Reflections on Itˆ
o Integral against Cylindrical Wiener
Processes defined on Hilbert Space
Sylvain Dotti∗ August 13, 2019
Abstract
The evolution equations taking into account continuous uncertainty need the theory of Itˆo integral against cylindrical Wiener processes. Based on the famous introductions of Da Prato Zabczyk [DZ92], Prevˆot R¨ockner [PR07], Gawarecki Man-drekar [GM10], I explain the idea of cylindrical Wiener process, the different ideas of constructions of the Itˆo integral, and give my opinion on the most suitable con-struction and definition.
Keywords: Cylindrical Wiener process, Elementary processes, Itˆo integral MSC Number: 60-01 (60G99, 60H05)
Contents
1 Introduction 1
2 Cylindrical Wiener process, a formal definition 2
3 Discussion on the construction of Itˆo integral against a Q-Wiener
Pro-cess 3
4 Discussion on the construction of Itˆo integral against a cylindrical
Wiener Process 5
5 Closing remarks 8
1
Introduction
The theory of Itˆo integral against cylindrical Wiener processes is a beautiful theory used in SPDE to quantify continuous uncertainty in evolution equations and give rig-orous definition of solutions. One can quote E Mazel Khanin Sinai [E+00], Debussche
∗
CEMOI, Universit´e de la R´eunion, 15 Avenue Ren´e Cassin, CS 92003, 97744 Saint-Denis Cedex 9, La R´eunion, France
Vovelle[DV10], Dotti Vovelle [DV18], Galimberti Karlsen [GK19], Pardoux [Par07] for examples of SPDE using cylindrical Wiener processes.
It is usefull to make its own intuition on such a mathematical tool. The idea of cylindri-cal Wiener process is quite similar to real Brownian motion. A common and practicylindri-cal vision of the real Brownian motion is the stock chart. What we call Brownian motion is the set of all possible stock charts.
For generalisation to finite, then infinite dimension, it is easier to see brownian motion as the set of all possible trajectories of a particle suspended in a liquid. To be more precise, let t be the time variable, and (x(t), y(t), z(t)) the space coordinates of the par-ticle, then the set of all possible path x(t) is the real brownian motion. The coordinates (x(t), y(t), z(t)) gives a good representation of the 3-dimensional Brownian motion (the trajectories of the path x(t), y(t) and z(t) are independent). It is quite fascinating that the properties of the set of all possible trajectories of one particle or of one coordinate of a particle or of a big amount of particle are the same. The set of all possible trajec-tories of a big amount of particle is the N-dimensional Brownian motion where N ∈ N∗. The mathematical idea of countless amount of particles is countable infinity of particles, thus the N-dimensional Brownian motion becomes cylindrical Wiener process when the particles are countless.
2
Cylindrical Wiener process, a formal definition
Let (Ω, F , P)) be a probability space, [0, T ] ⊂ R+ a bounded interval of time, and (Ft)t∈{0,T ] be a complete, right-continuous filtration associated with (Ω, F , P)). Let (βk)k∈N∗a sequence of independent real brownian motions adapted to the filtration (Ft).
A cylindrical Wiener process W is defined on an Hilbert space (H, h. , .iH), endowed with an orthonormal basis (ek)k∈N∗ by W : (ω, t) ∈ Ω × [0, T ] 7→ W (ω, t) := +∞ X k=1 βk(ω, t) ek Usually, the random variable ω is omitted. We follow this rule in the sequel. The previous definition is formal, W (t) doesn’t exist in H. Indeed,
E kW (t) k2H = E +∞ X k=1 βk(t)2 ! = +∞ X k=1 t = +∞.
To make it rigorous, we have to use J : H → U a Hilbert-Schmidt embedding, with U a Hilbert space. For exemple, let us take U = H and J ek= 1kek, we can thus define
J W (t) := +∞ X
k=1
which is not an element of J (H) (there is a slight abuse of notation) but an element of H. Note that J (H) is dense in H because (kJ ek)k∈N∗ is an orhtonormal basis of H .
J W is a Q-Wiener process on H with covariance operator Q = J ◦ J∗ = J2 (J∗ = J is the adjoint operator of J ). Indeed, ∀k, j ∈ N∗
hek, J∗(el)iH = hJ (ek) , eliH = 1
khek, eliH
The theory of integrals against such Q-Wiener processes is described in [PR07]. I give some ideas of its construction in the particular case of
J W (t) = +∞ X k=1 1 kβk(t) ek
where the integrand Φ belongs to L(H, R). In the general case Φ can belong to L(H, K) with K Hilbert space. My aim is to discuss on the ideas of construction, that’s why the scalar case is enough.
3
Discussion on the construction of Itˆ
o integral against a
Q-Wiener Process
I try to define properly the Itˆo integral of elementary processes Φ ∈ L2(H, R) against J W . The aim is to be able to define
Z T 0 Φ (ω, t) dW (t) := +∞ X k=1 1 k Z T 0 Φ (ω, t) ekdβk(t) with Φ : Ω × [0, T ] → L (H, R). First, each
Z T
0
Φ (ω, t) ek dβk(t) must exist. It is the case if
∀k ∈ N∗, E Z T 0 (Φ (ω, t) ek)2dt < +∞. (3.1) or if Z T 0 (Φ (ω, t) ek)2dt < +∞, a.s. (3.2) It means that the condition Φ : Ω × [0, T ] → L (H, R) is not sufficient to define the set of integrands, we also need (3.1) or (3.2). One of those assumptions will be satisfied if
E Z T 0 +∞ X k=1 1 k2 (Φ (ω, t) ek) 2dt ! < +∞ (3.3) or if Z T 0 +∞ X k=1 1 k2 (Φ (ω, t) ek) 2 dt < +∞, a.s. (3.4)
With the assumption (3.3), +∞ X k=1 1 k Z T 0 Φ (ω, t) ekdβk(t) converges in L2(Ω) because E M X k=N 1 k Z T 0 Φ (ω, t) ekdβk(t) !2 = E M X k=N 1 k2 Z T 0 (Φ (ω, t) ek)2dt ! .
It is the Itˆo isometry which will be proved using the elementary processes. With the assumption (3.4), +∞ X k=1 1 k Z T 0 Φ (ω, t) ekdβk(t)
converges almost surely, that follows the result under the assumption (3.3), and a so-called localization method. Da prato, Zabczyk [DZ92] or Prevˆot, R¨ockner [PR07] define naturally the integral
Z T
0
Φ (ω, t) dW (t)
starting from elementary processes (with 0 = t0< t1 < ... < ti< ... < tp = T ):
Φ (ω, t) = p−1 X
i=0
Φi(ω) 1(ti,ti+1](t)
then by extending the definition by density thanks to the Itˆo isometry.
We can notice that Φi : Ω → L (H, R) taking a finite number of values in L(H, R) is a sufficient condition to have ∀k ∈ N∗
E Z T 0 (Φ (ω, t) ek)2dt = E p−1 X i=0 (Φi(ω) ek)2(ti+1− ti) ! = p−1 X i=0 (ti+1− ti) E (Φi(ω) ek)2 < +∞.
The values of Φi are for exemple Φ1i, . . . , Φri on the subsets Ω1, . . . , Ωr de Ω , thus E (Φi(ω) ek)2 = r X j=1 Φjiek 2 P Ωj < +∞ .
Another condition could have been ∀i ∈ {0, . . . , p − 1}
E +∞ X k=1 1 k2 (Φi(ω) ek) 2 ! < +∞ or E kΦi(ω) k2L 2(H,R) < +∞ with Φi : Ω → L2(H, R) .
Gawarecki, Mandrekar in [GM10] p25, take the Φi uniformly bounded in L2(H, R) i.e. there exists a M ∈ [0, +∞) independent of i and ω ∈ Ω such that ∀i:
kΦi(ω) k2L2(H,R)≤ M < +∞, a.s.
Finally, the Itˆo integral against the Q-Wiener process
J W (t) = +∞ X k=1 1 kβk(t) ek will be defined on a space of integrands
Φ : Ω × [0, T ] → L (H, R) which verifiy E Z T 0 +∞ X k=1 1 k2 (Φ (ω, t) ek) 2 dt ! < +∞
4
Discussion on the construction of Itˆ
o integral against a
cylindrical Wiener Process
E Z T 0 +∞ X k=1 1 kΦ (ω, t) ek 2 dt ! < +∞
and remembering that the coefficients k12 where taken as an example of Hilbert-Schmidt
embedding for J , It is then easy to understand that to integrate against the cylindrical Wiener process W (t) = +∞ X k=1 βk(t) ek the integrands Φ : Ω × [0, T ] → L2(H, R) should verify
E Z T 0 +∞ X k=1 (Φ (ω, t) ek)2dt ! < +∞
Now, I want to come back to the way Da Prato and Zabczyk in [DZ92], Prevˆot and R¨ockner in [PR07] define the Itˆo integral against a cylindrical Wiener process, and to the Hilbert-Schmidt embedding J : H → J (H) ⊂ H, with J ek = αkek that permit to define J W as an element of H (I took αk= 1k).
Each Φ (ω, t) ◦ J−1 is defined on J (H), and must be linear continuous with ∀h ∈ H: Φ (ω, t) ◦ J−1(J (h)) = Φ (ω, t) (h)
to define the stochastic integral of Φ against the cylindrical Wiener process W by Z T 0 Φ (ω, t) dW (t) := Z T 0 Φ (ω, t) ◦ J−1dJ W (t)
J−1 is not continuous with the norm k.kH on J (H), but with the norm kJ (h) kJ (H) := khkH, ∀h ∈ H.
J (H) is a separable Hilbert space for the scalar product
hJ (h) , J (k)iJ (H):= hh, kiH, ∀h, k ∈ H and admit for orthonormal basis k1ek
k∈N∗
To overcome those difficulties, we could try to extend Φ ◦ J−1 on H by density of J (H) in H (for the topology of k.kH). As J W is not define on J (H) but on H, the canonical injection
I : (J (H) , k.kH) → J (H), k.kJ (H)
should be continuous to be able to define Z T 0 Φ (ω, t) dW (t) := Z T 0 Φ (ω, t) ◦ J−1dJ W (t)
Φ (ω, t) ◦ J−1 would be extended by continuity in ˜Φ (ω, t) defined on H. We would have ˜ Φ : Ω × [0, T ] → L (H, R) and +∞ X k=1 1 k2 ˜Φ (ω, t) ek 2 = +∞ X k=1 1 k2 ˜Φ (ω, t) J (kek) 2 = +∞ X k=1 1 k2(Φ (ω, t) (kek)) 2 = +∞ X k=1 (Φ (ω, t) (ek))2 < +∞ In that way, we could write (almost like in [PR07])
Z T 0 Φ (ω, t) dW (t) := Z T 0 ˜ Φ (ω, t) dJ W (t) and have an integral which exists for elementary processes Φ. Indeed:
Z T 0 Φ (ω, t) dW (t) = Z T 0 p−1 X i=0 Φi(ω) 1(ti,ti+1](t) dW (t) = p−1 X i=0 Z ti+1 ti Φi(ω) dW (t) = p−1 X i=0 ˜ Φi(ω) (J W (ti+1) − J W (ti)) = p−1 X i=0 (ΦiW (ti+1) − ΦiW (ti)) with the abuse of notation
ΦiW (ti+1) = +∞ X
k=1
βk(t) Φiek
But I is not continuous. I think that the Itˆo integral against cylindrical Wiener process is not well defined with the construction of [PR07]. On the other hand, we can let the cylindrical Wiener processes undefined (no need to use the Hilbert-Schmidt injection J ) because the Itˆo integral against cylindrical Wiener process can be constructed without it. We just need to notice that the equalities
Z T 0 Φ (ω, t) dW (t) = Z T 0 p−1 X i=0 Φi(ω) 1(ti,ti+1](t) dW (t) = p−1 X i=0 Z ti+1 ti Φi(ω) dW (t) = p−1 X i=0 (ΦiW (ti+1) − ΦiW (ti)) are well defined when each Φi ∈ L2(H, R), using the fact that ΦiW (ti) is defined by
ΦiW (ti) := +∞ X
k=1
5
Closing remarks
Remark 5.1. After all, a white noise dWdt doesn’t exist (at least as a time derivatie in the classical sense). What is important is to be able to integrate against dW , and eventually to define the weak in time solutions of SDE or SPDE. So, being able to integrate against a cylindrical Wiener process my be enough. The integral can be define with sums of integrals against independent real brownian motions or with integrals of elementary processes Z T 0 Φ (ω, t) dW (t) = p−1 X i=0 (ΦiW (ti+1) − ΦiW (ti)) then with the Itˆo isometry.
Remark 5.2. I didn’t talk about measurability of the integrands Φ : Ω × [0, T ] → L2(H, R) .
They have to be progressively measurable. It means measurable when L2(H, R) is en-dowed with the borelian σ-algebra and Ω, (Ft)t∈[0,T ] ×
[0, T ], (Bt)t∈[0,T ]
is endowed with the σ-algebra defined by
P rog := {Y : Ω×[0, T ] → L2(H, R) such that ∀t ∈ [0, T ] : Y|Ω×[0,t]is Ft⊗Bt-measurable} where Bt is the borelian σ-algebra of [0, t].
In [PR07] and most of articles solving SPDE the integrands are supposed to be pre-dictable. It means measurable when L2(H, R) is endowed with the borelian σ-algebra and Ω, (Ft)t∈[0,T ] ×
[0, T ], (Bt)t∈[0,T ]
is endowed with the σ-algebra defined by P := σ{Y : Ω×[0, T ] → L2(H, R) such that Y is adapted to (Ft)t∈[0,T ] and continuous} I recall that
P ⊂ P rog.
At first sight, it is more interesting to be able to integrate over a larger class of integrands. In fact, if we solve a SPDE, e.g. if we succeed to have a unique progressively measurable solution u (t, ω) to a Cauchy problem, then certainly with the same proof, there will be a unique predictable solution. It will be the same solution u (t, ω). As a solution of an SPDE, it is more precise to have a predictable solution than a progressively measurable solution.
References
[DZ92] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic equations in infinite di-mensions. Vol. v. 45. Encyclopedia of mathematics and its applications. Cam-bridge: Cambridge University Press, 1992. isbn: 0521385296 (hardback). url: http://www.loc.gov/catdir/description/cam026/93118317.html(cit. on pp.1,4,6).
[DV10] Arnaud Debussche and Julien Vovelle. “Scalar conservation laws with stochas-tic forcing”. In: Journal of Functional Analysis 259.4 (2010), pp. 1014–1042 (cit. on p.2).
[DV18] Sylvain Dotti and Julien Vovelle. “Convergence of approximations to stochastic scalar conservation laws”. In: Archive for Rational Mechanics and Analysis 230.2 (2018), pp. 539–591 (cit. on p.2).
[E+00] Weinan E, K Khanin, A Mazel, and Y Sinai. “Invariant measure for Burgers equation with stochastic forcing”. In: Annals of Mathematics-Second Series 151.3 (2000), pp. 877–960 (cit. on p.1).
[GK19] Luca Galimberti and Kenneth H Karlsen. “Well-posedness theory for stochas-tically forced conservation laws on Riemannian manifolds”. In: arXiv preprint arXiv:1904.03623 (2019) (cit. on p.2).
[GM10] Leszek Gawarecki and Vidyadhar Mandrekar. Stochastic differential equations in infinite dimensions: with applications to stochastic partial differential equa-tions. Springer Science & Business Media, 2010 (cit. on pp.1,5).
[Par07] E Pardoux. “Stochastic Partial Differential Equations, Lectures given in Fudan University, Shanghai”. In: Published by Marseille, France (2007) (cit. on p.2). [PR07] Claudia Pr´evˆot and Michael R¨ockner. A concise course on stochastic partial
