❤❡ ✉♥✐q✉❡ ✈❛r✐❛t✐♦♥❛❧ s♦❧✉t✐♦♥ t♦ ❇❚❲ ❛♥❞ ❧❡t τ✵(ω) := ✐♥❢{t ≥ ✵|❳t(ω) = ✵, ❢♦r ❛✳❡✳ ξ ∈ O}. ❚❤❡♥ ✜♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❤♦❧❞s✱ ✐✳❡✳ P[τ✵ < ∞] = ✶. ❋♦r ❡✈❡r② ♣ > ❞

✷ ∨ ✶✱ t❤❡ ❡①t✐♥❝t✐♦♥ t✐♠❡ τ✵(ω) ♠❛② ❜❡ ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❢♦r ①✵

❜♦✉♥❞❡❞ ✐♥ ▲♣(O)✳

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✹ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚r❛♥s❢♦r♠❛t✐♦♥

❘❡❝❛❧❧✿ ❞❳t = ∆s❣♥(❳t)+

◆

∑

❦=✶

❢❦❳t❞β ❦

t ,

❖✉r ❛♣♣r♦❛❝❤ t♦ ❋❚❊ ✇✐❧❧ ❜❡ ❜❛s❡❞ ♦♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥✿ ❙❡t µt := ∑◆

❦=✶ ❢❦β ❦ t ✱ ˜

µ := ∑◆

❦=✶ ❢ ✷ ❦ ❛♥❞ ❨t := ❡−µt❳t✳ ❆♥ ✐♥❢♦r♠❛❧ ❝❛❧❝✉❧❛t✐♦♥

s❤♦✇s ∂❨t ∈ ❡µt∆s❣♥(❨t)− ˜ µ❨t. ✭✯✮ ❈♦♠♣❛r❡ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ s❡tt✐♥❣✿ ∂❨t ∈ ∆s❣♥(❨t).

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✺ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚r❛♥s❢♦r♠❛t✐♦♥

❘❡❝❛❧❧✿ ❞❳t = ∆s❣♥(❳t)+

◆

∑

❦=✶

❢❦❳t❞β ❦

t ,

❖✉r ❛♣♣r♦❛❝❤ t♦ ❋❚❊ ✇✐❧❧ ❜❡ ❜❛s❡❞ ♦♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥✿ ❙❡t µt := ∑◆

❦=✶ ❢❦β ❦ t ✱ ˜

µ := ∑◆

❦=✶ ❢ ✷ ❦ ❛♥❞ ❨t := ❡−µt❳t✳ ❆♥ ✐♥❢♦r♠❛❧ ❝❛❧❝✉❧❛t✐♦♥

s❤♦✇s ∂❨t ∈ ❡µt∆s❣♥(❨t)− ˜ µ❨t. ✭✯✮ ❈♦♠♣❛r❡ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ s❡tt✐♥❣✿ ∂❨t ∈ ∆s❣♥(❨t).

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✺ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚r❛♥s❢♦r♠❛t✐♦♥

❘❡❝❛❧❧✿ ❞❳t = ∆s❣♥(❳t)+

◆

∑

❦=✶

❢❦❳t❞β ❦

t ,

❖✉r ❛♣♣r♦❛❝❤ t♦ ❋❚❊ ✇✐❧❧ ❜❡ ❜❛s❡❞ ♦♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥✿ ❙❡t µt := ∑◆

❦=✶ ❢❦β ❦ t ✱ ˜

µ := ∑◆

❦=✶ ❢ ✷ ❦ ❛♥❞ ❨t := ❡−µt❳t✳ ❆♥ ✐♥❢♦r♠❛❧ ❝❛❧❝✉❧❛t✐♦♥

s❤♦✇s ∂❨t ∈ ❡µt∆s❣♥(❨t)− ˜ µ❨t. ✭✯✮ ❈♦♠♣❛r❡ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ s❡tt✐♥❣✿ ∂❨t ∈ ∆s❣♥(❨t).

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✺ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢

❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✐♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♣r♦♦❢✿

✶

❆ ✉♥✐❢♦r♠ ❝♦♥tr♦❧ ♦♥ ❳t♣ ❢♦r ❛❧❧ ♣ ≥ ✶✳

✷

❆♥ ❡♥❡r❣② ✐♥❡q✉❛❧✐t② ❢♦r ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠✳

❖♥ ❛♥ ✐♥t✉✐t✐✈❡ ❧❡✈❡❧ t❤❡ ❛r❣✉♠❡♥ts ❜❡❝♦♠❡ ❝❧❡❛r ❜② ❛♣♣r♦①✐♠❛t✐♥❣ r [♠] := |r|♠−✶r → s❣♥, ❢♦r ♠ ↓ ✵. ❚♦ ♠❛❦❡ t❤❡s❡ ❛r❣✉♠❡♥ts r✐❣♦r♦✉s✱ ✐♥ ❢❛❝t ❛ ❞✐✛❡r❡♥t ✭♥♦♥✲s✐♥❣✉❧❛r✱ ♥♦♥✲ ❞❡❣❡♥❡r❛t❡✮ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s❣♥ ✐s ✉s❡❞✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡t ❨t ❜❡ ❛ s♦❧✉t✐♦♥ t♦ ∂t❨t ∈ ❡µt∆❨ [♠]

t

− ˜ µ❨t.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✻ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢

❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✐♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♣r♦♦❢✿

✶

❆ ✉♥✐❢♦r♠ ❝♦♥tr♦❧ ♦♥ ❳t♣ ❢♦r ❛❧❧ ♣ ≥ ✶✳

✷

❆♥ ❡♥❡r❣② ✐♥❡q✉❛❧✐t② ❢♦r ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠✳

❖♥ ❛♥ ✐♥t✉✐t✐✈❡ ❧❡✈❡❧ t❤❡ ❛r❣✉♠❡♥ts ❜❡❝♦♠❡ ❝❧❡❛r ❜② ❛♣♣r♦①✐♠❛t✐♥❣ r [♠] := |r|♠−✶r → s❣♥, ❢♦r ♠ ↓ ✵. ❚♦ ♠❛❦❡ t❤❡s❡ ❛r❣✉♠❡♥ts r✐❣♦r♦✉s✱ ✐♥ ❢❛❝t ❛ ❞✐✛❡r❡♥t ✭♥♦♥✲s✐♥❣✉❧❛r✱ ♥♦♥✲ ❞❡❣❡♥❡r❛t❡✮ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s❣♥ ✐s ✉s❡❞✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡t ❨t ❜❡ ❛ s♦❧✉t✐♦♥ t♦ ∂t❨t ∈ ❡µt∆❨ [♠]

t

− ˜ µ❨t.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✻ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢

❚❤❡r❡ ❛r❡ t✇♦ ♠❛✐♥ ✐♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♣r♦♦❢✿

✶

❆ ✉♥✐❢♦r♠ ❝♦♥tr♦❧ ♦♥ ❳t♣ ❢♦r ❛❧❧ ♣ ≥ ✶✳

✷

❆♥ ❡♥❡r❣② ✐♥❡q✉❛❧✐t② ❢♦r ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠✳

❖♥ ❛♥ ✐♥t✉✐t✐✈❡ ❧❡✈❡❧ t❤❡ ❛r❣✉♠❡♥ts ❜❡❝♦♠❡ ❝❧❡❛r ❜② ❛♣♣r♦①✐♠❛t✐♥❣ r [♠] := |r|♠−✶r → s❣♥, ❢♦r ♠ ↓ ✵. ❚♦ ♠❛❦❡ t❤❡s❡ ❛r❣✉♠❡♥ts r✐❣♦r♦✉s✱ ✐♥ ❢❛❝t ❛ ❞✐✛❡r❡♥t ✭♥♦♥✲s✐♥❣✉❧❛r✱ ♥♦♥✲ ❞❡❣❡♥❡r❛t❡✮ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ s❣♥ ✐s ✉s❡❞✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡t ❨t ❜❡ ❛ s♦❧✉t✐♦♥ t♦ ∂t❨t ∈ ❡µt∆❨ [♠]

t

− ˜ µ❨t.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✻ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✶✿ ■♥❢♦r♠❛❧ ▲♣ ❜♦✉♥❞

❙t❡♣ ✶✿ ❆ ✉♥✐❢♦r♠ ❝♦♥tr♦❧ ♦♥ ❳t♣ ❢♦r ❛❧❧ ♣ ≥ ✶✳ ❲❡ ♠❛② ✐♥❢♦r♠❛❧❧② ❝♦♠♣✉t❡ ❢♦r ❛❧❧ ♣ ≥ ✶✿ ∂t

• O |❨t|♣❞ξ =♣
• O ❨t

[♣−✶]❡µt∆❨t [♠]❞ξ

=− ✹(♣ −✶)♠♣ (♣ +♠ −✶)✷

• O ❡µt
• ∇|❨t|

♣+♠−✶ ✷

✷ ❞ξ + ♣♠ ♣ +♠ −✶

• O |❨t|♣+♠−✶∆❡µt❞ξ.

❚❛❦✐♥❣ ♣ > ✶ ❛♥❞ t❤❡♥ ♠ → ✵ ✇❡ ♠❛② ✏❞❡❞✉❝❡✑ ❢r♦♠ t❤✐s ∂t

• O |❨t|♣❞ξ ≤ ✵.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✼ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✶✿ ■♥❢♦r♠❛❧ ▲♣ ❜♦✉♥❞

❙t❡♣ ✶✿ ❆ ✉♥✐❢♦r♠ ❝♦♥tr♦❧ ♦♥ ❳t♣ ❢♦r ❛❧❧ ♣ ≥ ✶✳ ❲❡ ♠❛② ✐♥❢♦r♠❛❧❧② ❝♦♠♣✉t❡ ❢♦r ❛❧❧ ♣ ≥ ✶✿ ∂t

• O |❨t|♣❞ξ =♣
• O ❨t

[♣−✶]❡µt∆❨t [♠]❞ξ

=− ✹(♣ −✶)♠♣ (♣ +♠ −✶)✷

• O ❡µt
• ∇|❨t|

♣+♠−✶ ✷

✷ ❞ξ + ♣♠ ♣ +♠ −✶

• O |❨t|♣+♠−✶∆❡µt❞ξ.

❚❛❦✐♥❣ ♣ > ✶ ❛♥❞ t❤❡♥ ♠ → ✵ ✇❡ ♠❛② ✏❞❡❞✉❝❡✑ ❢r♦♠ t❤✐s ∂t

• O |❨t|♣❞ξ ≤ ✵.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✼ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✶✿ ■♥❢♦r♠❛❧ ▲♣ ❜♦✉♥❞

❙t❡♣ ✶✿ ❆ ✉♥✐❢♦r♠ ❝♦♥tr♦❧ ♦♥ ❳t♣ ❢♦r ❛❧❧ ♣ ≥ ✶✳ ❲❡ ♠❛② ✐♥❢♦r♠❛❧❧② ❝♦♠♣✉t❡ ❢♦r ❛❧❧ ♣ ≥ ✶✿ ∂t

• O |❨t|♣❞ξ =♣
• O ❨t

[♣−✶]❡µt∆❨t [♠]❞ξ

=− ✹(♣ −✶)♠♣ (♣ +♠ −✶)✷

• O ❡µt
• ∇|❨t|

♣+♠−✶ ✷

✷ ❞ξ + ♣♠ ♣ +♠ −✶

• O |❨t|♣+♠−✶∆❡µt❞ξ.

❚❛❦✐♥❣ ♣ > ✶ ❛♥❞ t❤❡♥ ♠ → ✵ ✇❡ ♠❛② ✏❞❡❞✉❝❡✑ ❢r♦♠ t❤✐s ∂t

• O |❨t|♣❞ξ ≤ ✵.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✼ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✷✿ ■♥❢♦r♠❛❧ ✏▲✶✑ ❜♦✉♥❞

❙t❡♣ ✷✿ ❆♥ ❡♥❡r❣② ✐♥❡q✉❛❧✐t② ❢♦r ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠✳ ∂t

• O |❨t|♣❞ξ = − ✹(♣ −✶)♠♣

(♣ +♠ −✶)✷

• O ❡µt
• ∇|❨t|

♣+♠−✶ ✷

✷ ❞ξ + ♣♠ ♣ +♠ −✶

• O |❨t|♣+♠−✶∆❡µt❞ξ, ♣ ≥ ✶.

❈❤♦♦s❡ ♣ = ♠ +✶ ❛♥❞ ❧❡t ♠ → ✵✳ ❲❡ ♦❜t❛✐♥ ∂t

• O |❨t|❞ξ = −
• O ❡µt (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O ∆❡µt❞ξ

❘❡❝❛❧❧✿ ❞❡t❡r♠✐♥✐st✐❝ ❝❛s❡ ∂t

• O |❨t|❞ξ = −
• O |∇s❣♥(❨t)|✷❞ξ.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✽ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✷✿ ■♥❢♦r♠❛❧ ✏▲✶✑ ❜♦✉♥❞

❙t❡♣ ✷✿ ❆♥ ❡♥❡r❣② ✐♥❡q✉❛❧✐t② ❢♦r ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠✳ ∂t

• O |❨t|♣❞ξ = − ✹(♣ −✶)♠♣

(♣ +♠ −✶)✷

• O ❡µt
• ∇|❨t|

♣+♠−✶ ✷

✷ ❞ξ + ♣♠ ♣ +♠ −✶

• O |❨t|♣+♠−✶∆❡µt❞ξ, ♣ ≥ ✶.

❈❤♦♦s❡ ♣ = ♠ +✶ ❛♥❞ ❧❡t ♠ → ✵✳ ❲❡ ♦❜t❛✐♥ ∂t

• O |❨t|❞ξ = −
• O ❡µt (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O ∆❡µt❞ξ

❘❡❝❛❧❧✿ ❞❡t❡r♠✐♥✐st✐❝ ❝❛s❡ ∂t

• O |❨t|❞ξ = −
• O |∇s❣♥(❨t)|✷❞ξ.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✽ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✷✿ ■♥❢♦r♠❛❧ ✏▲✶✑ ❜♦✉♥❞

❙t❡♣ ✷✿ ❆♥ ❡♥❡r❣② ✐♥❡q✉❛❧✐t② ❢♦r ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠✳ ∂t

• O |❨t|♣❞ξ = − ✹(♣ −✶)♠♣

(♣ +♠ −✶)✷

• O ❡µt
• ∇|❨t|

♣+♠−✶ ✷

✷ ❞ξ + ♣♠ ♣ +♠ −✶

• O |❨t|♣+♠−✶∆❡µt❞ξ, ♣ ≥ ✶.

❈❤♦♦s❡ ♣ = ♠ +✶ ❛♥❞ ❧❡t ♠ → ✵✳ ❲❡ ♦❜t❛✐♥ ∂t

• O |❨t|❞ξ = −
• O ❡µt (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O ∆❡µt❞ξ

❘❡❝❛❧❧✿ ❞❡t❡r♠✐♥✐st✐❝ ❝❛s❡ ∂t

• O |❨t|❞ξ = −
• O |∇s❣♥(❨t)|✷❞ξ.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✽ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✷✿ ■♥❢♦r♠❛❧ ✏▲✶✑ ❜♦✉♥❞

❙t❡♣ ✷✿ ❆♥ ❡♥❡r❣② ✐♥❡q✉❛❧✐t② ❢♦r ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠✳ ∂t

• O |❨t|♣❞ξ = − ✹(♣ −✶)♠♣

(♣ +♠ −✶)✷

• O ❡µt
• ∇|❨t|

♣+♠−✶ ✷

✷ ❞ξ + ♣♠ ♣ +♠ −✶

• O |❨t|♣+♠−✶∆❡µt❞ξ, ♣ ≥ ✶.

❈❤♦♦s❡ ♣ = ♠ +✶ ❛♥❞ ❧❡t ♠ → ✵✳ ❲❡ ♦❜t❛✐♥ ∂t

• O |❨t|❞ξ = −
• O ❡µt (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O ∆❡µt❞ξ

❘❡❝❛❧❧✿ ❞❡t❡r♠✐♥✐st✐❝ ❝❛s❡ ∂t

• O |❨t|❞ξ = −
• O |∇s❣♥(❨t)|✷❞ξ.

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✽ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✷✿ ■♥❢♦r♠❛❧ ✏▲✶✑ ❜♦✉♥❞

❑❡② tr✐❝❦✿ ❯s❡ ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠ ▲❡t ϕ ❜❡ t❤❡ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ ∆ϕ = −✶, ♦♥ O ϕ = ✶, ♦♥ ∂O. ◆♦t❡ ✶ ≤ ϕ ≤ ϕ∞ =: ❈ϕ✳ ❲❡ ✐♥❢♦r♠❛❧❧② ❝♦♠♣✉t❡ ∂t

• O ϕ|❨t|❞ξ =−
• O ϕ❡µt (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O ∆(ϕ❡µt)❞ξ.

◆♦t❡ ∆(ϕ❡µt) = −❡µt +✷∇ϕ ·∇❡µt +ϕ∆❡µt ❤❛s ❛ ♥❡❣❛t✐✈❡ s✐❣♥ ❢♦r s♠❛❧❧ t✐♠❡s ✭❡µt ≈ ✶✮✦ ❙❤✐❢t t❤❡ ✐♥✐t✐❛❧ t✐♠❡ ∂t

• O ❡−µsϕ|❨t|❞ξ = −
• O ❡µt−µsϕ (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O s❣♥(❨t)✷∆❡µt−µsϕ❞ξ

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✾ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✷✿ ■♥❢♦r♠❛❧ ✏▲✶✑ ❜♦✉♥❞

❑❡② tr✐❝❦✿ ❯s❡ ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠ ▲❡t ϕ ❜❡ t❤❡ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ ∆ϕ = −✶, ♦♥ O ϕ = ✶, ♦♥ ∂O. ◆♦t❡ ✶ ≤ ϕ ≤ ϕ∞ =: ❈ϕ✳ ❲❡ ✐♥❢♦r♠❛❧❧② ❝♦♠♣✉t❡ ∂t

• O ϕ|❨t|❞ξ =−
• O ϕ❡µt (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O ∆(ϕ❡µt)❞ξ.

◆♦t❡ ∆(ϕ❡µt) = −❡µt +✷∇ϕ ·∇❡µt +ϕ∆❡µt ❤❛s ❛ ♥❡❣❛t✐✈❡ s✐❣♥ ❢♦r s♠❛❧❧ t✐♠❡s ✭❡µt ≈ ✶✮✦ ❙❤✐❢t t❤❡ ✐♥✐t✐❛❧ t✐♠❡ ∂t

• O ❡−µsϕ|❨t|❞ξ = −
• O ❡µt−µsϕ (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O s❣♥(❨t)✷∆❡µt−µsϕ❞ξ

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✾ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✷✿ ■♥❢♦r♠❛❧ ✏▲✶✑ ❜♦✉♥❞

❑❡② tr✐❝❦✿ ❯s❡ ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠ ▲❡t ϕ ❜❡ t❤❡ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ ∆ϕ = −✶, ♦♥ O ϕ = ✶, ♦♥ ∂O. ◆♦t❡ ✶ ≤ ϕ ≤ ϕ∞ =: ❈ϕ✳ ❲❡ ✐♥❢♦r♠❛❧❧② ❝♦♠♣✉t❡ ∂t

• O ϕ|❨t|❞ξ =−
• O ϕ❡µt (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O ∆(ϕ❡µt)❞ξ.

◆♦t❡ ∆(ϕ❡µt) = −❡µt +✷∇ϕ ·∇❡µt +ϕ∆❡µt ❤❛s ❛ ♥❡❣❛t✐✈❡ s✐❣♥ ❢♦r s♠❛❧❧ t✐♠❡s ✭❡µt ≈ ✶✮✦ ❙❤✐❢t t❤❡ ✐♥✐t✐❛❧ t✐♠❡ ∂t

• O ❡−µsϕ|❨t|❞ξ = −
• O ❡µt−µsϕ (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O s❣♥(❨t)✷∆❡µt−µsϕ❞ξ

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✾ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❙t❡♣ ✷✿ ■♥❢♦r♠❛❧ ✏▲✶✑ ❜♦✉♥❞

❑❡② tr✐❝❦✿ ❯s❡ ❛ ✇❡✐❣❤t❡❞ ▲✶✲♥♦r♠ ▲❡t ϕ ❜❡ t❤❡ ❝❧❛ss✐❝❛❧ s♦❧✉t✐♦♥ t♦ ∆ϕ = −✶, ♦♥ O ϕ = ✶, ♦♥ ∂O. ◆♦t❡ ✶ ≤ ϕ ≤ ϕ∞ =: ❈ϕ✳ ❲❡ ✐♥❢♦r♠❛❧❧② ❝♦♠♣✉t❡ ∂t

• O ϕ|❨t|❞ξ =−
• O ϕ❡µt (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O ∆(ϕ❡µt)❞ξ.

◆♦t❡ ∆(ϕ❡µt) = −❡µt +✷∇ϕ ·∇❡µt +ϕ∆❡µt ❤❛s ❛ ♥❡❣❛t✐✈❡ s✐❣♥ ❢♦r s♠❛❧❧ t✐♠❡s ✭❡µt ≈ ✶✮✦ ❙❤✐❢t t❤❡ ✐♥✐t✐❛❧ t✐♠❡ ∂t

• O ❡−µsϕ|❨t|❞ξ = −
• O ❡µt−µsϕ (∇s❣♥(❨t))✷ ❞ξ + ✶

✷

• O s❣♥(❨t)✷∆❡µt−µsϕ❞ξ

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✷✾ ✴ ✸✵

❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❢♦r st♦❝❤❛st✐❝ ❇❚❲

❚❤❛♥❦s

❚❤❛♥❦s✦

❇✳ ●❡ss ✭❯♥✐✈❡rs✐tät ❇✐❡❧❡❢❡❧❞✮ ❋✐♥✐t❡ t✐♠❡ ❡①t✐♥❝t✐♦♥ ❛♥❞ ❙❖❈✳ ✸✵ ✴ ✸✵