Some linear SPDEs with fractional noiseaix1.uottawa.ca/~rbalan/Banff-slides.pdfOutline 1 Linear...
Transcript of Some linear SPDEs with fractional noiseaix1.uottawa.ca/~rbalan/Banff-slides.pdfOutline 1 Linear...
Some linear SPDEs with fractional noise
Raluca Balan
University of Ottawa
Workshop on Stochastic Analysis and SPDEsApril 2-6, 2012
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 1 / 21
Outline
1 Linear SPDEs with fractional noise
2 A parabolic equation
3 A hyperbolic equation
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 2 / 21
Linear SPDEs with fractional noise
Linear SPDEs
Let L be a second order pseudo-differential operator in (t , x). Assumethat the fundamental solution G of Lu = 0 exists. Consider the linearSPDE: {
Lu(t , x) = W (t , x), t > 0, x ∈ Rd
zero initial conditions(1)
Space-time white noise
{W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} is a centered Gaussian process with:
E [W (ϕ)W (ψ)] =
∫ ∞0
∫Rdϕ(t , x)ψ(t , x)dxdt
Isometry The map ϕ 7→W (ϕ) is extended to L2(R+ × Rd )
W (ϕ) :=
∫ ∞0
∫Rdϕ(t , x)W (dt ,dx) (stochastic integral)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 3 / 21
Linear SPDEs with fractional noise
Linear SPDEs
Let L be a second order pseudo-differential operator in (t , x). Assumethat the fundamental solution G of Lu = 0 exists. Consider the linearSPDE: {
Lu(t , x) = W (t , x), t > 0, x ∈ Rd
zero initial conditions(1)
Space-time white noise
{W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} is a centered Gaussian process with:
E [W (ϕ)W (ψ)] =
∫ ∞0
∫Rdϕ(t , x)ψ(t , x)dxdt
Isometry The map ϕ 7→W (ϕ) is extended to L2(R+ × Rd )
W (ϕ) :=
∫ ∞0
∫Rdϕ(t , x)W (dt ,dx) (stochastic integral)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 3 / 21
Linear SPDEs with fractional noise
Definition (Walsh, 1986)
The process {u(t , x); t ≥ 0, x ∈ Rd} defined by
u(t , x) =
∫ t
0
∫Rd
G(t − s, x − y)W (ds,dy) (2)
is a random field solution of (1), provided that the stochastic integralin the RHS of (2) is well-defined.
Remark If W is a space-time white noise, (1) has a random-fieldsolution iff G(t − ·, x − ·) ∈ L2(R+ × Rd ).
Example Let L = ∂∂t −
12∆. Then G(t , x) = 1
(2πt)d/2 exp{− |x |
2
2t
}and
∫ t
0
∫Rd
G2(t − s, x − y)dyds <∞⇐⇒ d = 1
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 4 / 21
Linear SPDEs with fractional noise
Definition (Walsh, 1986)
The process {u(t , x); t ≥ 0, x ∈ Rd} defined by
u(t , x) =
∫ t
0
∫Rd
G(t − s, x − y)W (ds,dy) (2)
is a random field solution of (1), provided that the stochastic integralin the RHS of (2) is well-defined.
Remark If W is a space-time white noise, (1) has a random-fieldsolution iff G(t − ·, x − ·) ∈ L2(R+ × Rd ).
Example Let L = ∂∂t −
12∆. Then G(t , x) = 1
(2πt)d/2 exp{− |x |
2
2t
}and
∫ t
0
∫Rd
G2(t − s, x − y)dyds <∞⇐⇒ d = 1
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 4 / 21
Linear SPDEs with fractional noise
Definition (Walsh, 1986)
The process {u(t , x); t ≥ 0, x ∈ Rd} defined by
u(t , x) =
∫ t
0
∫Rd
G(t − s, x − y)W (ds,dy) (2)
is a random field solution of (1), provided that the stochastic integralin the RHS of (2) is well-defined.
Remark If W is a space-time white noise, (1) has a random-fieldsolution iff G(t − ·, x − ·) ∈ L2(R+ × Rd ).
Example Let L = ∂∂t −
12∆. Then G(t , x) = 1
(2πt)d/2 exp{− |x |
2
2t
}and
∫ t
0
∫Rd
G2(t − s, x − y)dyds <∞⇐⇒ d = 1
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 4 / 21
Linear SPDEs with fractional noise
Colored Noise (Dalang and Frangos, 1998)
W = {W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} is a centered Gaussian process with:
E [W (ϕ)W (ψ)] =
∫ ∞0
∫Rd
∫Rdϕ(t , x)ψ(t , y)f (x − y)dxdydt =: J(ϕ,ψ)
J is non-negative definite iff ∃µ tempered measure with f = Fµ∫Rd
∫Rdϕ(x)ψ(y)f (x−y)dxdy =
∫RdFϕ(ξ)Fψ(ξ)µ(dξ), ∀ϕ,ψ ∈ S(Rd )
Examples
1. Riesz kernel µ(dξ) = |ξ|−αdξ, f (x) = cα,d |x |−(d−α), 0 < α < d2. Bessel kernel µ(dξ) = (1 + |ξ|2)−α/2dξ, α > 0
Isometry
ϕ 7→W (ϕ) is extended to a Hilbert space P, which may containdistributions in xRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 5 / 21
Linear SPDEs with fractional noise
Colored Noise (Dalang and Frangos, 1998)
W = {W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} is a centered Gaussian process with:
E [W (ϕ)W (ψ)] =
∫ ∞0
∫Rd
∫Rdϕ(t , x)ψ(t , y)f (x − y)dxdydt =: J(ϕ,ψ)
J is non-negative definite iff ∃µ tempered measure with f = Fµ∫Rd
∫Rdϕ(x)ψ(y)f (x−y)dxdy =
∫RdFϕ(ξ)Fψ(ξ)µ(dξ), ∀ϕ,ψ ∈ S(Rd )
Examples
1. Riesz kernel µ(dξ) = |ξ|−αdξ, f (x) = cα,d |x |−(d−α), 0 < α < d2. Bessel kernel µ(dξ) = (1 + |ξ|2)−α/2dξ, α > 0
Isometry
ϕ 7→W (ϕ) is extended to a Hilbert space P, which may containdistributions in xRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 5 / 21
Linear SPDEs with fractional noise
Colored Noise (Dalang and Frangos, 1998)
W = {W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} is a centered Gaussian process with:
E [W (ϕ)W (ψ)] =
∫ ∞0
∫Rd
∫Rdϕ(t , x)ψ(t , y)f (x − y)dxdydt =: J(ϕ,ψ)
J is non-negative definite iff ∃µ tempered measure with f = Fµ∫Rd
∫Rdϕ(x)ψ(y)f (x−y)dxdy =
∫RdFϕ(ξ)Fψ(ξ)µ(dξ), ∀ϕ,ψ ∈ S(Rd )
Examples
1. Riesz kernel µ(dξ) = |ξ|−αdξ, f (x) = cα,d |x |−(d−α), 0 < α < d2. Bessel kernel µ(dξ) = (1 + |ξ|2)−α/2dξ, α > 0
Isometry
ϕ 7→W (ϕ) is extended to a Hilbert space P, which may containdistributions in xRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 5 / 21
Linear SPDEs with fractional noise
Theorem 1 (Dalang, 1999)
Assume G(t , ·) ∈ S ′(Rd ) has rapid decrease (and satisfies someregularity conditions). Then G(t − ·, x − ·) belongs to the space P iff∫ t
0
∫Rd|FG(s, ·)(ξ)|2µ(dξ)ds <∞ (3)
Example
Let L = ∂∂t − L, where L is the L2-generator of a Lévy process (Xt )t≥0
with values in Rd . Let Ψ(ξ) be the characteristic exponent of (Xt )t≥0:
E(e−iξ·Xt ) = e−tΨ(ξ)
Assume that Xt has density pt . Then G(t , x) = pt (−x) and henceFG(t , ·)(ξ) = e−tΨ(ξ). Condition (3) holds iff
Υ(1) :=
∫Rd
11 + 2ReΨ(ξ)
µ(dξ) <∞Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 6 / 21
Linear SPDEs with fractional noise
Theorem 1 (Dalang, 1999)
Assume G(t , ·) ∈ S ′(Rd ) has rapid decrease (and satisfies someregularity conditions). Then G(t − ·, x − ·) belongs to the space P iff∫ t
0
∫Rd|FG(s, ·)(ξ)|2µ(dξ)ds <∞ (3)
Example
Let L = ∂∂t − L, where L is the L2-generator of a Lévy process (Xt )t≥0
with values in Rd . Let Ψ(ξ) be the characteristic exponent of (Xt )t≥0:
E(e−iξ·Xt ) = e−tΨ(ξ)
Assume that Xt has density pt . Then G(t , x) = pt (−x) and henceFG(t , ·)(ξ) = e−tΨ(ξ). Condition (3) holds iff
Υ(1) :=
∫Rd
11 + 2ReΨ(ξ)
µ(dξ) <∞Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 6 / 21
Linear SPDEs with fractional noise
Fractional Brownian Motion (FBM)
Let H ∈ (0,1). FBM is a centered Gaussian process (Bt )t≥0 with
E(BtBs) =12
(t2H + s2H − |t − s|2H) =: RH(t , s)
Important Remark
If H > 1/2, then RH(t , s) = αH∫ t
0
∫ s0 |u − v |2H−2dudv , αH = H(2H − 1)
B. and Tudor (2008)
Assume H > 1/2. Let {W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} be a centeredGaussian process with E [W (ϕ)W (ψ)] = 〈ϕ,ψ〉H
〈ϕ,ψ〉H = αH
∫R2
+
∫R2d
ϕ(u, x)ψ(v , y)|u − v |2H−2f (x − y)dxdydudv
Isometry: ϕ 7→W (ϕ) is extended to a Hilbert space H which maycontain distributions in t , and in xRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 7 / 21
Linear SPDEs with fractional noise
Fractional Brownian Motion (FBM)
Let H ∈ (0,1). FBM is a centered Gaussian process (Bt )t≥0 with
E(BtBs) =12
(t2H + s2H − |t − s|2H) =: RH(t , s)
Important Remark
If H > 1/2, then RH(t , s) = αH∫ t
0
∫ s0 |u − v |2H−2dudv , αH = H(2H − 1)
B. and Tudor (2008)
Assume H > 1/2. Let {W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} be a centeredGaussian process with E [W (ϕ)W (ψ)] = 〈ϕ,ψ〉H
〈ϕ,ψ〉H = αH
∫R2
+
∫R2d
ϕ(u, x)ψ(v , y)|u − v |2H−2f (x − y)dxdydudv
Isometry: ϕ 7→W (ϕ) is extended to a Hilbert space H which maycontain distributions in t , and in xRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 7 / 21
Linear SPDEs with fractional noise
Fractional Brownian Motion (FBM)
Let H ∈ (0,1). FBM is a centered Gaussian process (Bt )t≥0 with
E(BtBs) =12
(t2H + s2H − |t − s|2H) =: RH(t , s)
Important Remark
If H > 1/2, then RH(t , s) = αH∫ t
0
∫ s0 |u − v |2H−2dudv , αH = H(2H − 1)
B. and Tudor (2008)
Assume H > 1/2. Let {W (ϕ);ϕ ∈ C∞0 (R+ × Rd )} be a centeredGaussian process with E [W (ϕ)W (ψ)] = 〈ϕ,ψ〉H
〈ϕ,ψ〉H = αH
∫R2
+
∫R2d
ϕ(u, x)ψ(v , y)|u − v |2H−2f (x − y)dxdydudv
Isometry: ϕ 7→W (ϕ) is extended to a Hilbert space H which maycontain distributions in t , and in xRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 7 / 21
Linear SPDEs with fractional noise
Theorem 2 (B. and Tudor, 2010)
Assume G(t , ·) ∈ S ′(Rd ) and FG(t , ·) is a function (satisfying someregularity conditions). Then G(t − ·, x − ·) belongs to the space H iff
It =
∫Rd
∫ t
0
∫ t
0|r − s|2H−2FG(r , ·)(ξ)FG(s, ·)(ξ)drdsµ(dξ) <∞ (4)
Examples
1. L = ∂∂t + (−∆)β/2, β > 0. Condition (4) holds iff∫
Rd
(1
1 + |ξ|β
)2H
µ(dξ) <∞
2. L = ∂2
∂t2 + (−∆)β/2, β > 0. Condition (4) holds iff∫Rd
(1
1 + |ξ|β
)H+1/2
µ(dξ) <∞
3. L = ∂∂t − L, where L is the generator of a Lévy process with values
in Rd and characteristic exponent Ψ
(4)⇐⇒∫Rd
(1
1 + ReΨ(ξ)
)2H
µ(dξ) <∞
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 8 / 21
Linear SPDEs with fractional noise
Theorem 2 (B. and Tudor, 2010)
Assume G(t , ·) ∈ S ′(Rd ) and FG(t , ·) is a function (satisfying someregularity conditions). Then G(t − ·, x − ·) belongs to the space H iff
It =
∫Rd
∫ t
0
∫ t
0|r − s|2H−2FG(r , ·)(ξ)FG(s, ·)(ξ)drdsµ(dξ) <∞ (4)
Examples
1. L = ∂∂t + (−∆)β/2, β > 0. Condition (4) holds iff∫
Rd
(1
1 + |ξ|β
)2H
µ(dξ) <∞
2. L = ∂2
∂t2 + (−∆)β/2, β > 0. Condition (4) holds iff∫Rd
(1
1 + |ξ|β
)H+1/2
µ(dξ) <∞
3. L = ∂∂t − L, where L is the generator of a Lévy process with values
in Rd and characteristic exponent Ψ
(4)⇐⇒∫Rd
(1
1 + ReΨ(ξ)
)2H
µ(dξ) <∞
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 8 / 21
A parabolic equation
A parabolic equation
The equation
W is a Gaussian noise with E [W (ϕ)W (ψ)] = 〈ϕ,ψ〉H. We consider
∂u∂t
(t , x) = Lu(t , x) + W (t , x), t > 0, x ∈ Rd (5)
with zero initial conditions. L is the L2-generator of a d-dimensionalLévy process (Xt )t≥0 with characteristic exponent Ψ(ξ). We assumethat Xt has density pt .
The random-field solutionBy Theorem 2, (5) has a random-field solution iff
∫Rd Nt (ξ)µ(dξ) <∞,
Nt (ξ) = αH
∫ t
0
∫ t
0|r − s|2H−2e−rΨ(ξ)e−sΨ(ξ)drds
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 9 / 21
A parabolic equation
A parabolic equation
The equation
W is a Gaussian noise with E [W (ϕ)W (ψ)] = 〈ϕ,ψ〉H. We consider
∂u∂t
(t , x) = Lu(t , x) + W (t , x), t > 0, x ∈ Rd (5)
with zero initial conditions. L is the L2-generator of a d-dimensionalLévy process (Xt )t≥0 with characteristic exponent Ψ(ξ). We assumethat Xt has density pt .
The random-field solutionBy Theorem 2, (5) has a random-field solution iff
∫Rd Nt (ξ)µ(dξ) <∞,
Nt (ξ) = αH
∫ t
0
∫ t
0|r − s|2H−2e−rΨ(ξ)e−sΨ(ξ)drds
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 9 / 21
A parabolic equation
Theorem 3 (B. 2011)
For any t > 0, ξ ∈ Rd ,
c(1)H
(1
1/t + ReΨ(ξ)
)2H
≤ Nt (ξ) ≤ c(2)H
(1
1/t + ReΨ(ξ)
)2H
For the lower bound, we assume that there exists a constant K > 0such that
|ImΨ(ξ)| ≤ K ReΨ(ξ), for all ξ ∈ Rd . (6)
CorollaryIf (6) holds, then equation (5) has a random field solution iff
ΥH(1) :=
∫Rd
(1
1 + 2ReΨ(ξ)
)2H
µ(dξ) <∞ (7)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 10 / 21
A parabolic equation
Theorem 3 (B. 2011)
For any t > 0, ξ ∈ Rd ,
c(1)H
(1
1/t + ReΨ(ξ)
)2H
≤ Nt (ξ) ≤ c(2)H
(1
1/t + ReΨ(ξ)
)2H
For the lower bound, we assume that there exists a constant K > 0such that
|ImΨ(ξ)| ≤ K ReΨ(ξ), for all ξ ∈ Rd . (6)
CorollaryIf (6) holds, then equation (5) has a random field solution iff
ΥH(1) :=
∫Rd
(1
1 + 2ReΨ(ξ)
)2H
µ(dξ) <∞ (7)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 10 / 21
A parabolic equation
Example
(Xt )t≥0 is a symmetric β-stable Lévy process with values in Rd , forsome β ∈ (0,2]. In this case,
L = −(−∆)β/2 and Ψ(ξ) = |ξ|β
a) Assume that µ(dξ) = |ξ|−αdξ with α ∈ (0,d). Then (6) holds iff
2Hβ > d − α
b) Assume that µ(dξ) =∏d
i=1(cHi |ξi |−(2Hi−1))dξ with Hi ∈ (1/2,1).Then (6) holds iff
2Hβ > d −d∑
i=1
(2Hi − 1)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 11 / 21
A parabolic equation
Potential Theory
Let X t = Xt − Xt , where (Xt )t≥0 is an independent copy of (Xt )t≥0.
(P tφ)(x) =
∫Rdφ(y)pt (x − y)dy , t ≥ 0 semigroup of (X t )t≥0
(Rαφ)(x) =
∫ ∞0
e−αs(Psφ)(x)ds, α > 0 resolvent of (X t )t≥0
pt = pt ∗ pt is the density of X t , where pt (x) = pt (−x).
Maximum Principle (Foondun and Khoshnevisan, 2010)
If µ is a tempered measure in Rd which has a density g, and f = Fµ inS ′(Rd ), then for any α > 0
(Rαf )(0) = supx∈Rd
(Rαf )(x) = Υ(α) :=
∫Rd
1α + 2ReΨ(ξ)
µ(dξ)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 12 / 21
A parabolic equation
Potential Theory
Let X t = Xt − Xt , where (Xt )t≥0 is an independent copy of (Xt )t≥0.
(P tφ)(x) =
∫Rdφ(y)pt (x − y)dy , t ≥ 0 semigroup of (X t )t≥0
(Rαφ)(x) =
∫ ∞0
e−αs(Psφ)(x)ds, α > 0 resolvent of (X t )t≥0
pt = pt ∗ pt is the density of X t , where pt (x) = pt (−x).
Maximum Principle (Foondun and Khoshnevisan, 2010)
If µ is a tempered measure in Rd which has a density g, and f = Fµ inS ′(Rd ), then for any α > 0
(Rαf )(0) = supx∈Rd
(Rαf )(x) = Υ(α) :=
∫Rd
1α + 2ReΨ(ξ)
µ(dξ)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 12 / 21
A parabolic equation
Fractional Analogues
(Rα,Hφ)(x) = αH
∫ ∞0
∫ ∞0|r − s|2H−2e−α(r+s)(Pr+sφ)(x)drds
Υ∗H(α) = αH
∫Rd
∫ ∞0
∫ ∞0|r − s|2H−2e−(α+2ReΨ(ξ))(r+s)drdsµ(dξ)
Theorem 4 (B., 2011)
Let H > 1/2. If µ is a tempered measure in Rd which has a density g,and f = Fµ in S ′(Rd ), then for any α > 0
(Rα,H f )(0) = supx∈Rd
(Rα,H f )(x) = Υ∗H(α)
Remark: CH,αΥH(α) ≤ Υ∗H(α) ≤ CHΥH(α) where
ΥH(α) =
∫Rd
(1
α + 2ReΨ(ξ)
)2H
µ(dξ)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 13 / 21
A parabolic equation
Fractional Analogues
(Rα,Hφ)(x) = αH
∫ ∞0
∫ ∞0|r − s|2H−2e−α(r+s)(Pr+sφ)(x)drds
Υ∗H(α) = αH
∫Rd
∫ ∞0
∫ ∞0|r − s|2H−2e−(α+2ReΨ(ξ))(r+s)drdsµ(dξ)
Theorem 4 (B., 2011)
Let H > 1/2. If µ is a tempered measure in Rd which has a density g,and f = Fµ in S ′(Rd ), then for any α > 0
(Rα,H f )(0) = supx∈Rd
(Rα,H f )(x) = Υ∗H(α)
Remark: CH,αΥH(α) ≤ Υ∗H(α) ≤ CHΥH(α) where
ΥH(α) =
∫Rd
(1
α + 2ReΨ(ξ)
)2H
µ(dξ)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 13 / 21
A parabolic equation
Intersection local time
(X(1)t )t≥0 and (X
(2)t )t≥0 are two independent copies of (X t )t≥0
For any nonnegative measurable function φ, we define
Lt ,H(φ) = α
∫ t
0
∫ t
0|r − s|2H−2φ(X
(1)r − X
(2)s )drds
Lt ,H(φ) may be infinite.
Since X(1)r − X
(2)s
d= X r+s,
E [f (X(1)r − X
(2)s )] = E [f (X r+s)] = (Pr+sf )(0)
E [Lt ,H(f )] = αH
∫ t
0
∫ t
0|r − s|2H−2(Pr+sf )(0)drds
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 14 / 21
A parabolic equation
Intersection local time
(X(1)t )t≥0 and (X
(2)t )t≥0 are two independent copies of (X t )t≥0
For any nonnegative measurable function φ, we define
Lt ,H(φ) = α
∫ t
0
∫ t
0|r − s|2H−2φ(X
(1)r − X
(2)s )drds
Lt ,H(φ) may be infinite.
Since X(1)r − X
(2)s
d= X r+s,
E [f (X(1)r − X
(2)s )] = E [f (X r+s)] = (Pr+sf )(0)
E [Lt ,H(f )] = αH
∫ t
0
∫ t
0|r − s|2H−2(Pr+sf )(0)drds
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 14 / 21
A parabolic equation
Lemma (B. 2011)
If (Rα,H f )(0) <∞ for any α > 0, then Lt ,H(f ) <∞ for all t > 0 a.s.Moreover,
lim supt→∞
1t
log Lt ,H(f ) ≤ 0 a.s.
Example
(Xt )t≥0 is a symmetric β-stable process, β ∈ (0,2].f (x) = cα,d |x |−(d−α) with α ∈ (0,d).Since (Xt )t≥0 is self-similar with exponent 1/β,
X r+sd= (r + s)1/βX 1
E [f (X r+s)] = cα,dE |X r+s|−(d−α) = cα,d (r + s)−(d−α)/βE |X 1|−(d−α)
E [Lt ,H(f )] = cα,d ,H∫ t
0
∫ t
0|r − s|2H−2(r + s)−(d−α)/βdrds
E [Lt ,H(f )] <∞ iff 2Hβ > d − αRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 15 / 21
A parabolic equation
Lemma (B. 2011)
If (Rα,H f )(0) <∞ for any α > 0, then Lt ,H(f ) <∞ for all t > 0 a.s.Moreover,
lim supt→∞
1t
log Lt ,H(f ) ≤ 0 a.s.
Example
(Xt )t≥0 is a symmetric β-stable process, β ∈ (0,2].f (x) = cα,d |x |−(d−α) with α ∈ (0,d).Since (Xt )t≥0 is self-similar with exponent 1/β,
X r+sd= (r + s)1/βX 1
E [f (X r+s)] = cα,dE |X r+s|−(d−α) = cα,d (r + s)−(d−α)/βE |X 1|−(d−α)
E [Lt ,H(f )] = cα,d ,H∫ t
0
∫ t
0|r − s|2H−2(r + s)−(d−α)/βdrds
E [Lt ,H(f )] <∞ iff 2Hβ > d − αRaluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 15 / 21
A hyperbolic equation
A hyperbolic equation
The equation
∂2u∂t2 (t , x) = Lu(t , x) + W (t , x), t > 0, x ∈ Rd (8)
with zero initial conditions. L is the L2-generator of a d-dimensionalLévy process (Xt )t≥0 with characteristic exponent Ψ(ξ). Assume:
Ψ(·) is real-valued
Remark: We may not be able to identify the fundamental solution G.
Definition (Foondun, Khoshnevisan, E. Nualart, 2010)
A weak solution of (8) is the process {u(t , ϕ); t > 0, ϕ ∈ S(Rd )}:
u(t , ϕ) =1
(2π)d
∫ t
0
∫Rd
sin(√
Ψ(ξ)(t − s))√Ψ(ξ)
Fϕ(ξ) W (ds,dξ)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 16 / 21
A hyperbolic equation
A hyperbolic equation
The equation
∂2u∂t2 (t , x) = Lu(t , x) + W (t , x), t > 0, x ∈ Rd (8)
with zero initial conditions. L is the L2-generator of a d-dimensionalLévy process (Xt )t≥0 with characteristic exponent Ψ(ξ). Assume:
Ψ(·) is real-valued
Remark: We may not be able to identify the fundamental solution G.
Definition (Foondun, Khoshnevisan, E. Nualart, 2010)
A weak solution of (8) is the process {u(t , ϕ); t > 0, ϕ ∈ S(Rd )}:
u(t , ϕ) =1
(2π)d
∫ t
0
∫Rd
sin(√
Ψ(ξ)(t − s))√Ψ(ξ)
Fϕ(ξ) W (ds,dξ)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 16 / 21
A hyperbolic equation
The Fourier transform of W
Let E be the set of linear combinations of functions
h(t , x) = 1[0,a](t)ψ(x), a > 0, ψ ∈ S(Rd )
Let Fh(t , ξ) =∫Rd e−iξ·xh(t , x)dx . Endow E with the inner product
〈h1,h2〉H = 〈Fh1,Fh2〉H
For any h ∈ E , defineW (h) := W (Fh)
The Hilbert space H is the completion of E with respect to 〈·, ·〉H.The map h 7→ W (h) is extended to H
W (h) :=
∫ ∞0
∫Rd
h(t , ξ)W (dt ,dξ) (stochastic integral)
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 17 / 21
A hyperbolic equation
Proposition
Let h : R+ × Rd → C be such that h(t , ·) = 0 if t > T , h(t , ·) ∈ L2(Rd )for any t ∈ [0,T ] and g := Fh ∈ H. Then h ∈ H and
‖h‖2H = αH(2π)2d∫Rd
∫ T
0
∫ T
0|r − s|2H−2h(r , ξ)h(s, ξ)drdsµ(dξ)
ht ,ϕ(s, ξ) =1
(2π)d 1[0,t](s)sin(
√Ψ(ξ)(t − s))√
Ψ(ξ)Fϕ(ξ)
The weak solution of (8) exists since ht ,ϕ ∈ H for any t > 0, ϕ ∈ S(Rd )(by Theorem 2, gt ,ϕ := Fht ,ϕ ∈ H). Moreover,
E |u(t , ϕ)|2 = ‖ht ,ϕ‖2H =
∫Rd
Nt (ξ)|Fϕ(ξ)|2µ(dξ), where
Nt (ξ) =αH
Ψ(ξ)
∫ t
0
∫ t
0|r − s|2H−2 sin(r
√ψ(ξ)) sin(s
√Ψ(ξ))drds
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 18 / 21
A hyperbolic equation
Proposition
Let h : R+ × Rd → C be such that h(t , ·) = 0 if t > T , h(t , ·) ∈ L2(Rd )for any t ∈ [0,T ] and g := Fh ∈ H. Then h ∈ H and
‖h‖2H = αH(2π)2d∫Rd
∫ T
0
∫ T
0|r − s|2H−2h(r , ξ)h(s, ξ)drdsµ(dξ)
ht ,ϕ(s, ξ) =1
(2π)d 1[0,t](s)sin(
√Ψ(ξ)(t − s))√
Ψ(ξ)Fϕ(ξ)
The weak solution of (8) exists since ht ,ϕ ∈ H for any t > 0, ϕ ∈ S(Rd )(by Theorem 2, gt ,ϕ := Fht ,ϕ ∈ H). Moreover,
E |u(t , ϕ)|2 = ‖ht ,ϕ‖2H =
∫Rd
Nt (ξ)|Fϕ(ξ)|2µ(dξ), where
Nt (ξ) =αH
Ψ(ξ)
∫ t
0
∫ t
0|r − s|2H−2 sin(r
√ψ(ξ)) sin(s
√Ψ(ξ))drds
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 18 / 21
A hyperbolic equation
Random Field Solution
Defintion
Let M = ∩t>0Mt where Mt is the completion of S(Rd ) with respect to
‖ϕ‖t = E |u(t , ϕ)|2 =
∫Rd
Nt (ξ) |Fϕ(ξ)|2µ(dξ)
Equation (8) has a random-field solution u(t , x) := u(t , δx ) if
δx ∈ M for all x ∈ Rd
Let Z = ∩t>0Zt where Zt is the completion of S(Rd ) with respect to
|‖ϕ‖|t =
∫Rd
(1
1/t + Ψ(ξ)
)H+1/2
|Fϕ(ξ)|2µ(dξ)
Remark: For any s, t > 0, Zt = Zs.
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 19 / 21
A hyperbolic equation
Random Field Solution
Defintion
Let M = ∩t>0Mt where Mt is the completion of S(Rd ) with respect to
‖ϕ‖t = E |u(t , ϕ)|2 =
∫Rd
Nt (ξ) |Fϕ(ξ)|2µ(dξ)
Equation (8) has a random-field solution u(t , x) := u(t , δx ) if
δx ∈ M for all x ∈ Rd
Let Z = ∩t>0Zt where Zt is the completion of S(Rd ) with respect to
|‖ϕ‖|t =
∫Rd
(1
1/t + Ψ(ξ)
)H+1/2
|Fϕ(ξ)|2µ(dξ)
Remark: For any s, t > 0, Zt = Zs.
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 19 / 21
A hyperbolic equation
Theorem 5 (B. 2011)
For any t > 0, ξ ∈ Rd ,
c(1)H t
(1
1/t2 + Ψ(ξ)
)H+1/2
≤ Nt (ξ) ≤ c(2)H t
(1
1/t2 + Ψ(ξ)
)H+1/2
CorollaryWe have Mt = Zt2 for any t > 0, and M = Z. Equation (8) has arandom-field solution iff
δx ∈ Z for all x ∈ Rd .
A necessary and sufficient condition for this is:∫Rd
(1
1 + Ψ(ξ)
)H+1/2
µ(dξ) <∞.
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 20 / 21
A hyperbolic equation
Theorem 5 (B. 2011)
For any t > 0, ξ ∈ Rd ,
c(1)H t
(1
1/t2 + Ψ(ξ)
)H+1/2
≤ Nt (ξ) ≤ c(2)H t
(1
1/t2 + Ψ(ξ)
)H+1/2
CorollaryWe have Mt = Zt2 for any t > 0, and M = Z. Equation (8) has arandom-field solution iff
δx ∈ Z for all x ∈ Rd .
A necessary and sufficient condition for this is:∫Rd
(1
1 + Ψ(ξ)
)H+1/2
µ(dξ) <∞.
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 20 / 21
A hyperbolic equation
Thank you!
Raluca Balan (University of Ottawa) Linear SPDEs with fractional noise Banff (April 2-6, 2012) 21 / 21