SDEs and SPDEs with random set
coefficients
Michael Oberguggenberger
University of Innsbruck
International Conference on Controlled Deterministic and
Stochastic Systems, Universitatea “Alexandru Ioan Cuza”
Iaşi, July 3, 2012
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
1 / 22
Introduction
Plan of talk:
Introductory examples
Tools
SDEs with random set coefficients
SPDEs with random set coefficients
The 1D stochastic wave equation: classical
The nD stochastic wave: distributional
Hyperbolic systems with random field and random set coefficients:
beyond distributions
Joint work with B. Schmelzer and L. Wurzer (University of Innsbruck)
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
2 / 22
Example – Tuned Mass Dampers (1)
Description by linear system of SDEs with parameters
dxa (t) = F (a)xa (t)dt + G dW (t)
with noise excitation (white or filtered through the system) – risk.
Goal: control of earthquake-induced vibrations by tuning the
parameters a of the TMD.
Uncertainty of a modelled as a random set independent of W .
kd
md
cd
xd (t)
xs (t)
ms
cs
ks
xg(t)
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
3 / 22
Example – Tuned Mass Dampers (2)
Relative structural displacement: interval valued trajectory and
interval means w/o TMD:
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
4 / 22
Example – Elastically Bedded Beam (1)
EI w IV (x) + bc w (x) = q(x), −∞ < x < ∞ ,
w the displacement, q(x) the load (a random field), bc a bedding
parameter (modelled as a random set).
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
5 / 22
Example – Elastically Bedded Beam (2)
Interval valued trajectory of bending moment and p-box for maximal
bending moment:
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
6 / 22
Example – Stochastic Wave Equation
Linear stochastic wave equation as testing ground:
(
∂t2 u − c 2 ∆u = Ẇ ,
x ∈ Rn , t ≥ 0
u|{t < 0} = 0
Space-time white noise excitation Ẇ , propagation speed c as a
random set.
Dimension dependence – classical theory for n = 1.
Solution is a generalized stochastic process (trajectories distributions)
for n ≥ 2.
What is a random set with values in D0 (Rn+1 )?
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
7 / 22
Example – Seismic Waves in Random Media
ρ(x)
∂ 2 ul X ∂
∂ul
−
cijkl (x)
= fi (x, t),
2
∂t
∂x
∂x
j
k
j,k,l
i = 1, 2, 3
Density ρ(x), elastic coefficients cijkl (x) – random fields of low
regularity. Distributions may not suffice.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
8 / 22
Tools: Distribution Valued Processes
Let (Ω, Σ, P) be a probability space. A distribution valued process is a
map
Ω → D0 (Rn ),
ω → X (ω)
such that all maps ω → hX (ω), ϕi, ϕ ∈ D(Rn ), are measurable.
Central example: Gaussian white noise Ẇ with support in the closure
D of an open subset of Rn . Here Ω = S 0 (D), Σ the Borel σ-algebra
generated by the weak topology, and P given by the Bochner-Minlos
theorem as the unique measure satisfying
Z
e i hω,ϕi dP(ω) = exp − 21 kϕk2L2 (D)
for ϕ ∈ S(D). Then Ẇ : Ω → D0 (Rn ) is given by
hẆ (ω), ϕi = hω, ϕ|Di.
for ϕ ∈ D(Rn ).
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
9 / 22
Tools: Random Fields
We refer to a regular stochastic process (x, ω) → A(x, ω) as a
random field, especially when x denotes position in space.
If the random field is Gaussian, it is determined by the expectation
values E(A(x)) and the covariances COV(A(x), A(y )).
If, in addition, the random field is stationary and homogeneous, the
defining data are the expectation value µ ≡ E(A(x)) and the
autocovariance function C (ρ) = COV(A(x), A(y )) which depends
only on the distance ρ = |x − y |, often,
C (ρ) = σ 2 exp −
|ρ| L
or
C (ρ) = σ 2 exp −
ρ2 L
with the field variance σ 2 and the correlation length L as parameters.
Note that Gaussian fields of this type have continuous, but not
necessarily differentiable trajectories.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
10 / 22
Tools: Random Sets (1)
Let (Ω, Σ, P) be a probability space and E a complete, separable
metric (i.e., Polish) space with the Borel sigma algebra B(E). A
random set is a map X from Ω → P(E) such that the upper inverses
X − (B) = {ω ∈ Ω : X (ω) ∩ B 6= ∅}
are measurable for every Borel subset B of E.
Random compact sets: taking values in K(E), the set of non-empty
compact subsets of E. Equip K(E) with the Hausdorff metric.
Random compact sets are characterized as being measurable maps
Ω → K(E), thus they can be viewed as set-valued random variables.
Important functionals are the upper and lower probabilities of events
B ⊂ E, defined by
P(B) = P X − (B) ,
P(B) = P X− (B)
where X− (B) = {ω ∈ Ω : X (ω) ⊂ B} is the lower inverse of B.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
11 / 22
Tools: Random Sets (2)
Important tool: the fundamental measurability theorem stating the
equivalence of the measurability property of X − (B) for Borel, open,
and closed subsets B of E as well as the equivalence with the
existence of a Castaing representation.
A Castaing representation is a sequence of measurable selections
ξn (ω) ∈ X (ω) s.t. for almost all ω, {ξn (ω) : n ∈ N} is dense in X (ω).
A set-valued random variable such that X − (B) is measurable for
every open set B is called Effros-measurable.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
12 / 22
Tools: Random Sets (2)
Important tool: the fundamental measurability theorem stating the
equivalence of the measurability property of X − (B) for Borel, open,
and closed subsets B of E as well as the equivalence with the
existence of a Castaing representation.
A Castaing representation is a sequence of measurable selections
ξn (ω) ∈ X (ω) s.t. for almost all ω, {ξn (ω) : n ∈ N} is dense in X (ω).
A set-valued random variable such that X − (B) is measurable for
every open set B is called Effros-measurable.
Typical example – a random field whose parameters are intervals. For
example, if the correlation length is taken as an interval [L, L], the
assignment
ω → {AL (x, ω) : L ∈ [L, L]},
where x is a point in space and AL (x, ω) is a realization of the field
with correlation length L at point x, defines a random set.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
12 / 22
Stochastic ODEs (1)
SDE with parameters:
dxa (t) = f t, a, xa (t) dt + G t, a, xa (t) dW (t)
with initial data xa (t0 ) = x0 . Here t ∈ [t0 , t1 ], x(t) and f are
Rd -valued, G is a (d × m)-matrix, W (t) an m-dimensional Brownian
motion on the probability space Ω, and a is a k-dimensional
parameter.
Under suitable assumptions, the solutions form a family of stochastic
processes
x : [t0 , t1 ] × Rk × Ω → Rd ,
(t, a, ω) → xa (t, ω).
Modelling the uncertainty of the parameter a by a random set
Θ → K(Rk ), θ → A(θ) gives rise to a set-valued function
X : (t, θ, ω) → {xa (t, ω) : a ∈ A(θ)}
on the product probability space Θ ⊗ Ω with values in P(Rd ).
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
13 / 22
Stochastic ODEs (2)
Finally, one shows that X is a set-valued stochastic process, i.e.,
X (t) : Θ ⊗ Ω → K(Rd )
is a random compact set for every t.
The process as a random set in E = C([t0 , t1 ] : Rd ): Start with
x̃ : Rk × Ω → E,
(a, ω) → x̃a (ω)
where x̃a denotes the trajectory t → xa (t, ω). Then form
X̃ : Θ ⊗ Ω → P(E),
(θ, ω) → {x̃a (ω) : a ∈ A(θ)}
Then X̃ is a random compact set in E, and
πt (X̃ ) = X (t).
Further constructions: based on selections; sets of probability
measures. Theory of excursion times.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
14 / 22
The 1D Stochastic Wave Equation (1)
∂t2 u − c 2 ∂x2 u = Ẇ ,
u|{t < 0} = 0
x ∈ R, t ≥ 0
with space time Gaussian white noise Ẇ with support in t > 0.
Fundamental solution of deterministic problem:
Sc (x, t) =
1
H(t)H(ct − |x|)
2c
Pathwise solution is a rotated Brownian sheet:
ZZ
uc (x, t) = Sc ∗ Ẇ (x, t) =
dW (x, t)
Γc (x,t)
where Γc (x, t) is the backward lightcone emanating from (x, t). The
variance is t 2 /4c.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
15 / 22
The 1D Stochastic Wave Equation (2)
Interval valued propagation speed and the corresponding set-valued
process:
U(ω) = {uc (ω) : c ∈ [c, c]}
with values in C(R2 ).
Proposition. U is a random set in the Polish space E = C(R2 ).
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
16 / 22
The 1D Stochastic Wave Equation (2)
Interval valued propagation speed and the corresponding set-valued
process:
U(ω) = {uc (ω) : c ∈ [c, c]}
with values in C(R2 ).
Proposition. U is a random set in the Polish space E = C(R2 ).
Proof: The map c → uc (ω) is continuous with values in E; the image
of [c, c] is compact. Take a dense countable subset c1 , c2 , . . . of [c, c].
The sequence ucn forms a Castaing representation. Let O be an open
subset of E. Then
U− (B) = {ω : U(ω) ∩ O 6= ∅} =
∞
[
{ω : ucn (ω) ∈ O}
n=1
is measurable. The fundamental measurability theorem says that the
same is true when O is a Borel subset of E, that is, U is a random set
in E.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
16 / 22
The 1D Stochastic Wave Equation (3)
Similarly,
U(x, t, ω) = {uc (x, t, ω) : c ∈ [c, c]}
is a random set in R for every fixed (x, t).
Typical functionals of interest are the upper and lower excursion
probabilities of the set-valued solution at fixed (x, t)
P(B) = P(U ∩ B 6= ∅) = P {ω : supc∈[c,c] uc (x, t, ω) ≥ b}
P(B) = P(U ⊂ B) = P {ω : inf c∈[c,c] uc (x, t, ω) ≥ b}
for B = [b, ∞). The computation is aided by the observation:
Proposition. The process
(c, ω) → cuc (x, t, ω)
is a Wiener process with variance ct 2 /4, where (c, ω) → uc (x, t, ω) is
the solution process at fixed (x, t) as a function of c.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
17 / 22
The 1D Stochastic Wave Equation (4)
Proof: Observe that (c, ω) → uc (x, t, ω) is a mean zero Gaussian
process with covariance
E(uc1 (x, t)uc2 (x, t)) = min(c1 , c2 )t 2 /4c1 c2 .
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
18 / 22
The 1D Stochastic Wave Equation (4)
Proof: Observe that (c, ω) → uc (x, t, ω) is a mean zero Gaussian
process with covariance
E(uc1 (x, t)uc2 (x, t)) = min(c1 , c2 )t 2 /4c1 c2 .
Thus the excursion probabilities can be computed from the
well-known results about excursions of a Wiener process (Rayleigh
distribution).
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
18 / 22
The 1D Stochastic Wave Equation (4)
Proof: Observe that (c, ω) → uc (x, t, ω) is a mean zero Gaussian
process with covariance
E(uc1 (x, t)uc2 (x, t)) = min(c1 , c2 )t 2 /4c1 c2 .
Thus the excursion probabilities can be computed from the
well-known results about excursions of a Wiener process (Rayleigh
distribution).
Switch: The nD stochastic wave equation
∂t2 u − c 2 ∆u = Ẇ , x ∈ Rn , t ≥ 0
u|{t < 0} = 0
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
18 / 22
The nD Stochastic Wave Equation
If Sc ∈ D0 (Rn+1 is the fundamental solution of the deterministic
problem, the solution is again given by
uc = Sc ∗ Ẇ
but for n ≥ 2 it no longer a function, but a distribution in D0 (Rn+1 ).
One can form the set-valued function
U(ω) = {uc (ω) : c ∈ [c, c]}.
The same arguments as before show that there is a Castaing
representation, and so the map is Effros-measurable with values in the
space E = D0 (Rn+1 ). This space is separable and complete, but not
metrizable (bounded sets are metrizable with respect to the weak
topology, though).
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
19 / 22
The nD Stochastic Wave Equation
If Sc ∈ D0 (Rn+1 is the fundamental solution of the deterministic
problem, the solution is again given by
uc = Sc ∗ Ẇ
but for n ≥ 2 it no longer a function, but a distribution in D0 (Rn+1 ).
One can form the set-valued function
U(ω) = {uc (ω) : c ∈ [c, c]}.
The same arguments as before show that there is a Castaing
representation, and so the map is Effros-measurable with values in the
space E = D0 (Rn+1 ). This space is separable and complete, but not
metrizable (bounded sets are metrizable with respect to the weak
topology, though).
Open question: Is U a random set? Generalization of fundamental
measurability theorem?
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
19 / 22
Tools: Colombeau Algebras (1)
Notation: O – an open subset of Rn ; families (uε )ε∈(0,1] ⊂ C ∞ (O).
Moderate families, EM (O):
∀K b O ∀α ∈ Nn0 ∃p ≥ 0 : sup |∂ α uε (x)| = O(ε−p ) as ε → 0.
x∈K
Null or negligible families, N (O):
∀K b O ∀α ∈ Nn0 ∀q ≥ 0 : sup |∂ α uε (x)| = O(εq ) as ε → 0.
x∈K
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
20 / 22
Tools: Colombeau Algebras (1)
Notation: O – an open subset of Rn ; families (uε )ε∈(0,1] ⊂ C ∞ (O).
Moderate families, EM (O):
∀K b O ∀α ∈ Nn0 ∃p ≥ 0 : sup |∂ α uε (x)| = O(ε−p ) as ε → 0.
x∈K
Null or negligible families, N (O):
∀K b O ∀α ∈ Nn0 ∀q ≥ 0 : sup |∂ α uε (x)| = O(εq ) as ε → 0.
x∈K
Colombeau algebra: G(O) = EM (O)/N (O).
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
20 / 22
Tools: Colombeau Algebras (1)
Notation: O – an open subset of Rn ; families (uε )ε∈(0,1] ⊂ C ∞ (O).
Moderate families, EM (O):
∀K b O ∀α ∈ Nn0 ∃p ≥ 0 : sup |∂ α uε (x)| = O(ε−p ) as ε → 0.
x∈K
Null or negligible families, N (O):
∀K b O ∀α ∈ Nn0 ∀q ≥ 0 : sup |∂ α uε (x)| = O(εq ) as ε → 0.
x∈K
Colombeau algebra: G(O) = EM (O)/N (O).
Imbedding of E 0 (O) into G(O):
ι : E 0 (O) → G(O), ι(w ) = class of ((w ∗ ρε )|O )ε∈(0,1] ,
R
R
ρε (x) = ε−n ρ (x/ε) , ρ(x)dx = 1, x α ρ(x) = 0, |α| ≥ 1.
Extension to D0 (O) → G(O) using sheaf property of G.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
20 / 22
Tools: Colombeau Algebras (2)
Topology on G(O): Neighborhoods VK ,α,p of the zero function given
by the representatives (uε )ε∈(0,1] such that
sup |∂ α uε (x)| = O(ε−p ) as ε → 0.
x∈K
G(O) is a complete ultrametric differential algebra.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
21 / 22
Tools: Colombeau Algebras (2)
Topology on G(O): Neighborhoods VK ,α,p of the zero function given
by the representatives (uε )ε∈(0,1] such that
sup |∂ α uε (x)| = O(ε−p ) as ε → 0.
x∈K
G(O) is a complete ultrametric differential algebra.
Colombeau random generalized functions on a probability space
(Ω, Σ, P): Maps u : Ω → G(O) with representing nets uε (x, ω) s.t.
at fixed ε, uε (x, ω) is jointly measurable in O × Ω;
for almost all ω ∈ Ω, (uε (·, ω))ε belongs to EM (O) and is a
representative of u(ω).
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
21 / 22
Tools: Colombeau Algebras (2)
Topology on G(O): Neighborhoods VK ,α,p of the zero function given
by the representatives (uε )ε∈(0,1] such that
sup |∂ α uε (x)| = O(ε−p ) as ε → 0.
x∈K
G(O) is a complete ultrametric differential algebra.
Colombeau random generalized functions on a probability space
(Ω, Σ, P): Maps u : Ω → G(O) with representing nets uε (x, ω) s.t.
at fixed ε, uε (x, ω) is jointly measurable in O × Ω;
for almost all ω ∈ Ω, (uε (·, ω))ε belongs to EM (O) and is a
representative of u(ω).
Colombeau Gaussian white noise is represented pathwise by
Ẇε (x) = Ẇ ∗ ρε (x) = hẆ (y ), ρε (x − y )i.
Its variance is E|Ẇε (x)|2 = kρε k2L2 (Rn ) = ε−d kρk2L2 (Rn ) .
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
21 / 22
Hyperbolic Systems
Systems and higher order equation –
Pn
∂t u(t, x) =
j=1 Aj (t, x)∂xj u(t, x) + A0 (t, x)u(t.x) + f (t, x),
x ∈ Rn , t > 0,
u|{t < 0} = 0.
Issue: take Aj (t, x) as random fields. If paths are not regular enough,
classical solution theory is not applicable. But: solvability in the
Colombeau algebra as a random Colombeau function.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
22 / 22
Hyperbolic Systems
Systems and higher order equation –
Pn
∂t u(t, x) =
j=1 Aj (t, x)∂xj u(t, x) + A0 (t, x)u(t.x) + f (t, x),
x ∈ Rn , t > 0,
u|{t < 0} = 0.
Issue: take Aj (t, x) as random fields. If paths are not regular enough,
classical solution theory is not applicable. But: solvability in the
Colombeau algebra as a random Colombeau function.
Introducing random fields with interval valued parameters, we get the
solution as a set-valued process U with values in the Colombeau
algebra E = G(Rn+1 ). It is a complete (ultra-)metrizable space, but
not separable.
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
22 / 22
Hyperbolic Systems
Systems and higher order equation –
Pn
∂t u(t, x) =
j=1 Aj (t, x)∂xj u(t, x) + A0 (t, x)u(t.x) + f (t, x),
x ∈ Rn , t > 0,
u|{t < 0} = 0.
Issue: take Aj (t, x) as random fields. If paths are not regular enough,
classical solution theory is not applicable. But: solvability in the
Colombeau algebra as a random Colombeau function.
Introducing random fields with interval valued parameters, we get the
solution as a set-valued process U with values in the Colombeau
algebra E = G(Rn+1 ). It is a complete (ultra-)metrizable space, but
not separable.
Open question: Is U a random set? Generalization of fundamental
measurability theorem?
Michael Oberguggenberger (Innsbruck)
ICCDSS
Iaşi, July 3, 2012
22 / 22