❊ ❧ ❡ ❝ t r ♦♥ ✐ ❝
❏♦
✉r ♥ ❛ ❧ ♦ ❢
P r ♦
❜ ❛ ❜ ✐ ❧ ✐ t ② ❱♦❧✳ ✶✶ ✭✷✵✵✻✮✱ P❛♣❡r ♥♦✳ ✾✱ ♣❛❣❡s ✷✹✾✕✷✼✺✳
❏♦✉r♥❛❧ ❯❘▲
❤tt♣✿✴✴✇✇✇✳♠❛t❤✳✇❛s❤✐♥❣t♦♥✳❡❞✉✴∼❡❥♣❡❝♣✴
❖♥ r❛♥❞♦♠ ✇❛❧❦ s✐♠✉❧❛t✐♦♥ ♦❢ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧
❞✐✛✉s✐♦♥ ♣r♦❝❡ss❡s ✇✐t❤ ❞✐s❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts
P✐❡rr❡ ➱t♦ré
∗Pr♦❥❡t ❖▼❊●❆✱ ■❊❈◆✱ ❇P ✷✸✾✱ ❋✲✺✹✺✵✻ ❱❛♥❞÷✉✈r❡✲❧ès✲◆❛♥❝② ❈❊❉❊❳
P✐❡rr❡✳❊t♦r❡❅✐❡❝♥✳✉✲♥❛♥❝②✳❢r
❤tt♣✿✴✴✇✇✇✳✐❡❝♥✳✉✲♥❛♥❝②✳❢r✴⑦❡t♦r❡✴
❆❜str❛❝t
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ♣r♦✈✐❞❡ ❛ s❝❤❡♠❡ ❢♦r s✐♠✉❧❛t✐♥❣ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❝❡ss❡s ❣❡♥❡r❛t❡❞ ❜② ❞✐✈❡r❣❡♥❝❡ ♦r ♥♦♥✲❞✐✈❡r❣❡♥❝❡ ❢♦r♠ ♦♣❡r❛t♦rs ✇✐t❤ ❞✐s❝♦♥t✐♥✉♦✉s ❝♦❡✣❝✐❡♥ts✳ ❲❡ ✉s❡ ❛ s♣❛❝❡ ❜✐❥❡❝t✐♦♥ t♦ tr❛♥s❢♦r♠ s✉❝❤ ❛ ♣r♦❝❡ss ✐♥ ❛♥♦t❤❡r ♦♥❡ t❤❛t ❜❡❤❛✈❡s ❧♦❝❛❧❧② ❧✐❦❡ ❛ ❙❦❡✇ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ■♥❞❡❡❞ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❙❦❡✇ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❝❛♥ ❡❛s✐❧② ❜❡ ❛♣♣r♦❛❝❤❡❞ ❜② ❛♥ ❛s②♠♠❡tr✐❝ r❛♥❞♦♠ ✇❛❧❦✳
❑❡② ✇♦r❞s✿ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞s✱ r❛♥❞♦♠ ✇❛❧❦✱ ❙❦❡✇ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❝❡ss✱ ❞✐✈❡r❣❡♥❝❡ ❢♦r♠ ♦♣❡r❛t♦r✱ ❧♦❝❛❧ t✐♠❡✳
❆▼❙ ✷✵✵✵ ❙✉❜❥❡❝t ❈❧❛ss✐✜❝❛t✐♦♥✿ Pr✐♠❛r② ✻✻✵❏✻✵✱ ✻✺❈✳
❙✉❜♠✐tt❡❞ t♦ ❊❏P ♦♥ ▼❛② ✶✶✱ ✷✵✵✺✳ ❋✐♥❛❧ ✈❡rs✐♦♥ ❛❝❝❡♣t❡❞ ♦♥ ❋❡❜r✉❛r② ✷✶✱ ✷✵✵✻✳
∗❙✉♣♣♦rt ❛❝❦♥♦✇❧❡❞❣❡♠❡♥t✿ ❚❤✐s ✇♦r❦ ❤❛s ❜❡❡♥ s✉♣♣♦rt❡❞ ❜② t❤❡ ●❞❘ ▼❖▼❆❙✳
✇❤❡r❡ t❤❡βn
k✬s ❛r❡ ❡①♣❧✐❝✐t❡❧② ❦♥♦✇♥✳ ❚❤✉sYn ❜❡❤❛✈❡s ❛r♦✉♥❞ ❡❛❝❤ k/n❧✐❦❡ ❛ ❙❇▼ ♦❢ ♣❛r❛♠❡t❡r βkn
✭s❡❡ ❙✉❜s❡❝t✐♦♥✺✳✷❢♦r ❛ ❜r✐❡❢ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❙❇▼✮✳ ❚❤❛t ✐s✱ ❤❡✉r✐st✐❝❛❧❧②✱Yn✇❤❡♥ ✐♥k/n♠♦✈❡s ✉♣
✇✐t❤ ♣r♦❜❛❜✐❧✐t②(βn
k + 1)/2❛♥❞ ❞♦✇♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t②(βkn−1)/2✱ ❛♥❞ ❜❡❤❛✈❡s ❧✐❦❡ ❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥
♠♦t✐♦♥ ❡❧s❡✇❤❡r❡✳ ❚❤✉s ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ t❤❡ ❣r✐❞{k/n:k∈Z}❝❛♥ r❡✢❡❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢Yn❛s ✇❛s
s❤♦✇♥ ✐♥ ❬▲❡❣✽✺❪✳ ❲❡ ✉s❡ t❤✐s t♦ ✜♥❛❧❧② ❝♦♥str✉❝t ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥Ybn ♦❢Yn✳
❲❡ ♦❜t❛✐♥ ❛ ✈❡r② ❡❛s② t♦ ✐♠♣❧❡♠❡♥t ❛❧❣♦r✐t❤♠ t❤❛t ♦♥❧② r❡q✉✐r❡s s✐♠✉❧❛t✐♦♥s ♦❢ ❇❡r♥♦✉❧❧✐ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ❲❡ ❡st✐♠❛t❡ t❤❡ s♣❡❡❞ ♦❢ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦✉r ❛❧❣♦r✐t❤♠ ❜② ♠✐①✐♥❣ ❛♥ ❡st✐♠❛t❡ ♦❢ ❛ ✇❡❛❦ ❡rr♦r ❛♥❞ ❛♥ ❡st✐♠❛t❡ ♦❢ ❛ str♦♥❣ ❡rr♦r✳ ■♥❞❡❡❞ ❝♦♠♣✉t✐♥❣ t❤❡ str♦♥❣ ❡rr♦r ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ♣r❡s❡♥ts ❞✐✣❝✉❧t✐❡s ✇❡ ✇❡r❡ ♥♦t ❛❜❧❡ t♦ ♦✈❡r❝♦♠❡✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ❝♦♠♣✉t✐♥❣ ❞✐r❡❝t❧② t❤❡ ✇❡❛❦ ❡rr♦r ✇✐t❤♦✉t ✉s✐♥❣ ❛ str♦♥❣ ❡rr♦r ❡st✐♠❛t❡ ✇♦✉❧❞ ❧❡❛❞ t♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❝♦♠♣✉t❛t✐♦♥s ✇✐t❤♦✉t ❛♥② ✐♠♣r♦✈❡♠❡♥t ♦❢ t❤❡ s♣❡❡❞ ♦❢ ❝♦♥✈❡r❣❡♥❝❡✿ ❜❛s✐❝❛❧❧② ♦✉r ❛♣♣r♦❛❝❤ r❡❧✐❡s ♦♥ t❤❡ ❉♦♥s❦❡r t❤❡♦r❡♠ ❛♥❞ ✇❡ ❝❛♥♥♦t ❣❡t ❜❡tt❡r t❤❛♥ ❛♥ ❡rr♦r ✐♥O(n−1/2)✳ ▼♦r❡♦✈❡r t♦ ♠❛❦❡ s✉❝❤ ❝♦♠♣✉t❛t✐♦♥s ✇❡ s❤♦✉❧❞ r❡q✉✐r❡ ❛❞❞✐t✐♦♥♥❛❧
s♠♦♦t❤♥❡ss ♦♥ t❤❡ ❞❛t❛ ✭s❡❡ ❬▼❛r✵✹❪✮✳
❲❡ ✜♥❛❧❧② ♠❛❦❡ ♥✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts✿ t❤❡ ♣r♦♣♦s❡❞ s❝❤❡♠❡ ❛♣♣❡❛rs t♦ ❜❡ s❛t✐s❢②✐♥❣ ❝♦♠♣❛r❡❞ t♦ t❤❡ ♦♥❡s ♣r♦♣♦s❡❞ ✐♥ ❬▼❛r✵✹❪ ♦r ❬▲▼✵✻❪✳
❍②♣♦t❤❡s✐s✳ ❲❡ ♠❛❦❡ s♦♠❡ ❛ss✉♠♣t✐♦♥s ❢r♦♠ ♥♦✇ t✐❧❧ t❤❡ ❡♥❞ ♦❢ t❤❡ ♣❛♣❡r✱ ❢♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t② ❜✉t ✇✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✳
✭❆✶✮b= 0✳
■♥❞❡❡❞✱ ❛s ❡①♣❧❛✐♥❡❞ ✐♥ ❬▲▼✵✻❪ ❙❡❝t✐♦♥ ✷✱ ✐❢ ✇❡ ❝❛♥ tr❡❛t t❤❡ ❝❛s❡
L= ρ
2∇
a∇, ✭✶✳✸✮
✇❡ ❝❛♥ tr❡❛t t❤❡ ❝❛s❡ ✭✶✳✶✮ ❢♦r ❛♥② ♠❡❛s✉r❛❜❧❡ ❜♦✉♥❞❡❞ b❜② ❞❡✜♥✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥tsρ❛♥❞ a✐♥ ✭✶✳✸✮ ✐♥
t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♥♥❡r✿
If ab :=aexpΨ andρb :=ρexp−Ψ,
with Ψ(x) =
Z x
0
h(y)dy andh(x) = 2 b(x)
ρ(x)a(x),
then ρb
2∇
ab∇
= ρ
2∇
a∇+b∇.
✭❆✷✮ ▲❡t ❜❡G= (l, r)❛♥ ♦♣❡♥ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧ ♦❢R✳ ❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❡ ♣r♦❝❡ssX st❛rts ❢r♦♠x∈G
❛♥❞ ✐s ❦✐❧❧❡❞ ✇❤❡♥ r❡❛❝❤✐♥❣{l, r}✳ ❃❋r♦♠ ❛ P❉❊s ♣♦✐♥t ♦❢ ✈✐❡✇ t❤✐s ♠❡❛♥s t❤❡ ♣❛r❛❜♦❧✐❝ P❉❊s ✐♥✈♦❧✈✐♥❣ L ✇❡ ✇✐❧❧ st✉❞② ❛r❡ s✉❜♠✐tt❡❞ t♦ ✉♥✐❢♦r♠ ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❲❡ ❝♦✉❧❞ tr❡❛t ◆❡✉♠❛♥♥
❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ✭t❤❛♥❦s t♦ t❤❡ r❡s✉❧ts ♦❢ ❬❇❈✵✺❪ ❢♦r ✐♥st❛♥❝❡✮ ❛♥❞✱ ❜② ❧♦❝❛❧✐③❛t✐♦♥ ❛r❣✉♠❡♥ts✱ t❤❡ ❝❛s❡ ♦❢ ❛♥ ✉♥❜♦✉♥❞❡❞ ❞♦♠❛✐♥G✭s❡❡ ❬▲▼✵✻❪✮✳ ❇✉t t❤✐s ❛ss✉♠♣t✐♦♥ ✇✐❧❧ ♠❛❦❡ t❤❡ ♠❛t❡r✐❛❧ ♦❢ t❤❡ ♣❛♣❡r
s✐♠♣❧❡r ❛♥❞ ❝❧❡❛r❡r✳
❖✉t❧✐♥❡ ♦❢ t❤❡ ♣❛♣❡r✳ ■♥ ❙❡❝t✐♦♥ ✷ ✇❡ ❞❡✜♥❡ ♣r❡❝✐s❡❧② ❉✐✈❡r❣❡♥❝❡ ❋♦r♠ ❖♣❡r❛t♦rs ✭❉❋❖✮ ❛♥❞ r❡❝❛❧❧ s♦♠❡ ♦❢ t❤❡✐r ♣r♦♣❡rt✐❡s✳ ■♥ ❙❡❝t✐♦♥ ✸ ✇❡ s♣❡❛❦ ♦❢ ❙t♦❝❤❛st✐❝ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ▲♦❝❛❧ ❚✐♠❡ ✭❙❉❊▲❚✮✿ ✇❡ st❛t❡ ❛ ❣❡♥❡r❛❧ ❝❤❛♥❣❡ ♦❢ s❝❛❧❡ ❢♦r♠✉❧❛ ❛♥❞ r❡❝❛❧❧ s♦♠❡ ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts ❡st❛❜❧✐s❤❡❞ ❜② ❏✳❋✳ ▲❡ ●❛❧❧ ✐♥ ❬▲❡❣✽✺❪✳ ■♥ ❙❡❝t✐♦♥ ✹ ✇❡ ❧✐♥❦ ❉❋❖ ❛♥❞ ❙❉❊▲❚✿ ❛ ♣r♦❝❡ss ❣❡♥❡r❛t❡❞ ❜② ❛ ❉❋❖ ♦❢ ❝♦❡✣❝✐❡♥tsa❛♥❞ρ❤❛✈✐♥❣ ❝♦✉♥t❛❜❧❡ ❞✐s❝♦♥t✐♥✉✐t✐❡s ✇✐t❤♦✉t ❝❧✉st❡r ♣♦✐♥ts ✐s s♦❧✉t✐♦♥ ♦❢ ❛ ❙❉❊▲❚ ✇✐t❤
❝♦❡✣❝✐❡♥ts ❞❡t❡r♠✐♥❡❞ ❜② a ❛♥❞ρ✳ ■♥ ❙❡❝t✐♦♥ ✺ ✇❡ ♣r❡s❡♥t ♦✉r s❝❤❡♠❡✳ ■♥ ❙❡❝t✐♦♥ ✻ ✇❡ ❡st✐♠❛t❡ t❤❡
s♣❡❡❞ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤✐s s❝❤❡♠❡✳ ❙❡❝t✐♦♥ ✼ ✐s ❞❡✈♦t❡❞ t♦ ♥✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts✳
❙♦♠❡ ♥♦t❛t✐♦♥s✳ ❋♦r1≤p <∞✇❡ ❞❡♥♦t❡ ❜②Lp(G)t❤❡ s❡t ♦❢ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥sf ♦♥Gs✉❝❤ t❤❛t
kfkp:=
Z
G|
f(x)|pdx1/p<∞.
▲❡t ❜❡0< T <∞✜①❡❞✳ ❋♦r1≤p, q <∞✇❡ ❞❡♥♦t❡ ❜②Lq(0, T; Lp(G))t❤❡ s❡t ♦❢ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s
f ♦♥(0, T)×Gs✉❝❤ t❤❛t
|||f|||p,q :=
Z T
0 k fkqpdt
1/q
<∞.
❋♦r u∈ Lp(G) ✇❡ ❞❡♥♦t❡ ❜② du
dx t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ♦❢ u ✐♥ t❤❡ ❞✐str✐❜✉t✐♦♥❛❧ s❡♥s❡✳ ■t ✐s st❛♥❞❛r❞ t♦
❞❡♥♦t❡ ❜②W1,p(G)t❤❡ s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥su∈Lp(G)s✉❝❤ t❤❛t du dx ∈L
p(G)✱ ❛♥❞ ❜②W1,p
0 (G)t❤❡ ❝❧♦s✉r❡ ♦❢ C∞
c (G)✐♥W1,p(G)❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♥♦r♠(kuk p p+
du
dx
pp)1/p✳ ❲❡ ❞❡♥♦t❡ ❜②H1(G)t❤❡ s♣❛❝❡W1,2(G)✱
❛♥❞ ❜②H1
0(G)t❤❡ s♣❛❝❡W 1,2 0 (G)✳
❋♦r u ∈ L2(0, T; L2(G)) ✇❡ ❞❡♥♦t❡ ❜② ∂
tu t❤❡ ❞✐str✐❜✉t✐♦♥ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ϕ ∈ Cc∞((0, T)×G)✱ ✇❡
❤❛✈❡✱h∂tu, ϕi =−R0TRGu∂tϕ✳ ❲❡ st✐❧❧ ❞❡♥♦t❡ ❜② dudx t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ♦❢u✇✐t❤ r❡s♣❡❝t t♦ x✐♥ t❤❡
❞✐str✐❜✉t✐♦♥❛❧ s❡♥s❡✳
❲❡ ✇✐❧❧ ❝❧❛ss✐❝❛❧② ❞❡♥♦t❡ ❜②k.k∞ ❛♥❞|||.|||∞,∞ t❤❡ s✉♣r❡♠✉♠ ♥♦r♠s✳
❋♦r t❤❡ ✉s❡ ♦❢ ♣r♦❜❛❜✐❧✐t② ✇❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② C0(G)t❤❡ s❡t ♦❢ ❝♦♥t✐♥✉♦✉s ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ♦♥ G✳ ❚❤❡
s②♠❜♦❧∼=✇✐❧❧ ❞❡♥♦t❡ ❡q✉❛❧✐t② ✐♥ ❧❛✇✳
✷ ❖♥ ❞✐✈❡r❣❡♥❝❡ ❢♦r♠ ♦♣❡r❛t♦rs
❋♦r0< λ <Λ<∞❧❡t ✉s ❞❡♥♦t❡ ❜②Ell(λ,Λ)t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥sf ♦♥Gt❤❛t ❛r❡ ♠❡❛s✉r❛❜❧❡ ❛♥❞ s✉❝❤
t❤❛t
∀x∈G, λ≤f(x)≤Λ.
❋♦rρ∈Ell(λ,Λ)❧❡t ✉s ❞❡✜♥❡ t❤❡ ♠❡❛s✉r❡mρ(dx) :=ρ−1(x)dx✳
❋♦r ❛♥② ♠❡❛s✉r❡ m ✇✐t❤ ❛ ❜♦✉♥❞❡❞ ❞❡♥s✐t② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ✇❡ t❤❡♥ ❞❡♥♦t❡ ❜②
L2(G, m)t❤❡ ❍✐❧❜❡rt s♣❛❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✐♥L2(G)❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ s❝❛❧❛r ♣r♦❞✉❝t
(f, g)7−→
Z
G
f(x)g(x)m(dx).
❚❤✐s ✐s ❞♦♥❡ ✐♥ ♦r❞❡r t❤❛t t❤❡ ♦♣❡r❛t♦r ✇❡ ❞❡✜♥❡ ❜❡❧♦✇ ✐s s②♠♠❡tr✐❝ ♦♥L2(G, m
ρ)✳
❉❡✜♥✐t✐♦♥ ✷✳✶ ▲❡ta❛♥❞ρ❜❡ ✐♥Ell(λ,Λ)❢♦r s♦♠❡0< λ <Λ<∞✳ ❲❡ ❝❛❧❧ ❉✐✈❡r❣❡♥❝❡ ❢♦r♠ ♦♣❡r❛t♦r
♦❢ ❝♦❡✣❝✐❡♥ts a❛♥❞ρ✱ ❛♥❞ ✇❡ ♥♦t❡L(a, ρ)✱ t❤❡ ♦♣❡r❛t♦r(L, D(L))♦♥ L2(G, m
ρ)❞❡✜♥❡❞ ❜②
L=ρ 2
d dx
a d
dx
,
❛♥❞
D(L) ={u∈H10(G), Lu∈L2(G)}.
❆❝t✉❛❧❧② ✐❢a, ρ ∈Ell(λ,Λ)t❤❡ ♦♣❡r❛t♦rL(a, ρ)❤❛s s✉✣❝✐❡♥t ♣r♦♣❡rt✐❡s t♦ ❣❡♥❡r❛t❡ ❛ ❝♦♥t✐♥✉♦✉s ▼❛r❦♦✈
♣r♦❝❡ss✳ ❲❡ s✉♠ ✉♣ t❤❡s❡ ♣r♦♣❡rt✐❡s ✐♥ t❤❡ ♥❡①t t❤❡♦r❡♠✳
❚❤❡♦r❡♠ ✷✳✶ ▲❡ta❛♥❞ρ❜❡ ✐♥ Ell(λ,Λ)❢♦r s♦♠❡ 0< λ <Λ<∞✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿
✐✮ ❚❤❡ ♦♣❡r❛t♦rL(a, ρ)♦♥ L2(G, m
ρ)✐s ❝❧♦s❡❞ ❛♥❞ s❡❧❢✲❛❞❥♦✐♥t✱ ✇✐t❤ ❞❡♥s❡ ❞♦♠❛✐♥✳
✐✐✮ ❚❤✐s ♦♣❡r❛t♦r ✐s t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ ❛ str♦♥❣❧② ❝♦♥t✐♥✉♦✉s s❡♠✐❣r♦✉♣ ♦❢ ❝♦♥tr❛❝t✐♦♥(St)t≥0
♦♥L2(G, m
ρ)✳
✐✐✐✮ ▼♦r❡♦✈❡r (St)t≥0 ✐s ❛ ❋❡❧❧❡r s❡♠✐❣r♦✉♣✳ ❚❤✉s L(a, ρ) ✐s t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ ❛ ▼❛r❦♦✈
♣r♦❝❡ss(Xt, t≥0)✳
✐✈✮ ❚❤❡ ♣r♦❝❡ss(Xt, t≥0) ❤❛s ❝♦♥t✐♥✉♦✉s tr❛❥❡❝t♦r✐❡s✳
Pr♦♦❢✳ ❲❡ ❣✐✈❡ t❤❡ ❣r❡❛t ❧✐♥❡s ♦❢ t❤❡ ♣r♦♦❢ ❛♥❞ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❬▲❡❥✵✵❪ ❛♥❞ ❬❙tr✽✷❪ ❢♦r ❞❡t❛✐❧s✳ ❲❡ s❡t(L, D(L)) =L(a, ρ)✳ ❋✐rst ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❜✉✐❧❞ ❛ s②♠♠❡tr✐❝ ❜✐❧✐♥❡❛r ❢♦r♠E ♦♥L2(G, m
ρ)❞❡✜♥❡❞
❜②
E(u, v) =
Z
G
a
2
du dx
dv
dxdx, ∀(u, v)∈ D(E)×D(E), and D(E) = H
1 0(G),
t❤❛t ✈❡r✐✜❡s✱
E(u, v) =− hLu, viL2(G,mρ), ∀(u, v)∈D(L)×D(E).
❚❤✉s t❤❡ r❡s♦❧✈❡♥t ♦❢(L, D(L))❝❛♥ ❜❡ ❜✉✐❧t ❛♥❞ ✇❡ ❣❡t ✐✮✳ ❆♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❍✐❧❧❡✲❨♦s✐❞❛ t❤❡♦r❡♠
t❤❡♥ ❧❡❛❞s t♦ ✐✐✮✳
❋✉rt❤❡r ✐t ✐s ❛ ❝❧❛ss✐❝❛❧ r❡s✉❧t ♦❢ P❉❊s t❤❛t t❤❡ s❡♠✐❣r♦✉♣(Pt)t≥0 ❤❛s ❛ ❞❡♥s✐t②p(t, x, y)✇✐t❤ r❡s♣❡❝t t♦
t❤❡ ♠❡❛s✉r❡mρ s✉❝❤ t❤❛t
u(t, x) =
Z
G
p(t, x, y)f(y)ρ−1(y)dy ✭✷✳✶✮
✐s ❛ ❝♦♥t✐♥✉♦✉s ✈❡rs✐♦♥ ♦❢Ptf(x)✳ ❚❤❡♥ ❢♦rf ✐♥C0(G)✱Ptf ❜❡❧♦♥❣s t♦C0(G)✳ ❇② t❤❡ ✉s❡ ♦❢ t❤❡ ♠❛①✐♠✉♠
♣r✐♥❝✐♣❧❡ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t(Pt)t≥0 ✐s s❡♠✐✲♠❛r❦♦✈✐❛♥ ❛♥❞ ✇❡ ❣❡t ✐✐✐✮✳
❋✐♥❛❧❧② ❆r♦♥s♦♥ ❡st✐♠❛t❡s ♦♥ t❤❡ ❞❡♥s✐t②p(t, x, y)❝❛♥ ❜❡ ✉s❡❞ t♦ s❤♦✇ ❢♦r ❡①❛♠♣❧❡ t❤❛t
∀ε >0, ∀x∈G, lim
t↓0
1
t Z
|y−x|>ε
p(t, x, y)ρ−1(y)dy= 0,
❛♥❞ t❤✉s ✇❡ ❣❡t ✐✈✮ ✭s❡❡ Pr♦♣♦s✐t✐♦♥ ✷✳✾ ✐♥ ❝❤❛♣t❡r ✹ ♦❢ ❬❊❑✽✻❪✮✳
❲❡ ❤❛✈❡ ❛ ❝♦♥s✐st❡♥❝② t❤❡♦r❡♠✳
❚❤❡♦r❡♠ ✷✳✷ ▲❡t ❜❡ 0 < λ < Λ < ∞✳ ▲❡t a ❛♥❞ ρ ❜❡ ✐♥ Ell(λ,Λ) ❛♥❞ (an, ρn) ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢
Ell(λ,Λ)×Ell(λ,Λ)✳
▲❡t ✉s ❞❡♥♦t❡ ❜② S ❛♥❞ X r❡s♣❡❝t✐✈❡❧② t❤❡ s❡♠✐❣r♦✉♣ ❛♥❞ t❤❡ ♣r♦❝❡ss ❣❡♥❡r❛t❡❞ ❜② L(a, ρ) ❛♥❞ ❜②(Sn)
❛♥❞(Xn) t❤❡ s❡q✉❡♥❝❡s ♦❢ s❡♠✐❣r♦✉♣s ❛♥❞ ♣r♦❝❡ss❡s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡q✉❡♥❝❡ ♦❢ ♦♣❡r❛t♦rsL(an, ρn)✳
❆ss✉♠❡ t❤❛t
1
an
L2(G)
−−−−⇀
n→∞
1
a, and
1
ρn
L2(G)
−−−−⇀
n→∞
1
ρ.
❚❤❡♥ ❢♦r ❛♥②T >0 ❛♥❞ ❛♥② f ∈L2(G)✇❡ ❤❛✈❡ ✿
✐✮ ❚❤❡ ❢✉♥❝t✐♦♥Sn
tf(x)❝♦♥✈❡r❣❡s ✇❡❛❦❧② ✐♥L2(0, T; H01(G))t♦Stf(x)✳
✐✐✮ ❚❤❡ ❝♦♥t✐♥✉♦✉s ✈❡rs✐♦♥ ♦❢ Sn
tf(x) ❣✐✈❡♥ ❜② ✭✷✳✶✮ ✇✐t❤ pr❡♣❧❛❝❡❞ ❜② pn ❝♦♥✈❡r❣❡s ✉♥✐❢♦r♠❧② ♦♥ ❡❛❝❤
❝♦♠♣❛❝t ♦❢(0, T)×Gt♦ t❤❡ ❝♦♥t✐♥✉♦✉s ✈❡rs✐♦♥ ♦❢ Stf(x) ❣✐✈❡♥ ❜② ✭✷✳✶✮✳
✐✐✐✮
(Xtn, t≥0)
L
−−−−→
n→∞ (Xt, t≥0).
Pr♦♦❢✳ ❙❡❡ ✐♥ ❬▲▼✵✻❪ t❤❡ ♣r♦♦❢s ♦❢ Pr♦♣♦s✐t✐♦♥s ✸ ❛♥❞ ✹✳
✸ ❖♥ ❙❉❊ ✐♥✈♦❧✈✐♥❣ ❧♦❝❛❧ t✐♠❡
❋✐rst ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ❝❧❛ss ♦❢ ❝♦❡✣❝✐❡♥ts✳ ❋♦r0 < λ <Λ <∞ ✇❡ ❞❡♥♦t❡ ❜②Coeff(λ,Λ)t❤❡ s❡t ♦❢
t❤❡ ❡❧❡♠❡♥tsf ♦❢Ell(λ,Λ)t❤❛t ✈❡r✐❢②✿
i) f ✐s r✐❣❤t ❝♦♥t✐♥✉♦✉s ✇✐t❤ ❧❡❢t ❧✐♠✐ts ✭r✳❝✳❧✳❧✳✮✳
ii) f ❜❡❧♦♥❣s t♦C1(G\ I)✱ ✇❤❡r❡I ✐s ❛ ❝♦✉♥t❛❜❧❡ s❡t ✇✐t❤♦✉t ❝❧✉st❡r ♣♦✐♥t✳
▲❡t ✉s ❛❧s♦ ❞❡♥♦t❡ ❜②Mt❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞ ♠❡❛s✉r❡sν ♦♥Gs✉❝❤ t❤❛t|ν({x})|<1❢♦r ❛❧❧x✐♥G✳
❉❡✜♥✐t✐♦♥ ✸✳✶ ▲❡t σ ❜❡ ✐♥Coeff(λ,Λ) ❢♦r s♦♠❡ 0< λ <Λ<∞✱ ❛♥❞ ν ❜❡ ✐♥ M✳ ❲❡ ❝❛❧❧ ❙t♦❝❤❛st✐❝
❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥ ✇✐t❤ ▲♦❝❛❧ ❚✐♠❡ ♦❢ ❝♦❡✣❝✐❡♥tsσ❛♥❞ν✱ ❛♥❞ ✇❡ ♥♦t❡Sde(σ, ν)✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❙❉❊
Xt=X0+
Z t
0
σ(Xs)dWs+
Z
R
ν(dx)Lxt(X),
✇❤❡r❡Lx
t(X)✐s t❤❡ s②♠♠❡tr✐❝ ❧♦❝❛❧ t✐♠❡ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ♣r♦❝❡ssX✳
■♥ ❬▲❡❣✽✺❪ ❏✳❋✳ ▲❡ ●❛❧❧ st✉❞✐❡❞ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❙❉❊s ♦❢ t❤❡ t②♣❡Sde(σ, ν)✳ ❲❡ ✇✐❧❧ r❡❝❛❧❧ ❤❡r❡ s♦♠❡
r❡s✉❧ts ♦❢ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ ✉s❡ ✐♥ t❤❡ s❡q✉❡❧✳
❲❡ ✇✐❧❧ s❡❡ ❜❡❧♦✇ t❤❛t σ ∈ Coeff(λ,Λ) ❛♥❞ ν ∈ M✐s ❛ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ t♦ ❤❛✈❡ ❛ ✉♥✐q✉❡ str♦♥❣
s♦❧✉t✐♦♥ t♦Sde(σ, ν)✳ ❲❡ ✜rst ✜① s♦♠❡ ❛❞❞✐t✐♦♥♥❛❧ ♥♦t❛t✐♦♥s✳
❋♦rf ✐♥Coeff(λ,Λ)✇❡ ❞❡♥♦t❡ ❜②f′(dx)t❤❡ ❜♦✉♥❞❡❞ ♠❡❛s✉r❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ♦❢f
✐♥ t❤❡ ❣❡♥❡r❛❧✐③❡❞ s❡♥s❡✳ ❲❡ ❞❡♥♦t❡ ❜②f(x+) ❛♥❞f(x−)r❡s♣❡❝t✐✈❡❧② t❤❡ r✐❣❤t ❛♥❞ ❧❡❢t ❧✐♠✐ts ♦❢f ✐♥x✳
❲❡ ✇✐❧❧ ❛❧s♦ ❞❡♥♦t❡ ❜②f′(x)t❤❡ r✳❝✳❧✳❧✳ ❞❡♥s✐t② ♦❢ t❤❡ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ♣❛rt ♦❢f′(dx)✭t❤❛t ✐t ✐s t♦
s❛② ✇❡ t❛❦❡ ❢♦rf′(x)t❤❡ r✐❣❤t ❞❡r✐✈❛t✐✈❡ ♦❢ f ✐♥ x✮✳
❋♦rν ✐♥M✇❡ ❞❡♥♦t❡ ❜②νc t❤❡ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ♣❛rt ♦❢ν✳
✸✳✶ ❆ ❝❤❛♥❣❡ ♦❢ s❝❛❧❡ ❢♦r♠✉❧❛
▲❡t ✉s ❞❡✜♥❡ t❤❡ ❝❧❛ss ♦❢ ❜✐❥❡❝t✐♦♥s ✇❡ ✇✐❧❧ ✉s❡ ✐♥ ♦✉r ❝❤❛♥❣❡ ♦❢ s❝❛❧❡✳
❋♦r0< λ <Λ<∞✇❡ ❞❡♥♦t❡ ❜②T(λ,Λ)t❤❡ s❡t ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥sΦ♦♥Gt❤❛t ❤❛✈❡ ❛ ✜rst ❞❡r✐✈❛t✐✈❡Φ′
t❤❛t ❜❡❧♦♥❣s t♦Coeff(λ,Λ)✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♠❛❞❡ ♦♥ t❤❡ ❜✐❥❡❝t✐♦♥ ✐s ♠✐♥✐♠❛❧ ❛♥❞ ✇❡ ❝❛♥ t❤❡♥ st❛t❡ ❛
✈❡r② ❣❡♥❡r❛❧ ❝❤❛♥❣❡ ♦❢ s❝❛❧❡ ❢♦r♠✉❧❛✳
Pr♦♣♦s✐t✐♦♥ ✸✳✶ ▲❡tσ❜❡ ✐♥ Coeff(λ,Λ)❢♦r s♦♠❡0< λ <Λ<∞✳ ▲❡t
ν(dx) =b(x)dx+ X
xi∈I
cxiδxi(dx),
❜❡ ✐♥ M✭✐✳❡✳✱ b ✐s ♠❡❛s✉r❛❜❧❡ ❛♥❞ ❜♦✉♥❞❡❞✱ ❛♥❞ ❡❛❝❤|cxi|<1✮✳
▲❡t Φ❜❡ ✐♥T(λ′,Λ′)❢♦r s♦♠❡0< λ′<Λ′<∞❛♥❞ ❧❡tJ ❜❡ t❤❡ s❡t ♦❢ t❤❡ ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ Φ′✳
❚❤❡♥ t❤❡ ♥❡①t st❛t❡♠❡♥ts ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✐✮ ❚❤❡ ♣r♦❝❡ssX s♦❧✈❡s Sde(σ, ν)✳
✐✐✮ ❚❤❡ ♣r♦❝❡ss Y := Φ(X)s♦❧✈❡s Sde(γ, µ)✇✐t❤
γ(y) = (σΦ′)◦Φ−1(y),
❛♥❞
µ(dy) = Φ
′b+1 2(Φ′)′
(Φ′)2 ◦Φ
−1(y)dy+ X
xi∈I∪J
βxiδΦ(xi)(dy),
✇❤❡r❡✱
βx= Φ
′(x+)(1 +c
x)−Φ′(x−)(1−cx)
Φ′(x+)(1 +c
x) + Φ′(x−)(1−cx),
✇✐t❤cx= 0 ✐❢ x∈ J \ I✳
❘❡♠❛r❦ ✸✳✶ ◆♦t❡ t❤❛tγ♦❜✈✐♦✉s❧② ❜❡❧♦♥❣s t♦Coeff(λ′′,Λ′′)❢♦r s♦♠❡0< λ′′<Λ′′<∞✱ ❛♥❞ t❤❛tµ✐s
✐♥M✱ s♦ ✐t ♠❛❦❡s s❡♥s❡ t♦ s♣❡❛❦ ♦❢Sde(γ, µ)✳
❲❡ ❝❛♥ s❛② t❤❛t t❤❡ ❝❧❛ss ♦❢ ❙❉❊ ♦❢ t②♣❡Sde(σ, ν)✐s st❛❜❧❡ ❜② tr❛♥s❢♦r♠❛t✐♦♥ ❜② ❛ ❜✐❥❡❝t✐♦♥ ❜❡❧♦♥❣✐♥❣
t♦T(λ,Λ)❢♦r s♦♠❡ 0< λ <Λ<∞✳
Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✶✳ ❲❡ ♣r♦✈❡i)⇒ii)✳ ❚❤❡ ❝♦♥✈❡rs❡ ❝❛♥ ❜❡ ♣r♦✈❡♥ ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r q✉✐t❡
❜❡✐♥❣ t❡❝❤♥✐❝❛❧❧② ♠♦r❡ ❝✉♠❜❡rs♦♠❡✳
❇② t❤❡ s②♠♠❡tr✐❝ ■tô✲❚❛♥❛❦❛ ❢♦r♠✉❧❛ ✇❡ ✜rst ❣❡t✿
Yt= Φ(Xt) = Φ(X0) +R0t(σΦ′)(Xs)dWs+R0t(σ2bΦ′)(Xs)ds
+Pxi∈IΦ
′
(xi+)+Φ′(xi−) 2 cxiL
xi t (X)
+12R0t[σ2(Φ′)′](X
s)ds+Pxi∈J
Φ′(xi+)−Φ′(xi−)
2 L
xi t (X)
= Φ(X0) +R0t(σΦ′)◦Φ−1(Y
s)dWs
+R0t[σ2(Φ′b+1
2(Φ
′)′)]◦Φ−1(Y
s)ds+Pxi∈I∪JKxiLxit (X),
✇✐t❤Kx=cx(Φ′(x+) + Φ′(x−))/2 + (Φ′(x+)−Φ′(x−))/2.
❲❡ ❤❛✈❡ t❤❡♥ t♦ ❡①♣r❡ssLx
t(X)✐♥ ❢✉♥❝t✐♦♥ ♦❢L
Φ(x)
t (Y)❢♦rx∈ I ∪ J✳
❯s✐♥❣ ❈♦r♦❧❧❛r② ❱■✳✶✳✾ ♦❢ ❬❘❨✾✶❪ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t
LΦ(x)
±
t (Y) = Φ′(x±)Lxt±(X). ✭✸✳✶✮
❇❡s✐❞❡s t❤❡♦r❡♠ ❱■✳✶✳✼ ♦❢ ❬❘❨✾✶❪ ❧❡❛❞s t♦ (Lxt+(X)−Ltx−(X))/2 = cxLxt(X) ❛♥❞ ❝♦♠❜✐♥✐♥❣ ✇✐t❤
(Lxt+(X) +Lxt−(X))/2 =Lxt(X)✇❡ ❣❡t
Lxt+(X) = (1 +cx)Lxt(X). ✭✸✳✷✮
■♥ ❛ s✐♠✐❧❛r ♠❛♥♥❡r ✇❡ ❤❛✈❡(LΦ(t x)+(Y)−L
Φ(x)−
t (Y))/2 =KxLxt(X)❛♥❞ ✇❡ ❝❛♥ ❣❡t
KxLt(X) +LΦ(t x)(Y) =L
Φ(x)+
t (Y).
❚❤❡♥ ✉s✐♥❣ ✭✸✳✶✮ ❛♥❞ ✭✸✳✷✮ ✇❡ ❣❡t
LΦ(t x)(Y) = (Φ′(x+)(1 +cx)−Kx)Lxt(X),
❛♥❞ t❤❡ ❢♦r♠✉❧❛ ✐s ♣r♦✈❡❞✳
❚♦ ♣r♦✈❡ t❤❡ ♣r♦♣♦s✐t✐♦♥ ❜❡❧♦✇✱ ▲❡ ●❛❧❧ ✉s❡❞ ✐♥ ❬▲❡❣✽✺❪ ❛ s♣❛❝❡ ❜✐❥❡❝t✐♦♥ t❤❛t ❡♥t❡rs ✐♥ t❤❡ ❣❡♥❡r❛❧ s❡tt✐♥❣ ♦❢ Pr♦♣♦s✐t✐♦♥✸✳✶✳
Pr♦♣♦s✐t✐♦♥ ✸✳✷ ✭▲❡ ●❛❧❧ ✶✾✽✺✮ ▲❡t σ ❜❡ ✐♥ Coeff(λ,Λ) ❢♦r s♦♠❡ 0 < λ <Λ<∞ ❛♥❞ ν ❜❡ ✐♥ M✳
❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ str♦♥❣ s♦❧✉t✐♦♥ t♦Sde(σ, ν)✳
❲❡ ♥❡❡❞ t✇♦ ❧❡♠♠❛s✳
▲❡♠♠❛ ✸✳✶ ✭▲❡ ●❛❧❧ ✶✾✽✺✮ ▲❡t σ ❜❡ ✐♥ Coeff(λ,Λ) ❢♦r s♦♠❡ 0 < λ < Λ < ∞✳ ❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡
str♦♥❣ s♦❧✉t✐♦♥ t♦Sde(σ,0)✳
Pr♦♦❢✳ ❙❡❡ ❬▲❡❣✽✺❪✳
❚❤❡ ♥❡①t ❧❡♠♠❛ ✇✐❧❧ ♣❧❛② ❛ ❣r❡❛t r♦❧❡ ❢♦r ❝❛❧❝✉❧❛t✐♦♥s ✐♥ t❤❡ s❡q✉❡❧✳
▲❡♠♠❛ ✸✳✷ ▲❡t ν ❜❡ ✐♥ M✳ ❚❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥ fν ✐♥ Coeff(λ,Λ) ✭❢♦r s♦♠❡ 0 < λ < Λ < ∞✮✱
✉♥✐q✉❡ ✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✱ s✉❝❤ t❤❛t✿
fν′(dx) + (fν(x+) +fν(x−))ν(dx) = 0. ✭✸✳✸✮
Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✷✳ ■t s✉✣❝❡s t♦ s❡t
Φν(x) =
Z x
0
fν(y)dy.
❇② ▲❡♠♠❛✸✳✷Φν♦❜✈✐♦✉s❧② ❜❡❧♦♥❣s t♦T(λ,Λ)❢♦r s♦♠❡0< λ <Λ<∞✳ ❇② Pr♦♣♦s✐t✐♦♥✸✳✶❛♥❞ ▲❡♠♠❛ ✸✳✷ ✇❡ ❣❡t t❤❛tX s♦❧✈❡sSde(σ, ν)✐❢ ❛♥❞ ♦♥❧② ✐❢ Y := Φν(X) s♦❧✈❡sSde (σfν)◦Φ−ν1,0
✳ ❇② ▲❡♠♠❛
✸✳✶t❤❡ ♣r♦♦❢ ✐s ❝♦♠♣❧❡t❡❞✳
✸✳✷ ❈♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts
▲❡ ●❛❧❧ ♣r♦✈❡❞ t❤❡ ♥❡①t ❝♦♥s✐st❡♥❝② r❡s✉❧t ❢♦r ❡q✉❛t✐♦♥s ♦❢ t❤❡ t②♣❡Sde(σ, ν)✳
❚❤❡♦r❡♠ ✸✳✶ ✭▲❡ ●❛❧❧ ✶✾✽✺✮ ▲❡t ❜❡ t✇♦ s❡q✉❡♥❝❡s(σn)❛♥❞(νn)❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐st0< λ <Λ<∞✱
0< M <∞ ❛♥❞δ >0 s✉❝❤ t❤❛t
(H1) σn ∈Coeff(λ,Λ), ∀n∈N,
(H2) |νn({x})| ≤1−δ, ∀n∈N,∀x∈G.
(H3) |νn|(G)≤M, ∀n∈N,
s♦ t❤❛t ❡❛❝❤ νn ✐s ✐♥ M✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐st t✇♦ ❢✉♥❝t✐♦♥s σ ❛♥❞ f ✐♥ Coeff(λ′,Λ′) ✭❢♦r s♦♠❡ 0< λ′<Λ′ <∞✮ s✉❝❤ t❤❛t
σn L1
loc(R)
−−−−−→n→∞ σ ❛♥❞ fνn L1
loc(R)
−−−−−→n→∞ f,
❛♥❞ s❡t✿
ν(dx) =− f
′(dx)
f(x+) +f(x−). ✭✸✳✹✮
▲❡t (Ω,F,(Ft)t≥0,Px) ❜❡ ❛ ✜❧t❡r❡❞ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ ❝❛rr②✐♥❣ ❛♥ ❛❞❛♣t❡❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ W✳ ❖♥ t❤✐s
s♣❛❝❡✱ ❢♦r ❡❛❝❤ n∈N ❧❡t ❜❡ Xn t❤❡ str♦♥❣ s♦❧✉t✐♦♥ ♦❢ Sde(σn, νn)✱ ❛♥❞ ❧❡t ❜❡ X t❤❡ str♦♥❣ s♦❧✉t✐♦♥ ♦❢
Sde(σ, ν)✳ ❚❤❡♥✿
E[ sup
0≤s≤t|
Xsn−Xs|]−−−−→
n→∞ 0 ❛♥❞ (X
n t, t≥0)
L
−−−−→n→∞ (Xt, t≥0).
❘❡♠❛r❦ ✸✳✷ ■♥ t❤✐s t❤❡♦r❡♠νn❛♣♣r♦❛❝❤❡sν ✐♥ t❤❡ s❡♥s❡ t❤❛tfνnt❡♥❞s t♦fν ❢♦r t❤❡L1loc❝♦♥✈❡r❣❡♥❝❡✳
◆♦t❡ ✇❡ ❝❛♥ ❤❛✈❡νn ⇀ ν1✱ ❜✉tf
νn →fν2 ❢♦r t❤❡Lloc1 ❝♦♥✈❡r❣❡♥❝❡✱ ✇✐t❤ ν16=ν2 ✭❙❡❡ ❬▲❡❣✽✺❪ ♣ ✻✺ ❢♦r
❛♥ ❡①❛♠♣❧❡✮✳ ❚❤❡ t❤❡♦r❡♠ ❛ss❡rts t❤❛tXn t❡♥❞s t♦X t❤❛t s♦❧✈❡sSde(σ, ν2)❛♥❞ ♥♦tSde(σ, ν1)✦
▲❡ ●❛❧❧ ❛❧s♦ ♣r♦✈❡❞ ❛ ❉♦♥s❦❡r t❤❡♦r❡♠ ❢♦r s♦❧✉t✐♦♥ t♦ ❙❉❊s ♦❢ t❤❡ t②♣❡Sde(σ, ν)❢♦rσ≡1✳ ▲❡t ❜❡µ
✐♥M❛♥❞Y ❜❡ t❤❡ s♦❧✉t✐♦♥ t♦Sde(1, µ)✳
❲❡ ❞❡✜♥❡ s♦♠❡ ❝♦❡✣❝✐❡♥tsβn
k ❢♦r ❛❧❧k∈Z✱ ❛♥❞ ❛❧❧n∈N∗✱ ❜②✿
1−βkn
1 +βn
k
= exp −2µc(]k
n, k+ 1
n ])
Y
k n<y≤
k+1 n
1−µ({y})
1 +µ({y})
= fµ(
k+1
n )
fµ(kn)
. ✭✸✳✺✮
❲❡ ❞❡✜♥❡ ❛ s❡q✉❡♥❝❡(µn)♦❢ ♠❡❛s✉r❡s ✐♥M❜②
µn=Xβnkδk
n, ✭✸✳✻✮
❛♥❞ ❛ s❡q✉❡♥❝❡(Yn)♦❢ ♣r♦❝❡ss❡s s✉❝❤ t❤❛t ❡❛❝❤Yn s♦❧✈❡sSde(1, µn)✳
❋✐♥❛❧❧② ✇❡ ❞❡✜♥❡ ❢♦r ❛❧❧n∈N∗❛ s❡q✉❡♥❝❡ (τpn)p∈N♦❢ st♦♣♣✐♥❣ t✐♠❡s ❜②✱
τn
0 = 0
τn
p+1= inf{t > τpn :|Ytn−Yτnn
p|=
1
n}.
✭✸✳✼✮
❲❡ ❤❛✈❡ t❤❡ ♥❡①t t❤❡♦r❡♠✳
❚❤❡♦r❡♠ ✸✳✷ ✭▲❡ ●❛❧❧ ✶✾✽✺✮ ■♥ t❤❡ ♣r❡✈✐♦✉s ❝♦♥t❡①tSn
p :=nYτnn
p ❞❡✜♥❡s ❛ s❡q✉❡♥❝❡ ♦❢ r❛♥❞♦♠ ✇❛❧❦s
♦♥ t❤❡ ✐♥t❡❣❡rs s✉❝❤ t❤❛t✿ ✐✮
Sn
0 = 0, ∀n∈N∗,
P[Spn+1=k+ 1|Spn=k] = 1
2(1 +β
n
k), ∀n, p∈N∗,∀k∈Z,
P[Spn+1=k−1|Spn=k] = 1
2(1−β
n
k), ∀n, p∈N∗,∀k∈Z.
✐✐✮
❚❤❡ s❡q✉❡♥❝❡ ♦❢ ♣r♦❝❡ss❡s ❞❡✜♥❡❞ ❜② Yen
t := (1/n)S[nn2t]✱ ✇❤❡r❡ ⌊.⌋ st❛♥❞s ❢♦r t❤❡ ✐♥t❡❣❡r ♣❛rt ♦❢ ❛ ♥♦♥
♥❡❣❛t✐✈❡ r❡❛❧ ♥✉♠❜❡r✱ ✈❡r✐✜❡s ❢♦r ❛❧❧0< T <∞✿
E|Yetn−Yt| −−−−→
n→∞ 0, ∀t∈[0, T] and (Ye
n t , t≥0)
L
−−−−→n→∞ (Yt, t≥0).
✹ ▲✐♥❦ ❜❡t✇❡❡♥ ❉❋❖ ❛♥❞ ❙❉❊▲❚
❚❤✐s ❧✐♥❦ ✐s st❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳
Pr♦♣♦s✐t✐♦♥ ✹✳✶ ▲❡ta❛♥❞ρ❜❡ ✐♥Coeff(λ,Λ)❢♦r s♦♠❡ 0< λ <Λ<∞✳ ▲❡t ✉s ❞❡♥♦t❡ ❜②I t❤❡ s❡t ♦❢
t❤❡ ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t② ♦❢a✳ ❚❤❡♥L(a, ρ)✐s t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢ t❤❡ ✉♥✐q✉❡ str♦♥❣ s♦❧✉t✐♦♥
♦❢Sde(√aρ, ν)✇✐t❤✱
ν(dx) =a
′(x)
2a(x)
dx+ X
xi∈I
a(xi+)−a(xi−)
a(xi+) +a(xi−)
δxi(dx). ✭✹✳✶✮
■♥ ❬▲▼✵✻❪ t❤❡ ❛✉t❤♦rs ♣r♦✈❡❞ t❤❡ ♣r♦♣♦s✐t✐♦♥ ❛❜♦✈❡ ❜② t❤❡ ✉s❡ ♦❢ ❉✐r✐❝❤❧❡t ❢♦r♠s ❛♥❞ ❘❡✈✉③ ♠❡❛s✉r❡s✳ ❲❡ ❣✐✈❡ ❤❡r❡ ❛ ♠♦r❡ s✐♠♣❧❡ ♣r♦♦❢✱ ❜❛s❡❞ ♦♥ s♠♦♦t❤✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ❛♥❞ ✉s✐♥❣ t❤❡ ❝♦♥s✐st❡♥❝② t❤❡♦r❡♠s ♦❢ t❤❡ t✇♦ ♣r❡❝❡❡❞✐♥❣ s❡❝t✐♦♥s✳
Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳✶✳ ❆sa❛♥❞ρ❛r❡ ✐♥Coeff(λ,Λ)t❤❡ ❢✉♥❝t✐♦♥√ρa✐s ✐♥Coeff(λ,Λ)✳ ❇❡s✐❞❡s✱
❛s|a−b|/|a+b|<1❢♦r ❛♥②a✱b✐♥R∗+✱ t❤❡ ♠❡❛s✉r❡ν❞❡✜♥❡❞ ❜② ✭✹✳✶✮ ✐s ✐♥M✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ✉♥✐q✉❡
str♦♥❣ s♦❧✉t✐♦♥X t♦Sde(√aρ, ν)❢♦❧❧♦✇s ❢r♦♠ Pr♦♣♦s✐t✐♦♥✸✳✷✳
❲❡ t❤❡♥ ✐❞❡♥t✐❢② t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢X✳ ❲❡ ❝❛♥ ❜✉✐❧❞ t✇♦ s❡q✉❡♥❝❡s(an)❛♥❞(ρn)♦❢ ❢✉♥❝t✐♦♥s
✐♥Coeff(λ,Λ)∩ C∞(G)✱ s✉❝❤ t❤❛t
an −−−−→
n→∞ a a.e. ❛♥❞ ρ
n−−−−→ n→∞ ρ a.e.
❋♦r ❛♥②n✐♥N✇❡ ❞❡♥♦t❡ ❜②Xn t❤❡ ♣r♦❝❡ss ❣❡♥❡r❛t❡❞ ❜②L(an, ρn)✳
❖♥ ♦♥❡ ❤❛♥❞✱ ❜② ❞♦♠✐♥❛t❡❞ ❝♦♥✈❡r❣❡♥❝❡ t❤❡ ❤②♣♦t❤❡s❡s ♦❢ ❚❤❡♦r❡♠✷✳✷❛r❡ ❢✉❧✜❧❧❡❞✱ ❛♥❞ ✇❡ ❤❛✈❡✱
(Xtn, t≥0)
L
−−−−→n→∞ (Xet, t≥0), ✭✹✳✷✮
✇❤❡r❡ t❤❡ ♣r♦❝❡ssXe ✐s ❣❡♥❡r❛t❡❞ ❜②L(a, ρ)✳
❖♥ t❤❡ ♦t❤❡r ❤❛♥❞ ✇❡ ✇✐❧❧ s❤♦✇ ❜② ❚❤❡♦r❡♠✸✳✶t❤❛t
(Xtn, t≥0)
L
−−−−→n→∞ (Xt, t≥0). ✭✹✳✸✮
❚❤✉s t❛❦✐♥❣ ✐♥ ❝❛r❡ ✭✹✳✷✮ ❛♥❞ ✭✹✳✸✮ ✇❡ ✇✐❧❧ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ♦❢X ✐sL(a, ρ)✳
❆san ❛♥❞ρn ❛r❡C∞✱(Ln, D(Ln)) =L(an, ρn)❝❛♥ ❜❡ ✇r✐tt❡♥✱
Ln= ρ
n
2
h an′ d
dx +a
n d2
dx2 i
,
s♦ ✐t ✐s st❛♥❞❛r❞ t♦ s❛② t❤❛tXn s♦❧✈❡s
Xtn=x+
Z t
0 p
ρn(Xn
s)an(Xsn)dWs+
Z t
0
ρn(Xn s)an
′
(Xn
s)
2 ds. ✭✹✳✹✮
❆sdhXni
s=ρn(Xsn)an(Xsn)ds✱ ❜② t❤❡ ♦❝❝✉♣❛t✐♦♥ t✐♠❡ ❞❡♥s✐t② ❢♦r♠✉❧❛ ✇❡ ❝❛♥ r❡✇r✐t❡ ✭✹✳✹✮ ❛♥❞ ❛ss❡rt
t❤❛tXn s♦❧✈❡s✱
Xtn=x+
Z t
0 p
ρn(Xn
s)an(Xsn)dWs+
Z
R
νn(dx)Lxt(Xn),
✇❤❡r❡νn(dx) = (an ′
(x)/2an(x))λ(dx)✳
❚❤❡♥ ❡❧❡♠❡♥t❛r② ❝❛❧❝✉❧❛t✐♦♥s s❤♦✇ t❤❛t t❤❡ ❢✉♥❝t✐♦♥fνn ❛ss♦❝✐❛t❡❞ t♦νn ❜② ▲❡♠♠❛✸✳✷ ✐s ♦❢ t❤❡ ❢♦r♠
fνn(x) = K/an(x) ✇✐t❤ K ❛ r❡❛❧ ♥✉♠❜❡r✳ ❚❤✐s ♦❜✈✐♦✉s❧② t❡♥❞s t♦ K/a(x) =: f(x) ❢♦r t❤❡ L1loc(R)✲
❝♦♥✈❡r❣❡♥❝❡✳ ❲❡ t❤❡♥ ❞❡t❡r♠✐♥❡ t❤❡ ♠❡❛s✉r❡ ν ❛ss♦❝✐❛t❡❞ t♦ f ❜② ✭✸✳✹✮✳ ❋✐rst ✇❡ ❝❤❡❝❦ t❤❛tνc(dx) =
(a′(x)/2(a(x))λ(dx)✳ ❙❡❝♦♥❞✱ t❤❡ s❡t{x∈G: ν({x})6= 0}✐s ❡q✉❛❧ t♦I✱ ❛♥❞ ✇❡ ❤❛✈❡ ❢♦r ❛❧❧x∈ I✱
ν({x}) =−ff((xx+)+) +−ff((xx−) −) =
a(x+)−a(x−)
a(x+) +a(x−).
❙♦ t❤❡ ♠❡❛s✉r❡ν ✐s ❡q✉❛❧ t♦ t❤❡ ♦♥❡ ❞❡✜♥❡❞ ❜② ✭✹✳✶✮✳ ❆s ✐t ✐s ♦❜✈✐♦✉s t❤❛t
√ ρnan L
1 loc(R)
−−−−−→n→∞ √ρa,
❛♥❞ t❤❛t t❤❡ ❤②♣♦t❤❡s❡s(H1)−(H3)♦❢ ❚❤❡♦r❡♠✸✳✶❛r❡ ❢✉❧✜❧❧❡❞✱ ✇❡ ❝❛♥ s❛② t❤❛t ✭✹✳✸✮ ❤♦❧❞s✳ ❚❤❡ ♣r♦♦❢
✐s ❝♦♠♣❧❡t❡❞✳
✺ ❘❛♥❞♦♠ ✇❛❧❦ ❛♣♣r♦①✐♠❛t✐♦♥
✺✳✶ ▼♦♥t❡ ❈❛r❧♦ ❆♣♣r♦①✐♠❛t✐♦♥
❋r♦♠ ♥♦✇ t❤❡ ❤♦r✐③♦♥ 0 < T <∞ ✐s ✜①❡❞✳ ❋♦r ❛♥② a, ρ ∈Coeff(λ,Λ) ❛♥❞ ❛♥② ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥f ✇❡
❞❡♥♦t❡ ❜②(P)(a, ρ, f)t❤❡ ♣❛r❛❜♦❧✐❝ P❉❊
(P)(a, ρ, f)
∂u(t,x)
∂t =Lu(t, x), ❢♦r(t, x)∈[0, T]×G,
u(t, l) =u(t, r) = 0 ❢♦rt∈[0, T],
u(0, x) =f(x) ❢♦rx∈G,
✇✐t❤(L, D(L)) =L(a, ρ)✳
▲❡t ❜❡0< λ <Λ<∞✳ ❋r♦♠ ♥♦✇ t✐❧❧ t❤❡ ❡♥❞ ♦❢ t❤✐s ♣❛♣❡r ✇❡ ❛ss✉♠❡ t❤❛t a❛♥❞ρ❛r❡ ✐♥Coeff(λ,Λ)✳
❲❡ ❞❡♥♦t❡ ❜②I={xi}i∈I t❤❡ s❡t ♦❢ t❤❡ ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t② ♦❢a✭I={0≤i≤k1} ⊂Z✐s ✜♥✐t❡✮✳ ❲❡
s❡tX t♦ ❜❡ t❤❡ ♣r♦❝❡ss ❣❡♥❡r❛t❡❞ ❜②L(a, ρ)✳
❲❡ s❡❡❦ ❢♦r ❛ ♣r♦❜❛❜✐❧✐st✐❝ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ (P)(a, ρ, f)✳ ❇② ❚❤❡♦r❡♠
✷✳✶❛♥❞ s♦♠❡ st❛♥❞❛r❞ P❉❊s r❡✜♥❡♠❡♥ts ✇❡ ❦♥♦✇ t❤❛t ❢♦r ❛❧❧f ∈L2(G)✱(P)(a, ρ, f)❤❛s ❛ ✉♥✐q✉❡ ✇❡❛❦
s♦❧✉t✐♦♥u(t, x)✐♥C([0, T],L2(G, mρ))∩L2(0, T; H10(G))✳ ❲❡ ❦♥♦✇ t❤❛tEx[f(Xt)]✐s ❛ ❝♦♥t✐♥✉♦✉s ✈❡rs✐♦♥
♦❢u(t, x)✳
❖✉r ❣♦❛❧ ✐s t♦ ❜✉✐❧❞ ❛ ♣r♦❝❡ssXbn ❡❛s② t♦ s✐♠✉❧❛t❡ ❛♥❞ s✉❝❤ t❤❛t
(Xbn
t, t≥0)
L
−−−−→
n→∞ (Xt, t≥0). ✭✺✳✶✮
❚❤✉sEx[f(Xbtn)]→Ex[f(Xt)]❢♦r ❛♥②t∈[0, T]✱ ❛♥❞ t❤❡ str♦♥❣ ❧❛✇ ♦❢ ❧❛r❣❡ ♥✉♠❜❡rs ❛ss❡rts t❤❛t✱
1
N
N
X
i=1
f(Xbtn,(i)) n→∞
−−−−→
N→∞ u(t, x), ✭✺✳✷✮
✇❤❡r❡ ❢♦r ❡❛❝❤i✱Xbn,(i) ✐s ❛ r❡❛❧✐s❛t✐♦♥ ♦❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡Xbn✳
✺✳✷ ❙❦❡✇ ❇r♦✇♥✐❛♥ ▼♦t✐♦♥
❚❤❡ ❙❦❡✇ ❇r♦✇♥✐❛♥ ▼♦t✐♦♥ ✭❙❇▼✮ ♦❢ ♣❛r❛♠❡t❡rβ ∈(−1,1)st❛rt✐♥❣ ❢r♦♠ y✱ ✇❤✐❝❤ ✇❡ ❞❡♥♦t❡ ❜②Yβ,y
✐s ❦♥♦✇♥ t♦ s♦❧✈❡✿
Ytβ,y=y+Wt+βLyt(Yβ,y), ✭✺✳✸✮
✐✳❡✳ Yβ,y s♦❧✈❡sSde(1, βδ0)✭s❡❡ ❬❍❙✽✶❪✮✳
■t ✇❛s ✜rst ❝♦♥str✉❝t❡❞ ❜② ■tô ❛♥❞ ▼❝❑❡❛♥ ✐♥ ❬■▼✼✹❪ ✭Pr♦❜❧❡♠ ✶ ♣✶✶✺✮ ❜② ✢✐♣♣✐♥❣ t❤❡ ❡①❝✉rs✐♦♥s ♦❢ ❛ r❡✢❡❝t❡❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t②α= (β+ 1)/2✳ ❖♥ ❙❇▼ s❡❡ ❛❧s♦ ❬❲❛❧✼✽❪ ✳ ■t ❜❡❤❛✈❡s ❧✐❦❡ ❛
❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ❡①❝❡♣t ✐♥y ✇❤❡r❡ ✐ts ❜❡❤❛✈✐♦✉r ✐s ♣❡rt✉❜❛t❡❞✱ s♦ t❤❛t
P(Ytβ,y > y) =α, ∀t >0. ✭✺✳✹✮
❲❡ ❞❡♥♦t❡ ❜② T(∆) t❤❡ ❧❛✇ ♦❢ t❤❡ st♦♣♣✐♥❣ t✐♠❡ τ = inf{t ≥ 0, |Wt| = ∆} ✇❤❡r❡ W ✐s ❛ st❛♥❞❛r❞
❇r♦✇♥✐❛♥ ♠♦t✐♦♥ st❛rt✐♥❣ ❛t ③❡r♦✳ ❋♦r t❤❡ ❙❇▼ Yβ,0 ❛♥❞ ∆ ✐♥ R∗
+ ✇❡ ❞❡✜♥❡ t❤❡ st♦♣♣✐♥❣ t✐♠❡ τ∆ =
inf{t≥0, Ytβ,0∈ {∆,−∆}}✳ ❖✉r ❛♣♣r♦❛❝❤ r❡❧✐❡s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✳
▲❡♠♠❛ ✺✳✶ ▲❡ty ❛♥❞x❜❡ ✐♥ R✱∆ ✐♥ R∗+ ❛♥❞β ✐♥ (−1,1)✳ ❙❡tα= (β+ 1)/2✳ ❚❤❡♥
✐✮Yβ,y+x∼=Yβ,y+x✱ ❛♥❞ ✐✐✮(τ∆, Yβ,0
τ∆ )∼=T(∆)⊗Ber(α).
Pr♦♦❢✳ ❚❤✐s ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❙❇▼ ❜② ■tô ❛♥❞ ▼❝❑❡❛♥ ✐♥ ❬■▼✼✹❪✳
✺✳✸ ❙♦♠❡ ♣♦ss✐❜❧❡ ❛♣♣r♦❛❝❤❡s
❘❡❝❡♥t❧② t✇♦ ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ ❜✉✐❧❞Xbn s❛t✐s❢②✐♥❣ ✭✺✳✶✮✳
■♥ ❬▼❛r✵✹❪ ▼✳ ▼❛rt✐♥❡③ ♣r♦♣♦s❡❞ t♦ ✉s❡ ❛♥ ❊✉❧❡r s❝❤❡♠❡✳ ❲❡ ❦♥♦✇ ❜② Pr♦♣♦s✐t✐♦♥✹✳✶ t❤❛t X s♦❧✈❡s
Sde(√aρ, ν) ✇✐t❤ ν ❞❡✜♥❡❞ ❜② ✭✹✳✶✮✳ ❲❡ ❤❛✈❡ s❡❡♥ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳✷ t❤❛t ✐❢ ✇❡ ❞❡✜♥❡
Φ(x) =R0xfν(y)t❤❡♥ Y = Φ(X)s♦❧✈❡sSde(γ,0) ✇✐t❤γ ✐♥ s♦♠❡ Coeff(m, M)✳ ❚❤✉s ❛♥ ❊✉❧❡r s❝❤❡♠❡
❛♣♣r♦①✐♠❛t✐♦♥ Ybn ♦❢ Y ❝❛♥ ❜❡ ❜✉✐❧t ❛♥❞ ❜② s❡tt✐♥❣ Xbn = Φ−1(Ybn) ✇❡ ❣❡t ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ X✳
❇❡❝❛✉s❡ t❤❡ ❝♦❡✣❝✐❡♥tγ✐s ♥♦t ▲✐♣s❝❤✐t③ ✐❢a❛♥❞ρ❛r❡ ♥♦t✱ ❡✈❛❧✉❛t✐♥❣ t❤❡ s♣❡❡❞ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s✉❝❤
❛ s❝❤❡♠❡ ✐s ♥♦t ❡❛s②✳
■♥ ❬▲▼✵✻❪✱ ❆✳ ▲❡❥❛② ❛♥❞ ▼✳ ▼❛rt✐♥❡③ ♣r♦♣♦s❡❞ t♦ ✉s❡ t❤❡ ❙❇▼✳ ❚❤❡② ✜rst ❜✉✐❧❞ ❛ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t ❛♣♣r♦①✐♠❛t✐♦♥ (an, ρn) ♦❢ (a, ρ) ✐♥ ♦r❞❡r t❤❛t t❤❡ ♣r♦❝❡ss Xn ❣❡♥❡r❛t❡❞ ❜② L(an, ρn) s♦❧✈❡s Sde(√anρn, νn)✇✐t❤ νn s❛t✐s❢②✐♥❣(νn)c = 0✳ ❙❡❝♦♥❞ ❜② ❛ ♣r♦♣❡r ❜✐❥❡❝t✐♦♥ Φn ∈ T(m, M)✱ t❤❡② ❣❡t
t❤❛t Yn = Φn(Xn) s♦❧✈❡s Sde(1, µn) ✇✐t❤ µn = Pβ
kδyk✱ ✐✳❡✳ Yn ❜❡❤❛✈❡s ❧♦❝❛❧❧② ❧✐❦❡ ❛ ❙❇▼✳ ❚❤✐r❞
t❤❡② ♣r♦♣♦s❡❞ ❛ s❝❤❡♠❡Ybn ❢♦rYn ❜❛s❡❞ ♦♥ ▲❡♠♠❛✺✳✶❛♥❞ s✐♠✉❧❛t✐♦♥s ♦❢ ❡①✐t t✐♠❡s ♦❢ t❤❡ ❙❇▼ ❛♥❞
t❤❡② ✜♥❛❧❧② s❡tXbn= (Φn)−1(Ybn)✳
❖✉r ♠❡t❤♦❞ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ✈❛r✐❛t✐♦♥ ♦❢ t❤✐s ❧❛st ❛♣♣r♦❛❝❤ ❜❡❝❛✉s❡ ✐t ❛❧s♦ ❞❡❡♣❧② r❡❧✐❡s ♦♥ ❣❡tt✐♥❣ s✉❝❤ ❛Yn ❛♥❞ ✉s✐♥❣ ▲❡♠♠❛ ✺✳✶✳ ❇✉t ✇❡ t❤❡♥ ✉s❡ r❛♥❞♦♠ ✇❛❧❦s ✐♥st❡❛❞ ♦❢ t❤❡ s❝❤❡♠❡ ♣r♦♣♦s❡❞ ✐♥ ❬▲▼✵✻❪✳
✺✳✹ ❚❤❡ ❜❛s✐❝ ✐❞❡❛ ♦❢ ♦✉r ❛♣♣r♦❛❝❤
❲❡ ❢♦❝✉s ♦♥ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ ♣r♦♣♦s❡ ❛ t❤r❡❡✲st❡♣ ❛♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡ ❞✐✛❡r✐♥❣ s❧✐❣❤t❧② ❢r♦♠ t❤❡ ♦♥❡ ♣r♦♣♦s❡❞ ❜② ❚❤❡♦r❡♠✸✳✷✳
❲❡ ✜①n∈N∗✱ ❛♥❞1/n ✇✐❧❧ ❜❡ t❤❡ s♣❛t✐❛❧ ❞✐s❝r❡t✐③❛t✐♦♥ st❡♣ s✐③❡✳
❙❚❊P ✶✳ ❲❡ ❜✉✐❧❞(an, ρn)✐♥ Coeff(λ,Λ)×Coeff(λ,Λ)s✉❝❤ t❤❛t✿
✐✮ ❚❤❡ ❢✉♥❝t✐♦♥s an ❛♥❞ ρn ❛r❡ ♣✐❡❝❡✇✐s❡ ❝♦♥st❛♥t✳ ❚❤❡ ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ ❡✐t❤❡r an ❛♥❞ ρn
❛r❡ ✐♥❝❧✉❞❡❞ ✐♥ s♦♠❡ s❡t In✳ ❲❡ ❛ss✉♠❡ In = {xn
k}k∈In ❢♦r In = {0 ≤ k ≤ k1n} ⊂ Z ✜♥✐t❡ ❛♥❞
xn
k < xnk+1, ∀k∈In✳
✐✐✮ ❋♦r ❡❛❝❤xn
k ∈ In ✇❡ ❤❛✈❡an(xnk) =a(xnk)❛♥❞ρn(xnk) =ρ(xnk)✳
✐✐✐✮ ❈♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥
Φn(x) =
kn,x−1 X
k=0 xn
k+1−xnk
p a(xn
k)ρ(xnk)
+ x−x
n kn,x
q a(xn
kn,x)ρ(x n kn,x)
, ✭✺✳✺✮
✇❤❡r❡ t❤❡ ✐♥t❡❣❡rkn,x✈❡r✐✜❡sxnkn,x ≤x≤xnkn,x+1✳
❚❤❡ s❡tIn s❛t✐s✜❡sΦn(In) = {k/n, k∈Z} ∩Φn(G)✳ ❋r♦♠ ♥♦✇ ✇❡ ❛ss✉♠❡ xn
k ✐s t❤❡ ♣♦✐♥t ♦❢ In s✉❝❤
t❤❛tΦn(xn
k) =k/n✳
❘❡♠❛r❦ ✺✳✶ ■♥ ❢❛❝t t❤❡ ✜rst t❤✐♥❣ t♦ ❞♦ ✐s t♦ ❝♦♥str✉❝t t❤❡ ❣r✐❞ In s❛t✐s❢②✐♥❣ ✐✐✐✮✳ ■t ✐s ✈❡r② ❡❛s② ❛♥❞
♦♥❧② r❡q✉✐r❡s t♦ ❦♥♦✇ t❤❡ ❝♦❡✣❝✐❡♥tsa❛♥❞ρ✭s❡❡ ♣♦✐♥t ✶ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐♥ ❙✉❜s❡❝t✐♦♥✺✳✺✮✳ ❚❤❡♥an
❛♥❞ρn ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞✳
❘❡♠❛r❦ ✺✳✷ ❚❤❡ s❡tsI ❛♥❞In ♠❛② ❤❛✈❡ ♥♦ ❝♦♠♠♦♥ ♣♦✐♥ts✳
❲❡ t❛❦❡Xn t♦ ❜❡ t❤❡ ♣r♦❝❡ss ❣❡♥❡r❛t❡❞ ❜②L(an, ρn)✳
❘❡♠❛r❦ ✺✳✸ ■t ❝❛♥ ❜❡ s❤♦✇♥ ❜② ❚❤❡♦r❡♠✷✳✷t❤❛tXn ❝♦♥✈❡r❣❡s ✐♥ ❧❛✇ t♦X✳
❙❚❊P ✷✳ ❇② Pr♦♣♦s✐t✐♦♥✹✳✶t❤❡ ♣r♦❝❡ssXn s♦❧✈❡sSde(√anρn, νn)✇✐t❤
νn= X
xn k∈In
an(xn
k+)−an(xnk−)
an(xn
k+) +an(xnk−)
δxn k.
❚❤❡ ❢✉♥❝t✐♦♥Φn ❞❡✜♥❡❞ ❜② ✭✺✳✺✮ ❜❡❧♦♥❣s t♦T(1/Λ,1/λ)✳ ❚❤❡ ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t② ♦❢Φn′
❛r❡ t❤♦s❡ ✐♥
In✱ ❛♥❞(Φn′
)′= 0✱ s♦ ❜② Pr♦♣♦s✐t✐♦♥✸✳✶t❤❡ ♣r♦❝❡ssYn= Φn(Xn)s♦❧✈❡s
Ytn=Y0n+Wt+
X
xn k∈In
βknL k/n
t (Yn), ✭✺✳✻✮
✇❤❡r❡
βnk =
p a(xn
k)/ρ(xnk)−
q a(xn
k−1)/ρ(xnk−1) p
a(xn
k)/ρ(xnk) +
q a(xn
k−1)/ρ(xnk−1)
. ✭✺✳✼✮
❚♦ ✇r✐t❡ t❤❡s❡ ❝♦❡✣❝✐❡♥ts ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ❢❛❝t t❤❛tan❛♥❞ρn❛r❡ r✳❝✳❧✳❧✳ ❛♥❞ t❤❛t ❢♦r ✐♥st❛♥❝❡an(xn k+) =
a(xnk)❛♥❞an(xkn−) =a(xnk−1)✳
❘❡♠❛r❦ ✺✳✹ ❲❡ ❤❛✈❡ ❣♦tYn t❤❛t s♦❧✈❡sSde(1,Pβn
kδk/n)✐♥ ❛ ❞✐✛❡r❡♥t ✇❛② t❤❛♥ t❤❡ ♦♥❡ ✉s❡❞ ❜② ▲❡
●❛❧❧ ✐♥ ❚❤❡♦r❡♠✸✳✷✳ ❲❡ ♥♦✇ ✉s❡ ❤✐s ♠❡t❤♦❞ t♦ ❣❡tYen t❤❛t ✈❡r✐✜❡s
E|Yetn−Ytn| −−−−→
n→∞ 0, ∀t∈[0, T]. ✭✺✳✽✮
❙❚❊P ✸✳ ▲✐❦❡ ✐♥ ✭✸✳✼✮ ✇❡ ❞❡✜♥❡ ❛ s❡q✉❡♥❝❡(τn
p)p∈N♦❢ st♦♣♣✐♥❣ t✐♠❡s ❜②✱
τ0n= 0, ❛♥❞ τpn+1= inf{t > τpn:|Ytn−Yτnn
p|= 1/n}.
❚❤❛♥❦s t♦ t❤❡ ✉♥✐❢♦r♠✐t② ♦❢ t❤❡ ❣r✐❞{k/n, k∈In}✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✳
▲❡♠♠❛ ✺✳✷ ✐✮ ❋♦r ❛❧❧k∈Z❛♥❞ ❛❧❧p∈N✱(Yτnn p+u−Y
n τn
p,0≤u≤τ
n
p+1−τpn)❦♥♦✇✐♥❣ t❤❛t{Yτnn
p =k/n}
❤❛s t❤❡ s❛♠❡ ❧❛✇ ❛s(Yβkn,0
t ,0≤t≤τ1/n)✳
✐✐✮
∀p∈N, σnp :=n2 τpn−τpn−1∼=T(1),
❛♥❞ t❤❡σn
p✬s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳
Pr♦♦❢✳ ❚❤❡ st❛t❡♠❡♥t ✐✮ ❢♦❧❧♦✇s s✐♠♣❧② ❢r♦♠ ♣♦✐♥t ✐✮ ♦❢ ▲❡♠♠❛✺✳✶❛♥❞ t❤❡ ❝♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ✭✺✳✸✮ ❛♥❞ ✭✺✳✻✮✳ ❋r♦♠ ✐✮✱ t❤❡ str♦♥❣ ▼❛r❦♦✈ ♣r♦♣❡rt②✱ ❛♥❞ ♣♦✐♥t ✐✐✮ ♦❢ ▲❡♠♠❛✺✳✶✱ ✇❡ ❣❡t t❤❛t(τn
p−τpn−1)∼=T(1/n2)✱
❛♥❞ t❤❡ st❛t❡♠❡♥t ✐✐✮ ❢♦❧❧♦✇s ❜② s❝❛❧✐♥❣✳ ❯s✐♥❣ ❛❣❛✐♥ t❤❡ ▼❛r❦♦✈ ♣r♦♣❡rt② ✇❡ ❣❡t t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡σn
p✬s✳
▲❡ ●❛❧❧ ✉s❡❞ t❤✐s ❧❡♠♠❛ ✐♥ ❤✐s ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠✸✳✷✳ ■♥❞❡❡❞ ✐t ✐s ♦❜✈✐♦✉s ❜② t❤❡ ✐✮ ♦❢ ▲❡♠♠❛ ✺✳✷ ❛♥❞ t❤❡ ✐✐✮ ♦❢ ▲❡♠♠❛✺✳✶t❤❛tSn
p :=nYτnn
p s❛t✐s✜❡s
Sn
0 = 0,
P[Spn+1=k+ 1|Spn=k] = 1
2(1 +β
n
k) =:αnk, ∀p∈N∗,∀k∈In,
P[Spn+1=k−1|Spn=k] = 1
2(1−β
n
k) = 1−αnk, ∀p∈N∗,∀k∈In.
✭✺✳✾✮
▼♦r❡♦✈❡r t❤❡ ✐✐✮ ♦❢ ▲❡♠♠❛✺✳✷❛❧❧♦✇s t♦ s❤♦✇ t❤❛tYen
t := (1/n)S[nn2t]=Yτnn
[n2t] s❛t✐s✜❡s ✭✺✳✽✮✳
❚❤✉s t❤❡ ✐❞❡❛ ✐s t♦ t❛❦❡
b
Xtn:= (Φn)−1
1
nSb
n
[n2t]
, ✭✺✳✶✵✮
✇❤❡r❡ Sbn ✐s ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ t❤❡ ✐♥t❡❣❡rs ❞❡✜♥❡❞ ❜② ✭✺✳✾✮✳ ❚❤❡ ♣r♦❝❡ssXbn ✐s ❛ r❛♥❞♦♠ ✇❛❧❦ ♦♥ t❤❡
❣r✐❞ In✳ ■♥ ❢❛❝t t❤✐s ❣r✐❞ ✐s ♠❛❞❡ ✐♥ ♦r❞❡r t❤❛t Xbn s♣❡♥❞s t❤❡ s❛♠❡ ❛✈❡r❛❣❡ t✐♠❡ ✐♥ ❡❛❝❤ ♦❢ ✐ts ❝❡❧❧s✳
❈♦♠❜✐♥✐♥❣ r❡♠❛r❦✺✳✸❛♥❞ t❤❡♦r❡♠✸✳✷✇❡ s❤♦✉❧❞ ❤❛✈❡ ✭✺✳✶✮✳ ❚♦ s✉♠ ✉♣ t❤✐s s❡❝t✐♦♥ ✇❡ ✇r✐t❡ ♦✉r s❝❤❡♠❡ ✐♥ t❤❡ ❛❧❣♦r✐t❤♠ ❢♦r♠✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❡st✐♠❛t❡ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ♦❢ ♦✉r s❝❤❡♠❡✳
✺✳✺ ❚❤❡ ❛❧❣♦r✐t❤♠
◆♦t❡ t❤❛t ❜② ❝♦♥str✉❝t✐♦♥(Φn)−1 k/n=xn
k ❢♦r ❛❧❧k∈In✳
❲❡ ❞❡✜♥❡ ❛ ❢✉♥❝t✐♦♥ALGO✐♥ t❤❡ ♥❡①t ♠❛♥♥❡r✿
■◆P❯❚ ❉❆❚❆✿ t❤❡ ❝♦❡✣❝✐❡♥tsa❛♥❞ρ✱ t❤❡ st❛rt✐♥❣ ♣♦✐♥tx✱ t❤❡ ♣r❡❝✐s✐♦♥ ♦r❞❡rn❛♥❞ t❤❡ ✜♥❛❧ t✐♠❡t✳
❖❯❚P❯❚ ❉❆❚❆✿ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ❧❛✇Xbn ♦❢X ❛t t✐♠❡ t✳
✶✳ ❙❡txn
0 ←l✳
✇❤✐❧❡xn k ≤r
s❡t xn k ←
p a(xn
k)ρ(xnk)(1/n) +xnk ❛♥❞k←k+ 1✳
❡♥❞✇❤✐❧❡
✷✳ ❈♦♠♣✉t❡ t❤❡αn
k = (1 +βkn)/2 ✇✐t❤βkn ❞❡✜♥❡❞ ❜② ✭✺✳✼✮✳
✸✳ ❙❡ty←Φn(x)✳
✐❢(n y− ⌊n y⌋)<0.5
s❡t s0← ⌊n y⌋✳
❡❧s❡
s❡t s0← ⌊n y⌋+ 1✳
❡♥❞✐❢
✹✳ ❢♦ri= 0 t♦i=⌊n2t⌋=:N
✐❢xn
si∈R\(l, r)
❘❡t✉r♥xn si✳
❡♥❞✐❢
❲❡ ❤❛✈❡ si=k ❢♦r s♦♠❡k∈Ik✳ ❙✐♠✉❧❛t❡ ❛ r❡❛❧✐③❛t✐♦♥B ♦❢Ber(αnk)✳
❚❤❡♥ s❡t si←si+B✳
❡♥❞❢♦r ✺✳ ❘❡t✉r♥xn
sN✳
✻ ❙♣❡❡❞ ♦❢ ❝♦♥✈❡r❣❡♥❝❡
■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳
❚❤❡♦r❡♠ ✻✳✶ ❆ss✉♠❡ t❤❛ta, ρ∈Coeff(λ,Λ)❢♦r s♦♠❡ 0< λ≤Λ<∞✳ ▲❡t ❜❡0< T <∞ ❛♥❞X t❤❡
♣r♦❝❡ss ❣❡♥❡r❛t❡❞ ❜②L(a, ρ)✳ ❋♦rn∈N ❝♦♥s✐❞❡r t❤❡ ♣r♦❝❡ssXbn st❛rt✐♥❣ ❢r♦♠x❞❡✜♥❡❞ ❜②✱
∀t∈[0, T], Xbtn=ALGO(a, ρ, x, n, t).
❋♦r ❛❧❧f ∈W10,∞(G)∩ C0(G)✱ ❛❧❧ε >0✱ ❛♥❞ ❛❧❧γ∈(0,1/2)t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥tC ❞❡♣❡♥❞✐♥❣ ♦♥ε✱γ✱ T✱λ✱Λ✱G✱ka′k
∞✱kρ′k∞ kfk∞✱kdf /dxk2✱kdf /dxk∞✱supi∈I1/(xi+1−xi)✱ ❛♥❞ t❤❡ t✇♦ ✜rst ♠♦♠❡♥ts
♦❢T(1) s✉❝❤ t❤❛t✱ ❢♦rn❧❛r❣❡ ❡♥♦✉❣❤✱
sup
(t,x)∈[ε,T]×G¯
Exf(Xt)−Exf(Xbtn)
≤Cn−γ.
❲❡ ❤❛✈❡✱
|Exf(Xt)−Exf(Xbtn)| ≤ |Exf(Xt)−Exf(Xtn)|+|Exf(Xtn)−Exf(Xbtn)|
=: e1(t, x, n) +e2(t, x, n).
✭✻✳✶✮
❲❡ ✇✐❧❧ ❡st✐♠❛t❡e1(t, x, n)❜② P❉❊s t❡❝❤♥✐q✉❡s ❛♥❞e2(t, x, n)❜② ✈❡r② s✐♠♣❧❡ ♣r♦❜❛❜✐❧✐st✐❝ t❡❝❤♥✐q✉❡s✳
✻✳✶ ❊st✐♠❛t❡ ♦❢ ❛ ✇❡❛❦ ❡rr♦r
■♥ t❤✐s s✉❜s❡❝t✐♦♥ ✇❡ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥✳
Pr♦♣♦s✐t✐♦♥ ✻✳✶ ❆ss✉♠❡f❜❡❧♦♥❣s t♦H1
0(G)∩C0(G)✳ ▲❡t ❜❡u(t, x)❛♥❞un(t, x)r❡s♣❡❝t✐✈❡❧② t❤❡ s♦❧✉t✐♦♥s
♦❢(P)(a, ρ, f)❛♥❞(P)(an, ρn, f)✱ ✇✐t❤an ❛♥❞ρn ❧✐❦❡ ✐♥ ❙✉❜s❡❝t✐♦♥ ✺✳✹✱ ❙t❡♣ ✶✳ ❚❤❡♥ ❢♦r ❛❧❧ ε >0t❤❡r❡
✐s ❛ ❝♦♥st❛♥tC1 ❞❡♣❡♥❞✐♥❣ ♦♥ε✱ T✱ λ✱ Λ✱G✱kfk∞✱kdf /dxk2✱ka′k∞✱kρ′k∞✱ ❛♥❞ supi∈I1/(xi+1−xi)
s✉❝❤ t❤❛t ❢♦rn ❧❛r❣❡ ❡♥♦✉❣❤✱
sup
(t,x)∈[ε,T]×G¯
|u(t, x)−un(t, x)| ≤ C1√1n.
❆s ✇❡ ✇✐❧❧ s❡❡ ✐♥ Pr♦♣♦s✐t✐♦♥✻✳✷✱ ✐❢ ✇❡ ❤❛❞I ⊂ In ✇❡ ❝♦✉❧❞ ♦❜t❛✐♥ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r|||u−un|||
∞,∞
♦❢ t❤❡ ❢♦r♠K(ka−ank2
∞+kρ−ρnk∞)✳ ❇✉t t❤✐s ✐s ♥♦t ♥❡❝❡ss❛r② t❤❡ ❝❛s❡ ✭s❡❡ ❘❡♠❛r❦✺✳✷✮✳ ❍♦✇❡✈❡r
✐t ✐s ♣♦ss✐❜❧❡ t♦ ♠♦❞✐❢②a❛♥❞ ρ✐♥ ♦r❞❡r t♦ r❡✜♥❞ ✉s ✐♥ ❛ s✐t✉❛t✐♦♥ ❝❧♦s❡ t♦ t❤✐s ♦♥❡✱ ❛♥❞ ✇❡ ✇✐❧❧ ❞♦ t❤❛t
t♦ ♣r♦✈❡ Pr♦♣♦s✐t✐♦♥✻✳✶✳
Pr♦♣♦s✐t✐♦♥ ✻✳✷ ▲❡t ❜❡f ∈H10(G)∩C0(G)✳ ▲❡t ❜❡a1, ρ1, a2, ρ2∈Coeff(λ,Λ)✱ ❛♥❞I1❛♥❞I2r❡s♣❡❝t✐✈❡❧②
t❤❡ s❡t ♦❢ ♣♦✐♥ts ♦❢ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ a1 ❛♥❞ρ1 ❛♥❞a2 ❛♥❞ρ2✳ ❆ss✉♠❡ I1⊂ I2✳ ▲❡t ❜❡ u1(t, x) t❤❡ ✇❡❛❦
s♦❧✉t✐♦♥ ♦❢ (P)(a1, ρ1, f) ❛♥❞ u2(t, x) t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥ ♦❢ (P)(a2, ρ2, f)✳ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t C1e
❞❡♣❡♥❞✐♥❣ ♦♥T✱λ✱Λ✱G✱kfk∞✱ ❛♥❞kdf /dxk2✱ s✉❝❤ t❤❛t✱
|||u1−u2|||∞,∞≤ C1e
ka1−a2k2∞+kρ1−ρ2k∞
.
❋✐♥❛❧❧②✱ ✇❡ ❦♥♦✇ t❤❛tu(t, x)✐s ❝♦♥t✐♥✉♦✉s ♦♥[0, T]×G¯ ❛♥❞ ♦❢ ❝❧❛ssC1♦♥ ❡❛❝❤[ε, T]×(x
i, xi+1)✭s❡❡ ✐♥
❬▲❘❯✻✽❪ ❚❤❡♦r❡♠s ✻ ❛♥❞ ✼✮✳ ❙♦ ✐❢x❛♥❞φn(x)❜❡❧♦♥❣ t♦ t❤❡ s❛♠❡ ✐♥t❡r✈❛❧(xi, xi+1)✇❡ ❤❛✈❡✱
|u(t, x)−u(t, φn(x))| ≤ sup
(t,x)∈[ε,T]×(xi,xi+1)
|dudx(t, x)| · |x−φn(x)|.
▲❡t ✉s s❡t
M = sup
i∈I (t,x)∈[ε,Tsup]×(xi,xi+1)
|dudx(t, x)|.
■❢ ❢♦r ✐♥st❛♥❝❡x∈(xi−1, xi)❛♥❞φn(x)∈(xi, xi+1)✇❡ ❤❛✈❡✱
|u(t, x)−u(t, φn(x))| ≤ |u(t, x)−u(t, xi)|+|u(t, xi)−u(t, φn(x))| ≤2Mkid−φnk∞. ✭✻✳✶✷✮
❲❡ ✇✐❧❧ s❡❡ ❜❡❧♦✇ t❤❛tkid−φnk∞ →0 ❛sn→ ∞✱ s♦ ❢♦r n❧❛r❣❡ ❡♥♦✉❣❤ ✇❡ ❛r❡ ❛❧✇❛②s ❛t ❧❡❛st ✐♥ t❤❡
❧❛st s✐t✉❛t✐♦♥✳
❙t❡♣ ✹✳ ❇② ❝♦♥str✉❝t✐♦♥ ✭s❡❡ ♣♦✐♥t ✶ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✮ t❤❡ ❣r✐❞ In s❛t✐s✜❡s |xn
k+1−xnk| ≤ Λ/n✱ ❢♦r ❛❧❧
k∈In✳
❙♦ ❡❧❡♠❡♥t❛r② ❝♦♠♣✉t❛t✐♦♥s s❤♦✇ t❤❛t
kid−φnk∞≤ 3Λ
n and 1−φ1′
n
∞
≤2Λ sup
i∈I
1
xi+1−xi
.1 n.
❆s ✇❡ ❤❛✈❡ s❛✐❞ ❛❜♦✈❡✱ ❢♦rn❧❛r❣❡ ❡♥♦✉❣❤ ✭✻✳✶✷✮ ✐s ✈❛❧✐❞ ❛♥❞ ✇❡ t❤❡♥ ❤❛✈❡✱
|u(t, x)−u(t, φn(x))| ≤6MΛ1
n. ✭✻✳✶✸✮
■t r❡♠❛✐♥s t♦ ❡✈❛❧✉❛t❡k˜an−ank
∞ ❛♥❞kρ˜n−ρnk∞✳ ❖♥ ❡❛❝❤ (xi, xi+1)✱a✐s ♦❢ ❝❧❛ssC1❛♥❞ a′ ✐s r✳❝✳❧✳❧✳✱
s♦ ✐t ♠❛❦❡s s❡♥s❡ t♦ s♣❡❛❦ ♦❢ ka′k
∞✳ ▼♦r❡♦✈❡r ❡❛❝❤φn([xnk, xnk+1))✐s ✐♥❝❧✉❞❡❞ ✐♥ s♦♠❡ [xi, xi+1] t❤❛t
❝♦♥t❛✐♥sxn
k✱ s♦ ✇❡ ❤❛✈❡✱
ka˜n−ank
∞ ≤ supk∈Insupx∈[xn
k,xnk+1)|a(φn(x))/φ
′
n(x)−an(x)|
≤ ka′k
∞supx∈[xn
k,xnk+1)|φn(x)−x n
k|+ Λ supx∈[xn
k,xnk+1)|1−
1
φ′ n(x)|.
■♥ ❛❞❞✐t✐♦♥|φn(x)−xkn| ≤ |φn(x)−x|+|x−xnk| ≤4Λ/n✱ ❢♦r ❛❧❧x∈[xnk, xnk+1)✱ ❛♥❞ ✇❡ ❝❛♥ ❣❡t ❛ s✐♠✐❧❛r
❜♦✉♥❞ ❢♦r|1−φ′1
n(x)|s♦ ✜♥❛❧❧② t❤❡r❡ ❡①✐stsK1s✉❝❤ t❤❛t
k˜an−ank∞≤ K1
1
n. ✭✻✳✶✹✮
■♥ ❛ s✐♠✐❧❛r ♠❛♥♥❡r ✇❡ ❣❡tK2s✉❝❤ t❤❛t✱
kρ˜n−ρnk∞≤ K2
1
n. ✭✻✳✶✺✮
❚❤✉s✱ ❝♦♠❜✐♥✐♥❣ ✭✻✳✶✵✮✱ ✭✻✳✶✶✮✱ ✭✻✳✶✸✮✱ ✭✻✳✶✹✮ ❛♥❞ ✭✻✳✶✺✮✱ ✇❡ ❝♦♠♣❧❡t❡ t❤❡ ♣r♦♦❢✳
