Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2013, Article ID 798549, 25 pages
http://dx.doi.org/10.1155/2013/798549
Review Article
Foundations of the Theory of Semilinear Stochastic
Partial Differential Equations
Stefan Tappe
Leibniz Universität Hannover, Institut für Mathematische Stochastik, Welfengarten 1, 30167 Hannover, Germany
Correspondence should be addressed to Stefan Tappe; tappe@stochastik.uni-hannover.de
Received 28 May 2013; Accepted 16 August 2013
Academic Editor: Hong-Kun Xu
Copyright © 2013 Stefan Tappe. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The goal of this review article is to provide a survey about the foundations of semilinear stochastic partial differential equations. In
particular, we provide a detailed study of the concepts of strong, weak, and mild solutions, establish their connections, and review
a standard existence and uniqueness result. The proof of the existence result is based on a slightly extended version of the Banach
fixed point theorem.
1. Introduction
Semilinear stochastic partial differential equations (SPDEs)
have a broad spectrum of applications including natural
sciences and economics. The goal of this review article is to
provide a survey on the foundations of SPDEs, which have
been presented in the monographs [1–3]. It may be beneficial
for students who are already aware about stochastic calculus
in finite dimensions and who wish to have survey material
accompanying the aforementioned references. In particular,
we review the relevant results from functional analysis about
unbounded operators in Hilbert spaces and strongly continuous semigroups.
A large part of this paper is devoted to a detailed study of
the concepts of strong, weak, and mild solutions to SPDEs,
to establish their connections and to review and prove a
standard existence and uniqueness result. The proof of the
existence result is based on a slightly extended version of the
Banach fixed point theorem.
In the last part of this paper, we study invariant manifolds
for weak solutions to SPDEs. This topic does not belong to the
general theory of SPDEs, but it uses and demonstrates many
of the results and techniques of the previous sections. It arises
from the natural desire to express the solutions of SPDEs,
which generally live in an infinite dimensional state space,
by means of a finite dimensional state process and thus to
ensure larger analytical tractability.
This paper should also serve as an introductory study
to the general theory of SPDEs, and it should enable the
reader to learn about further topics and generalizations in this
field. Possible further directions are the study of martingale
solutions (see, e.g., [1, 3]), SPDEs with jumps (see, e.g., [4] for
SPDEs driven by the Lévy processes and [5–8] for SPDEs
driven by Poisson random measures), and support theorems
as well as further invariance results for SPDEs; see, for
example, [9, 10].
The remainder of this paper is organized as follows: In
Sections 2 and 3, we review the required results from functional analysis. In particular, we collect the relevant material
about unbounded operators and strongly continuous semigroups. In Section 4 we review stochastic processes in infinite
dimension. In particular, we recall the definition of a trace
class Wiener process and outline the construction of the
Itô integral. In Section 5 we present the solution concepts for
SPDEs and study their various connections. In Section 6 we
review results about the regularity of stochastic convolution
integrals, which is essential for the study of mild solutions
to SPDEs. In Section 7 we review a standard existence and
uniqueness result. Finally, in Section 8 we deal with invariant
manifolds for weak solutions to SPDEs.
2. Unbounded Operators in the Hilbert Spaces
In this section, we review the relevant properties about
unbounded operators. We will start with operators in Banach
2
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spaces and focus on operators in Hilbert spaces later on. The
reader can find the proofs of the upcoming results in any
textbook about functional analysis, such as [11] or [12].
Let 𝑋 and 𝑌 be Banach spaces. For a linear operator 𝐴 :
𝑋 ⊃ D(𝐴) → 𝑌, defined on some subspace D(𝐴) of 𝑋, we
call D(𝐴) the domain of 𝐴.
Definition 1. A linear operator 𝐴 : 𝑋 ⊃ D(𝐴) → 𝑌 is called
closed, if for every sequence (𝑥𝑛 )𝑛∈N ⊂ D(𝐴), such that the
limits 𝑥 = lim𝑛 → ∞ 𝑥𝑛 ∈ 𝑋 and 𝑦 = lim𝑛 → ∞ 𝐴𝑥𝑛 ∈ 𝑌 exist,
one has 𝑥 ∈ D(𝐴) and 𝐴𝑥 = 𝑦.
Let 𝑦 ∈ D(𝐴∗ ) be arbitrary. By virtue of the extension result
for linear operators (Proposition 4), the operator
D (𝐴) 󳨃󳨀→ R,
𝑥 󳨃󳨀→ ⟨𝐴𝑥, 𝑦⟩ ,
(5)
has a unique extension to a linear functional 𝑧󸀠 ∈ 𝐻󸀠 . By the
representation theorem of Fréchet-Riesz (Theorem 5), there
exists a unique element 𝑧 ∈ 𝐻 with ⟨𝑧󸀠 , ∙⟩ = ⟨𝑧, ∙⟩. This
implies that
⟨𝐴𝑥, 𝑦⟩ = ⟨𝑥, 𝑧⟩
∀𝑥 ∈ D (𝐴) .
(6)
Definition 2. A linear operator 𝐴 : 𝑋 ⊃ D(𝐴) → 𝑌 is
called densely defined, if its domain D(𝐴) is dense in 𝑋; that
is, D(𝐴) = 𝑋.
Setting 𝐴∗ 𝑦 := 𝑧, this defines a linear operator 𝐴∗ : 𝐻 ⊃
D(𝐴∗ ) → 𝐻, and one has
Definition 3. Let 𝐴 : 𝑋 ⊃ D(𝐴) → 𝑋 be a linear operator.
Definition 6. The operator 𝐴∗ : 𝐻 ⊃ D(𝐴∗ ) → 𝐻 is called
the adjoint operator of 𝐴.
(1) The resolvent set of 𝐴 is defined as
𝜌 (𝐴) := {𝜆 ∈ C : 𝜆 − 𝐴 : D (𝐴) 󳨀→ 𝑋 is bijective and
(1)
(2) The spectrum of 𝐴 is defined as 𝜎(𝐴) := C \ 𝜌(𝐴).
(3) For 𝜆 ∈ 𝜌(𝐴), one defines the resolvent 𝑅(𝜆, 𝐴) ∈
𝐿(𝑋) as
(2)
Now, we will introduce the adjoint operator of a densely
defined operator in a Hilbert space. Recall that for a bounded
linear operator 𝑇 ∈ 𝐿(𝐻1 , 𝐻2 ), mapping between two Hilbert
spaces 𝐻1 and 𝐻2 , the adjoint operator is the unique bounded
linear operator 𝑇∗ ∈ 𝐿(𝐻2 , 𝐻1 ) such that
⟨𝑇𝑥, 𝑦⟩𝐻2 = ⟨𝑥, 𝑇∗ 𝑦⟩𝐻1
∀𝑥 ∈ 𝐻1 , 𝑦 ∈ 𝐻2 .
(3)
In order to extend this definition to unbounded operators,
one recalls the following extension result for linear operators.
Proposition 4. Let 𝑋 be a normed space, let 𝑌 be a Banach
space, let 𝐷 ⊂ 𝑋 be a dense subspace, and let Φ : 𝐷 → 𝑌 be a
continuous linear operator. Then there exists a unique continû : 𝑋 → 𝑌, that is, a continuous linear operator
ous extension Φ
̂ = ‖Φ‖.
̂
with Φ|𝐷 = Φ. Moreover, one has ‖Φ‖
Now, let 𝐻 be a Hilbert space. We recall the representation
theorem of Fréchet-Riesz. In the sequel, the space 𝐻󸀠 denotes
the dual space of 𝐻.
󸀠
󸀠
Theorem 5. For every 𝑥 ∈ 𝐻 , there exists a unique element
𝑥 ∈ 𝐻 with ⟨𝑥󸀠 , ∙⟩ = ⟨𝑥, ∙⟩. In addition, one has ‖𝑥‖ = ‖𝑥󸀠 ‖.
Let 𝐴 : 𝐻 ⊃ D(𝐴) → 𝐻 be a densely defined operator.
One defines the subspace
D (𝐴∗ ) := {𝑦 ∈ 𝐻 : 𝑥 󳨃󳨀→ ⟨𝐴𝑥, 𝑦⟩ is continuous
on D (𝐴) } .
∀𝑥 ∈ D (𝐴) , 𝑦 ∈ D (𝐴∗ ) .
(7)
Proposition 7. Let 𝐴 : 𝐻 ⊃ D(𝐴) → 𝐻 be densely defined
and closed. Then 𝐴∗ is densely defined and one has 𝐴 = 𝐴∗∗ .
(𝜆 − 𝐴)−1 ∈ 𝐿 (𝑋)} .
𝑅 (𝜆, 𝐴) := (𝜆 − 𝐴)−1 .
⟨𝐴𝑥, 𝑦⟩ = ⟨𝑥, 𝐴∗ 𝑦⟩
(4)
Lemma 8. Let 𝐻 be a separable Hilbert space and let 𝐴 :
𝐻 ⊃ D(𝐴) → 𝐻 be a closed operator. Then the domain
(D(𝐴), ‖ ⋅ ‖D(𝐴) ) endowed with the graph norm
1/2
‖𝑥‖D(𝐴) = (‖𝑥‖2 + ‖𝐴𝑥‖2 )
(8)
is a separable Hilbert space, too.
3. Strongly Continuous Semigroups
In this section, we present the required results about strongly
continuous semigroups. Concerning the proofs of the
upcoming results, the reader is referred to any textbook about
functional analysis, such as [11] or [12]. Throughout this section, let 𝑋 be a Banach space.
Definition 9. Let (𝑆𝑡 )𝑡≥0 be a family of continuous linear
operators 𝑆𝑡 : 𝑋 → 𝑋, 𝑡 ≥ 0.
(1) The family (𝑆𝑡 )𝑡≥0 is a called a strongly continuous
semigroup (or 𝐶0 -semigroup), if the following conditions are satisfied:
(i) 𝑆0 = Id,
(ii) 𝑆𝑠+𝑡 = 𝑆𝑠 𝑆𝑡 for all 𝑠, 𝑡 ≥ 0,
(iii) lim𝑡 → 0 𝑆𝑡 𝑥 = 𝑥 for all 𝑥 ∈ 𝑋.
(2) The family (𝑆𝑡 )𝑡≥0 is called a norm continuous semigroup, if the following conditions are satisfied:
(i) 𝑆0 = Id,
(ii) 𝑆𝑠+𝑡 = 𝑆𝑠 𝑆𝑡 for all 𝑠, 𝑡 ≥ 0,
(iii) lim𝑡 → 0 ‖𝑆𝑡 − Id‖ = 0.
Note that every norm continuous semigroup is also a 𝐶0 semigroup. The following growth estimate (9) will often be
used when dealing with SPDEs.
International Journal of Stochastic Analysis
3
Lemma 10. Let (𝑆𝑡 )𝑡≥0 be a 𝐶0 -semigroup. Then there are
constants 𝑀 ≥ 1 and 𝜔 ∈ R such that
󵄩󵄩 󵄩󵄩
𝜔𝑡
󵄩󵄩𝑆𝑡 󵄩󵄩 ≤ 𝑀𝑒
∀𝑡 ≥ 0.
(9)
Definition 11. Let (𝑆𝑡 )𝑡≥0 be a 𝐶0 -semigroup.
(1) The semigroup (𝑆𝑡 )𝑡≥0 is called a semigroup of contractions (or contractive), if
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑆𝑡 󵄩󵄩 ≤ 1 ∀𝑡 ≥ 0;
(2) The semigroup (𝑆𝑡 )𝑡≥0 is called a semigroup of pseudocontractions (or pseudocontractive), if there exists a
constant 𝜔 ∈ R such that
∀𝑡 ≥ 0;
(11)
that is, the growth estimate (9) is satisfied with 𝑀 = 1.
If (𝑆𝑡 )𝑡≥0 is a semigroup of pseudocontractions with
growth estimate (11), then (𝑇𝑡 )𝑡≥0 given by
−𝜔𝑡
𝑇𝑡 := 𝑒
𝑆𝑡 ,
𝑡 ≥ 0,
We proceed with some examples of 𝐶0 -semigroups and
their generators.
Example 15. For every bounded linear operator 𝐴 ∈ 𝐿(𝑋) the
family (𝑒𝑡𝐴 )𝑡≥0 given by
∞ 𝑛
(10)
that is, the growth estimate (9) is satisfied with 𝑀 = 1
and 𝜔 = 0.
󵄩󵄩 󵄩󵄩
𝜔𝑡
󵄩󵄩𝑆𝑡 󵄩󵄩 ≤ 𝑒
Proposition 14. The infinitesimal generator 𝐴 : 𝑋 ⊃
D(𝐴) → 𝑋 of a 𝐶0 -semigroup (𝑆𝑡 )𝑡≥0 is densely defined and
closed.
𝑡 𝐴𝑛
𝑛=0 𝑛!
𝑒𝑡𝐴 := ∑
is a norm continuous semigroup with generator 𝐴. In particular, one has D(𝐴) = 𝑋.
Example 16. We consider the separable Hilbert space 𝑋 =
𝐿2 (R). Let (𝑆𝑡 )𝑡≥0 be the shift semigroup that is defined as
𝑆𝑡 𝑓 := 𝑓 (𝑡 + ∙) ,
Lemma 12. Let (𝑆𝑡 )𝑡≥0 be a 𝐶0 -semigroup. Then the following
statements are true.
R+ × 𝑋 󳨀→ 𝑋,
(𝑡, 𝑥) 󳨃󳨀→ 𝑆𝑡 𝑥,
(18)
D (𝐴) = {𝑓 ∈ 𝐿2 (R) : 𝑓 is absolutely continuous
and 𝑓󸀠 ∈ 𝐿2 (R)} ,
(19)
𝐴𝑓 = 𝑓󸀠 .
Example 17. On the separable Hilbert space 𝑋 = 𝐿2 (R𝑑 ) we
define the heat semigroup (𝑆𝑡 )𝑡≥0 by 𝑆0 := Id and
(𝑆𝑡 𝑓) (𝑥) :=
(1) The mapping
𝑡 ≥ 0.
Then (𝑆𝑡 )𝑡≥0 is a semigroup of contractions with generator 𝐴 :
𝐿2 (R) ⊃ D(𝐴) → 𝐿2 (R) given by
(12)
is a semigroup of contractions. Hence, every pseudocontractive semigroup can be transformed into a semigroup of
contractions, which explains the term pseudocontractive.
(17)
1
(4𝜋𝑡)𝑑/2
󵄨2
󵄨󵄨
󵄨𝑥 − 𝑦󵄨󵄨󵄨
) 𝑓 (𝑦) 𝑑𝑦,
∫ exp (− 󵄨
4𝑡
R𝑑
(13)
𝑡 > 0;
is continuous.
(20)
(2) For all 𝑥 ∈ 𝑋 and 𝑇 ≥ 0, the mapping
[0, 𝑇] 󳨀→ 𝑋,
𝑡 󳨃󳨀→ 𝑆𝑡 𝑥,
(14)
is uniformly continuous.
Definition 13. Let (𝑆𝑡 )𝑡≥0 be a 𝐶0 -semigroup. The infinitesimal
generator (in short generator) of (𝑆𝑡 )𝑡≥0 is the linear operator
𝐴 : 𝑋 ⊃ D(𝐴) → 𝑋, which is defined on the domain
D (𝐴) := {𝑥 ∈ 𝑋 : lim
𝑡→0
𝑆𝑡 𝑥 − 𝑥
exists}
𝑡
(15)
𝑆𝑡 𝑥 − 𝑥
.
𝑡→0
𝑡
D (𝐴) = 𝑊2 (R𝑑 ) ,
𝐴𝑓 = Δ𝑓.
(16)
Note that the domain D(𝐴) is indeed a subspace of 𝑋.
The following result gives some properties of the infinitesimal
generator of a 𝐶0 -semigroup. Recall that we have provided the
required concepts in Definitions 1 and 2.
(21)
Here, 𝑊2 (R𝑑 ) denotes the Sobolev space
𝑊2 (R𝑑 ) = {𝑓 ∈ 𝐿2 (R𝑑 ) : 𝐷(𝛼) 𝑓 ∈ 𝐿2 (R𝑑 ) exists
∀𝛼 ∈ N𝑑0 with |𝛼| ≤ 2}
and given by
𝐴𝑥 := lim
that is, 𝑆𝑡 𝑓 arises as the convolution of 𝑓 with the density of
the normal distribution 𝑁(0, 2𝑡). Then (𝑆𝑡 )𝑡≥0 is a semigroup
of contractions with generator 𝐴 : 𝐿2 (R𝑑 ) ⊃ D(𝐴) →
𝐿2 (R𝑑 ) given by
(22)
and Δ the Laplace operator
𝑑
𝜕2
.
2
𝑖=1 𝜕𝑥𝑖
Δ=∑
(23)
We proceed with some results regarding calculations with
strongly continuous semigroups and their generators.
4
International Journal of Stochastic Analysis
Lemma 18. Let (𝑆𝑡 )𝑡≥0 be a 𝐶0 -semigroup with infinitesimal
generator 𝐴. Then the following statements are true.
(1) For every 𝑥 ∈ D(𝐴), the mapping
R+ 󳨀→ 𝑋,
𝑡 󳨃󳨀→ 𝑆𝑡 𝑥,
(24)
belongs to class 𝐶1 (R+ ; 𝑋), and for all 𝑡 ≥ 0, one has
𝑆𝑡 𝑥 ∈ D(𝐴) and
𝑑
𝑆 𝑥 = 𝐴𝑆𝑡 𝑥 = 𝑆𝑡 𝐴𝑥.
𝑑𝑡 𝑡
(25)
𝑡
(2) For all 𝑥 ∈ 𝑋 and 𝑡 ≥ 0, one has ∫0 𝑆𝑠 𝑥 𝑑𝑠 ∈ D(𝐴) and
𝑡
𝐴 (∫ 𝑆𝑠 𝑥 𝑑𝑠) = 𝑆𝑡 𝑥 − 𝑥.
0
(26)
In particular, we obtain the following characterization of
the generators of semigroups of contractions.
Corollary 22. For a linear operator 𝐴 : 𝑋 ⊃ D(𝐴) → 𝑋 the
following statements are equivalent.
(1) 𝐴 is the generator of a semigroup (𝑆𝑡 )𝑡≥0 of contractions.
(2) 𝐴 is densely defined and closed, and one has (0, ∞) ⊂
𝜌(𝐴) and
‖𝑅 (𝜆, 𝐴)‖ ≤
1
𝜆
∀𝜆 ∈ (0, ∞) .
Proposition 23. Let (𝑆𝑡 )𝑡≥0 be a 𝐶0 -semigroup on 𝑋 with
generator 𝐴. Then the family (𝑆𝑡 |D(𝐴) )𝑡≥0 is a 𝐶0 -semigroup
on (D(𝐴), ‖ ⋅ ‖D(𝐴) ) with generator 𝐴 : D(𝐴2 ) ⊂ D(𝐴) →
D(𝐴2 ), where the domain is given by
D (𝐴2 ) = {𝑥 ∈ D (𝐴) : 𝐴𝑥 ∈ D (𝐴)} .
(3) For all 𝑥 ∈ D(𝐴) and 𝑡 ≥ 0, one has
(29)
(30)
(27)
Recall that we have introduced the adjoint operator for
operators in the Hilbert spaces in Definition 6.
The following result shows that the strongly continuous
semigroup (𝑆𝑡 )𝑡≥0 associated with generator 𝐴 is unique. This
explains the term generator.
Proposition 24. Let 𝐻 be a Hilbert space and let (𝑆𝑡 )𝑡≥0
be a 𝐶0 -semigroup on 𝐻 with generator 𝐴. Then the family
of adjoint operators (𝑆𝑡∗ )𝑡≥0 is a 𝐶0 -semigroup on 𝐻 with
generator 𝐴∗ .
Proposition 19. Two 𝐶0 -semigroups (𝑆𝑡 )𝑡≥0 and (𝑇𝑡 )𝑡≥0 with
the same infinitesimal generator 𝐴 coincide; that is, one has
𝑆𝑡 = 𝑇𝑡 for all 𝑡 ≥ 0.
4. Stochastic Processes in Infinite Dimension
𝑡
∫ 𝑆𝑠 𝐴𝑥 𝑑𝑠 = 𝑆𝑡 𝑥 − 𝑥.
0
The next result characterizes all norm continuous semigroups in terms of their generators.
Proposition 20. Let (𝑆𝑡 )𝑡≥0 be a 𝐶0 -semigroup with infinitesimal generator 𝐴. Then the following statements are equivalent.
(1) The semigroup (𝑆𝑡 )𝑡≥0 is norm continuous.
(2) The operator 𝐴 is continuous.
(3) The domain of 𝐴 is given by D(𝐴) = 𝑋.
If the previous conditions are satisfied, then one has 𝑆𝑡 = 𝑒𝑡𝐴
for all 𝑡 ≥ 0.
Now, we are interested in characterizing all linear operators 𝐴 which are the infinitesimal generator of some strongly
continuous semigroup (𝑆𝑡 )𝑡≥0 . The following theorem of
Hille-Yosida gives a characterization in terms of the resolvent,
which we have introduced in Definition 3.
Theorem 21 (Hille-Yosida theorem). Let 𝐴 : 𝑋 ⊃ D(𝐴) →
𝑋 be a linear operator and let 𝑀 ≥ 1, 𝜔 ∈ R be constants. Then
the following statements are equivalent.
(1) 𝐴 is the generator of a 𝐶0 -semigroup (𝑆𝑡 )𝑡≥0 with
growth estimate (9).
(2) 𝐴 is densely defined and closed and one has (𝜔, ∞) ⊂
𝜌(𝐴) and
󵄩󵄩
𝑛󵄩
−𝑛
󵄩󵄩𝑅(𝜆, 𝐴) 󵄩󵄩󵄩 ≤ 𝑀(𝜆 − 𝜔)
∀𝜆 ∈ (𝜔, ∞) , 𝑛 ∈ N.
(28)
In this section, we recall the required foundations about
stochastic processes in infinite dimension. In particular, we
recall the definition of a trace class Wiener process and outline the construction of the Itô integral.
In the sequel, (Ω, F, (F𝑡 )𝑡≥0 , P) denotes a filtered probability space satisfying the usual conditions. Let H be a
separable Hilbert space and let 𝑄 ∈ 𝐿(H) be a nuclear, selfadjoint, positive definite linear operator.
Definition 25. A H-valued, adapted, continuous process 𝑊
is called a 𝑄-Wiener process, if the following conditions are
satisfied.
(i) One has 𝑊0 = 0.
(ii) The random variable 𝑊𝑡 − 𝑊𝑠 and the 𝜎-algebra F𝑠
are independent for all 0 ≤ 𝑠 ≤ 𝑡.
(iii) One has 𝑊𝑡 − 𝑊𝑠 ∼ 𝑁(0, (𝑡 − 𝑠)𝑄) for all 0 ≤ 𝑠 ≤ 𝑡.
In Definition 25, the distribution 𝑁(0, (𝑡 − 𝑠)𝑄) is a
Gaussian measure with mean 0 and covariance operator (𝑡 −
𝑠)𝑄; see, for example, [1, Section 2.3.2]. The operator 𝑄 is also
called the covariance operator of the Wiener process 𝑊. As
𝑄 is a trace class operator, we also call 𝑊 a trace class Wiener
process.
Now, let 𝑊 be a 𝑄-Wiener process. Then, there exist an
orthonormal basis (𝑒𝑗 )𝑗∈N of H and a sequence (𝜆 𝑗 )𝑗∈N ⊂
(0, ∞) with ∑𝑗∈N 𝜆 𝑗 < ∞ such that
𝑄𝑢 = ∑ 𝜆 𝑗 ⟨𝑢, 𝑒𝑗 ⟩H 𝑒𝑗 ,
𝑗∈N
𝑢 ∈ H.
(31)
International Journal of Stochastic Analysis
5
Namely, the 𝜆 𝑗 are the eigenvalues of 𝑄, and each 𝑒𝑗 is an
eigenvector corresponding to 𝜆 𝑗 . The space H0 := 𝑄1/2 (H),
equipped with the inner product
⟨𝑢, V⟩H0 := ⟨𝑄−1/2 𝑢, 𝑄−1/2 V⟩H ,
(32)
is another separable Hilbert space, and (√𝜆 𝑗 𝑒𝑗 )
𝑗∈N
is an
orthonormal basis. According to [1, Proposition 4.1], the
sequence of stochastic processes (𝛽𝑗 )𝑗∈N defined as
𝛽𝑗 :=
1
√𝜆 𝑗
⟨𝑊, 𝑒𝑗 ⟩H ,
𝑗 ∈ N,
Now, let us briefly sketch the construction of the Itô integral
with respect to the Wiener process 𝑊. Further details can be
found in [1, 3]. We denote by 𝐿02 (𝐻) := 𝐿 2 (H0 , 𝐻) the space
of Hilbert-Schmidt operators from H0 into 𝐻, endowed with
the Hilbert-Schmidt norm
1/2
‖Φ‖𝐿02 (𝐻)
󵄩
󵄩2
:= (∑ 𝜆 𝑗 󵄩󵄩󵄩󵄩Φ𝑒𝑗 󵄩󵄩󵄩󵄩 )
,
𝑗∈N
Φ ∈ 𝐿02 (𝐻) ,
(35)
(40)
𝑡
The Itô integral (∫0 𝑋𝑠 𝑑𝑊𝑠 )𝑡≥0 is an 𝐻-valued, continuous,
local martingale, and we have the series expansion
(33)
(34)
𝑗∈N
𝑡
󵄩 󵄩2
P (∫ 󵄩󵄩󵄩Φ𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠 < ∞) = 1 ∀𝑡 ≥ 0.
2
0
𝑡
𝑡
0
𝑗∈N 0
∫ 𝑋𝑠 𝑑𝑊𝑠 = ∑ ∫ 𝑋𝑠𝑗 𝑑𝛽𝑠𝑗 ,
is a sequence of real-valued independent standard Wiener
processes, and one has the expansion
𝑊 = ∑ √𝜆 𝑗 𝛽𝑗 𝑒𝑗 .
𝑡
(3) By localization, we extend the Itô integral ∫0 𝑋𝑠 𝑑𝑊𝑠
for every predictable 𝐿02 (𝐻)-valued process 𝑋 satisfying
𝑡 ≥ 0,
(41)
where 𝑋𝑗 := √𝜆 𝑗 𝑋𝑒𝑗 for each 𝑗 ∈ N. An indispensable tool
for stochastic calculus in infinite dimensions is Itô’s formula,
which we will recall here.
Theorem 26 (Itô’s formula). Let 𝐸 be another separable
Hilbert space, let 𝑓 ∈ 𝐶𝑏1,2,loc (R+ × 𝐻; 𝐸) be a function, and
let 𝑋 be an 𝐻-valued Itô process of the form
𝑡
𝑡
0
0
𝑋𝑡 = 𝑋0 + ∫ 𝑌𝑠 𝑑𝑠 + ∫ 𝑍𝑠 𝑑𝑊𝑠 ,
𝑡 ≥ 0.
(42)
Then (𝑓(𝑡, 𝑋𝑡 ))𝑡≥0 is an 𝐸-valued Itô process, and one has Palmost surely
𝑓 (𝑡, 𝑋𝑡 )
𝑡
which itself is a separable Hilbert space. The construction of
the Itô integral is divided into three steps as follows.
= 𝑓 (0, 𝑋0 ) + ∫ (𝐷𝑠 𝑓 (𝑠, 𝑋𝑠 ) + 𝐷𝑥 𝑓 (𝑠, 𝑋𝑠 ) 𝑌𝑠
0
1
+ ∑ 𝐷𝑥𝑥 𝑓 (𝑠, 𝑋𝑠 ) (𝑍𝑠𝑗 , 𝑍𝑠𝑗 )) 𝑑𝑠
2 𝑗∈N
(1) For every 𝐿(H, 𝐻)-valued simple process of the form
𝑛
𝑋 = 𝑋0 1{0} + ∑𝑋𝑖 1(𝑡𝑖 ,𝑡𝑖+1 ] ,
(36)
𝑖=1
with 0 = 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑛+1 = 𝑇 and F𝑡𝑖 -measurable
random variables 𝑋𝑖 : Ω → 𝐿(H, 𝐻) for 𝑖 = 1, . . . , 𝑛,
we set
𝑡
𝑛
0
𝑖=1
∫ 𝑋𝑠 𝑑𝑊𝑠 := ∑𝑋𝑖 (𝑊𝑡∧𝑡𝑖+1 − 𝑊𝑡∧𝑡𝑖 ) .
(37)
(2) For every predictable 𝐿02 (𝐻)-valued process 𝑋 satisfying
𝑇
󵄩 󵄩2
E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] < ∞,
2
0
(38)
𝑡
we extend the Itô integral ∫0 𝑋𝑠 𝑑𝑊𝑠 by an extension
argument for linear operators. In particular, we obtain
the Itô isometry as follows:
󵄩󵄩2
󵄩󵄩 𝑇
𝑇
󵄩
󵄩
󵄩 󵄩2
E [󵄩󵄩󵄩󵄩∫ 𝑋𝑠 𝑑𝑊𝑠 󵄩󵄩󵄩󵄩 ] = E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] .
2
󵄩󵄩
󵄩󵄩 0
0
(39)
𝑡
+ ∫ 𝐷𝑥 𝑓 (𝑠, 𝑋𝑠 ) 𝑍𝑠 𝑑𝑊𝑠 ,
0
𝑡 ≥ 0,
(43)
where one uses the notation 𝑍𝑗 := √𝜆 𝑗 𝑍𝑒𝑗 for each 𝑗 ∈ N.
Proof. This result is a consequence of [3, Theorem 2.9].
5. Solution Concepts for SPDEs
In this section, we present the concepts of strong, mild, and
weak solutions to SPDEs and discuss their relations.
Let 𝐻 be a separable Hilbert space, and let (𝑆𝑡 )𝑡≥0 be a
𝐶0 -semigroup on 𝐻 with infinitesimal generator 𝐴. Furthermore, let 𝑊 be a trace class Wiener process on some separable
Hilbert space H. We consider the SPDE:
𝑑𝑋𝑡 = (𝐴𝑋𝑡 + 𝛼 (𝑡, 𝑋𝑡 )) 𝑑𝑡 + 𝜎 (𝑡, 𝑋𝑡 ) 𝑑𝑊𝑡
𝑋0 = ℎ0 .
(44)
Here 𝛼 : R+ × 𝐻 → 𝐻 and 𝜎 : R+ × 𝐻 → 𝐿02 (𝐻) are
measurable mappings.
6
International Journal of Stochastic Analysis
Definition 27. Let ℎ0 : Ω → 𝐻 be a F0 -measurable random
variable, and let 𝜏 > 0 be a strictly positive stopping time. Furthermore, let 𝑋 = 𝑋(ℎ0 ) be an 𝐻-valued, continuous, adapted
process such that
P (∫
𝑡∧𝜏
0
󵄩 󵄩 󵄩
󵄩 󵄩
󵄩2
(󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝛼 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩 + 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) ) 𝑑𝑠 < ∞)
2
= 1 ∀𝑡 ≥ 0.
Proof. Let 𝑋 be a local strong solution to (44) with lifetime
𝜏. Furthermore, let 𝜁 ∈ D(𝐴∗ ) be arbitrary. Then we have
P-almost surely for all 𝑡 ≥ 0 the identities
⟨𝜁, 𝑋𝑡∧𝜏 ⟩ = ⟨𝜁, ℎ0 + ∫
𝑡∧𝜏
+∫
𝑡∧𝜏
0
(45)
𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ⟩
= ⟨𝜁, ℎ0 ⟩ + ∫
P (∫
𝑡∧𝜏
0
∀𝑡 ≥ 0, P-almost surely,
󵄩󵄩
󵄩
󵄩󵄩𝐴𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠 < ∞) = 1 ∀𝑡 ≥ 0,
(46)
(47)
𝑡∧𝜏
0
+∫
𝑡∧𝜏
0
𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
0
+∫
𝑡∧𝜏
0
𝑡 ≥ 0.
(⟨𝐴∗ 𝜁, 𝑋𝑠 ⟩ + ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩) 𝑑𝑠
⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠 ,
𝑡 ≥ 0.
(49)
(3) 𝑋 is called a local mild solution to (44), if P-almost
surely one has
𝑋𝑡∧𝜏 = 𝑆𝑡∧𝜏 ℎ0 + ∫
𝑡∧𝜏
0
+∫
𝑡∧𝜏
0
0
⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
= ⟨𝜁, ℎ0 ⟩ + ∫
𝑡∧𝜏
0
𝑡∧𝜏
⟨𝜁, 𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑠
(⟨𝐴∗ 𝜁, 𝑋𝑠 ⟩ + ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩) 𝑑𝑠
⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩𝑑𝑊𝑠 ,
(51)
(𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
(48)
𝑡∧𝜏
𝑡∧𝜏
0
(2) 𝑋 is called a local weak solution to (44), if for all 𝜁 ∈
D(𝐴∗ ) the following equation is fulfilled P-almost
surely:
⟨𝜁, 𝑋𝑡∧𝜏 ⟩ = ⟨𝜁, ℎ0 ⟩ + ∫
+∫
+∫
and P-almost surely one has
𝑋𝑡∧𝜏 = ℎ0 + ∫
𝑡∧𝜏
0
(1) 𝑋 is called a local strong solution to (44), if
𝑋𝑡∧𝜏 ∈ D (𝐴)
(𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
0
Proposition 30. Let 𝑋 be a stochastic process with 𝑋0 = ℎ0 .
Then the following statements are equivalent.
(1) The process 𝑋 is a local strong solution to (44) with
lifetime 𝜏.
(2) The process 𝑋 is a local weak solution to (44) with
lifetime 𝜏, and one has (46), (47).
Proof. (1)⇒(2): This implication is a direct consequence of
Proposition 29.
(2)⇒(1): Let 𝜁 ∈ D(𝐴∗ ) be arbitrary. Then we have Palmost surely for all 𝑡 ≥ 0 the identities
⟨𝜁, 𝑋𝑡∧𝜏 ⟩ = ⟨𝜁, ℎ0 ⟩ + ∫
𝑆(𝑡∧𝜏)−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
𝑆(𝑡∧𝜏)−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
showing that 𝑋 is also a local weak solution to (44) with
lifetime 𝜏.
𝑡∧𝜏
0
(50)
𝑡 ≥ 0.
One calls 𝜏 the lifetime of 𝑋. If 𝜏 ≡ ∞, then one calls 𝑋 a
strong, weak or mild solution to (44), respectively.
Remark 28. Note that the concept of a strong solution is
rather restrictive, because condition (46) has to be fulfilled.
For what follows, we fix a F0 -measurable random variable ℎ0 : Ω → 𝐻 and a strictly positive stopping time 𝜏 > 0.
Proposition 29. Every local strong solution 𝑋 to (44) with
lifetime 𝜏 is also a local weak solution to (44) with lifetime 𝜏.
+∫
𝑡∧𝜏
0
⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
= ⟨𝜁, ℎ0 ⟩ + ∫
𝑡∧𝜏
0
+∫
𝑡∧𝜏
0
𝑡∧𝜏
0
𝑡∧𝜏
0
⟨𝜁, 𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑠
⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
= ⟨𝜁, ℎ0 + ∫
+∫
(⟨𝐴∗ 𝜁, 𝑋𝑠 ⟩ + ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩) 𝑑𝑠
(𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ⟩ .
(52)
International Journal of Stochastic Analysis
7
By Proposition 7, the domain D(𝐴∗ ) is dense in 𝐻, and hence
we obtain P-almost surely
𝑡
= 𝑆𝑡 ℎ0 + ∫ (−𝐴𝑆𝑡−𝑠 𝑋𝑠 + 𝑆𝑡−𝑠 (𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 ))) 𝑑𝑠
0
𝑡
𝑋𝑡∧𝜏 = ℎ0 + ∫
𝑡∧𝜏
0
+∫
𝑡∧𝜏
0
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
(𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
0
(53)
𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
𝑡 ≥ 0.
0
𝑡
Consequently, the process 𝑋 is also a local strong solution to
(44) with lifetime 𝜏.
Corollary 31. Let M ⊂ D(𝐴) be a subset such that 𝐴 is
continuous on M, and let 𝑋 be a local weak solution to (44)
with lifetime 𝜏 such that
𝑋𝑡∧𝜏 ∈ M
∀𝑡 ≥ 0, P-𝑎𝑙𝑚𝑜𝑠𝑡 𝑠𝑢𝑟𝑒𝑙𝑦.
(54)
Then 𝑋 is also a local strong solution to (44) with lifetime 𝜏.
Proof. Since M ⊂ D(𝐴), condition (54) implies that (46) is
fulfilled. Moreover, by the continuity of 𝐴 on M, the sample
paths of the process 𝐴𝑋 are P-almost surely continuous; and
hence, we obtain (47). Consequently, using Proposition 30,
the process 𝑋 is also a local strong solution to (44) with
lifetime 𝜏.
Proposition 32. Every strong solution 𝑋 to (44) is also a mild
solution to (44).
Proof. According to Lemma 8, the domain (D(𝐴), ‖ ⋅ ‖D(𝐴) )
endowed with the graph norm is a separable Hilbert space,
too. Hence, by Lemma 18, for all 𝑡 ≥ 0, the function
𝑓 : [0, 𝑡] × D (𝐴) 󳨀→ 𝐻,
𝑓 (𝑠, 𝑥) := 𝑆𝑡−𝑠 𝑥,
(55)
belongs to the class 𝐶𝑏1,2,loc ([0, 𝑡] × D(𝐴); 𝐻) with partial
derivatives
𝐷𝑡 𝑓 (𝑡, 𝑥) = −𝐴𝑆𝑡−𝑠 𝑥,
𝐷𝑥 𝑓 (𝑡, 𝑥) = 𝑆𝑡−𝑠 ,
Hence, by Itô’s formula (see Theorem 26) and Lemma 18, we
obtain P-almost surely:
𝑋𝑡 = 𝑓 (𝑡, 𝑋𝑡 )
0
(57)
Thus, 𝑋 is also a mild solution to (44).
We recall the following technical auxiliary result without
proof and refer, for example, to [3, Section 3.1].
Lemma 33. Let 𝑇 ≥ 0 be arbitrary. Then the linear space
𝑈𝑇 := lin {𝑔𝜁 : 𝑔 ∈ 𝐶1 ([0, 𝑇] ; R) , 𝜁 ∈ D (𝐴∗ )}
(58)
is dense in 𝐶1 ([0, 𝑇], D(𝐴∗ )), where (D(𝐴∗ ), ‖ ⋅ ‖D(𝐴∗ ) ) is
endowed with the graph norm.
Lemma 34. Let 𝑋 be a weak solution to (44). Then for all 𝑇 ≥
0 and all 𝑓 ∈ 𝐶1 ([0, 𝑇], D(𝐴∗ )), one has P-almost surely
⟨𝑓 (𝑡) , 𝑋𝑡 ⟩
= ⟨𝑓 (0) , ℎ0 ⟩
𝑡
+ ∫ (⟨𝑓󸀠 (𝑠) + 𝐴∗ 𝑓 (𝑠) , 𝑋𝑠 ⟩ + ⟨𝑓 (𝑠) , 𝛼 (𝑠, 𝑋𝑠 )⟩) 𝑑𝑠
0
𝑡
+ ∫ ⟨𝑓 (𝑠) , 𝜎 (𝑠, 𝑋𝑠 )⟩𝑑𝑊𝑠 ,
0
𝑡 ∈ [0, 𝑇] .
(59)
Proof. By virtue of Lemma 33, it suffices to prove formula (59)
for all 𝑓 ∈ 𝑈𝑇 . Let 𝑓 ∈ 𝑈𝑇 be arbitrary. Then there are
𝑔1 , . . . , 𝑔𝑛 ∈ 𝐶1 ([0, 𝑇]; R) and 𝜁1 , . . . , 𝜁𝑛 ∈ D(𝐴∗ ) for some
𝑛 ∈ N such that
𝑓 (𝑡) = ∑𝑔𝑖 (𝑡) 𝜁𝑖 ,
𝑡 ∈ [0, 𝑇] .
(60)
𝑖=1
We define the function
𝐹 : [0, 𝑇] × R𝑛 󳨀→ R,
𝑛
𝐹 (𝑡, 𝑥) := ∑𝑔𝑖 (𝑡) 𝑥𝑖 .
(61)
𝑖=1
Then, we have 𝐹 ∈ 𝐶1,2 ([0, 𝑇]×R𝑛 ; R) with partial derivatives
𝑡
= 𝑓 (0, ℎ0 ) + ∫ (𝐷𝑠 𝑓 (𝑠, 𝑋𝑠 )
0
+𝐷𝑥 𝑓 (𝑠, 𝑋𝑠 ) (𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 ))) 𝑑𝑠
+ ∫ 𝐷𝑥 𝑓 (𝑠, 𝑋𝑠 ) 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
0
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 .
𝑛
(56)
𝐷𝑥𝑥 𝑓 (𝑡, 𝑥) = 0.
𝑡
𝑡
= 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
𝑛
𝐷𝑡 𝐹 (𝑡, 𝑥) = ∑𝑔𝑖󸀠 (𝑡) 𝑥𝑖 ,
𝑖=1
𝐷𝑥 𝐹 (𝑡, 𝑥) = ⟨𝑔 (𝑡) , ∙⟩R𝑛 ,
𝐷𝑥𝑥 𝐹 (𝑡, 𝑥) = 0.
(62)
8
International Journal of Stochastic Analysis
Since 𝑋 is a weak solution to (44), the R𝑛 -valued process
⟨𝜁, 𝑋⟩ := ⟨𝜁𝑖 , 𝑋⟩𝑖=1,...,𝑛 ,
𝑡
= ⟨𝑓 (0) , ℎ0 ⟩ + ∫ (⟨𝑓󸀠 (𝑠) + 𝐴∗ 𝑓 (𝑠) , 𝑋𝑠 ⟩
0
(63)
+ ⟨𝑓 (𝑠) , 𝛼 (𝑠, 𝑋𝑠 )⟩ ) 𝑑𝑠
is an Itô process with representation
𝑡
𝑡
+ ∫ ⟨𝑓 (𝑠) , 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠 ,
⟨𝜁, 𝑋𝑡 ⟩ = ⟨𝜁, ℎ0 ⟩ + ∫ (⟨𝐴∗ 𝜁, 𝑋𝑠 ⟩ + ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩) 𝑑𝑠
0
0
𝑡 ∈ [0, 𝑇] .
(64)
𝑡
+ ∫ ⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠 ,
0
(66)
𝑡 ≥ 0.
By Itô’s formula (Theorem 26), we obtain P-almost surely
𝑛
𝑛
𝑖=1
𝑖=1
This concludes the proof.
Proposition 35. Every weak solution 𝑋 to (44) is also a mild
solution to (44).
⟨𝑓 (𝑡) , 𝑋𝑡 ⟩ = ⟨∑𝑔𝑖 (𝑡) 𝜁𝑖 , 𝑋𝑡 ⟩ = ∑𝑔𝑖 (𝑡) ⟨𝜁𝑖 , 𝑋𝑡 ⟩
= 𝐹 (𝑡, ⟨𝜁, 𝑋𝑡 ⟩) = 𝐹 (0, ⟨𝜁, ℎ0 ⟩)
𝑡
+ ∫ (𝐷𝑠 𝐹 (𝑠, ⟨𝜁, 𝑋𝑠 ⟩) + 𝐷𝑥 𝐹 (𝑠, ⟨𝜁, 𝑋𝑠 ⟩)
0
× (⟨𝐴∗ 𝜁, 𝑋𝑠 ⟩ + ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩)) 𝑑𝑠
𝑡
Proof. By Proposition 24, the family (𝑆𝑡∗ )𝑡≥0 is a 𝐶0 -semigroup with generator 𝐴∗ . Thus, Proposition 23 yields that
the family of restrictions (𝑆𝑡∗ |D(𝐴∗ ) )𝑡≥0 is a 𝐶0 -semigroup
on (D(𝐴∗ ), ‖ ⋅ ‖D(𝐴∗ ) ) with generator 𝐴∗ : D((𝐴∗ )2 ) ⊂
D(𝐴∗ ) → D((𝐴∗ )2 ).
Now, let 𝑡 ≥ 0 and 𝜁 ∈ D((𝐴∗ )2 ) be arbitrary. We define
the function
+ ∫ 𝐷𝑥 𝐹 (𝑠, ⟨𝜁, 𝑋𝑠 ⟩) ⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
0
𝑓 : [0, 𝑡] 󳨀→ D (𝐴∗ ) ,
𝑛
= ∑𝑔𝑖 (0) ⟨𝜁𝑖 , ℎ0 ⟩
𝑖=1
𝑡
𝑛
0
𝑖=1
∗
𝑓󸀠 (𝑠) = −𝐴∗ 𝑆𝑡−𝑠
𝜁 = −𝐴∗ 𝑓 (𝑠) .
𝑛
+ ∑𝑔𝑖 (𝑡) (⟨𝐴∗ 𝜁𝑖 , 𝑋𝑠 ⟩
+ ⟨𝜁𝑖 , 𝛼 (𝑠, 𝑋𝑠 )⟩) ) 𝑑𝑠
0
𝑖=1
(68)
Using Lemma 34, we obtain P-almost surely
𝑖=1
𝑛
(67)
By Lemma 18 we have 𝑓 ∈ 𝐶1 ([0, 𝑡]; D(𝐴∗ )) with derivative
+ ∫ (∑𝑔𝑖󸀠 (𝑡) ⟨𝜁𝑖 , 𝑋𝑠 ⟩
𝑡
∗
𝑓 (𝑠) := 𝑆𝑡−𝑠
𝜁.
⟨𝜁, 𝑋𝑡 ⟩ = ⟨𝑓 (𝑡) , 𝑋𝑡 ⟩
𝑡
= ⟨𝑓 (0) , ℎ0 ⟩ + ∫ ⟨𝑓 (𝑠) , 𝛼 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑠
+ ∫ (∑𝑔𝑖 (𝑠) ⟨𝜁𝑖 , 𝜎 (𝑠, 𝑋𝑠 )⟩) 𝑑𝑊𝑠 ,
0
𝑡
+ ∫ ⟨𝑓 (𝑠) , 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
𝑡 ∈ [0, 𝑇] ,
(65)
0
𝑡
∗
= ⟨𝑆𝑡∗ 𝜁, ℎ0 ⟩ + ∫ ⟨𝑆𝑡−𝑠
𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑠
and hence
0
⟨𝑓 (𝑡) , 𝑋𝑡 ⟩
𝑡
∗
𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
+ ∫ ⟨𝑆𝑡−𝑠
𝑛
0
= ⟨∑𝑔𝑖 (0) 𝜁𝑖 , ℎ0 ⟩
𝑡
𝑖=1
𝑛
𝑡
𝑛
+ ∫ (⟨∑𝑔𝑖󸀠 (𝑠) 𝜁𝑖 , 𝑋𝑠 ⟩ + ⟨𝐴∗ (∑𝑔𝑖 (𝑠) 𝜁𝑖 ) , 𝑋𝑠 ⟩
0
𝑖=1
𝑖=1
𝑛
+ ⟨∑𝑔𝑖 (𝑠) 𝜁𝑖 , 𝛼 (𝑠, 𝑋𝑠 )⟩) 𝑑𝑠
𝑖=1
𝑡
𝑛
0
𝑖=1
+ ∫ ⟨∑𝑔𝑖 (𝑠) 𝜁𝑖 , 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
= ⟨𝜁, 𝑆𝑡 ℎ0 ⟩ + ∫ ⟨𝜁, 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑠
0
𝑡
+ ∫ ⟨𝜁, 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
0
𝑡
= ⟨𝜁, 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
0
𝑡
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ⟩ .
0
(69)
International Journal of Stochastic Analysis
9
Since, by Proposition 14, the domain D((𝐴∗ )2 ) is dense in
(D(𝐴∗ ), ‖ ⋅ ‖D(𝐴∗ ) ), we get P-almost surely for all 𝜁 ∈ D(𝐴∗ )
the identity
0
0
𝑆𝑢 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑢)⟩ 𝑑𝑠
𝑡
⟨𝜁, 𝑋𝑡 ⟩ = ⟨𝜁, 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
0
(70)
𝑡
𝑡−𝑠
= ∫ ⟨𝜁, 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑠
𝑡
0
𝑡
= ∫ ⟨𝜁, 𝐴 (∫
𝑡
𝑡
0
0
= ⟨𝜁, ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠⟩ − ∫ ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩𝑑𝑠.
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ⟩ .
0
(74)
Since, by Proposition 14, the domain D(𝐴∗ ) is dense in 𝐻, we
obtain P-almost surely
𝑡
𝑡
0
0
𝑋𝑡 = 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 , (71)
Due to assumption (72), we may use Fubini’s theorem for
stochastic integrals (see [3, Theorem 2.8]), which, together
with Lemma 18, gives us P-almost surely
proving that 𝑋 is a mild solution to (44).
Remark 36. Now, the proof of Proposition 32 is an immediate
consequence of Propositions 29 and 35.
𝑡
𝑠
0
0
∫ ⟨𝐴∗ 𝜁, ∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑊𝑢 ⟩ 𝑑𝑠
𝑡
𝑠
0
0
𝑡
𝑡
0
𝑢
We have just seen that every weak solution to (44) is also a
mild solution. Under the following regularity condition (72),
the converse of this statement holds true as well.
= ⟨𝐴∗ 𝜁, ∫ (∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑊𝑢 ) 𝑑𝑠⟩
Proposition 37. Let 𝑋 be a mild solution to (44) such that
= ⟨𝐴∗ 𝜁, ∫ (∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑠) 𝑑𝑊𝑢 ⟩
𝑇
󵄩
󵄩2
E [∫ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] < ∞ ∀𝑇 ≥ 0.
2
0
(72)
Then 𝑋 is also a weak solution to (44).
Proof. Let 𝑡 ≥ 0 and 𝜁 ∈ D(𝐴∗ ) be arbitrary. Using Lemma 18,
we obtain P-almost surely
𝑡
𝑡
𝑡
0
𝑢
= ∫ ⟨𝐴∗ 𝜁, ∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑠⟩ 𝑑𝑊𝑢
𝑡
𝑡−𝑠
= ∫ ⟨𝐴∗ 𝜁, ∫ 𝑆𝑢 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑢⟩ 𝑑𝑊𝑠
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
0
0
∈D(𝐴)
∫ ⟨𝐴∗ 𝜁, 𝑆𝑠 ℎ0 ⟩ 𝑑𝑠
0
𝑡
𝑡−𝑠
0
0
= ∫ ⟨𝜁, 𝐴 (∫
𝑡
𝑡
= ⟨𝐴∗ 𝜁, ∫ 𝑆𝑠 ℎ0 𝑑𝑠⟩ = ⟨𝜁, 𝐴 (∫ 𝑆𝑠 ℎ0 𝑑𝑠)⟩
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
0
0
(73)
∈D(𝐴)
By Fubini’s theorem for Bochner integrals (see [3, Section 1.1,
page 21]) and Lemma 18. we obtain P-almost surely
𝑠
0
0
∫ ⟨𝐴∗ 𝜁, ∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑢⟩ 𝑑𝑠
𝑡
𝑠
0
0
𝑡
𝑡
0
𝑢
= ⟨𝐴∗ 𝜁, ∫ (∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑢) 𝑑𝑠⟩
∗
= ⟨𝐴 𝜁, ∫ (∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑠) 𝑑𝑢⟩
𝑡
𝑡
0
𝑢
= ∫ ⟨𝐴∗ 𝜁, ∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑠⟩ 𝑑𝑢
𝑡
∗
𝑡
= ∫ ⟨𝜁, 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑋𝑠 )⟩𝑑𝑊𝑠
0
= ⟨𝜁, 𝑆𝑡 ℎ0 − ℎ0 ⟩ = ⟨𝜁, 𝑆𝑡 ℎ0 ⟩ − ⟨𝜁, ℎ0 ⟩.
𝑡
𝑆𝑢 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑢)⟩ 𝑑𝑊𝑠
𝑡
𝑡
0
0
= ⟨𝜁, ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ⟩ − ∫ ⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠 .
(75)
Therefore, and since 𝑋 is a mild solution to (44), we obtain
P-almost surely
⟨𝜁, 𝑋𝑡 ⟩
𝑡
𝑡
0
0
= ⟨𝜁, 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ⟩
𝑡
= ⟨𝜁, 𝑆𝑡 ℎ0 ⟩ + ⟨𝜁, ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠⟩
0
𝑡−𝑠
= ∫ ⟨𝐴 𝜁, ∫ 𝑆𝑢 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑢⟩ 𝑑𝑠
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
0
0
∈D(𝐴)
𝑡
+ ⟨𝜁, ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ⟩
0
10
International Journal of Stochastic Analysis
𝑡
= ⟨𝜁, ℎ0 ⟩ + ∫ ⟨𝐴∗ 𝜁, 𝑆𝑠 ℎ0 ⟩ 𝑑𝑠
0
𝑡
𝑠
0
0
𝑡
𝑠
0
0
+ ∫ ⟨𝐴∗ 𝜁, ∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑊𝑢 ⟩ 𝑑𝑠
𝑡
+ ∫ ⟨𝐴∗ 𝜁, ∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑢⟩ 𝑑𝑠
+ ∫ ⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠 ,
0
(76)
𝑡
+ ∫ ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑠
0
and hence
𝑡
𝑠
𝑠
⟨𝜁, 𝑋𝑡 ⟩ = ⟨𝜁, ℎ0 ⟩ + ∫ ⟨𝐴∗ 𝜁, 𝑆𝑠 ℎ0 + ∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑢 + ∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑊𝑢 ⟩ 𝑑𝑠
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
0
0
0
=𝑋𝑠
𝑡
𝑡
0
0
(77)
+ ∫ ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑠 + ∫ ⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠
𝑡
𝑡
0
0
= ⟨𝜁, ℎ0 ⟩ + ∫ (⟨𝐴∗ 𝜁, 𝑋𝑠 ⟩ + ⟨𝜁, 𝛼 (𝑠, 𝑋𝑠 )⟩) 𝑑𝑠 + ∫ ⟨𝜁, 𝜎 (𝑠, 𝑋𝑠 )⟩ 𝑑𝑊𝑠 .
Consequently, the process 𝑋 is also a weak solution to (44).
𝑡
𝑡
0
𝑢
𝑡
𝑠
0
0
= ∫ (∫ 𝐴𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑠) 𝑑𝑢
Next, we provide conditions which ensure that a mild
solution to (44) is also a strong solution.
= ∫ (∫ 𝐴𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑢) 𝑑𝑠
Proposition 38. Let 𝑋 be a mild solution to (44) such that Palmost surely one has
= ∫ 𝐴 (∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑢) 𝑑𝑠.
𝑋𝑠 , 𝛼 (𝑠, 𝑋𝑠 ) ∈ D (𝐴) ,
𝜎 (𝑠, 𝑋𝑠 ) ∈ 𝐿02 (D (𝐴))
𝑡
𝑠
0
0
(82)
∀𝑠 ≥ 0,
(78)
as well as
𝑡
󵄩 󵄩
󵄩
󵄩
P (∫ (󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩D(𝐴) + 󵄩󵄩󵄩𝛼 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩D(𝐴) ) 𝑑𝑠 < ∞) = 1
0
𝑇
󵄩
󵄩2
E [∫ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (D(𝐴)) 𝑑𝑠] < ∞
2
0
∀𝑇 ≥ 0.
∀𝑡 ≥ 0,
(79)
(80)
𝑡
∫ (𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑋𝑠 )) 𝑑𝑊𝑠
Then 𝑋 is also a strong solution to (44).
0
Proof. By hypotheses (78) and (79), we have (46) and (47).
Let 𝑡 ≥ 0 be arbitrary. By Lemma 18, we have
𝑡
𝑆𝑡 ℎ0 − ℎ0 = ∫ 𝐴𝑆𝑠 ℎ0 𝑑𝑠.
0
(81)
Furthermore, by Lemma 18 and Fubini’s theorem for Bochner
integrals (see [3, Section 1.1, page 21]) we have P-almost surely
𝑡
∫ (𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
0
𝑡
𝑡−𝑠
0
0
= ∫ (∫
𝐴𝑆𝑢 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑢) 𝑑𝑠
Due to assumption (80), we may use Fubini’s theorem for
stochastic integrals (see [3, Theorem 2.8]), which, together
with Lemma 18, gives us P-almost surely
𝑡
𝑡−𝑠
0
0
𝑡
𝑡
0
𝑢
𝑡
𝑠
0
0
= ∫ (∫
𝐴𝑆𝑢 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑢) 𝑑𝑊𝑠
= ∫ (∫ 𝐴𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑠) 𝑑𝑊𝑢
= ∫ (∫ 𝐴𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑊𝑢 ) 𝑑𝑠
𝑡
𝑠
0
0
= ∫ 𝐴 (∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑊𝑢 ) 𝑑𝑠.
(83)
International Journal of Stochastic Analysis
11
Since 𝑋 is a mild solution to (44), we have P-almost surely
𝑡
𝑡
0
0
and, hence, combining the latter identities, we obtain Palmost surely
𝑋𝑡 = 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
𝑡
𝑡
0
𝑡
= ℎ0 + ∫ 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + ∫ 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
0
𝑡
𝑋𝑡 = ℎ0 + ∫ 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + ∫ 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
0
𝑡
0
𝑠
0
0
+ ∫ 𝐴𝑆𝑠 ℎ0 𝑑𝑠 + ∫ 𝐴 (∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑢) 𝑑𝑠 (85)
(84)
𝑡
𝑡
0
+ (𝑆𝑡 ℎ0 − ℎ0 ) + ∫ (𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
0
𝑡
𝑠
0
0
+ ∫ 𝐴 (∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑊𝑢 ) 𝑑𝑠,
𝑡
+ ∫ (𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑋𝑠 )) 𝑑𝑊𝑠 ,
which implies that
0
𝑡
𝑡
0
0
𝑋𝑡 = ℎ0 + ∫ 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + ∫ 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
𝑡
𝑠
𝑠
+ ∫ 𝐴(𝑆𝑠 ℎ0 + ∫ 𝑆𝑠−𝑢 𝛼 (𝑢, 𝑋𝑢 ) 𝑑𝑢 + ∫ 𝑆𝑠−𝑢 𝜎 (𝑢, 𝑋𝑢 ) 𝑑𝑊𝑢 )𝑑𝑠
0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
0
0
(86)
=𝑋𝑠
𝑡
𝑡
0
0
= ℎ0 + ∫ (𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠 + ∫ 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 .
𝑡
This proves that 𝑋 is also a strong solution to (44).
= 𝑒𝑡𝐴 ℎ0 + 𝑒𝑡𝐴 ∫ 𝑒−𝑠𝐴 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
0
𝑡
The following result shows that for norm continuous
semigroups, the concepts of strong, weak, and mild solutions
are equivalent. In particular, this applies for finite dimensional state spaces.
Proposition 39. Suppose that the semigroup (𝑆𝑡 )𝑡≥0 is norm
continuous. Let 𝑋 be a stochastic process with 𝑋0 = ℎ0 . Then
the following statements are equivalent.
(1) The process 𝑋 is a strong solution to (44).
(2) The process 𝑋 is a weak solution to (44).
(3) The process 𝑋 is a mild solution to (44).
Proof. (1)⇒(2): This implication is a consequence of
Proposition 29.
(2)⇒(3): This implication is a consequence of
Proposition 35.
(3)⇒(1): By Proposition 20, we have 𝐴 ∈ 𝐿(𝐻) and 𝑆𝑡 =
𝑒𝑡𝐴 , 𝑡 ≥ 0. Furthermore, the family (𝑒𝑡𝐴)𝑡∈R is a 𝐶0 -group on
𝐻. Therefore, and since 𝑋 is a mild solution to (44), we have
P-almost surely
𝑡 ≥ 0.
(87)
Let 𝑌 be the Itô process:
𝑡
𝑡
0
0
𝑌𝑡 := ∫ 𝑒−𝑠𝐴 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + ∫ 𝑒−𝑠𝐴 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
𝑡 ≥ 0.
(88)
Then, we have P-almost surely
𝑋𝑡 = 𝑒𝑡𝐴 (ℎ0 + 𝑌𝑡 ) ,
𝑡 ≥ 0,
(89)
𝑒𝑡𝐴ℎ0 − ℎ0 = ∫ 𝐴𝑒𝑠𝐴 ℎ0 𝑑𝑠.
(90)
and, by Lemma 18, we have
𝑡
0
Defining the function
𝑓 : R+ × 𝐻 󳨀→ 𝐻,
𝑓 (𝑠, 𝑦) := 𝑒𝑠𝐴𝑦,
(91)
by Lemma 18, we have 𝑓 ∈ 𝐶𝑏1,2,loc (R+ × 𝐻; 𝐻) with partial
derivatives
𝐷𝑠 𝑓 (𝑠, 𝑦) = 𝐴𝑒𝑠𝐴 𝑦,
0
𝐷𝑦 𝑓 (𝑠, 𝑦) = 𝑒𝑠𝐴 ,
+ ∫ 𝑒(𝑡−𝑠)𝐴 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
0
0
𝑡
𝑋𝑡 = 𝑒𝑡𝐴 ℎ0 + ∫ 𝑒(𝑡−𝑠)𝐴 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
𝑡
+ 𝑒𝑡𝐴 ∫ 𝑒−𝑠𝐴 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
𝐷𝑦𝑦 𝑓 (𝑠, 𝑦) = 0.
(92)
12
International Journal of Stochastic Analysis
By Itô’s formula (Theorem 26), we get P-almost surely
Proof. Let 𝑡 ∈ R+ be arbitrary. It suffices to prove that 𝐹 is
right-continuous and left-continuous in 𝑡.
𝑒𝑡𝐴𝑌𝑡 = 𝑓 (𝑡, 𝑌𝑡 )
= 𝑓 (0, 0)
𝑡
+ ∫ (𝐷𝑠 𝑓 (𝑠, 𝑌𝑠 ) + 𝐷𝑦 𝑓 (𝑠, 𝑌𝑠 ) 𝑒−𝑠𝐴 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
0
𝑡
+ ∫ 𝐷𝑦 𝑓 (𝑠, 𝑌𝑠 ) 𝑒−𝑠𝐴 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
0
𝑡
𝑡
0
0
= ∫ (𝐴𝑒𝑠𝐴 𝑌𝑠 + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠 + ∫ 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 .
(93)
Combining the previous identities, we obtain P-almost surely
𝑋𝑡 = 𝑒𝑡𝐴 (ℎ0 + 𝑌𝑡 )
= ℎ0 + (𝑒𝑡𝐴 ℎ0 − ℎ0 ) + 𝑒𝑡𝐴 𝑌𝑡
𝑡
𝑡
0
0
+ ∫ 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
0
(94)
𝑡
𝑠𝐴
= ℎ0 + ∫ (𝐴𝑒⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
(ℎ0 + 𝑌𝑠 ) + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
(97)
𝑡
0
𝑡
𝑡
0
0
= ℎ0 + ∫ (𝐴𝑋𝑠 + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠 + ∫ 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
𝑡 ≥ 0,
proving that 𝑋 is a strong solution to (44).
(98)
is continuous. Thus, taking into account estimate (9)
from Lemma 10, by Lebesgue’s dominated convergence theorem we obtain
󵄩
󵄩󵄩
(99)
󵄩󵄩𝐹 (𝑡) − 𝐹 (𝑡𝑛 )󵄩󵄩󵄩 󳨀→ 0 for 𝑛 󳨀→ ∞.
𝑡𝑛
𝑡
󵄩
󵄩
󵄩
󵄩
≤ ∫ 󵄩󵄩󵄩󵄩𝑆𝑡−𝑠 𝑓 (𝑠) − 𝑆𝑡𝑛 −𝑠 𝑓 (𝑠)󵄩󵄩󵄩󵄩 𝑑𝑠 + ∫ 󵄩󵄩󵄩𝑆𝑡−𝑠 𝑓 (𝑠)󵄩󵄩󵄩 𝑑𝑠.
𝑡
0
𝑛
6. Stochastic Convolution Integrals
(100)
In this section, we deal with the regularity of stochastic
convolution integrals, which occur when dealing with mild
solutions to SPDEs of the type (44).
Let 𝐸 be a separable Banach space, and let (𝑆𝑡 )𝑡≥0 be a 𝐶0 semigroup on 𝐸. We start with the drift term.
Lemma 40. Let 𝑓 : R+ → 𝐸 be a measurable mapping such
that
𝑡
󵄩
󵄩
∫ 󵄩󵄩󵄩𝑓 (𝑠)󵄩󵄩󵄩 𝑑𝑠 < ∞ ∀𝑡 ≥ 0.
0
(𝑢, 𝑥) 󳨃󳨀→ 𝑆𝑢 𝑥,
(2) Let (𝑡𝑛 )𝑛∈N ⊂ R+ be a sequence such that 𝑡𝑛 ↑ 𝑡. Then
for every 𝑛 ∈ N we have
󵄩󵄩
󵄩
󵄩󵄩𝐹 (𝑡) − 𝐹 (𝑡𝑛 )󵄩󵄩󵄩
𝑡𝑛
󵄩󵄩󵄩 𝑡
󵄩󵄩󵄩
= 󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝑓 (𝑠) 𝑑𝑠 − ∫ 𝑆𝑡𝑛 −𝑠 𝑓 (𝑠) 𝑑𝑠󵄩󵄩󵄩
0
󵄩󵄩 0
󵄩󵄩
󵄩󵄩 𝑡𝑛
󵄩󵄩
𝑡
𝑡𝑛
󵄩
󵄩
= 󵄩󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝑓 (𝑠) 𝑑𝑠 − ∫ 𝑆𝑡−𝑠 𝑓 (𝑠) 𝑑𝑠 − ∫ 𝑆𝑡𝑛 −𝑠 𝑓 (𝑠) 𝑑𝑠󵄩󵄩󵄩󵄩
󵄩󵄩 0
󵄩󵄩
𝑡𝑛
0
=𝑋𝑠
+ ∫ 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠
(95)
Then the mapping
is continuous.
𝑡𝑛
󵄩
󵄩
+ ∫ 󵄩󵄩󵄩󵄩𝑆𝑡𝑛 −𝑠 𝑓 (𝑠)󵄩󵄩󵄩󵄩 𝑑𝑠.
𝑡
R+ × 𝐸 󳨀→ 𝐸,
𝑡
𝐹 : R+ 󳨀→ 𝐸,
𝑡
󵄩
󵄩
≤ ∫ 󵄩󵄩󵄩󵄩𝑆𝑡−𝑠 𝑓 (𝑠) − 𝑆𝑡𝑛 −𝑠 𝑓 (𝑠)󵄩󵄩󵄩󵄩 𝑑𝑠
0
By Lemma 12, the mapping
= ℎ0 + ∫ 𝐴𝑒𝑠𝐴 ℎ0 𝑑𝑠 + ∫ (𝐴𝑒𝑠𝐴 𝑌𝑠 + 𝛼 (𝑠, 𝑋𝑠 )) 𝑑𝑠
0
(1) Let (𝑡𝑛 )𝑛∈N ⊂ R+ be a sequence such that 𝑡𝑛 ↓ 𝑡. Then
for every 𝑛 ∈ N we have
󵄩󵄩
󵄩
󵄩󵄩𝐹 (𝑡) − 𝐹 (𝑡𝑛 )󵄩󵄩󵄩
󵄩󵄩 𝑡
󵄩󵄩
𝑡𝑛
󵄩
󵄩
= 󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝑓 (𝑠) 𝑑𝑠 − ∫ 𝑆𝑡𝑛 −𝑠 𝑓 (𝑠) 𝑑𝑠󵄩󵄩󵄩
0
󵄩󵄩 0
󵄩󵄩
󵄩󵄩 𝑡
󵄩󵄩
𝑡
𝑡𝑛
󵄩
󵄩
= 󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝑓 (𝑠) 𝑑𝑠 − ∫ 𝑆𝑡𝑛 −𝑠 𝑓 (𝑠) 𝑑𝑠 − ∫ 𝑆𝑡𝑛 −𝑠 𝑓 (𝑠) 𝑑𝑠󵄩󵄩󵄩
0
𝑡
󵄩󵄩 0
󵄩󵄩
Proceeding as in the previous situation, by Lebesgue’s
dominated convergence theorem we obtain
󵄩󵄩
󵄩
(101)
󵄩󵄩𝐹 (𝑡) − 𝐹 (𝑡𝑛 )󵄩󵄩󵄩 󳨀→ 0 for 𝑛 󳨀→ ∞.
This completes the proof.
Proposition 41. Let 𝑋 be a progressively measurable process
satisfying
𝑡
󵄩 󵄩
P (∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠 < ∞) = 1 ∀𝑡 ≥ 0.
0
(102)
Then the process 𝑌 defined as
𝑡
𝐹 (𝑡) := ∫ 𝑆𝑡−𝑠 𝑓 (𝑠) 𝑑𝑠,
0
(96)
𝑡
𝑌𝑡 := ∫ 𝑆𝑡−𝑠 𝑋𝑠 𝑑𝑠,
0
is continuous and adapted.
𝑡 ≥ 0,
(103)
International Journal of Stochastic Analysis
13
Proof. The continuity of 𝑌 is a consequence of Lemma 40.
Moreover, 𝑌 is adapted, because 𝑋 is progressively measurable.
The following result is the well-known Banach fixed point
theorem. Its proof can be found, for example, in [13, Theorem
3.48].
Now, we will deal with stochastic convolution integrals
driven by the Wiener processes. Let 𝐻 be a separable Hilbert
space, and let (𝑆𝑡 )𝑡≥0 be a 𝐶0 -semigroup on 𝐻. Moreover, let
𝑊 be a trace class Wiener process on some separable Hilbert
space H.
Theorem 45 (The Banach fixed point theorem). Let 𝐸 be a
complete metric space, and let Φ : 𝐸 → 𝐸 be a contraction.
Then the mapping Φ has a unique fixed point.
Definition 42. Let 𝑋 be a 𝐿02 (𝐻)-valued predictable process
such that
𝑡
󵄩 󵄩2
P (∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠 < ∞) = 1
2
0
∀𝑡 ≥ 0.
(104)
One defines the stochastic convolution 𝑋 ⋆ 𝑊 as
𝑡
(𝑋 ⋆ 𝑊)𝑡 := ∫ 𝑆𝑡−𝑠 𝑋𝑠 𝑑𝑊𝑠 ,
0
𝑡 ≥ 0.
(105)
One recalls the following result concerning the regularity
of stochastic convolutions.
𝐿02 (𝐻)-valued
predictable process
Proposition 43. Let 𝑋 be a
such that one of the following two conditions is satisfied.
(1) There exists a constant 𝑝 > 1 such that
𝑡
󵄩 󵄩2𝑝
E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] < ∞ ∀𝑡 ≥ 0.
2
0
(106)
(2) The semigroup (𝑆𝑡 )𝑡≥0 is a semigroup of pseudocontractions, and one has
𝑡
󵄩 󵄩2
E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] < ∞ ∀𝑡 ≥ 0.
2
0
(107)
Then the stochastic convolution 𝑋 ⋆ 𝑊 has a continuous
version.
Proof. See [3, Lemma 3.3].
7. Existence and Uniqueness Results for SPDEs
In this section, we will present results concerning existence
and uniqueness of solutions to the SPDE (44).
First, we recall the Banach fixed point theorem, which will
be a basic result for proving the existence of mild solutions to
(44).
Definition 44. Let (𝐸, 𝑑) be a metric space, and let Φ : 𝐸 → 𝐸
be a mapping.
(1) The mapping Φ is called a contraction, if for some constant 0 ≤ 𝐿 < 1 one has
𝑑 (Φ (𝑥) , Φ (𝑦)) ≤ 𝐿 ⋅ 𝑑 (𝑥, 𝑦)
∀𝑥, 𝑦 ∈ 𝐸.
(108)
(2) An element 𝑥 ∈ 𝐸 is called a fixed point of Φ, if one
has
Φ (𝑥) = 𝑥.
(109)
In this text, we will use the following slight extension of
the Banach fixed point theorem.
Corollary 46. Let 𝐸 be a complete metric space, and let Φ :
𝐸 → 𝐸 be a mapping such that for some 𝑛 ∈ N the mapping
Φ𝑛 is a contraction. Then the mapping Φ has a unique fixed
point.
Proof. According to the Banach fixed point theorem
(Theorem 45) the mapping Φ𝑛 has a unique fixed point; that
is, there exists a unique element 𝑥 ∈ 𝐸 such that Φ𝑛 (𝑥) = 𝑥.
Therefore, we have
Φ (𝑥) = Φ (Φ𝑛 (𝑥)) = Φ𝑛 (Φ (𝑥)) ,
(110)
showing that Φ(𝑥) is a fixed point of Φ𝑛 . Since Φ𝑛 has a unique
fixed point, we deduce that Φ(𝑥) = 𝑥, showing that 𝑥 is a fixed
point of Φ.
In order to prove uniqueness, let 𝑦 ∈ 𝐸 be another fixed
point of Φ; that is, we have Φ(𝑦) = 𝑦. By induction, we obtain
Φ𝑛 (𝑦) = Φ𝑛−1 (Φ (𝑦)) = Φ𝑛−1 (𝑦) = ⋅ ⋅ ⋅ = Φ (𝑦) = 𝑦, (111)
showing that 𝑦 is a fixed point of Φ𝑛 . Since the mapping Φ𝑛
has exactly one fixed point, we obtain 𝑥 = 𝑦.
An indispensable tool for proving uniqueness of mild
solutions to (44) will be the following version of Gronwall’s
inequality; see, for example, [14, Theorem 5.1].
Lemma 47 (Gronwall’s inequality). Let 𝑇 ≥ 0 be fixed, let
𝑓 : [0, 𝑇] → R+ be a nonnegative continuous mapping, and
let 𝛽 ≥ 0 be a constant such that
𝑡
𝑓 (𝑡) ≤ 𝛽 ∫ 𝑓 (𝑠) 𝑑𝑠 ∀𝑡 ∈ [0, 𝑇] .
0
(112)
Then one has 𝑓 ≡ 0.
The following result shows that local Lipschitz continuity
of 𝛼 and 𝜎 ensures the uniqueness of mild solutions to the
SPDE (44).
Theorem 48. One supposes that for every 𝑛 ∈ N there exists a
constant 𝐿 𝑛 ≥ 0 such that
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝛼 (𝑡, ℎ1 ) − 𝛼 (𝑡, ℎ2 )󵄩󵄩󵄩 ≤ 𝐿 𝑛 󵄩󵄩󵄩ℎ1 − ℎ2 󵄩󵄩󵄩 ,
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝜎 (𝑡, ℎ1 ) − 𝜎 (𝑡, ℎ2 )󵄩󵄩󵄩𝐿0 (𝐻) ≤ 𝐿 𝑛 󵄩󵄩󵄩ℎ1 − ℎ2 󵄩󵄩󵄩 ,
2
(113)
for all 𝑡 ≥ 0 and all ℎ1 , ℎ2 ∈ 𝐻 with ‖ℎ1 ‖, ‖ℎ2 ‖ ≤ 𝑛. Let ℎ0 , 𝑔0 :
Ω → 𝐻 be two F0 -measurable random variables, let 𝜏 > 0 be
a strictly positive stopping time, and let 𝑋, 𝑌 be two local mild
14
International Journal of Stochastic Analysis
solutions to (44) with initial conditions ℎ0 , 𝑔0 and lifetime 𝜏.
Then one has up to indistinguishability
𝑋𝜏 1{ℎ0 =𝑔0 } = 𝑌𝜏 1{ℎ0 =𝑔0 }
(114)
and hence, by the Cauchy-Schwarz inequality, the Itô isometry (39), the growth estimate (9) from Lemma 10, and the
local Lipschitz conditions (113) we obtain
𝑓 (𝑡)
≤ 3𝑇E [∫
(Two processes 𝑋 and 𝑌 are called indistinguishable if the set
{𝜔 ∈ Ω : 𝑋𝑡 (𝜔) ≠ 𝑌𝑡 (𝜔) for some 𝑡 ∈ R+ } is a P-nullset.)
𝑡∧𝜏𝑛
󵄩󵄩
󵄩2
󵄩󵄩1Γ 𝑆(𝑡∧𝜏𝑛 )−𝑠 (𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑌𝑠 ))󵄩󵄩󵄩 𝑑𝑠]
󵄩
󵄩
0
𝑡∧𝜏𝑛
+ 3E [∫
0
Proof. Defining the stopping times (𝜏𝑛 )𝑛∈N as
󵄩󵄩
󵄩2
󵄩󵄩1Γ 𝑆(𝑡∧𝜏𝑛 )−𝑠 (𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑌𝑠 ))󵄩󵄩󵄩 0 𝑑𝑠]
󵄩
󵄩𝐿 2 (𝐻)
2
≤ 3𝑇(𝑀𝑒𝜔𝑇 ) E [∫
󵄩 󵄩
󵄩 󵄩
𝜏𝑛 := 𝜏 ∧ inf {𝑡 ≥ 0 : 󵄩󵄩󵄩𝑋𝑡 󵄩󵄩󵄩 ≥ 𝑛} ∧ inf {𝑡 ≥ 0 : 󵄩󵄩󵄩𝑌𝑡 󵄩󵄩󵄩 ≥ 𝑛} ,
(115)
we have P(𝜏𝑛 → 𝜏) = 1. Let 𝑛 ∈ N and 𝑇 ≥ 0 be arbitrary,
and set
𝑡∧𝜏𝑛
0
2
+ 3(𝑀𝑒𝜔𝑇 ) E [∫
𝑡∧𝜏𝑛
0
󵄩
󵄩2
1Γ 󵄩󵄩󵄩𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑌𝑠 )󵄩󵄩󵄩 𝑑𝑠]
󵄩
󵄩2
1Γ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑌𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠]
2
𝑡
2
󵄩
󵄩2
≤ 3 (𝑇 + 1) (𝑀𝑒𝜔𝑇 ) 𝐿2𝑛 ∫ E [1Γ 󵄩󵄩󵄩󵄩𝑋𝑠∧𝜏𝑛 − 𝑌𝑠∧𝜏𝑛 󵄩󵄩󵄩󵄩 ] 𝑑𝑠
0
2
𝑡
= 3 (𝑇 + 1) (𝑀𝑒𝜔𝑇 ) 𝐿2𝑛 ∫ 𝑓 (𝑠) 𝑑𝑠.
0
Γ := {ℎ0 = 𝑔0 } ∈ F0 .
(116)
(119)
Using Gronwall’s inequality (see Lemma 47) we deduce that
𝑓 ≡ 0. Thus, by the continuity of the sample paths of 𝑋 and
𝑌, we obtain
The mapping
󵄩
󵄩2
𝑓 (𝑡) := E [1Γ 󵄩󵄩󵄩󵄩𝑋𝑡∧𝜏𝑛 − 𝑌𝑡∧𝜏𝑛 󵄩󵄩󵄩󵄩 ] ,
𝑓 : [0, 𝑇] 󳨀→ R,
P (⋂ {𝑋𝑡∧𝜏𝑛 1Γ = 𝑌𝑡∧𝜏𝑛 1Γ }) = 1 ∀𝑛 ∈ N,
(120)
𝑡≥0
(117)
is nonnegative, and it is continuous by Lebesgue’s dominated
convergence theorem. For all 𝑡 ∈ [0, 𝑇] we have
and hence, by the continuity of the probability measure P, we
conclude that
P (⋂ {𝑋𝑡∧𝜏 1Γ = 𝑌𝑡∧𝜏 1Γ })
𝑡≥0
󵄩
󵄩2
𝑓 (𝑡) = E [1Γ 󵄩󵄩󵄩󵄩𝑋𝑡∧𝜏𝑛 − 𝑌𝑡∧𝜏𝑛 󵄩󵄩󵄩󵄩 ]
󵄩
󵄩2
≤ 3E [1Γ 󵄩󵄩󵄩󵄩𝑆𝑡∧𝜏𝑛 (ℎ0 − 𝑔0 )󵄩󵄩󵄩󵄩 ]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
=0
󵄩󵄩2
󵄩󵄩 𝑡∧𝜏𝑛
󵄩
󵄩
+ 3E [1Γ 󵄩󵄩󵄩∫
𝑆(𝑡∧𝜏𝑛 )−𝑠 (𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑌𝑠 )) 𝑑𝑠󵄩󵄩󵄩 ]
󵄩󵄩
󵄩󵄩 0
󵄩󵄩 𝑡∧𝜏𝑛
󵄩󵄩2
󵄩
󵄩
+ 3E [1Γ 󵄩󵄩󵄩∫
𝑆(𝑡∧𝜏𝑛 )−𝑠 (𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑌𝑠 )) 𝑑𝑊𝑠 󵄩󵄩󵄩 ]
󵄩󵄩 0
󵄩󵄩
󵄩󵄩 𝑡∧𝜏𝑛
󵄩󵄩2
󵄩
󵄩
= 3E [󵄩󵄩󵄩∫
1Γ 𝑆(𝑡∧𝜏𝑛 )−𝑠 (𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑌𝑠 )) 𝑑𝑠󵄩󵄩󵄩 ]
󵄩󵄩 0
󵄩󵄩
󵄩󵄩 𝑡∧𝜏𝑛
󵄩󵄩2
󵄩
󵄩
+ 3E [󵄩󵄩󵄩∫
1Γ 𝑆(𝑡∧𝜏𝑛 )−𝑠 (𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑌𝑠 )) 𝑑𝑊𝑠 󵄩󵄩󵄩 ] ,
󵄩󵄩 0
󵄩󵄩
(118)
= P (⋂ ⋂ {𝑋𝑡∧𝜏𝑛 1Γ = 𝑌𝑡∧𝜏𝑛 1Γ })
(121)
𝑛∈N 𝑡≥0
= lim P (⋂ {𝑋𝑡∧𝜏𝑛 1Γ = 𝑌𝑡∧𝜏𝑛 1Γ }) = 1,
𝑛→∞
𝑡≥0
which completes the proof.
The local Lipschitz conditions (113) are, in general, not
sufficient in order to ensure the existence of mild solutions to
the SPDE (44). Now, we will prove that the existence of mild
solutions follows from global Lipschitz and linear growth
conditions on 𝛼 and 𝜎. For this, we recall an auxiliary result
which extends the Itô isometry (39).
Lemma 49. Let 𝑇 ≥ 0 be arbitrary, and let 𝑋 = (𝑋𝑡 )𝑡∈[0,𝑇] be
a 𝐿02 (𝐻)-valued, predictable process such that
𝑇
󵄩 󵄩2
E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] < ∞.
2
0
(122)
International Journal of Stochastic Analysis
15
Then, for every 𝑝 ≥ 1 one has
𝑝
󵄩󵄩 𝑇
󵄩󵄩2𝑝
𝑇
󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩2
󵄩
󵄩
E [󵄩󵄩∫ 𝑋𝑠 𝑑𝑊𝑠 󵄩󵄩 ] ≤ 𝐶𝑝 E[∫ 󵄩󵄩𝑋𝑠 󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] ,
2
󵄩󵄩 0
󵄩󵄩
0
(123)
𝑝
𝑇
󵄩
󵄩
E [∫ 󵄩󵄩󵄩𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩 𝑑𝑠]
0
where the constant 𝐶𝑝 > 0 is given by
𝐶𝑝 = (𝑝 (2𝑝 − 1)) (
Then the process Φ𝑋 is well defined. Indeed, by the growth
estimate (9), the linear growth condition (127), and Hölder’s
inequality we have
𝑇
2𝑝2
2𝑝
)
2𝑝 − 1
.
(124)
󵄩
󵄩
≤ 𝑀𝑒𝜔𝑇 E [∫ 󵄩󵄩󵄩𝛼 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩 𝑑𝑠]
0
𝑇
Proof. See [3, Lemma 3.1].
󵄩 󵄩
≤ 𝑀𝑒𝜔𝑇 𝐾E [∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩) 𝑑𝑠]
Theorem 50. Suppose that there exists a constant 𝐿 ≥ 0 such
that
󵄩 󵄩
= 𝑀𝑒𝜔𝑇 𝐾 (𝑇 + E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠])
0
0
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝛼 (𝑡, ℎ1 ) − 𝛼 (𝑡, ℎ2 )󵄩󵄩󵄩 ≤ 𝐿 󵄩󵄩󵄩ℎ1 − ℎ2 󵄩󵄩󵄩 ,
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝜎 (𝑡, ℎ1 ) − 𝜎 (𝑡, ℎ2 )󵄩󵄩󵄩𝐿0 (𝐻) ≤ 𝐿 󵄩󵄩󵄩ℎ1 − ℎ2 󵄩󵄩󵄩 ,
2
(125)
≤ 𝑀𝑒𝜔𝑇 𝐾 (𝑇 + 𝑇1−1/2𝑝
(126)
𝑇
󵄩 󵄩2𝑝
× E[∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠]
0
for all 𝑡 ≥ 0 and all ℎ1 , ℎ2 ∈ 𝐻, and suppose that there exists a
constant 𝐾 ≥ 0 such that
‖𝛼 (𝑡, ℎ)‖ ≤ 𝐾 (1 + ‖ℎ‖) ,
(127)
‖𝜎 (𝑡, ℎ)‖𝐿02 (𝐻) ≤ 𝐾 (1 + ‖ℎ‖) ,
(128)
for all 𝑡 ≥ 0 and all ℎ ∈ 𝐻. Then, for every F0 -measurable
random variable ℎ0 : Ω → 𝐻, there exists a (up to
indistinguishability) unique mild solution 𝑋 to (44).
Proof. The uniqueness of mild solutions to (44) is a direct
consequence of Theorem 48, and hence, we may concentrate
on the existence proof, which we divide into the following
several steps.
Step 1. First, we suppose that the initial condition ℎ0 satisfies
E[‖ℎ0 ‖2𝑝 ] < ∞ for some 𝑝 > 1. Let 𝑇 ≥ 0 be arbitrary. We
define the Banach space
2𝑝
𝐿 𝑇 (𝐻) := 𝐿2𝑝 (Ω × [0, 𝑇] , P𝑇 , P ⊗ 𝑑𝑡; 𝐻) ,
(129)
𝑋𝑡 = 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
(130)
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
0
𝑇
󵄩
󵄩2
E [∫ 󵄩󵄩󵄩𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠]
2
0
𝑇
2
󵄩
󵄩2
≤ (𝑀𝑒𝜔𝑇 ) E [∫ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠]
2
0
𝑇
2
󵄩 󵄩2
≤ (𝑀𝑒𝜔𝑇 𝐾) E [∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩) 𝑑𝑠]
0
𝑇
2
󵄩 󵄩2
≤ 2(𝑀𝑒𝜔𝑇 𝐾) E [∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 ) 𝑑𝑠]
0
𝑇
2
󵄩 󵄩2
= 2(𝑀𝑒𝜔𝑇 𝐾) (𝑇 + E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠])
0
2
𝜔𝑇
2𝑝
Step 1.1. For 𝑋 ∈
≤ 2(𝑀𝑒 𝐾) (𝑇 + 𝑇
we define the process Φ𝑋 by
) < ∞.
Step 1.2. Next, we show that the mapping Φ maps 𝐿 𝑇 (𝐻) into
2𝑝
2𝑝
itself; that is, we have Φ : 𝐿 𝑇 (𝐻) → 𝐿 𝑇 (𝐻). Indeed, let 𝑋 ∈
2𝑝
𝐿 𝑇 (𝐻) be arbitrary. Defining the processes Φ𝛼 𝑋 and Φ𝜎 𝑋 as
𝑡
(Φ𝜎 𝑋)𝑡 := ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
0
(131)
𝑡 ∈ [0, 𝑇] .
1/𝑝
The previous two estimates show that Φ is a well-defined
2𝑝
mapping on 𝐿 𝑇 (𝐻).
𝑡
0
0
󵄩 󵄩2𝑝
E[∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 ]
0
0
(Φ𝑋)𝑡 = 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
𝑇
(Φ𝛼 𝑋)𝑡 := ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠,
𝑡
𝑡
1−1/𝑝
2𝑝
𝑡 ∈ [0, 𝑇] ,
has a unique solution in the space 𝐿 𝑇 (𝐻). This is done in the
following three steps.
2𝑝
𝐿 𝑇 (𝐻)
) < ∞.
(133)
𝑡
𝑡
1/2𝑝
Furthermore, by the growth estimate (9), the linear growth
condition (128), and Hölder’s inequality we have
and prove that the variation of constants equation
0
(132)
𝑇
𝑡 ∈ [0, 𝑇] ,
(134)
𝑡 ∈ [0, 𝑇] ,
we have
(Φ𝑋)𝑡 = 𝑆𝑡 ℎ0 + (Φ𝛼 𝑋)𝑡 + (Φ𝜎 𝑋)𝑡 ,
𝑡 ∈ [0, 𝑇] .
(135)
16
International Journal of Stochastic Analysis
By the growth estimate (9), we have
and hence, by the linear growth condition (128) and Hölder’s
inequality, we obtain
𝑇
2𝑝
󵄩
󵄩2𝑝
󵄩 󵄩2𝑝
E [∫ 󵄩󵄩󵄩𝑆𝑡 ℎ0 󵄩󵄩󵄩 𝑑𝑡] ≤ (𝑀𝑒𝜔𝑇 ) 𝑇E [󵄩󵄩󵄩ℎ0 󵄩󵄩󵄩 ] < ∞.
0
(136)
𝑇
󵄩
󵄩2𝑝
E [∫ 󵄩󵄩󵄩(Φ𝜎 𝑋)𝑡 󵄩󵄩󵄩 𝑑𝑡]
0
𝑇
By Hölder’s inequality and the growth estimate (9), we have
󵄩
󵄩2𝑝
E [∫ 󵄩󵄩󵄩(Φ𝛼 𝑋)𝑡 󵄩󵄩󵄩 𝑑𝑡]
0
≤𝑡
𝑡
0
0
(137)
𝑡
󵄩
󵄩2𝑝
E [∫ ∫ 󵄩󵄩󵄩𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩 𝑑𝑠 𝑑𝑡]
0 0
𝑇
𝑇
𝑡
0
0
2𝑝
󵄩 󵄩2𝑝
≤ 𝐶𝑝 (𝑀𝑒𝜔𝑇 𝐾) 𝑇𝑝−1 22𝑝−1 E [∫ ∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 ) 𝑑𝑠 𝑑𝑡]
󵄩󵄩2𝑝
󵄩
󵄩
= E [∫ 󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠󵄩󵄩󵄩 𝑑𝑡]
󵄩󵄩
0 󵄩
󵄩 0
𝑇󵄩
󵄩 𝑡
𝑇
𝑇
2𝑝
󵄩 󵄩 2𝑝
≤ 𝐶𝑝 (𝑀𝑒𝜔𝑇 𝐾) 𝑇𝑝−1 E [∫ ∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩) 𝑑𝑠 𝑑𝑡]
𝑇
2𝑝−1
𝑡
2𝑝
󵄩
󵄩2𝑝
≤ 𝐶𝑝 (𝑀𝑒𝜔𝑇 ) 𝑡𝑝−1 E [∫ ∫ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠 𝑑𝑡]
2
0 0
2𝑝
≤ 𝐶𝑝 (𝑀𝑒𝜔𝑇 𝐾) 2𝑝 (2𝑇)𝑝−1
×(
𝑡
2𝑝
󵄩
󵄩2𝑝
≤ 𝑇2𝑝−1 (𝑀𝑒𝜔𝑇 ) E [∫ ∫ 󵄩󵄩󵄩𝛼 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩 𝑑𝑠 𝑑𝑡] ,
0 0
𝑇
𝑇2
󵄩 󵄩2𝑝
+ 𝑇E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠]) < ∞.
2
0
(140)
and hence, by the linear growth condition (127) and Hölder’s
inequality, we obtain
The previous three estimates show that Φ𝑋 ∈ 𝐿2𝑝 (𝐻).
2𝑝
Consequently, the mapping Φ maps 𝐿 𝑇 (𝐻) into itself.
Step 1.3. Now, we show that for some index 𝑛 ∈ N the map2𝑝
2𝑝
ping Φ𝑛 is a contraction on 𝐿 𝑇 (𝐻). Let 𝑋, 𝑌 ∈ 𝐿 𝑇 (𝐻), and
𝑡 ∈ [0, 𝑇] be arbitrary. By Hölder’s inequality, the growth
estimate (9), and the Lipschitz condition (125) we have
𝑇
󵄩
󵄩2𝑝
E [∫ 󵄩󵄩󵄩(Φ𝛼 𝑋)𝑡 󵄩󵄩󵄩 𝑑𝑡]
0
𝑇
𝑡
0
0
2𝑝
󵄩 󵄩 2𝑝
≤ 𝑇2𝑝−1 (𝑀𝑒𝜔𝑇 𝐾) E [∫ ∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩) 𝑑𝑠 𝑑𝑡]
𝑇
𝑡
0
0
󵄩2𝑝
󵄩
E [󵄩󵄩󵄩(Φ𝛼 𝑋)𝑡 − (Φ𝛼 𝑌)𝑡 󵄩󵄩󵄩 ]
2𝑝
󵄩 󵄩2𝑝
≤ 𝑇2𝑝−1 (𝑀𝑒𝜔𝑇 𝐾) 22𝑝−1 E [∫ ∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 ) 𝑑𝑠 𝑑𝑡]
2𝑝
≤ (2𝑇)2𝑝−1 (𝑀𝑒𝜔𝑇 𝐾) (
𝑇
𝑇2
󵄩 󵄩2𝑝
+ 𝑇E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠]) < ∞.
2
0
(138)
󵄩󵄩2𝑝
󵄩󵄩 𝑡
𝑡
󵄩
󵄩
= E [󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 − ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑌𝑠 ) 𝑑𝑠󵄩󵄩󵄩 ]
󵄩󵄩
󵄩󵄩 0
0
󵄩󵄩 𝑡
󵄩󵄩2𝑝
󵄩
󵄩
= E [󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 (𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑌𝑠 )) 𝑑𝑠󵄩󵄩󵄩 ]
󵄩󵄩 0
󵄩󵄩
𝑡
Furthermore, by Lemma 49 and the growth estimate (9), we
have
𝑇
󵄩
󵄩2𝑝
E [∫ 󵄩󵄩󵄩(Φ𝜎 𝑋)𝑡 󵄩󵄩󵄩 𝑑𝑡]
0
𝑡
󵄩2𝑝
󵄩
E [󵄩󵄩󵄩(Φ𝜎 𝑋)𝑡 − (Φ𝜎 𝑌)𝑡 󵄩󵄩󵄩 ]
𝑝
𝑡
󵄩
󵄩2
≤ 𝐶𝑝 ∫ E[∫ 󵄩󵄩󵄩𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] 𝑑𝑡
2
0
0
𝑡
(141)
Furthermore, by Lemma 49, the growth estimate (9), the
Lipschitz condition (126), and Hölder’s inequality we obtain
󵄩󵄩 𝑡
󵄩󵄩2𝑝
󵄩
󵄩
= ∫ E [󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 󵄩󵄩󵄩 ] 𝑑𝑡
󵄩󵄩
0
󵄩󵄩 0
𝑇
𝑇
𝑡
2𝑝
󵄩
󵄩2𝑝
≤ 𝑇2𝑝−1 (𝑀𝑒𝜔𝑇 ) E [∫ 󵄩󵄩󵄩𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑌𝑠 )󵄩󵄩󵄩 𝑑𝑠]
0
2𝑝
󵄩2𝑝
󵄩
≤ 𝑇2𝑝−1 (𝑀𝑒𝜔𝑇 𝐿) ∫ E [󵄩󵄩󵄩𝑋𝑠 − 𝑌𝑠 󵄩󵄩󵄩 ] 𝑑𝑠.
0
𝑇󵄩
󵄩󵄩 𝑡
󵄩󵄩󵄩2𝑝
= E [∫ 󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 󵄩󵄩󵄩 𝑑𝑡]
󵄩󵄩
0 󵄩
󵄩 0
𝑇
󵄩
󵄩2𝑝
≤ 𝑡2𝑝−1 E [∫ 󵄩󵄩󵄩𝑆𝑡−𝑠 (𝛼 (𝑠, 𝑋𝑠 ) − 𝛼 (𝑠, 𝑌𝑠 ))󵄩󵄩󵄩 𝑑𝑠]
0
󵄩󵄩 𝑡
𝑡
󵄩󵄩󵄩2𝑝
󵄩
= E [󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 − ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑌𝑠 ) 𝑑𝑊𝑠 󵄩󵄩󵄩 ]
󵄩󵄩 0
󵄩󵄩
0
𝑝
2𝑝
󵄩
󵄩2
≤ 𝐶𝑝 (𝑀𝑒𝜔𝑇 ) ∫ E[∫ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠] 𝑑𝑡,
2
0
0
(139)
󵄩󵄩 𝑡
󵄩󵄩2𝑝
󵄩
󵄩
= E [󵄩󵄩󵄩∫ 𝑆𝑡−𝑠 (𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑌𝑠 )) 𝑑𝑊𝑠 󵄩󵄩󵄩 ]
󵄩󵄩 0
󵄩󵄩
International Journal of Stochastic Analysis
17
𝑡
󵄩
󵄩2
≤ 𝐶𝑝 E[∫ 󵄩󵄩󵄩𝑆𝑡−𝑠 (𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑌𝑠 ))󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠]
2
0
𝑝
𝑡
2𝑝
󵄩
󵄩2
≤ 𝐶𝑝 (𝑀𝑒𝜔𝑇 ) E[∫ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 ) − 𝜎 (𝑠, 𝑌𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠]
2
0
𝑝
𝑡
2𝑝
󵄩2𝑝
󵄩
≤ 𝐶𝑝 (𝑀𝑒𝜔𝑇 𝐿) ∫ E [󵄩󵄩󵄩𝑋𝑠 − 𝑌𝑠 󵄩󵄩󵄩 ] 𝑑𝑠.
0
(142)
Consequently, there exists an index 𝑛 ∈ N such that Φ𝑛 is
a contraction, and hence, according to the extension of the
Banach fixed point theorem (see Corollary 46), the mapping
2𝑝
Φ has a unique fixed point 𝑋 ∈ 𝐿 𝑇 (𝐻). This fixed point 𝑋 is
a solution to the variation of constants equation (130). Since
𝑇 ≥ 0 was arbitrary, there exists a process 𝑋 which is a solution of the variation of constants equation:
Therefore, defining the constant
2𝑝−1
𝐶 := 2
2𝑝−1
(𝑇
𝑡
0
0
𝜔𝑇
(𝑀𝑒 𝐿)
2𝑝
+ 𝐶𝑝 (𝑀𝑒 𝐿) ) ,
(143)
by Hölder’s inequality, we get
Step 1.4. In order to prove that 𝑋 is a mild solution to (44),
it remains to show that 𝑋 has a continuous version. By
Lemma 12, the process
󵄩
󵄩2𝑝
E [󵄩󵄩󵄩(Φ𝑋)𝑡 − (Φ𝑌)𝑡 󵄩󵄩󵄩 ]
𝑡 󳨃󳨀→ 𝑆𝑡 ℎ0 ,
󵄩2𝑝
󵄩
+E [󵄩󵄩󵄩(Φ𝜎 𝑋)𝑡 − (Φ𝜎 𝑌)𝑡 󵄩󵄩󵄩 ])
Thus, by induction for every 𝑛 ∈ N, we obtain
𝑇
1/2𝑝
󵄩 󵄩 2𝑝
≤ 𝐾2𝑝 E [∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩) 𝑑𝑠]
0
󵄩󵄩
≤ (𝐶 ∫ (∫ E [ 󵄩󵄩󵄩(Φ𝑛−1 𝑋)𝑡
󵄩
2
0
0
𝑡1
𝑡1
𝑡𝑛−1
0
0
0
2𝑝 2𝑝−1
≤𝐾 2
󵄩 󵄩2𝑝
E [∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 ) 𝑑𝑠]
0
𝑇
𝑡
𝑛
󵄩2𝑝
󵄩
(∫ E [󵄩󵄩󵄩𝑋𝑠 − 𝑌𝑠 󵄩󵄩󵄩 ] 𝑑𝑠)
0
Thus, by Proposition 43 the stochastic convolution 𝜎 ⋆ 𝑊
given by
𝑡
(𝜎 ⋆ 𝑊)𝑡 = ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
0
× 𝑑𝑡𝑛 ⋅ ⋅ ⋅ 𝑑𝑡2 𝑑𝑡1 )
𝑡1
(149)
𝑇
󵄩 󵄩2𝑝
= 𝐾(2𝐾)2𝑝−1 (𝑇 + E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠]) < ∞.
0
1/2𝑝
󵄩󵄩2𝑝
𝑌)𝑡 󵄩󵄩󵄩 ] 𝑑𝑡2 ) 𝑑𝑡1 )
2󵄩
1/2𝑝
𝑇
(148)
𝑇
󵄩2𝑝
󵄩
= (∫ E [󵄩󵄩󵄩󵄩(Φ𝑛 𝑋)𝑡1 − (Φ𝑛 𝑌)𝑡1 󵄩󵄩󵄩󵄩 ] 𝑑𝑡1 )
0
𝑇
𝑡 ≥ 0,
󵄩
󵄩2𝑝
E [∫ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) 𝑑𝑠]
2
0
󵄩󵄩 𝑛
𝑛 󵄩
󵄩󵄩Φ 𝑋 − Φ 𝑌󵄩󵄩󵄩𝐿2𝑝 (𝐻)
𝑇
≤ ⋅ ⋅ ⋅ ≤ (𝐶𝑛 ∫ ∫ ⋅ ⋅ ⋅ ∫
(147)
is continuous, too. Moreover, for every 𝑇 ≥ 0, we have, by the
linear growth condition (128), Hölder’s inequality, and since
2𝑝
𝑋 ∈ 𝐿 𝑇 (𝐻), the following estimate:
0
−(Φ
∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠,
0
𝑡
𝑛−1
𝑡
(144)
󵄩2𝑝
󵄩
≤ 𝐶 ∫ E [󵄩󵄩󵄩𝑋𝑠 − 𝑌𝑠 󵄩󵄩󵄩 ] 𝑑𝑠.
𝑇
𝑡 ≥ 0,
is continuous, and by Proposition 41, the process
󵄩2𝑝
󵄩
≤ 22𝑝−1 (E [󵄩󵄩󵄩(Φ𝛼 𝑋)𝑡 − (Φ𝛼 𝑌)𝑡 󵄩󵄩󵄩 ]
𝑇
(146)
𝑡 ≥ 0.
2𝑝
𝜔𝑇
𝑡
𝑋𝑡 = 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠 + ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
𝑡 ≥ 0,
(150)
has a continuous version, and consequently, the process 𝑋 has
a continuous version, too. This continuous version is a mild
solution to (44).
𝑡𝑛−1
≤ (𝐶𝑛 (∫ ∫ ⋅ ⋅ ⋅ ∫ 1𝑑𝑡𝑛 ⋅ ⋅ ⋅ 𝑑𝑡2 𝑑𝑡1 )
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
0 0
0
Step 2. Now let ℎ0 : Ω → 𝐻 be an arbitrary F0 -measurable
random variable. We define the sequence (ℎ𝑛 )𝑛∈N of F0 measurable random variables as
=𝑇𝑛 /𝑛!
1/2𝑝
𝑇
󵄩
󵄩2𝑝
× E [∫ 󵄩󵄩󵄩𝑋𝑠 − 𝑌𝑠 󵄩󵄩󵄩 𝑑𝑠] )
0
ℎ0𝑛 := ℎ0 1{‖ℎ0 ‖≤𝑛} ,
𝑛 ∈ N.
(151)
Let 𝑛 ∈ N be arbitrary. Then, as ℎ0𝑛 is bounded, we have
2𝑝
E[‖ℎ0𝑛 ‖ ] < ∞ for all 𝑝 > 1. By Step 1 the SPDE
1/2𝑝
(𝐶𝑇)𝑛
=(
) ‖𝑋 − 𝑌‖𝐿2𝑝 (𝐻) .
𝑇
𝑛!
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑑𝑋𝑡𝑛 = (𝐴𝑋𝑡𝑛 + 𝛼 (𝑡, 𝑋𝑡𝑛 )) 𝑑𝑡 + 𝜎 (𝑡, 𝑋𝑡𝑛 ) 𝑑𝑊𝑡 ,
→ 0 for 𝑛 → ∞
(145)
𝑋0𝑛 = ℎ0𝑛 ,
(152)
18
International Journal of Stochastic Analysis
has a mild solution 𝑋𝑛 . We define the sequence (Ω𝑛 )𝑛∈N ⊂ F0
as
󵄩 󵄩
Ω𝑛 := {󵄩󵄩󵄩ℎ0 󵄩󵄩󵄩 ≤ 𝑛} ,
𝑛 ∈ N.
(153)
Then, we have Ω𝑛 ⊂ Ω𝑚 for 𝑛 ≤ 𝑚, Ω = ⋃𝑛∈N Ω𝑛 , and
Ω𝑛 ⊂ {ℎ0𝑛 = ℎ0𝑚 } ⊂ {ℎ0𝑛 = ℎ0 }
∀𝑛 ≤ 𝑚.
(154)
Thus, by Theorem 48 we have (up to indistinguishability)
𝑋𝑛 1Ω𝑛 = 𝑋𝑚 1Ω𝑛
∀𝑛 ≤ 𝑚.
(155)
Consequently, the process
𝑋 := lim 𝑋𝑛 1Ω𝑛
(156)
𝑛→∞
is a well-defined, continuous, and adapted process, and we
have
𝑋𝑛 1Ω𝑛 = 𝑋𝑚 1Ω𝑛 = 𝑋1Ω𝑛
∀𝑛 ≤ 𝑚.
(157)
Furthermore, we obtain P-almost surely
Corollary 53. Suppose that conditions (125)–(128) are fulfilled. Let ℎ0 : Ω → 𝐻 be a F0 -measurable random variable
such that E[‖ℎ0 ‖2𝑝 ] < ∞ for some 𝑝 > 1. Then there exists a
(up to indistinguishability) unique weak solution 𝑋 to (44).
Proof. According to Proposition 35, every weak solution 𝑋 to
(44) is also a mild solution to (44). Therefore, the uniqueness
of weak solutions to (44) is a consequence of Theorem 48.
It remains to prove the existence of a weak solution to
(44). Let 𝑇 ≥ 0 be arbitrary. By Theorem 50 and its proof,
2𝑝
there exists a mild solution 𝑋 ∈ 𝐿 𝑇 (𝐻) to (44). By the linear
growth condition (128) and Hölder’s inequality we obtain
󵄩
󵄩2
E [∫ 󵄩󵄩󵄩𝜎 (𝑠, 𝑋𝑠 )󵄩󵄩󵄩𝐿0 (𝐻) ]
2
0
𝑛→∞
𝑡
= lim 1Ω𝑛 (𝑆𝑡 ℎ0𝑛 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠𝑛 ) 𝑑𝑠
𝑛→∞
𝑇
󵄩 󵄩2
≤ 𝐾2 E [∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩) 𝑑𝑠]
0
0
𝑡
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠𝑛 ) 𝑑𝑊𝑠 )
𝑇
󵄩 󵄩2
≤ 2𝐾2 E [∫ (1 + 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 ) 𝑑𝑠]
0
𝑇
= lim (𝑆𝑡 (1Ω𝑛 ℎ0𝑛 ) + ∫ 1Ω𝑛 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠𝑛 ) 𝑑𝑠
𝑛→∞
󵄩 󵄩2
= 2𝐾2 (𝑇 + E [∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠])
0
0
𝑡
+ ∫ 1Ω𝑛 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠𝑛 ) 𝑑𝑊𝑠 )
𝑇
󵄩 󵄩2𝑝
≤ 2𝐾2 (𝑇 + 𝑇1−1/𝑝 E[∫ 󵄩󵄩󵄩𝑋𝑠 󵄩󵄩󵄩 𝑑𝑠]
0
0
𝑡
= lim (𝑆𝑡 (1Ω𝑛 ℎ0 ) + ∫ 1Ω𝑛 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
𝑛→∞
0
(158)
𝑡
+ ∫ 1Ω𝑛 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 )
0
𝑡
= lim 1Ω𝑛 (𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
𝑛→∞
0
𝑡
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 )
0
𝑡
= 𝑆𝑡 ℎ0 + ∫ 𝑆𝑡−𝑠 𝛼 (𝑠, 𝑋𝑠 ) 𝑑𝑠
0
+ ∫ 𝑆𝑡−𝑠 𝜎 (𝑠, 𝑋𝑠 ) 𝑑𝑊𝑠 ,
(159)
0
𝑡
0
We close this section with a consequence about the
existence of weak solutions.
𝑇
𝑋𝑡 = lim 𝑋𝑡𝑛 1Ω𝑛
𝑡
Remark 52. A recent method for proving existence and
uniqueness of mild solutions to the SPDE (44) is the method
of the moving frame presented in [6]; see also [8]. It allows
to reduce SPDE problems to the study of SDEs in infinite
dimension. In order to apply this method, we need that the
semigroup (𝑆𝑡 )𝑡≥0 is a semigroup of pseudocontractions.
𝑡 ≥ 0,
proving that 𝑋 is a mild solution to (44).
Remark 51. For the proof of Theorem 50, we have used
Corollary 46, which is a slight extension of the Banach fixed
point theorem. Such an idea has been applied, for example, in
[15].
1/𝑝
) < ∞,
showing that condition (72) is fulfilled. Thus, by
Proposition 37, the process 𝑋 is also a weak solution to
(44).
8. Invariant Manifolds for Weak
Solutions to SPDEs
In this section, we deal with invariant manifolds for timehomogeneous SPDEs of the type (44). This topic arises from
the natural desire to express the solutions of the SPDE (44),
which generally live in the infinite dimensional Hilbert space
𝐻, by means of a finite dimensional state process and thus
to ensure larger analytical tractability. Our goal is to find
conditions on the generator 𝐴 and the coefficients 𝛼, 𝜎 such
that for every starting point of a finite dimensional submanifold the solution process stays on the submanifold.
We start with the required preliminaries about finite
dimensional submanifolds in Hilbert spaces. In the sequel, let
𝐻 be a separable Hilbert space.
Definition 54. Let 𝑚, 𝑘 ∈ N be positive integers. A subset
M ⊂ 𝐻 is called an 𝑚-dimensional 𝐶𝑘 -submanifold of 𝐻,
International Journal of Stochastic Analysis
19
if for every ℎ ∈ M there exist an open neighborhood 𝑈 ⊂ 𝐻
of ℎ, an open set 𝑉 ⊂ R𝑚 , and a mapping 𝜙 ∈ 𝐶2 (𝑉; 𝐻) such
that
(1) the mapping 𝜙 : 𝑉 → 𝑈 ∩ M is a homeomorphism;
(2) for all 𝑦 ∈ 𝑉 the mapping 𝐷𝜙(𝑦) is injective.
The mapping 𝜙 is called a parametrization of M around ℎ.
In what follows, let M be an 𝑚-dimensional 𝐶𝑘 submanifold of 𝐻.
Lemma 55. Let 𝜙𝑖 : 𝑉𝑖 → 𝑈𝑖 ∩ M, 𝑖 = 1, 2 be two
parametrizations with 𝑊 := 𝑈1 ∩𝑈2 ∩M ≠ 0. Then the mapping
𝜙1−1 ∘ 𝜙2 : 𝜙2−1 (𝑊) 󳨀→ 𝜙1−1 (𝑊)
(160)
is a 𝐶𝑘 -diffeomorphism.
Corollary 56. Let ℎ ∈ M be arbitrary, and let 𝜙𝑖 : 𝑉𝑖 →
𝑈𝑖 ∩ M, 𝑖 = 1, 2 be two parametrizations of M around ℎ. Then
one has
𝐷𝜙1 (𝑦1 ) (R𝑚 ) = 𝐷𝜙2 (𝑦2 ) (R𝑚 ) ,
Proof. Since 𝑈1 and 𝑈2 are open neighborhoods of ℎ, we have
𝑊 := 𝑈1 ∩ 𝑈2 ∩ M ≠ 0. Thus, by Lemma 55 the mapping
∘ 𝜙2 :
(𝑊) 󳨀→
𝜙1−1
(𝑊)
is a 𝐶 -diffeomorphism. Using the chain rule, we obtain
(165)
where one uses the notation ⟨𝜁, ℎ⟩ := (⟨𝜁1 , ℎ⟩, . . . , ⟨𝜁𝑚 , ℎ⟩) ∈
R𝑚 .
Proof. See [16, Proposition 6.1.2].
(1) The elements 𝜁1 , . . . , 𝜁𝑚 are linearly independent in 𝐻.
(2) For every ℎ ∈ 𝑈 ∩ M, one has the direct sum decomposition
⊥
𝐻 = 𝑇ℎ M ⊕ ⟨𝜁1 , . . . , 𝜁𝑚 ⟩ .
𝐷𝜙2 (𝑦2 ) (R )
(3) For every ℎ ∈ 𝑈 ∩ M the mapping
Πℎ = 𝐷𝜙 (𝑦) (⟨𝜁, ∙⟩) : 𝐻 󳨀→ 𝑇ℎ M,
Π2ℎ = Πℎ ,
Πℎ ∈ 𝐿 (𝐻) ,
= 𝐷 (𝜙1 ∘ (𝜙1−1 ∘ 𝜙2 )) (𝑦2 ) (R𝑚 )
(163)
𝑚
∘ 𝜙2 ) (𝑦2 ) (R )
(166)
𝑤ℎ𝑒𝑟𝑒 𝑦 = ⟨𝜁, ℎ⟩,
(167)
is the corresponding projection according to (166) from
𝐻 onto 𝑇ℎ M, that is, we have
𝑚
=
𝜙 (⟨𝜁, ℎ⟩) = ℎ ∀ℎ ∈ 𝑈 ∩ M,
(162)
𝑘
𝐷𝜙1 (𝑦1 ) 𝐷 (𝜙1−1
Proposition 61. Let 𝐷 ⊂ 𝐻 be a dense subset. For every ℎ0 ∈
M there exist 𝜁1 , . . . , 𝜁𝑚 ∈ 𝐷 and a parametrization 𝜙 : 𝑉 →
𝑈 ∩ M around ℎ0 such that
(161)
where 𝑦𝑖 = 𝜙𝑖−1 (ℎ) for 𝑖 = 1, 2.
𝜙2−1
Remark 60. By Proposition 59 we may assume that any
parametrization 𝜙 : 𝑉 → 𝑈 ∩ M has an extension 𝜙 ∈
𝐶𝑏𝑘 (R𝑚 ; 𝐻).
Proposition 62. Let 𝜙 : 𝑉 → 𝑈 ∩ M be a parametrization
as in Proposition 61. Then the following statements are true.
Proof. See [16, Lemma 6.1.1].
𝜙1−1
Proof. See [16, Remark 6.1.1].
⊂ 𝐷𝜙1 (𝑦1 ) (R𝑚 ) ,
ran (Πℎ ) = 𝑇ℎ M,
⊥
ker (Πℎ ) = ⟨𝜁1 , . . . , 𝜁𝑚 ⟩ .
(168)
Proof. See [16, Lemma 6.1.3].
𝑚
and, analogously, we prove that 𝐷𝜙1 (𝑦1 )(R )
𝐷𝜙2 (𝑦2 )(R𝑚 ).
⊂
From now on, we assume that M is an 𝑚-dimensional
𝐶2 -submanifold of 𝐻.
Definition 57. Let ℎ ∈ M be arbitrary. The tangent space of M
to ℎ is the subspace
Proposition 63. Let 𝜙 : 𝑉 → 𝑈 ∩ M be a parametrization
as in Proposition 61. Furthermore, let 𝜎 ∈ 𝐶1 (𝐻) be a mapping
such that
𝑇ℎ M := 𝐷𝜙 (𝑦) (R𝑚 ) ,
(164)
−1
where 𝑦 = 𝜙 (ℎ) and 𝜙 : 𝑉 → 𝑈 ∩ M denotes a
parametrization of M around ℎ.
Remark 58. Note that, according to Corollary 56, the
Definition 57 of the tangent space 𝑇ℎ M does not depend on
the choice of the parametrization 𝜙 : 𝑉 → 𝑈 ∩ M.
Proposition 59. Let ℎ ∈ M be arbitrary, and let 𝜙 : 𝑉 →
𝑈 ∩ M be a parametrization of M around ℎ. Then there exist
an open set 𝑉0 ⊂ 𝑉, an open neighborhood 𝑈0 ⊂ 𝑈 of ℎ, and
̂ such that 𝜙| :
a mapping 𝜙̂ ∈ 𝐶𝑏𝑘 (R𝑚 ; 𝐻) with 𝜙|𝑉0 = 𝜙|
𝑉0
𝑉0
𝑉0 → 𝑈0 ∩ M is a parametrization of M around ℎ, too.
𝜎 (ℎ) ∈ 𝑇ℎ M
∀ℎ ∈ 𝑈 ∩ M.
(169)
Then, for every ℎ ∈ 𝑈 ∩ M the direct sum decomposition of
𝐷𝜎(ℎ)𝜎(ℎ) according to (166) is given by
𝐷𝜎 (ℎ) 𝜎 (ℎ)
= 𝐷𝜙 (𝑦) (⟨𝜁, 𝐷𝜎 (ℎ) 𝜎 (ℎ)⟩)
+ 𝐷2 𝜙 (𝑦) (⟨𝜁, 𝜎 (ℎ)⟩ , ⟨𝜁, 𝜎 (ℎ)⟩) ,
where 𝑦 = 𝜙−1 (ℎ).
(170)
20
International Journal of Stochastic Analysis
Proof. Since 𝑉 is an open subset of R𝑚 , there exists 𝜖 > 0 such
that
−1
𝑦 + 𝑡𝐷𝜙(𝑦) 𝜎 (ℎ) ∈ 𝑉
∀𝑡 ∈ (−𝜖, 𝜖) .
(171)
Proof. By (178), for all ℎ1 , ℎ2 ∈ 𝐻 we have
󵄩󵄩
󵄩
󵄩󵄩𝜎 (ℎ1 ) − 𝜎 (ℎ2 )󵄩󵄩󵄩𝐿0 (𝐻)
2
Therefore, the curve
𝑐 : (−𝜖, 𝜖) 󳨀→ 𝑈 ∩ M,
Proposition 64. For every ℎ0 ∈ 𝐻 there exists a (up to
indistinguishability) unique weak solution to (176).
−1
𝑐 (𝑡) := 𝜙 (𝑦 + 𝑡𝐷𝜙(𝑦) 𝜎 (ℎ)) ,
(172)
is well defined, and we have 𝑐 ∈ 𝐶1 ((−𝜖, 𝜖); 𝐻) with 𝑐(0) = ℎ
and 𝑐󸀠 (0) = 𝜎(ℎ). Hence, we have
󵄨󵄨
𝑑
󵄨
𝜎 (𝑐 (𝑡))󵄨󵄨󵄨 = 𝐷𝜎 (ℎ) 𝜎 (ℎ) .
(173)
󵄨󵄨𝑡=0
𝑑𝑡
Moreover, by condition (169) and Proposition 62, we have
󵄨󵄨
󵄨󵄨
𝑑
𝑑
󵄨
󵄨
𝜎 (𝑐 (𝑡))󵄨󵄨󵄨 =
Π𝑐(𝑡) 𝜎 (𝑐 (𝑡))󵄨󵄨󵄨
󵄨󵄨𝑡=0
󵄨󵄨𝑡=0 𝑑𝑡
𝑑𝑡
󵄩
󵄩2
= (∑ 󵄩󵄩󵄩󵄩𝜎𝑗 (ℎ1 ) − 𝜎𝑗 (ℎ2 )󵄩󵄩󵄩󵄩 )
1/2
𝑗∈N
1/2
≤
(∑ 𝜅𝑗2 )
𝑗∈N
(180)
󵄩󵄩
󵄩
󵄩󵄩ℎ1 − ℎ2 󵄩󵄩󵄩 .
Moreover, by (177), for every ℎ ∈ 𝐻 we have
‖𝛼 (ℎ)‖ ≤ ‖𝛼 (ℎ) − 𝛼 (0)‖ + ‖𝛼 (0)‖
≤ 𝐿 ‖ℎ‖ + ‖𝛼 (0)‖
󵄨󵄨
𝑑
󵄨
=
𝐷𝜙 (⟨𝜁, 𝑐 (𝑡)⟩) (⟨𝜁, 𝜎 (𝑐 (𝑡))⟩)󵄨󵄨󵄨
󵄨󵄨𝑡=0 (174)
𝑑𝑡
(181)
≤ max {𝐿, ‖𝛼 (0)‖} (1 + ‖ℎ‖) ,
and, by (179) we obtain
= 𝐷𝜙 (𝑦) (⟨𝜁, 𝐷𝜎 (ℎ) 𝜎 (ℎ)⟩)
+ 𝐷2 𝜙 (𝑦) (⟨𝜁, 𝜎 (ℎ)⟩ , ⟨𝜁, 𝜎 (ℎ)⟩) .
The latter two identities prove the desired decomposition
(170).
‖𝜎 (ℎ)‖𝐿02 (𝐻)
󵄩2
󵄩
= (∑ 󵄩󵄩󵄩󵄩𝜎𝑗 (ℎ)󵄩󵄩󵄩󵄩 )
1/2
𝑗∈N
(182)
1/2
After these preliminaries, we will study invariant manifolds for time-homogeneous SPDEs of the form
≤ (∑ 𝜅𝑗2 )
𝑑𝑋𝑡 = (𝐴𝑋𝑡 + 𝛼 (𝑋𝑡 )) 𝑑𝑡 + 𝜎 (𝑋𝑡 ) 𝑑𝑊𝑡
Therefore, conditions (125)–(128) are fulfilled, and hence,
applying Corollary 53 completes the proof.
𝑋0 = ℎ0 ,
(175)
with measurable mappings 𝛼 : 𝐻 → 𝐻 and 𝜎 : 𝐻 →
𝐿02 (𝐻). As in the previous sections, the operator 𝐴 is the
infinitesimal generator of a 𝐶0 -semigroup (𝑆𝑡 )𝑡≥0 on 𝐻. Note
that, by (41), the SPDE (175) can be rewritten equivalently as
𝑗
𝑑𝑋𝑡 = (𝐴𝑋𝑡 + 𝛼 (𝑋𝑡 )) 𝑑𝑡 + ∑ 𝜎𝑗 (𝑋𝑡 ) 𝑑𝛽𝑡 ,
𝑗∈N
(176)
𝑋0 = ℎ0 ,
𝑗
where (𝛽 )𝑗∈N denotes the sequence of real-valued independent standard Wiener processes defined in (33) and the
mappings 𝜎𝑗 : 𝐻 → 𝐻, 𝑗 ∈ N are given by 𝜎𝑗 = √𝜆 𝑗 𝜎𝑒𝑗 .
For the rest of this section, we assume that there exist a
constant 𝐿 ≥ 0 such that
󵄩
󵄩󵄩
󵄩
󵄩
(177)
󵄩󵄩𝛼 (ℎ1 ) − 𝛼 (ℎ2 )󵄩󵄩󵄩 ≤ 𝐿 󵄩󵄩󵄩ℎ1 − ℎ2 󵄩󵄩󵄩 , ℎ1 , ℎ2 ∈ 𝐻,
and a sequence (𝜅𝑗 )𝑗∈N ⊂ R+ with ∑𝑗∈N 𝜅𝑗2 < ∞ such that for
every 𝑗 ∈ N we have
󵄩󵄩 𝑗
󵄩
󵄩󵄩𝜎 (ℎ1 ) − 𝜎𝑗 (ℎ2 )󵄩󵄩󵄩 ≤ 𝜅𝑗 󵄩󵄩󵄩󵄩ℎ1 − ℎ2 󵄩󵄩󵄩󵄩 , ℎ1 , ℎ2 ∈ 𝐻,
󵄩
󵄩
󵄩󵄩 𝑗 󵄩󵄩
󵄩󵄩𝜎 (ℎ)󵄩󵄩 ≤ 𝜅𝑗 (1 + ‖ℎ‖) , ℎ ∈ 𝐻.
󵄩
󵄩
(178)
(179)
(1 + ‖ℎ‖) .
𝑗∈N
Recall that M denotes a finite dimensional 𝐶2 submanifold of 𝐻.
Definition 65. The submanifold M is called locally invariant
for (176), if for every ℎ0 ∈ M there exists a local weak solution
𝑋 to (176) with some lifetime 𝜏 > 0 such that
𝑋𝑡∧𝜏 ∈ M ∀𝑡 ≥ 0, P-almost surely.
(183)
In order to investigate local invariance of M, we will
assume, from now on, that 𝜎𝑗 ∈ 𝐶1 (𝐻) for all 𝑗 ∈ N.
Lemma 66. The following statements are true.
(1) For every ℎ ∈ 𝐻 one has
󵄩
󵄩
∑ 󵄩󵄩󵄩󵄩𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)󵄩󵄩󵄩󵄩 < ∞.
𝑗∈N
(184)
(2) The mapping
𝐻 󳨀→ 𝐻,
ℎ 󳨃󳨀→ ∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ) ,
𝑗∈N
is continuous.
(185)
International Journal of Stochastic Analysis
21
and a sequence (̃
𝜅𝑗 )𝑗∈N ⊂ R+ with ∑𝑗∈N 𝜅̃𝑗2 < ∞ such that for
every 𝑗 ∈ N we have
Proof. By (178) and (179), for every ℎ ∈ 𝐻 we have
󵄩
󵄩
∑ 󵄩󵄩󵄩󵄩𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)󵄩󵄩󵄩󵄩
𝑗∈N
󵄩󵄩
󵄩
󵄩
≤ ∑ 󵄩󵄩󵄩󵄩𝐷𝜎𝑗 (ℎ)󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝜎𝑗 (ℎ)󵄩󵄩󵄩󵄩
(186)
𝑗∈N
≤ (1 + ‖ℎ‖) ∑ 𝜅𝑗2 < ∞,
𝑗∈N
showing (184). Moreover, for every 𝑗 ∈ N the mapping
𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ) ,
𝐻 󳨃󳨀→ 𝐻,
(187)
is continuous, because for all ℎ1 , ℎ2 ∈ 𝐻 we have
󵄩󵄩󵄩𝐷𝜎𝑗 (ℎ ) 𝜎𝑗 (ℎ ) − 𝐷𝜎𝑗 (ℎ ) 𝜎𝑗 (ℎ )󵄩󵄩󵄩
󵄩󵄩
1
1
2
2 󵄩
󵄩
󵄩󵄩 𝑗
󵄩
≤ 󵄩󵄩󵄩𝐷𝜎 (ℎ1 ) 𝜎𝑗 (ℎ1 ) − 𝐷𝜎𝑗 (ℎ1 ) 𝜎𝑗 (ℎ2 )󵄩󵄩󵄩󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩󵄩𝐷𝜎𝑗 (ℎ1 ) 𝜎𝑗 (ℎ2 ) − 𝐷𝜎𝑗 (ℎ2 ) 𝜎𝑗 (ℎ2 )󵄩󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝐷𝜎𝑗 (ℎ1 )󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝜎𝑗 (ℎ1 ) − 𝜎𝑗 (ℎ2 )󵄩󵄩󵄩󵄩
󵄩
󵄩󵄩
󵄩
+ 󵄩󵄩󵄩󵄩𝐷𝜎𝑗 (ℎ1 ) − 𝐷𝜎𝑗 (ℎ2 )󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝜎𝑗 (ℎ2 )󵄩󵄩󵄩󵄩 .
N
𝑗∈N
Hence, because of the estimate
󵄩
󵄩󵄩 𝑗
󵄩󵄩𝐷𝜎 (ℎ) 𝜎𝑗 (ℎ)󵄩󵄩󵄩 ≤ (1 + ‖ℎ‖) 𝜅𝑗2 ,
󵄩
󵄩
ℎ ∈ 𝐻, 𝑗 ∈ N,
Theorem 68. The following statements are equivalent.
(188)
M ⊂ D (𝐴) ,
(189)
(190)
For a mapping 𝜙 ∈ 𝐶𝑏2 (R𝑚 ; 𝐻) and elements 𝜁1 , . . . , 𝜁𝑚 ∈
𝑗
D(𝐴∗ ) we define the mappings 𝛼𝜙,𝜁 : R𝑚 → R𝑚 and 𝜎𝜙,𝜁 :
R𝑚 → R𝑚 , 𝑗 ∈ N as
𝛼𝜙,𝜁 (𝑦) := ⟨𝐴∗ 𝜁, 𝜙 (𝑦)⟩ + ⟨𝜁, 𝛼 (𝜙 (𝑦))⟩ ,
(191)
𝑗
Proposition 67. Let 𝜙 ∈ 𝐶𝑏2 (R𝑚 ; 𝐻) and 𝜁1 , . . . , 𝜁𝑚 ∈ D(𝐴∗ )
be arbitrary. Then, for every 𝑦0 ∈ R𝑚 there exists a (up to
indistinguishability) unique strong solution to the SDE:
𝑗
𝑗
𝑑𝑌𝑡 = 𝛼𝜙,𝜁 (𝑌𝑡 ) 𝑑𝑡 + ∑ 𝜎𝜙,𝜁 (𝑌𝑡 ) 𝑑𝛽𝑡 ,
𝑗∈N
(1) The submanifold M is locally invariant for (176).
(2) One has
the continuity of the mapping (185) is a consequence of
Lebesgue’s dominated convergence theorem.
𝜎𝜙,𝜁 (𝑦) := ⟨𝜁, 𝜎𝑗 (𝜙 (𝑦))⟩ .
Therefore, by Proposition 64, for every 𝑦0 ∈ R𝑚 there exists
a (up to indistinguishability) unique weak solution to (192),
which, according to Proposition 39 is also a strong solution
to (192). The uniqueness of strong solutions to (192) is a consequence of Proposition 39 and Theorem 48.
Now, we are ready to formulate and prove our main result
of this section.
Let ] be the counting measure on (N, P(N)), which is given
by ]({𝑗}) = 1 for all 𝑗 ∈ N. Then we have
∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ) = ∫ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ) ] (𝑑𝑗) .
󵄩󵄩 𝑗
󵄩
󵄩󵄩𝜎 (𝑦1 ) − 𝜎𝑗 (𝑦2 )󵄩󵄩󵄩 ≤ 𝜅̃𝑗 󵄩󵄩󵄩𝑦1 − 𝑦2 󵄩󵄩󵄩 𝑚 , 𝑦1 , 𝑦2 ∈ R𝑚 ,
󵄩
󵄩R
󵄩󵄩 𝜙,𝜁
󵄩󵄩R𝑚
𝜙,𝜁
󵄩󵄩 𝑗
󵄩󵄩
󵄩 󵄩
𝑚
󵄩󵄩󵄩𝜎𝜙,𝜁 (𝑦)󵄩󵄩󵄩 𝑚 ≤ 𝜅̃𝑗 (1 + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩R𝑚 ) , 𝑦 ∈ R .
󵄩
󵄩R
(194)
(192)
𝑌0 = 𝑦0 .
Proof. By virtue of the assumption 𝜙 ∈ 𝐶𝑏2 (R𝑚 ; 𝐻) and (177)–
̃ ≥ 0 such that
(179), there exist a constant 𝐿
󵄩󵄩󵄩𝛼 (𝑦 ) − 𝛼 (𝑦 )󵄩󵄩󵄩 ≤ 𝐿
̃ 󵄩󵄩󵄩𝑦1 − 𝑦2 󵄩󵄩󵄩 𝑚 , 𝑦1 , 𝑦2 ∈ R𝑚 ,
󵄩󵄩 𝜙,𝜁 1
𝜙,𝜁
2 󵄩
󵄩R
󵄩
󵄩R 𝑚
(193)
𝜎𝑗 (ℎ) ∈ 𝑇ℎ M
𝐴ℎ + 𝛼 (ℎ) −
∀ℎ ∈ M, 𝑎𝑙𝑙 𝑗 ∈ N,
(195)
(196)
1
∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ) ∈ 𝑇ℎ M ∀ℎ ∈ M. (197)
2 𝑗∈N
(3) The operator 𝐴 is continuous on M, and for each ℎ0 ∈
M there exists a local strong solution 𝑋 to (176) with
some lifetime 𝜏 > 0 such that
𝑋𝑡∧𝜏 ∈ M
∀𝑡 ≥ 0, P-𝑎𝑙𝑚𝑜𝑠𝑡 𝑠𝑢𝑟𝑒𝑙𝑦.
(198)
Proof. (1)⇒(2): Let ℎ ∈ M be arbitrary. By Proposition 61
and Remark 60 there exist elements 𝜁1 , . . . , 𝜁𝑚 ∈ D(𝐴∗ ) and
a parametrization 𝜙 : 𝑉 → 𝑈 ∩ M around ℎ such that the
inverse 𝜙−1 : 𝑈 ∩ M → 𝑉 is given by 𝜙−1 = ⟨𝜁, ∙⟩, and 𝜙
has an extension 𝜙 ∈ 𝐶𝑏2 (R𝑚 ; 𝐻). Since the submanifold M is
locally invariant for (176), there exists a local weak solution 𝑋
to (176) with initial condition ℎ and some lifetime 󰜚 > 0 such
that
𝑋𝑡∧󰜚 ∈ M
∀𝑡 ≥ 0, P-almost surely.
(199)
Since 𝑈 is an open neighborhood of ℎ, there exists 𝜖 > 0 such
that 𝐵𝜖 (ℎ) ⊂ 𝑈, where 𝐵𝜖 (ℎ) denotes the open ball:
󵄩
󵄩
𝐵𝜖 (ℎ) = {𝑔 ∈ 𝐻 : 󵄩󵄩󵄩𝑔 − ℎ󵄩󵄩󵄩 < 𝜖} .
(200)
We define the stopping time
𝜏 := 󰜚 ∧ inf {𝑡 ≥ 0 : 𝑋𝑡 ∉ 𝐵𝜖 (ℎ)} .
(201)
Since the process 𝑋 has continuous sample paths and satisfies
𝑋0 = ℎ, we have 𝜏 > 0 and P-almost surely
𝑋𝑡∧𝜏 ∈ 𝑈 ∩ M
∀𝑡 ≥ 0.
(202)
22
International Journal of Stochastic Analysis
Defining the R𝑚 -valued process 𝑌 := ⟨𝜁, 𝑋⟩ we have Palmost surely
𝑌𝑡∧𝜏 ∈ 𝑉
∀𝑡 ≥ 0.
(203)
Moreover, since 𝑋 is a weak solution to (176) with initial
condition ℎ, setting 𝑦 := ⟨𝜁, ℎ⟩ ∈ 𝑉 we have P-almost surely
𝑡∧𝜏
𝑌𝑡∧𝜏 = ⟨𝜁, ℎ⟩ + ∫
0
+∑∫
𝑡∧𝜏
𝑡∧𝜏
= ⟨𝜁, ℎ⟩ + ∫
0
+∑∫
𝑡∧𝜏
𝑗∈N 0
=𝑦+∫
𝑡∧𝜏
0
+∑∫
+∑∫
𝐵 + 𝑀 = 0,
(204)
𝑗
0
(208)
where the processes 𝐵 and 𝑀 are defined as
𝑡∧𝜏
0
𝛼𝜙,𝜁 (𝑌𝑠 ) 𝑑𝑠
(⟨𝐴∗ 𝜉, 𝑋𝑠 ⟩ + ⟨𝜉, 𝛼 (𝑋𝑠 ) − 𝐷𝜙 (𝑌𝑠 ) 𝛼𝜙,𝜁 (𝑌𝑠 )
−
𝑡 ≥ 0,
1
∑ 𝐷2 𝜙 (𝑌𝑠 )
2 𝑗∈N
𝑗
𝑗
× (𝜎𝜙,𝜁 (𝑌𝑠 ) , 𝜎𝜙,𝜁 (𝑌𝑠 ))⟩) 𝑑𝑠,
𝑡 ≥ 0,
𝑀𝑡 := ∑ ∫
𝑋𝑡∧𝜏 = 𝜙 (𝑌𝑡∧𝜏 )
𝑡∧𝜏
⟨𝜉, 𝜎𝑗 (𝑋𝑠 )⟩ 𝑑𝛽𝑠𝑗 .
Combining (206) and (207) yields up to indistinguishability
showing that 𝑌 is a local strong solution to (192) with initial
condition 𝑦. By Itô’s formula (Theorem 26) we obtain Palmost surely
=ℎ+∫
(⟨𝐴∗ 𝜉, 𝑋𝑠 ⟩ + ⟨𝜉, 𝛼 (𝑋𝑠 )⟩) 𝑑𝑠
(207)
𝜎𝜙,𝜁 (⟨𝜁, 𝑋𝑠 ⟩) 𝑑𝛽𝑠𝑗
𝑗
𝑡∧𝜏
𝑗∈N 0
𝛼𝜙,𝜁 (⟨𝜁, 𝑋𝑠 ⟩) 𝑑𝑠
𝜎𝜙,𝜁 (𝑌𝑠 ) 𝑑𝛽𝑠𝑗 ,
𝑡∧𝜏
0
𝐵𝑡 := ∫
𝑡∧𝜏
𝑗∈N 0
⟨𝜉, 𝑋𝑡∧𝜏 ⟩ = ⟨𝜉, ℎ⟩ + ∫
(⟨𝐴∗ 𝜁, 𝑋𝑠 ⟩ + ⟨𝜁, 𝛼 (𝑋𝑠 )⟩) 𝑑𝑠
⟨𝜁, 𝜎𝑗 (𝑋𝑠 )⟩ 𝑑𝛽𝑠𝑗
𝑗∈N 0
On the other hand, since 𝑋 is a local weak solution to (176)
with initial condition ℎ and lifetime 𝜏, we have P-almost
surely for all 𝑡 ≥ 0 the identity
𝑡∧𝜏
𝑗∈N 0
𝑗
⟨𝜉, 𝜎𝑗 (𝑋𝑠 ) − 𝐷𝜙 (𝑌𝑠 ) 𝜎𝜙,𝜁 (𝑌𝑠 )⟩ 𝑑𝛽𝑠𝑗 ,
𝑡 ≥ 0.
(𝐷𝜙 (𝑌𝑠 ) 𝛼𝜙,𝜁 (𝑌𝑠 )
(209)
1
𝑗
𝑗
+ ∑ 𝐷2 𝜙 (𝑌𝑠 ) (𝜎𝜙,𝜁 (𝑌𝑠 ) , 𝜎𝜙,𝜁 (𝑌𝑠 ))) 𝑑𝑠
2 𝑗∈N
𝑡∧𝜏
+∑∫
𝑗∈N 0
𝑗
𝐷𝜙 (𝑌𝑠 ) 𝜎𝜙,𝜁 (𝑌𝑠 ) 𝑑𝛽𝑠𝑗 ,
𝑡 ≥ 0.
The process 𝐵 + 𝑀 is a continuous semimartingale with
canonical decomposition (208). Since the canonical decomposition of a continuous semimartingale is unique up to
indistinguishability, we deduce that 𝐵 = 𝑀 = 0 up to
indistinguishability. Using the Itô isometry (39) we obtain Palmost surely
(205)
∗
Now, let 𝜉 ∈ D(𝐴 ) be arbitrary. Then we have P-almost
surely
∫
𝑡∧𝜏
0
( ⟨𝐴∗ 𝜉, 𝑋𝑠 ⟩
⟨𝜉, 𝑋𝑡∧𝜏 ⟩
+ ⟨𝜉, 𝛼 (𝑋𝑠 ) − 𝐷𝜙 (𝑌𝑠 ) 𝛼𝜙,𝜁 (𝑌𝑠 )
= ⟨𝜉, ℎ⟩
+∫
𝑡∧𝜏
0
−
⟨𝜉, 𝐷𝜙 (𝑌𝑠 ) 𝛼𝜙,𝜁 (𝑌𝑠 )
1
𝑗
𝑗
+ ∑ 𝐷2 𝜙 (𝑌𝑠 ) (𝜎𝜙,𝜁 (𝑌𝑠 ) , 𝜎𝜙,𝜁 (𝑌𝑠 ))⟩ 𝑑𝑠
2 𝑗∈N
𝑡∧𝜏
+∑∫
𝑗∈N 0
𝑗
⟨𝜉, 𝐷𝜙 (𝑌𝑠 ) 𝜎𝜙,𝜁 (𝑌𝑠 )⟩𝑑𝛽𝑠𝑗 ,
= 0,
∫
𝑡∧𝜏
0
𝑡 ≥ 0.
1
𝑗
𝑗
∑ 𝐷2 𝜙 (𝑌𝑠 ) (𝜎𝜙,𝜁 (𝑌𝑠 ) , 𝜎𝜙,𝜁 (𝑌𝑠 ))⟩) 𝑑𝑠
2 𝑗∈N
𝑡 ≥ 0,
󵄨󵄨
󵄨󵄨2
𝑗
∑ 󵄨󵄨󵄨⟨𝜉, 𝜎𝑗 (𝑋𝑠 ) − 𝐷𝜙 (𝑌𝑠 ) 𝜎𝜙,𝜁 (𝑌𝑠 )⟩󵄨󵄨󵄨 𝑑𝑠 = 0,
󵄨
󵄨
𝑗∈N
𝑡 ≥ 0.
(206)
(210)
International Journal of Stochastic Analysis
23
By the continuity of the processes 𝑋 and 𝑌 we obtain for all
𝜉 ∈ D(𝐴∗ ) the following identities:
𝐴ℎ + 𝛼 (ℎ) −
⟨𝐴∗ 𝜉, ℎ⟩ + ⟨𝜉, 𝛼 (ℎ) − 𝐷𝜙 (𝑦) 𝛼𝜙,𝜁 (𝑦)
1
𝑗
𝑗
− ∑ 𝐷2 𝜙 (𝑦) (𝜎𝜙,𝜁 (𝑦) , 𝜎𝜙,𝜁 (𝑦))⟩ = 0,
2 𝑗∈N
𝑗
⟨𝜉, 𝜎𝑗 (ℎ) − 𝐷𝜙 (𝑦) 𝜎𝜙,𝜁 (𝑦)⟩ = 0,
Consequently, the mapping 𝜉 󳨃→ ⟨𝐴∗ 𝜉, ℎ⟩ is continuous on
D(𝐴∗ ), and hence we have ℎ ∈ D(𝐴∗∗ ) by the definition
of the domain provided in (4). By Proposition 7 we have
𝐴 = 𝐴∗∗ , and thus we obtain ℎ ∈ D(𝐴), proving (195). By
Proposition 7, the domain D(𝐴∗ ) is dense in 𝐻, and thus
𝑗
𝜎𝑗 (ℎ) = 𝐷𝜙 (𝑦) 𝜎𝜙,𝜁 (𝑦) ∈ 𝑇ℎ M, 𝑗 ∈ N,
(213)
1
𝑗
𝑗
− ∑ 𝐷2 𝜙 (𝑦) (𝜎𝜙,𝜁 (𝑦) , 𝜎𝜙,𝜁 (𝑦))⟩ = 0.
2 𝑗∈N
Since the domain D(𝐴∗ ) is dense in 𝐻, together with
Proposition 63, we obtain
1
𝐴ℎ + 𝛼 (ℎ) − ∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)
2 𝑗∈N
+ ⟨𝜁, 𝛼 (ℎ) −
− 𝛼 (ℎ) +
1
∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)⟩)
2 𝑗∈N
1
∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ) .
2 𝑗∈N
(216)
𝜎𝑗 (ℎ) = 𝐷𝜙 (𝑦) 𝜎𝜙,𝜁 (ℎ)
for every 𝑗 ∈ N.
(217)
Moreover, by (195), (197), and Propositions 62 and 63, we
obtain
1
𝐴ℎ + 𝛼 (ℎ) − ∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)
2 𝑗∈N
= 𝐷𝜙 (𝑦) (⟨𝜁, 𝐴ℎ + 𝛼 (ℎ)
= 𝐷𝜙 (𝑦) (⟨𝐴∗ 𝜁, ℎ⟩ + ⟨𝜁, 𝛼 (ℎ)⟩)
(𝑦)))
1
∑ 𝐷𝜙 (𝑦) (⟨𝜁, 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)⟩)
2 𝑗∈N
= 𝐷𝜙 (𝑦) (𝛼𝜙,𝜁 (𝑦) −
𝐴ℎ = 𝐷𝜙 (𝑦) ( ⟨𝐴∗ 𝜁, ℎ⟩
1
− ∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)⟩)
2 𝑗∈N
1
= 𝐴ℎ + 𝛼 (ℎ) − ∑ (𝐷𝜙 (𝑦) (⟨𝜁, 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)⟩)
2 𝑗∈N
= 𝐷𝜙 (𝑦) 𝛼𝜙,𝜁 (𝑦) −
(215)
𝑗
𝑗
(𝑦) , 𝜎𝜙,𝜁
1
∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)⟩) ,
2 𝑗∈N
Together with Lemma 66, this proves the continuity of 𝐴 on
𝑈 ∩ M. Since ℎ0 ∈ M was arbitrary, this proves that 𝐴 is continuous on M.
Furthermore, by (196) and Proposition 62 we have
⟨𝜉, 𝐴ℎ + 𝛼 (ℎ) − 𝐷𝜙 (𝑦) 𝛼𝜙,𝜁 (𝑦)
+𝐷
= 𝐷𝜙 (𝑦) (⟨𝜁, 𝐴ℎ + 𝛼 (ℎ) −
(212)
showing (196). Moreover, for all 𝜉 ∈ D(𝐴∗ ) we have
𝑗
𝜙 (𝑦) (𝜎𝜙,𝜁
1
∑ 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)
2 𝑗∈N
and thus
𝑗 ∈ N.
(211)
2
arbitrary, and set 𝑦 := ⟨𝜁, ℎ⟩ ∈ 𝑉. By relations (195), (197)
and Proposition 62, we obtain
1
∑ ⟨𝜁, 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)⟩) ∈ 𝑇ℎ M,
2 𝑗∈N
−
1
∑ 𝐷𝜙 (𝑦) ⟨𝜁, 𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)⟩
2 𝑗∈N
= 𝐷𝜙 (𝑦) 𝛼𝜙,𝜁 (𝑦)
+
1
𝑗
𝑗
∑ (𝐷2 𝜙 (𝑦) (𝜎𝜙,𝜁 (𝑦) , 𝜎𝜙,𝜁 (𝑦))
2 𝑗∈N
(214)
which proves (197).
(2)⇒(1): Let ℎ0 ∈ M be arbitrary. By Proposition 61
and Remark 60 there exist 𝜁1 , . . . , 𝜁𝑚 ∈ D(𝐴∗ ) and a
parametrization 𝜙 : 𝑉 → 𝑈 ∩ M around ℎ0 such that the
inverse 𝜙−1 : 𝑈 ∩ M → 𝑉 is given by 𝜙−1 = ⟨𝜁, ∙⟩, and
𝜙 has an extension 𝜙 ∈ 𝐶𝑏2 (R𝑚 ; 𝐻). Let ℎ ∈ 𝑈 ∩ M be
(218)
−𝐷𝜎𝑗 (ℎ) 𝜎𝑗 (ℎ)) .
This gives us
𝐴ℎ + 𝛼 (ℎ) = 𝐷𝜙 (𝑦) 𝛼𝜙,𝜁 (𝑦)
+
1
𝑗
𝑗
∑ 𝐷2 𝜙 (𝑦) (𝜎𝜙,𝜁 (𝑦) , 𝜎𝜙,𝜁 (𝑦)) .
2 𝑗∈N
(219)
24
International Journal of Stochastic Analysis
Now, let 𝑌 be the strong solution to (192) with initial
condition 𝑦0 := ⟨𝜁, ℎ0 ⟩ ∈ 𝑉. Since 𝑉 is open, there exists 𝜖 > 0
such that 𝐵𝜖 (𝑦0 ) ⊂ 𝑉. We define the stopping time
𝜏 := inf {𝑡 ≥ 0 : 𝑌𝑡 ∉ 𝐵𝜖 (𝑦0 )} .
(220)
Since the process 𝑌 has continuous sample paths and satisfies
𝑌0 = 𝑦0 , we have 𝜏 > 0 and P-almost surely
𝑌𝑡∧𝜏 ∈ 𝑉
∀𝑡 ≥ 0.
(221)
Therefore, defining the 𝐻-valued process 𝑋 := 𝜙(𝑌), we have
P-almost surely
𝑋𝑡∧𝜏 ∈ 𝑈 ∩ M ∀𝑡 ≥ 0.
(222)
Moreover, using Itô’s formula (Theorem 26) and incorporating (217), (219), we obtain P-almost surely
𝑋𝑡∧𝜏 = 𝜙 (𝑦0 )
+∫
𝑡∧𝜏
(𝐷𝜙 (𝑌𝑠 ) 𝛼𝜙,𝜁 (𝑌𝑠 )
0
+
1
∑ 𝐷2 𝜙 (𝑌𝑠 ) 𝜙 (𝑌𝑠 )
2 𝑗∈N
𝑗
𝑗
× (𝜎𝜙,𝜁 (𝑌𝑠 ) , 𝜎𝜙,𝜁 (𝑌𝑠 ))) 𝑑𝑠
+∑∫
𝑡∧𝜏
𝑗
𝐷𝜙 (𝑌𝑠 ) 𝜎𝜙,𝜁 (𝑌𝑠 ) 𝑑𝛽𝑠𝑗
𝑗∈N 0
(223)
𝑡∧𝜏
= 𝜙 (𝑦0 ) + ∫
0
+∑∫
𝑡∧𝜏
𝑗∈N 0
𝑡∧𝜏
= ℎ0 + ∫
0
+∑∫
𝜎𝑗 (𝜙 (𝑌𝑠 )) 𝑑𝛽𝑠𝑗
(𝐴𝑋𝑠 + 𝛼 (𝑋𝑠 )) 𝑑𝑠
𝑡∧𝜏
𝑗∈N 0
(𝐴𝜙 (𝑌𝑠 ) + 𝛼 (𝜙 (𝑌𝑠 ))) 𝑑𝑠
𝜎𝑗 (𝑋𝑠 ) 𝑑𝛽𝑠𝑗 ,
𝑡 ≥ 0,
showing that 𝑋 is a local strong solution to (44) with lifetime
𝜏.
(3)⇒(1): This implication is a direct consequence of
Proposition 29.
The results from this section are closely related to the existence of finite dimensional realizations, that is, the existence
of invariant manifolds for each starting point ℎ0 , and we point
out the papers [17–22] regarding this topic. Furthermore, we
mention that Theorem 68 has been extended in [23] to SPDEs
with jumps.
Acknowledgments
The author is grateful to Daniel Gaigall, Georg Grafendorfer,
Florian Modler, and Thomas Salfeld for the valuable comments and discussions.
References
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