The stochastic heat equation with a fractional-colored noise: existence of solution

Transcription

1 The stochastic heat equation with a fractional-colored noise: existence of solution Raluca Balan (Ottawa) Ciprian Tudor (Paris 1) June 11-12, 27 aluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 1 / 25

2 Outline 1 Statement of the Problem 2 The colored noise 3 The fractional noise 4 The fractional-colored noise Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 2 / 25

3 Statement of the Problem The Framework The noise W = {W (ϕ); ϕ H} is a zero-mean Gaussian noise with covariance E(W (ϕ)w (ψ)) = ϕ, ψ H H= Hilbert space of time-space distributions ϕ(t, x) on [, T ] R d The linear equation u t = u + Ẇ, (t, x) [, T ] Rd u(, ) = Let G be the fundamental solution of L = / t, i.e. G(t, x) = (4πt) d/2 e x 2 /(4t), t >, x R d Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 3 / 25

4 Statement of the Problem Solution of the SPDE Mild solution: u(t, x) = W (g tx ) where g tx = G(t, x ), i.e. u(t, x) = R d G(t s, x y)w (ds, dy) Distribution solution: u(η) = W (η G), η D((, T ) R d ) The problem Find an appropriate Hilbert space H such that u(t, x) and u(η) are well-defined, i.e. g tx H, η G H, η D((, T ) R d ), and u(η) = R d X(t, x)η(t, x)dxdt η, a.s where X = {X(t, x)} is a jointly measurable version of u = {u(t, x)}. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 4 / 25

5 The colored noise The colored noise Let f = Fµ, where µ is a tempered measure on R d Riesz kernel f (x) = γ α,d x α Bessel kernel f (x) = γ α s(α d)/2 1 e s x 2 /(4s) ds Heat kernel f (x) = γ α,d e x 2 /(4α) Poisson kernel f (x) = γ α,d ( x 2 + α 2 ) (d+1)/2 Space of deterministic integrands in x P(R d ) is the completion of D(R d ) (or E(R d )) w.r.t. ϕ, ψ P(R d ) = ϕ(t)f (x y)ψ(y)dydx = Fϕ(ξ)Fψ(ξ)µ(dξ) R d R d R d Remark: If f is the Riesz or the Bessel kernel, then P(R d ) is included in W k,2 (R d ), the fractional Sobolev space of index k = α/2. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 5 / 25

6 The colored noise The Hilbert space P is the completion of D([, T ] R d ) w.r.t. ϕ, ψ P = = R d R d ϕ(t, x)f (x y)ψ(t, y)dydxdt R d Fϕ(t, ξ)fψ(t, ξ)µ(dξ)dt Remark: P is also the completion of E w.r.t., P, where E is the class of all linear combinations of 1 [,t] A, A bounded Borel set in R d The noise M = {M(ϕ); ϕ P} is a zero-mean Gaussian noise with covariance E(M(ϕ)M(ψ)) = ϕ, ψ P M is white in time and colored in space Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 6 / 25

7 Dalang (1999) The colored noise Lemma A. If 1 µ(dξ) <, (1) R d 1 + ξ 2 then: (a) g tx P for all (t, x); (b) η G P for all η D((, T ) R d ). Theorem A. In order that there exists a measurable version X = {X(t, x)} of u = {u(t, x)} such that u(η) = R d X(t, x)η(t, x)dxdt, it is necessary and sufficient that (1) holds. η, a.s. Remark: If f is the Riesz or Bessel kernel, (1) holds iff α > d 2. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 7 / 25

8 The fractional noise The fractional noise (H > 1/2) Space of deterministic integrands in t H(, T ) is the completion of D(, T ) (or E(, T )) w.r.t. Lemma 1. ϕ, ψ H(,T ) = α H ϕ, ψ H(,T ) = α H c H R ϕ(u) u v 2H 2 ψ(v)dvdu F,T ϕ(τ)f,t ψ(τ) τ (2H 1) dτ where F,T ϕ(τ) = e itτ ϕ(t)dt. Here α H = H(2H 1) Remark H(, T ) is included in W k,2 (R), the fractional Sobolev space of index k = (H 1/2). Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 8 / 25

9 Proof of Lemma 1 The fractional noise Let ϕ, ψ L 2 (, T ) be arbitrary. Let α = 2H 1. a) Using classical ideas (e.g. Stein s book, 1971) we prove that b a t (1 α) ϕ(t)dt = c α R τ α F a,b ϕ(τ)dτ for every ϕ L 2 (a, b). Here F a,b ϕ(τ) = b a e itτ ϕ(t)dt. b) Let ψ u = ψ(u ). Using part a), we get u v (1 α) ψ(v)dv = c α = c α R R τ α F u T,u ψ u (τ)dτ τ α e iτu F,T ψ(τ)dτ Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 9 / 25

10 The fractional noise The Hilbert space H is the completion of D([, T ] R d ) (or E) w.r.t. The noise ϕ, ψ H = α H = α H c H R d R R d ϕ(t, x) t s 2H 2 ψ(s, x)dxdsdt F,T ϕ(τ, x)f,t ψ(τ, x) τ (2H 1) dτdx F = {F(ϕ); ϕ P} is a zero-mean Gaussian noise with covariance E(F(ϕ)F(ψ)) = ϕ, ψ H F is fractional in time and white in space Remark Maslowksi and Nualart (23) considered the same type of noise for the stochastic heat equation on [, 1] [, 1] (with d = 1.) Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 1 / 25

11 The fractional noise Theorem 1. Let ϕ : [, T ] R d R be such that: (i) ϕ(, x) L 2 (, T ) for every x R d ; (ii) (τ, x) F,T ϕ(τ, x) is measurable; (iii) ϕ 2 H = α Hc H R R F d,t ϕ(τ, x) 2 τ (2H 1) dxdτ <. Then ϕ H. Proof Let H be the set of ϕ s which satisfy (i)-(iii). Let Λ be the set of ϕ s with ϕ 2 Λ := c H [I H 1/2 R d T (u H 1/2 ϕ(u, x))(s)] 2 s (2H 1) dxds <. Here (IT α f )(s) = 1 Γ(α) s (u s)α 1 f (u)du. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 11 / 25

12 The fractional noise Proof of Theorem 1 (cont d) Step 1. We prove that: H Λ and ϕ H = ϕ Λ, ϕ H Use the fact that K H : H L2 ((, T ) R d ) is an isometry: (KH ϕ)(s, x) := c H s (H 1/2) I H 1/2 T (u H 1/2 ϕ(u, x))(s) Step 2. We prove that: E is dense in Λ Pipiras and Taqqu (21): ε > ψ=simple function such that R d [1 [a,b] (s) I H 1/2 T (u H 1/2 ψ(u))(s)] 2 s (2H 1) dxds < ε. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 12 / 25

13 The fractional noise Theorem 2. If H > d 4, (2) then: (a) g tx H for all (t, x); (b) η G H for all η D((, T ) R d ). Moreover, g tx H < for all (t, x) iff (2) holds. Proof We check that g tx and η G satisfy conditions (i)-(iii) of Theorem 1. g tx 2 H = α H c H F,T g tx (τ, y) 2 τ (2H 1) dydτ R R d t t = α H s r R 2H 2 g tx (s, y)g tx (r, y)dydrds d = α H t t s r 2H 2 (2t s r) d/2 drds Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 13 / 25

14 The fractional noise Set u(t, x) = F(g tx ) and u(η) = F(η G). (They are well-defined, due to Theorem 2.) Theorem 3. In order that there exists a measurable version X = {X(t, x)} of u = {u(t, x)} such that u(η) = R d X(t, x)η(t, x)dxdt, it is necessary and sufficient that (2) holds. Remark Condition (2) forces d < 4. η, a.s. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 14 / 25

15 Proof of Theorem 3 The fractional noise 1) We prove that u = {u(t, x)} is L 2 (Ω)-continuous. Hence, it has a measurable version X. 2) We prove that E X(t, x)η(t, x)dxdt R d For this we use: 2 = E ( u(η) = E u(η) 2. R d X(t, x)η(t, x)dxdt E(X(t, x)x(s, y)) = E(F(g tx )F(g sy )) = g tx, g sy H E(u(η)X(t, x)) = E(F(η G)F (g tx )) = η G, g tx H. ) Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 15 / 25

16 The fractional-colored noise The fractional-colored noise The Hilbert space HP is the completion of D([, T ] R d ) (or E) w.r.t. ϕ, ψ HP = α H [,T ] 2 = α H c H R (R d ) 2 ϕ(t, x) t s 2H 2 f (x y)ψ(s, y)dydxdsdt τ (2H 1) F,T ϕ(τ, )F,T ψ(τ, ) P(R d ) dτ The noise B = {B(ϕ); ϕ HP} is a zero-mean Gaussian noise with covariance E(B(ϕ)B(ψ)) = ϕ, ψ HP B is fractional in time and colored in space Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 16 / 25

17 The fractional-colored noise Remark Quer-Sardanyons and Tindel (26) considered the same type of noise for the stochastic wave equation, when f (x) = x ν, ν (, 1) and d = 1. In this case, B = {B(t, x)} is a 2-dimensional fbf of Hurst indices H and H = 1 ν/2. Theorem 4. Let ϕ : [, T ] R d R be such that: (i) ϕ(, x) L 2 (, T ) for every x R d ; (ii) (τ, x) F,T ϕ(τ, x) is measurable; (iii) R (R d ) F 2,T ϕ(τ, x)f (x y)f,t ϕ(τ, y) τ (2H 1) dydxdτ <. Then ϕ HP. Proof The basic idea is similar to the proof of Theorem 1. The details are different. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 17 / 25

18 Proof of Theorem 4 The fractional-colored noise Step 1. Let HP be the set of all ϕ s which satisfy (i)-(iii). Let Λ be the set of ϕ s such that ϕ 2 Λ := c H One proves that (R d ) 2 I H 1/2 T (u H 1/2 ϕ(u, x))(s) f (x y) I H 1/2 T (u H 1/2 ϕ(u, y))(s) s (2H 1) dydxds < HP Λ and ϕ HP = ϕ Λ, ϕ HP Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 18 / 25

19 The fractional-colored noise Proof of Theorem 4 (cont d) Step 2. We prove that E is dense in Λ. We approximate a(s, x) = I H 1/2 T (u H 1/2 ϕ(u, x))(s) by g E: (R d ) 2 [a(s, x) g(s, x)]f (x y)[a(s, y) g(s, y)] s (2H 1) dydxds < ε. We approximate g E by b(s, x) = I H 1/2 T (u H 1/2 ψ(u, x))(s), where ψ E: (R d ) 2 [g(s, x) b(s, x)]f (x y)[g(s, y) b(s, y)] s (2H 1) dydxds < ε. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 19 / 25

20 The fractional-colored noise Theorem 5. Suppose that f is one of the Riesz, Bessel, heat, or Poisson kernels. Define α if f is the Riesz kernel α f = if f is the Bessel kernel or the heat kernel 1 if f is the Poisson kernel If H > d α f 4 then: (a) g tx HP for all (t, x); (b) η G HP for all η D((, T ) R d ). Moreover, Proof g tx HP < for all (t, x) iff (3) holds., (3) We check that g tx and η G satisfy conditions (i)-(iii) of Theorem 4. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 2 / 25

21 Proof of Theorem 5 The fractional-colored noise We illustrate only part (a). Using Lemma 1, we have g tx 2 HP =α H c H τ (R (2H 1) F,T g tx (τ, y)f (y z) d ) 2 R F,T g tx (τ, z)dzdydτ =α H t t s r 2H 2 (R d ) 2 g tx (s, y)f (y z)g tx (r, z)dzdydrds We show that there exist some constants A f, B f > such that A f (2t r s) (d α f )/2 I f (r, s) B f (2t r s) (d α f )/2 (4) where I f (r, s) = g tx (s, y)f (y z)g tx (s, z)dzdy (R d ) 2 Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 21 / 25

22 The fractional-colored noise Proof of Theorem 5 (cont d) To check (4) We use the form of G, the fundamental solution of L = / t : I f (r, s) = 1 (4π) d [(t s)(t r)] d/2 f (y z)e x y 2 4(t s) x z 2 4(t r) dydz (R d ) 2 = E[f ( 2(t s)y 2(t r)z )] = E[f (U)] where Y i, Z i are i.i.d. N(, 1) and U = 2(t s)y 2(t r)z : U 2 = d Ui 2 = 2(2t r s)w, where W χ 2 d i=1 If f is the Riesz kernel of order α I f (r, s) = γ α,d E U (d α) = C α,d (2t r s) (d α)/2 Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 22 / 25

23 The fractional-colored noise Proof of Theorem 5 (cont d) If f is the Bessel kernel of order α I f (r, s) = γ α w (α d)/2 1 e w E[e U 2 /(4w) ]dw = γ α If f is the heat kernel of order α ( w (α d)/2 1 e w 1 + 2t r s ) d/2 dw w I f (r, s) = γ α,d E[e U 2 /(4α) ] = γ α,d If f is the Poisson kernel of order α ( 1 + 2t r s ) d/2. α I f (r, s) = γ α,d E [( U 2 + α 2 ) (d+1)/2]. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 23 / 25

24 The fractional-colored noise Set u(t, x) = B(g tx ) and u(η) = B(η G). (They are well-defined, due to Theorem 5.) Theorem 6. In order that there exists a measurable version X = {X(t, x)} of u = {u(t, x)} such that u(η) = R d X(t, x)η(t, x)dxdt, it is necessary and sufficient that (3) holds. η, a.s. Remark 1. If f is the Riesz kernel, condition (3) does not impose any restrictions on d. Remark 2. If d = 1 and f (x) = x 2H 2 (i.e. B = {B(t, x)} is a 2-dim. fbf of Hurst indices H, H > 1/2), then (3) is satisfied. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 24 / 25

25 The fractional-colored noise Main References Dalang, R.C. (1999). Extending martingale measure stochastic integral with application to spatially homogenous s.p.d.e. s. Electr. J. Probab. 4, Maslovski, B. and Nualart, D. (23). Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 22, Pipiras, V. and Taqqu, M. (2). Integrations questions related to fractional Brownian motion. Probab. Theory Rel. Fields 118, Pipiras, V. and Taqqu, M. (21). Are classes of deterimistic integrands for the fractional Brownian motion on a finite interval complete? Bernoulli 7, Quer-Sardanyons, L. and Tindel, S. (26). The 1-d stochastic wave equation driven by a fractional Brownian motion. Preprint. Raluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic () Heat Equation with Colored Noise Stochastic Dynamics, Paris 1 25 / 25