Linear SPDEs driven by stationary random distributions

Linear SPDEs driven by stationary random distributions aluca Balan University of Ottawa Workshop on Stochastic Analysis and Applications June 4-8, 2012 aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 1 / 21

Outline 1 Stationary random distributions 2 Application to SPDEs 3 Some examples aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 2 / 21

Stationary random distributions Stationary andom Distributions Definition 1. (Itô 1954) Let {X(ϕ); ϕ D C ( )} be a random distribution, i.e. a collection of C-valued random variables such that E[X(ϕ)] = 0, E X(ϕ) 2 < and ϕ X(ϕ) L 2 C (Ω) is linear and continuous. {X(ϕ)} is stationary if for any ϕ, ψ D C ( ) and h E[X(τ h ϕ)x(τ h ψ)] = E[X(ϕ)X(ψ)], where (τ h ϕ)(t) = ϕ(t + h), t. emark A random distribution {X(ϕ)} is called real if for any ϕ D C ( ), X(ϕ) = X(ϕ). aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 3 / 21

Stationary random distributions Example Let {X(t); t } be a (zero-mean) L 2 C (Ω)-continuous stationary random field with covariance function ρ, i.e. E[X(t)X(s)] = ρ(t s). Then X(ϕ) = ϕ(t)x(t)dt, ϕ D C ( ) is a stationary random distribution with covariance: E[X(ϕ)X(ψ)] = (ϕ ψ)(t)ρ(t)dt. Theorem 1. (Yaglom 1957) For any stationary random distribution {X(ϕ)} there exists a unique distribution ρ D C (d ) such that, for any ϕ, ψ D C ( ), E[X(ϕ)X(ψ)] = ρ(ϕ ψ). By Bochner-Schwartz Theorem, ρ = Fµ in S ( ) where µ is a tempered measure on (called the spectral measure of {X(ϕ)}). aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 4 / 21

Stationary random distributions Example {X(t); t } is a stationary random field with spectral measure µ, i.e. ρ(t) = e iτ t µ(dτ). {X(t)} has the spectral representation: X(t) = e iτ t M(dτ) where M = {M(A)} is a complex random measure with control measure µ, i.e. E[M(A)M(B)] = µ(a B). Then X(ϕ) = ϕ(t)x(t)dt = Fϕ(τ)M(dτ). Theorem 2. (Yaglom 1957) For any stationary random distribution {X(ϕ)} with spectral measure µ there exists a complex random measure M with control measure µ such that: X(ϕ) = Fϕ(τ)M(dτ). aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 5 / 21

Stationary random distributions Stochastic Integral {X(ϕ)} is a stationary random distribution with spectral measure µ. H is the completion of D C ( ) with respect to the inner product: ϕ, ψ H = FϕFψdµ The isometry ϕ X(ϕ) L 2 C (Ω) is extended to H. For any ϕ H, emark X(ϕ) = ϕ(t)x(dt) = Fϕ(τ)M(dτ) If A is a bounded Borel set with F1 A L 2 C (d, µ) then 1 A H X(A) := X(1 A ) = F1 A dm If F1 [0,t] L 2 C (d, µ) then X(t) := X([0, t]) = F1 [0,t] dm. aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 6 / 21

Stationary random distributions Definition 2. A random field {X(t); t } has stationary increments if E[X(A + h)x(b + h)] = E[X(A)X(B)] for any rectangles A, B in. A rectangle is a set of the form (a, b]. The increment over A = (a, b] is defined by X(A) := X(b) X(a). Theorem 3. (Yaglom 1957) Any (zero-mean) random field {X(t); t } with stationary increments and continuous covariance function has the spectral representation X(t) = (e iτ t 1) M(dτ) where M is a complex random measure with control measure µ satisfying: d τ 2 µ(dτ) <. 1 + τ 2 emark: E[X(t)X(s)] = (e iτ t 1)(e iτ s 1) µ(dτ) aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 7 / 21

Stationary random distributions Theorem 4. (Itô 1954) Assume d = 1. {X(ϕ); ϕ D C ()} is a stationary random distribution whose spectral measure µ satisfies 1 µ(dτ) < (1) 1 + τ 2 iff there exists a process {Y (t); t } with stationary increments such that X(ϕ) = ϕ (t)y (t)dt =: Y (ϕ). emark If µ satisfies (1) then F1 [0,t] (τ) = (e iτt 1)/( iτ) lies in L 2 C (, µ) and X(t) := X(1 [0,t] ) = e iτt 1 dm iτ is well-defined in L 2 C (Ω). In fact, Y (t) = A + X(t) and hence Y (ϕ) = X (ϕ) for all ϕ D C (). aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 8 / 21

Application to SPDE Application to SPDEs The noise Let {X(ϕ); ϕ D C (+1 )} be a (real) stationary random distribution with spectral random measure M and spectral measure Π: X(ϕ) = Fϕ(τ, ξ)m(dτ, dξ), +1 τ, ξ ϕ, ψ H := E[X(ϕ)X(ψ)] = ρ(ϕ ψ) = FϕFψdΠ +1 The space of deterministic integrands Let H be the completion of D C (+1 ) with respect to, H. The isometry ϕ X(ϕ) is extended to H. For any ϕ H, we denote X(ϕ) = ϕ(t, x)x(dt, dx), +1 t, x aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 9 / 21

Application to SPDEs Example Assume that {X(ϕ)} is Gaussian and Π = ν µ, where µ is a tempered measure on and ν satisfies (1), i.e. 1 1 + τ 2 ν(dτ) < For any ϕ D C ( ), F(1 [0,t] ϕ) L 2 C (d+1, Π) and hence, 1 [0,t] ϕ H. Define X t (ϕ) := X(1 [0,t] ϕ). Then E[X t (ϕ)x s (ψ)] = (t, s) Fϕ(ξ)Fψ(ξ)µ(dξ), (t, s) = where (e iτt 1)(e iτs 1) 1 ν(dτ) = E[Z (t)z (s)] τ 2 {Z (t)} t is a Gaussian process with stationary increments. emark {Z (t)} is a fbm of index H (0, 1) if ν(dτ) = τ (2H 1) dτ. aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 10 / 21

Application to SPDEs Basic observation For any ϕ D C (+1 ) and for any t fixed, let φ ξ (t) := Fϕ(t, )(ξ) = e iξ x ϕ(t, x)dx, ξ be the Fourier transform of ϕ in the x-variable. Then ) Fϕ(τ, ξ) = e ( iτt e iξ x ϕ(t, x)dx dt = Fφ ξ (τ) d where Fφ ξ denotes the Fourier transform of the function t φ ξ (t). Alternative calculation of the inner product in H For any ϕ (1), ϕ (2) D C (+1 ) ϕ (1), ϕ (2) H = Fφ (1) ξ (τ)fφ (2) ξ (τ)π(dτ, dξ) =: ϕ (1), ϕ (2) 0 +1 aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 11 / 21

Application to SPDEs Theorem 5. (B. 2012) Suppose that ϕ(t, ) S (+1 ) for any t and (i) Fϕ(t, ) is a function for any t (ii) (t, ξ) φ ξ (t) := Fϕ(t, )(ξ) is measurable on (iii) φ ξ(t) dt < for any ξ. Denote Fφ ξ (τ) = e iτt φ ξ (t)dt, τ. If ϕ 2 0 := +1 Fφ ξ (τ) 2 Π(dτ, dξ) < then ϕ H, ϕ 2 H = ϕ 2 0 and X(ϕ) = ϕ(t, x)x(dt, dx) = +1 Fφ ξ (τ)m(dτ, dξ). +1 aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 12 / 21

Application to SPDEs The equation Let L be a second-order pseudo-differential operator in (t, x). Assume that the fundamental solution G of Lu = 0 exists. Consider the SPDE: { Lu(t, x) = Ẋ(t, x), t > 0, x (2) zero initial conditions Definition 3. (Walsh, 1986) The process {u(t, x); t 0, x } defined by u(t, x) = t 0 G(t s, x y)x(ds, dy) (3) is called a random field solution of (2), provided that the stochastic integral in the HS of (3) is well-defined, i.e. g t,x := 1 [0,t] G(t, x ) H. aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 13 / 21

Application to SPDEs Consequence of Theorem 5 Suppose that G(t, ) S ( ) for any t > 0 and (i) FG(t, ) is a function for any t > 0 (ii) (t, ) H ξ (t) := FG(t, )(ξ) is measurable on + (ii) t 0 H ξ(s) ds < for any ξ and t > 0. Denote F 0,t H ξ (τ) = t 0 e iτs H ξ (s)ds, τ. Equation (2) has a random field solution iff for any t > 0 I t := F 0,t H ξ (τ) 2 Π(dτ, dξ) <. +1 In this case, the solution is given by u(t, x) = e iτt e iξ x F 0,t H ξ (τ)m(dτ, dξ). +1 aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 14 / 21

Application to SPDEs Assume that Π = ν µ I t = N t (ξ)µ(dξ) where N t (ξ) = F 0,t H ξ (τ) 2 ν(dτ) To see when I t <, it suffices to find upper/lower bounds for N t (ξ). For any t > 0 fixed, {u(t, x); x } is a stationary random field with spectral measure µ t : E[u(t, x)u(s, y)] = e iξ (x y) µ t (dξ), µ t (A) = N t (ξ)µ(dξ) A Spectral epresentation: If µ has density g, then u(t, x) = e iξ x N t (ξ) 1/2 g(ξ) 1/2 M 0 (dξ) M 0 is a complex random measure with Lebesque control measure aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 15 / 21

First example Some examples A parabolic equation u t (t, x) = ( )β/2 u(t, x) + Ẋ(t, x), u(0, x) = 0, x t > 0, x d (4) with β > 0. In this case, H ξ (t) = FG(t, )(ξ) = e tψ(ξ) with Ψ(ξ) = ξ β 1 N t (ξ) = τ 2 + Ψ(ξ) 2 {sin2 (τt) + [e tψ(ξ) cos(τt)] 2 }ν(dτ) Lemma 1. (B. 2012) For some constants C (1) t, C (2) C (2) t 1 τ 2 + Ψ(ξ) 2 + 1 ν(dτ) N t(ξ) C (1) t t > 0, 1 τ 2 + Ψ(ξ) 2 + 1 ν(dτ) aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 16 / 21

Some examples Consequence of Lemma 1 The parabolic equation (4) has a random field solution if and only if 1 τ 2 + Ψ(ξ) 2 ν(dτ)µ(dξ) < (5) + 1 Particular Cases (a) Let ν(dτ) = τ γ dτ for some γ ( 1, 1). Then (5) holds iff ( ) 1 1+γ µ(dξ) < 1 + Ψ(ξ) (b) Let ν(dτ) = (1 + τ 2 ) γ/2 dτ for some γ > 1. Then (5) holds iff ( ) 1 c(γ) µ(dξ) <, where c(γ) = 1 + Ψ(ξ) { 1 + γ if γ < 1 2 if γ > 1 aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 17 / 21

Second example Some examples A hyperbolic equation 2 u t 2 (t, x) = ( )β/2 u(t, x) + Ẋ(t, x), u(0, x) = 0 and u (0, x) = 0, t x with β > 0. In this case, d t > 0, x d (6) H ξ (t) = FG(t, )(ξ) = sin(t Ψ(ξ)) Ψ(ξ) with Ψ(ξ) = ξ β N t (ξ) = 1 (τ 2 Ψ(ξ)) 2 [ sin(τt) [ cos(τt) cos(t ] } 2 Ψ(ξ)) ν(dτ) τ Ψ(ξ) sin(t Ψ(ξ))] 2 + aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 18 / 21

Some examples Lemma 2. (B. 2012) Assume that ν(dτ) = η( τ )dτ where η : + + satisfies some regularity conditions. (a) There exist some constants C (1) t, C (2) t > 0 such that where N(ξ) = C (2) t N(ξ) N t (ξ) C (1) t N(ξ) 1 Ψ(ξ) + 1 1 τ 2 + Ψ(ξ) + 1 ν(dτ). (b) There exist some constants D (1) t, D (2) t > 0 such that D (2) η( Ψ(ξ) + 1) t Ψ(ξ) + 1 N t (ξ) D (1) t η( Ψ(ξ) + 1). Ψ(ξ) + 1 aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 19 / 21

Some examples Consequence of Lemma 2 The hyperbolic equation (6) has a random field solution if and only if η( Ψ(ξ) + 1) µ(dξ) < (7) Ψ(ξ) + 1 Particular Cases Assume that η(τ) = τ γ or η(τ) = (1 + τ 2 ) γ/2 for some γ ( 1, 1). Then (7) holds iff ( ) 1 1+γ/2 µ(dξ) < 1 + Ψ(ξ) aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 20 / 21

Some examples Thank you! aluca Balan (University of Ottawa) Linear SPDEs with stationary noise Lausanne (June 4-8, 2012) 21 / 21