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✐♦♥ ♦❢ ❙❊❊ ✐❢

  < 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 < ∞, _{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 > ✵ ✇✐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✐❝❧❡ ■❉ ✼✽✾✺✹✾✱ ✷✵✶✸✳

  ❚❤❛♥❦ ❨♦✉✦✦