# An Introduction to Stochastic Partial Di erential Equations

❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦

❙t♦❝❤❛st✐❝ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s

❍❡rr② Pr✐❜❛✇❛♥t♦ ❙✉r②❛✇❛♥ ❉❡♣t✳ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❙❛♥❛t❛ ❉❤❛r♠❛ ❯♥✐✈❡rs✐t②✱ ❨♦❣②❛❦❛rt❛

✷✾✳ ❆✉❣✉st ✷✵✶✹

❖✉t❧✐♥❡

❙P❉❊ ❊①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ♦❢ ❙♦❧✉t✐♦♥ ❆ ❙t♦❝❤❛st✐❝ ❍❡❛t ❊q✉❛t✐♦♥

❙P❉❊

❙P❉❊ ✐s ❛♥ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r② ❛r❡❛ ❛t t❤❡ ❝r♦ssr♦❛❞s ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ❛♥❞ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❲❛✈❡ ❡q✉❛t✐♦♥

✷ ✷

∂ u ∂ u (t, x) (t, x)

= κ + F (t, x), t ≥ ✵, ✵ ≤ x ≤ L ✭✶✮

✷ ✷

∂t ∂x ■❢ F ✐s ❛ r❛♥❞♦♠ ♥♦✐s❡✱ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ❣✉✐t❛r ✐♥ t❤❡ ❞❡s❡rt✳ ❋♦r ❡①❛♠♣❧❡✱ F (t, x) = W (t, x) ✐s t❤❡ s♣❛❝❡✲t✐♠❡ ✇❤✐t❡ ♥♦✐s❡✳ ❍❡✉r✐st✐❝❛❧❧②✱ W

(t, x) ✐s ❛ ✭●❛✉ss✐❛♥✮ r❛♥❞♦♠ ✜❡❧❞ s✉❝❤ t❤❛t E (W (t, x)W (s, y )) = δ(t − s)δ(x − y ).

■♥ t❤✐s ❝❛s❡✱ ❞♦❡s ♥♦t ❤❛✈❡ ❝❧❛ss✐❝❛❧ ♠❡❛♥✐♥❣ ❛♥❞ ♠✉st ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛♥ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✳

❖✉r t♦♣✐❝✿ ❙❡♠✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s ❞r✐✈❡♥ ❜② ❛❞❞✐t✐✈❡ ❇r♦✇♥✐❛♥ ♥♦✐s❡✳ ❖✉r ❛♣♣r♦❛❝❤✿ ❍✐❧❜❡rt s♣❛❝❡ ❛♣♣r♦❛❝❤ ❜❛s❡❞ ♦♥ t❤❡ t❤❡♦r② ♦❢ ♦♣❡r❛t♦r s❡♠✐❣r♦✉♣✳ ❙❡❡ ❬❉❛Pr❛t♦✲❩❛❜❝③②❦❪ ❛♥❞ ❬Pr❡✈♦t✲❘ö❝❦♥❡r❪✳ ❚❤✐s ✐s ❛♥ ✭❡❧❡♠❡♥t❛r②✮ ✐♥tr♦❞✉❝t✐♦♥ s✐♥❝❡ ✇❡ ❞♦ ♥♦t ❝♦♥s✐❞❡r ❊q✉❛t✐♦♥s ✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ♥♦✐s❡

❊q✉❛t✐♦♥s ❞r✐✈❡♥ ❜② ❢r❛❝t✐♦♥❛❧ ❇r♦✇♥✐❛♥ ♥♦✐s❡ ✭●❛✉ss✐❛♥✱ ♥♦♥✲▼❛r❦♦✈✱ ♥♦♥✲s❡♠✐♠❛rt✐♥❣❛❧❡✮ ❊q✉❛t✐♦♥s ❞r✐✈❡♥ ❜② ♥♦♥✲●❛✉ss✐❛♥ ♥♦✐s❡ ✭❡✳❣✳ ▲❡✈② ♥♦✐s❡✱ ❛❧♣❤❛✲st❛❜❧❡ ♥♦✐s❡✮ ❊q✉❛t✐♦♥s ✇✐t❤ r♦✉❣❤ ✭♥♦♥✲▲✐♣s❝❤✐t③✮ ♥♦♥❧✐♥❡❛r✐t✐❡s ❱❛r✐❛t✐♦♥❛❧ s♦❧✉t✐♦♥ ✐♥ ●❡❧❢❛♥❞ tr✐♣❧❡ts ❍②♣❡r❜♦❧✐❝ ❛♥❞ ❊❧❧✐♣t✐❝ ♣r♦❜❧❡♠s ▼❛❧❧✐❛✈✐❛♥ ❝❛❧❝✉❧✉s ❛♣♣r♦❛❝❤ ❛♥❞ ❞❡♥s✐t✐❡s ♦❢ s♦❧✉t✐♦♥ ❍✐❞❛ ❝❛❧❝✉❧✉s ❛♣♣r♦❛❝❤ ✭❲✐❝❦ t②♣❡ ❡q✉❛t✐♦♥s✮ ✶✵ ❙♦❧✉t✐♦♥s ✈✐❛ ❉✐r✐❝❤❧❡t ❋♦r♠s ✶✶ ◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ❡t❝

❖✉r ❛♣♣r♦❛❝❤✿ ❆♥ ❙P❉❊ ✐s tr❛♥s❧❛t❡❞ ✐♥t♦ ❛ st♦❝❤❛st✐❝ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✭❈❛✉❝❤② ♣r♦❜❧❡♠✮ ✐♥ s♦♠❡ ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳

❚❤❡r❡ ❛r❡ s♦♠❡ ❝r✉❝✐❛❧ ♣r♦❜❧❡♠s ❞✉❡ t♦ ∞✲❞✐♠❡♥s✐♦♥✦ ■♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡ ❞♦❡s ♥♦t ❡①✐st✦

❊s❝❛♣❡ ❢r♦♠ ♣r♦❜❧❡♠✿ ●❛✉ss✐❛♥ ♠❡❛s✉r❡✦

❚❤❡♦r❡♠ ✭▼✐♥❧♦s✲❙❛③❛♥♦✈✮

▲❡t H ❜❡ ❛ s❡♣❛r❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡✳ ▲❡t Q ❜❡ ❛ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡✱ s②♠♠❡tr✐❝✱ tr❛❝❡✲❝❧❛ss ♦♣❡r❛t♦r ✐♥ H ❛♥❞ ❧❡t m ∈ H✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ●❛✉ss✐❛♥ ♠❡❛s✉r❡ µ = N (m, Q)

♦♥ (H, B(H)) ❣✐✈❡♥ ✈✐❛ Z i hh,ui i hm,hi− hQh,hi µ ˆ (h) := e µ (du) = e , h ∈ H. H ■♠♣♦rt❛♥❝❡✿ t♦ ❞❡✜♥❡ ❍✐❧❜❡rt✲s♣❛❝❡ ✈❛❧✉❡❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ B ❛♥❞✱ ❤❡♥❝❡✱ t♦ ❝♦♥str✉❝t ❍✐❧❜❡rt✲s♣❛❝❡ ✈❛❧✉❡❞ ■tô ✐♥t❡❣r❛❧ ✇✐t❤ r❡s♣❡❝t t♦ B

❊①✐st❡♥❝❡ r❡s✉❧t ✐s ❛ ♣r✐♦r✐ ♥♦t ❝❧❡❛r✦ ❚❤❡♦r❡♠ ✭P❡❛♥♦✮

❋♦r ❡❛❝❤ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f : R × B → B ❞❡✜♥❡❞ ♦♥ s♦♠❡ ♦♣❡♥ s❡t V , x

⊂ R × B ❛♥❞ ❢♦r ❡❛❝❤ ♣♦✐♥t (t ) ∈ V ✵ ✵ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ x x

(t) = f (t, x(t)), (t ) = x

✵ ✵

❤❛s ❛ s♦❧✉t✐♦♥ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ♦♥ s♦♠❡ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ t ✳

✵ ❚❤❡♦r❡♠ ✭●♦❞✉♥♦✈✮

❊❛❝❤ ❇❛♥❛❝❤ s♣❛❝❡ ✐♥ ✇❤✐❝❤ P❡❛♥♦✬s t❤❡♦r❡♠ ✐s tr✉❡ ✐s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✳

❚❤❡ ❙t♦❝❤❛st✐❝ ❊✈♦❧✉t✐♦♥ ❊q✉❛t✐♦♥

❙❡tt✐♥❣✿ H

❛♥❞ U ❛r❡ t✇♦ s❡♣❛r❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡s (Ω, F, P) ✐s ❛ ❝♦♠♣❧❡t❡ ♣r♦❜❛❜✐❧✐t② s♣❛❝❡ B : [

✵, T ] × Ω → U ✐s ❛ tr❛❝❡✲❝❧❛ss ❲✐❡♥❡r ♣r♦❝❡ss ♦♥ U ❛❞❛♣t❡❞ t♦ ❛ ♥♦r♠❛❧ ✜❧tr❛t✐♦♥ (F t ) t ∈[

✵,T ]

A : ❞♦♠(A) ⊂ H → H ✐s ❛ ❞❡♥s❡❧② ❞❡✜♥❡❞✱ s❡❧❢✲❛❞❥♦✐♥t ❛♥❞ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡

❧✐♥❡❛r ♦♣❡r❛t♦r ✇✐t❤ ❝♦♠♣❛❝t ✐♥✈❡rs❡✳ ❆✐♠✿ ❡①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ❛ ♣r❡❞✐❝t❛❜❧❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss X : [✵, T ] × Ω → H ✇❤✐❝❤ s♦❧✈❡s t❤❡ s❡♠✐❧✐♥❡❛r st♦❝❤❛st✐❝ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ❞r✐✈❡♥ ❜② t❤❡ ❲✐❡♥❡r ♣r♦❝❡ss B dX

(t) + (AX (t) + f (t, X (t))) dt = g (t, X (t)) dB(t), ✵ ≤ t ≤ T

X (

✵) = X ✵ ,

❢♦r s♦♠❡ ♥✐❝❡ ❢✉♥❝t✐♦♥s f ❛♥❞ g✳

❊①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ♦❢ ❙♦❧✉t✐♦♥

❙t♦❝❤❛st✐❝ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✭❙❊❊✮✿ dX

(t) + (AX (t) + f (t, X (t))) dt = g (t, X (t)) dB(t), ✵ ≤ t ≤ T

X , (

✵) = X ✵

❉❡✜♥✐t✐♦♥ ✭♦❢ ♠✐❧❞ s♦❧✉t✐♦♥ ♦❢ ❙❊❊✮

▲❡t p ≥ ✷✳ ❆ ♣r❡❞✐❝t❛❜❧❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss X : [✵, T ] × Ω → H ✐s ❝❛❧❧❡❞ ❛ p✲❢♦❧❞ ✐♥t❡❣r❛❜❧❡ ♠✐❧❞ s♦❧✉t✐♦♥ ♦❢ ❙❊❊ ✐❢

&lt; kX (t)k p ∞ t ∈[ s✉♣ L (Ω;H)

✵,T ]

❛♥❞✱ ❢♦r ❛❧❧ t ∈ [✵, T ]✱ ✐t ❤♦❧❞s P✲❛✳s✳ Z t Z t X − E E

(t) = E (t)X (t − s)f (s, X (s)) ds + (t − s)g (s, X (s)) dB(s),

✵ ✵ ✵

✇❤❡r❡ (E(t)) ✐s t❤❡ ❛♥❛❧②t✐❝ s❡♠✐❣r♦✉♣ ♦♥ H ❣❡♥❡r❛t❡❞ ❜② −A✱ t❤❡ ✜rst t ∈[

✵,∞)

✐♥t❡❣r❛❧ ✐s ❛ ❇♦❝❤♥❡r ✐♥t❡❣r❛❧ ❛♥❞ t❤❡ s❡❝♦♥❞ ✐♥t❡❣r❛❧ ✐s t❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ✈❛❧✉❡❞ ■tô ✐♥t❡❣r❛❧✳

❊①✐st❡♥❝❡✲❯♥✐q✉❡♥❡ss ♦❢ ▼✐❧❞ ❙♦❧✉t✐♦♥ ❚❤❡♦r❡♠ ✭❉❛Pr❛t♦✲❩❛❜❝③②❦✮ p

❯♥❞❡r s♦♠❡ ♠❡❛s✉r❛❜✐❧✐t②✱ L ✲r❡❣✉❧❛r✐t② ❛♥❞ ❧✐♥❡❛r ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s ♦♥ X ✵ ✱ f ❛♥❞ g ✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ p✲❢♦❧❞ ✐♥t❡❣r❛❜❧❡ ♠✐❧❞ s♦❧✉t✐♦♥ X : [✵, T ] × Ω → H t♦ ❙❊❊ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② t ∈ [✵, T ] ❛♥❞ ❡✈❡r② s ∈ [✵, ✶) ✐t ❤♦❧❞s t❤❛t s

X (t) ∈ ˙ H = P ✶ ✇✐t❤ kX (t)k p s &lt; ∞, t ∈[ s✉♣ L H (Ω; ˙ ) s s ✵,T ] / ✷

) ✇❤❡r❡ ˙H ✐s t❤❡ ❍✐❧❜❡rt s♣❛❝❡ ❣✐✈❡♥ ❜② ❞♦♠(A ✳ ❋✉rt❤❡r♠♦r❡✱ ❢♦r ❡✈❡r②

✶

δ ∈ ( )

✵, t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t CY &gt; ✵ ✇✐t❤

✷ δ

kX (t ) − X (t )k p ≤ C |t − t |

✶ ✷ L ✶ ✷ (Ω;H)

, t ∈ [ ❢♦r ❛❧❧ t ✵, T ]✳

✶ ✷

❯♥✐q✉❡♥❡ss ❤❡r❡ ✐s ✐♥ t❤❡ s❡♥s❡ ♦❢ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s✳

❆ ❙t♦❝❤❛st✐❝ ❍❡❛t ❊q✉❛t✐♦♥

❙t♦❝❤❛st✐❝ ❤❡❛t ❡q✉❛t✐♦♥ ✇✐t❤ ❛❞❞✐t✐✈❡ ♥♦✐s❡ ♦♥ t❤❡ ✉♥✐t ✐♥t❡r✈❛❧

❙❡tt✐♥❣✿

✷

H = L ([

✵, ✶], B([✵, ✶]), dx; R) B

✐s ❛ tr❛❝❡✲❝❧❛ss ❲✐❡♥❡r ♣r♦❝❡ss ♦♥ H✳ Pr♦❜❧❡♠✿ ❋✐♥❞ ❛ ♠❡❛s✉r❛❜❧❡ ♠❛♣♣✐♥❣ X : [✵, T ] × Ω → R s✉❝❤ t❤❛t

✷

∂ dX (t, x) = X (t, x) dt + dB(t, x)

❢♦r ❛❧❧ t ∈ (✵, T ], x ∈ [✵, ✶]

✷

∂x X (t,

✵) = X (t, ✶) = ✵ ❢♦r ❛❧❧ t ∈ (✵, T ] X ( (x)

✵, x) = X ❢♦r ❛❧❧ x ∈ [✵, ✶],

✵

: Ω × [ (ω, ·) ✇❤❡r❡ X ✵, ✶] → R ✐s s✉❝❤ t❤❛t ❢♦r ❛❧♠♦st ❛❧❧ ω ∈ Ω✱ X ✐s ❛

✵ ✵

s✉✣❝✐❡♥t❧② s♠♦♦t❤ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❛❧s♦ s❛t✐s✜❡s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳

❲❡ tr❛♥s❧❛t❡ ✐♥t♦ ❛♥ ❛❜str❛❝t st♦❝❤❛st✐❝ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ♦♥ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ H

✿ dX (t) + AX (t) dt = dB(t), ❢♦r ❛❧❧ t ∈ [✵, T ],

X ( , ✵) = X

✇❤❡r❡

✷

∂ A

:= −

✷

∂x

✶ ✷

( ( ✇✐t❤ ❞♦♠(A) = H ✵, ✶) T H ✵, ✶)✳

✵

❚❤❡ ♠✐❧❞ s♦❧✉t✐♦♥ ✐s t❤❡♥ ❣✐✈❡♥ ❜② t❤❡ st♦❝❤❛st✐❝ ♣r♦❝❡ss X : [✵, T ] × Ω → H ✇✐t❤ Z t

E (t) = E (t)X (t − s) dB(s)

• X

✵ ✵

❘❡❢❡r❡♥❝❡s

❘✳ ❉❛❧❛♥❣ ❡t ❛❧✳ ❆ ▼✐♥✐❝♦✉rs❡ ♦♥ ❙t♦❝❤❛st✐❝ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱

❙♣r✐♥❣❡r✱ ✷✵✵✽✳ ▲✳ ●❛✇❛r❡❝❦✐ ❛♥❞ ❱✳ ▼❛♥❞r❡❦❛r✳ ❙t♦❝❤❛st✐❝ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✐♥ ■♥✜♥✐t❡

❉✐♠❡♥s✐♦♥s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❙t♦❝❤❛st✐❝ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱

❙♣r✐♥❣❡r✱ ✷✵✶✶✳

●✳ ❞❛ Pr❛t♦ ❛♥❞ ❏✳ ❩❛❜❝③②❦✳ ❙t♦❝❤❛st✐❝ ❊q✉❛t✐♦♥s ✐♥ ■♥✜♥✐t❡ ❉✐♠❡♥s✐♦♥s✱

❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✷✵✵✶✳ ❈✳ Pr❡✈♦t ❛♥❞ ▼✳ ❘ö❝❦♥❡r✳ ❆ ❈♦♥❝✐s❡ ❈♦✉rs❡ ♦♥ ❙t♦❝❤❛st✐❝ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ✷✵✵✼✳ ❙✳ ❚❛♣♣❡✳ ✑❋♦✉♥❞❛t✐♦♥ ♦❢ t❤❡ t❤❡♦r② ♦❢ s❡♠✐❧✐♥❡❛r st♦❝❤❛st✐❝ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧

❡q✉❛t✐♦♥s✑✱ ■♥t✳ ❏✳ ❙t♦❝❤✳ ❆♥❛❧✳ ❱♦❧✉♠❡ ✷✵✶✸✳ ❆rt✐❝❧❡ ■❉ ✼✽✾✺✹✾✱ ✷✵✶✸✳

❚❤❛♥❦ ❨♦✉✦✦