This paper studies the spatial Sobolev regularity of solutions to stochastic Burgers equations with additive trace class noise, using bootstrap methods and nonlinearity analysis.
Contribution
It provides new insights into the regularity properties of solutions to stochastic Burgers equations with trace class noise.
Findings
01
Established Sobolev regularity results for mild solutions.
02
Analyzed the impact of nonlinearity on solution regularity.
03
Combined bootstrap techniques with nonlinearity analysis.
Abstract
In this article we investigate the spatial Sobolev regularity of mild solutions to stochastic Burgers equations with additive trace class noise. Our findings are based on a combination of suitable bootstrap-type arguments and a detailed analysis of the nonlinearity in the equation.
Equations566
Xt=etAξ+∫0te(t−s)AF(Xs)ds+∫0te(t−s)ABdWs.
Xt=etAξ+∫0te(t−s)AF(Xs)ds+∫0te(t−s)ABdWs.
\mathcal{C}=\Big{\{}C\in({\mathcal{B}}([a,b])\otimes{\mathcal{F}})\colon\Big{(}\Omega\ni\omega\mapsto\int_{a}^{b}{\mathbbm{1}}_{C}(s,\omega)\,ds\in{\mathbbm{R}}\Big{)}\text{ is }{\mathcal{F}}/{\mathcal{B}}({\mathbbm{R}})\text{-measurable}\Big{\}},
\mathcal{C}=\Big{\{}C\in({\mathcal{B}}([a,b])\otimes{\mathcal{F}})\colon\Big{(}\Omega\ni\omega\mapsto\int_{a}^{b}{\mathbbm{1}}_{C}(s,\omega)\,ds\in{\mathbbm{R}}\Big{)}\text{ is }{\mathcal{F}}/{\mathcal{B}}({\mathbbm{R}})\text{-measurable}\Big{\}},
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Full text
Spatial Sobolev regularity for stochastic Burgers
equations with additive trace class noise
Arnulf Jentzen1, Felix Lindner2, and Primož Pušnik3
1 Seminar for Applied Mathematics, Department of Mathematics,
In this article we investigate the spatial Sobolev regularity of mild solutions to stochastic Burgers equations with additive trace class noise. Our findings are based on a combination of suitable bootstrap-type arguments and a detailed analysis of the nonlinearity in the equation.
In the literature, there are nowadays various results on existence, uniqueness, and regularity of solutions
to
stochastic Burgers equations.
In particular, existence and uniqueness results for mild solutions to stochastic Burgers equations with additive space-time white noise and zero Dirichlet boundary conditions on the unit interval (0,1)
taking values in the space
Lp((0,1),\mathbbmR) for p∈[2,∞),
in the space
C([0,1],\mathbbmR) of continuous functions,
and in L2((0,1),\mathbbmR)-Sobolev-type spaces
of order up to \nicefrac12
can be found, e.g., in
Da Prato et al. [7],
Blömker & Jentzen [4],
Jentzen et al. [23],
and Mazzonetto & Salimova [31].
Results on existence, uniqueness, and regularity of solutions to stochastic Burgers equations with multiplicative space-time white noise and zero Dirichlet boundary conditions on the unit interval have been established, e.g., in
Da Prato & Gatarek [8]
and Gyöngy [15].
Existence, uniqueness, and regularity results for solutions to stochastic Burgers equations on the whole real line can be found, e.g., in Bertini et al. [3],
Gyöngy & Nualart [16],
Kim [25],
and
Lewis & Nualart [28].
Results on existence, uniqueness, and regularity of mild solutions to stochastic Burgers equations driven by Lévy noise are presented, e.g., in Dong & Xu [12] and Hausenblas & Giri [17].
We also refer to
Brzeźniak et al. [6],
Da Prato & Zabczyk [10, Section 14],
Da Prato & Zabczyk [11, Section 13.9],
Röckner et al. [34],
and the references mentioned therein for further existence, uniqueness,
and regularity results for stochastic Burgers-type
equations.
In this paper, we
present a higher order regularity result for stochastic Burgers equations with additive trace-class
noise and zero Dirichlet boundary conditions on the unit interval (0,1).
More specifically,
in Theorem 5.10, which is the main result of
this article, we establish the unique existence of mild solutions taking values in L2((0,1),\mathbbmR)-Sobolev-type
spaces of order up to 2. A slightly simplified version of our main result is given in the following
theorem.
Theorem 1.1**.**
Let
(H,⟨⋅,⋅⟩H,∥⋅∥H)
be the \mathbbmR-Hilbert space
of equivalence classes
of Lebesgue-Borel square-integrable functions
from (0,1) to \mathbbmR,
let
A:D(A)⊆H→H
be the
Laplacian with zero Dirichlet boundary conditions on H,
let
(Hr,⟨⋅,⋅⟩Hr,∥⋅∥Hr),
r∈\mathbbmR, be a family of interpolation spaces associated to −A,
let
β∈(−\nicefrac14,∞),
γ∈(\nicefrac14,min{1,\nicefrac12+β}),
T∈(0,∞),
ξ∈H1,
B∈HS(H,Hβ),
let
(Ω,F,P)
be a probability space,
and
let
(Wt)t∈[0,T]
be an
IdH-cylindrical
Wiener process.
Then
(i)
there exists a unique
continuous function
F:H\nicefrac18→H−\nicefrac12
which satisfies for every v∈H\nicefrac12
that F(v)=−v′v
and
2. (ii)
there exists an up to indistinguishability
unique stochastic process
X:[0,T]×Ω→Hγ
with continuous sample paths
which satisfies that
for every
t∈[0,T]
it holds
P-a.s. that
[TABLE]
Theorem 1.1
is a direct consequence of Theorem 5.10
(with
T=T,
ε=1−γ,
c0=1,
c1=−1,
β=β,
γ=γ,
A=A,
Hr=Hr,
(Ω,F,P)=(Ω,F,P),
(Wt)t∈[0,T]=(Wt)t∈[0,T],
B=B,
ξ=(Ω∋ω↦ξ∈H1)
for
r∈\mathbbmR,
γ∈(\nicefrac14,min{1,\nicefrac12+β})
in the notation of
Theorem 5.10)
in
Section 5
below.
Note that the assumption in Theorem 1.1 above that
(Hr,⟨⋅,⋅⟩Hr,∥⋅∥Hr),
r∈\mathbbmR,
is a family of interpolation spaces associated to
−A ensures that for every
r∈[0,∞)
it holds that
(Hr,⟨⋅,⋅⟩Hr,∥⋅∥Hr)=(D((−A)r),⟨(−A)r(⋅),(−A)r(⋅)⟩H,∥(−A)r(⋅)∥H).
The equation in (1)
above is referred to as stochastic evolution equation (SEE)
or stochastic partial differential equation (SPDE) in the scientific literature
and,
roughly speaking,
there are mainly three common approaches for
describing and
analyzing solutions of
SPDEs:
(i) the martingale measure approach (cf., e.g., Walsh [39]),
(ii) the variational (weak solution) approach
(cf., e.g.,
Grecksch & Tudor [14],
Liu & Röckner [29],
Prévôt & Röckner [33],
and
Rozovskiĭ [35]),
and
(iii) the semigroup (mild solution)
approach
(cf., e.g.,
Da Prato & Zabczyk [10, 11],
Grecksch & Tudor [14],
and Liu & Röckner [29])
in the literature.
Theorem 1.1
and most of the other results in this article
are formulated within the semigroup approach.
The proof of
Theorem 1.1
and
Theorem 5.10,
respectively,
is mainly based on
combining
Corollary 2.4,
Lemma 4.16,
Corollary 4.18,
Lemma 5.3,
and
Lemma 5.8.
Corollary 2.4
establishes the unique existence of suitable spatial spectral Galerkin approximations of stochastic Burgers equations
(see the proof of
Lemma 5.9
and (263)
in the proof of Theorem 5.10 below).
An existence and uniqueness result for stochastic differential equations (SDEs) similar to
Corollary 2.4
can be found, e.g., in
Liu & Röckner [29, Theorem 3.1.1].
Lemma 4.16
and
Corollary 4.18
(cf., e.g., Blömker & Jentzen [4, Lemma 4.7])
prove
that the involved nonlinearity F
(see item (i)
in Theorem 1.1 above)
satisfies specific local Lipschitz conditions
(see (273)
in the proof of Theorem 5.10 below).
Lemma 5.3
establishes appropriate
pathwise uniform a priori bounds
for the
spatial spectral Galerkin approximations of the considered stochastic Burgers equation
(see (280) in
the proof of Theorem 5.10 below).
Its proof
is based on consecutive applications of
suitable bootstrap-type arguments
in Section 3
to establish appropriate a priori
bounds for the solution processes
of the considered SDEs
in higher order smoothness spaces.
Related bootstrap-type arguments
can be found, e.g.,
in
Jentzen & Pušnik [21, Section 3],
Jentzen & Röckner [22, Theorem 1],
and
Zhang [40, Section 3].
Lemma 5.8
(cf., e.g., Blömker & Jentzen [4, Lemma 4.3])
demonstrates
pathwise uniform
convergence rates of
spatial spectral Galerkin approximations
of the considered stochastic integral
(see (277) in
the proof of Theorem 5.10 below).
Its proof is essentially based on an application of the factorization method for stochastic convolutions
in Lemma 5.6.
Combining these mentioned results
with
the existence and uniqueness result in Blömker & Jentzen [4, Theorem 3.1]
proves Theorem 5.10.
The remainder of this article is structured as follows.
In Section 2
we recall some elementary
existence and uniqueness results for random
ordinary differential equations (ODEs).
In
Section 3
we
employ bootstrap-type arguments to establish
suitable a priori bounds for certain approximation processes.
In Subsection 4.1
we recall some elementary properties of
Sobolev-Slobodeckij
and interpolation
spaces.
In Subsection 4.2 we
recall and derive several auxiliary results on the regularity properties
of the nonlinearity appearing in the
stochastic Burgers equation.
In Section 5
we combine the results
in
Sections 2–4
to establish the main result of this article
in Theorem 5.10
below.
1.1 General setting
Throughout this article the following
setting is frequently used.
Setting 1.2**.**
For every measurable space
(Ω1,F1) and every
measurable space
(Ω2,F2)
let M(F1,F2) be
the set of all F1/F2-measurable functions from Ω1 to Ω2,
let
(H,⟨⋅,⋅⟩H,∥⋅∥H)
be a separable \mathbbmR-Hilbert space,
let
H⊆H
be a non-empty orthonormal basis of H,
let
v:H→\mathbbmR
be a function which
satisfies
suph∈Hvh<0,
let
A:D(A)⊆H→H
be the linear operator which satisfies
D(A)={v∈H:∑h∈H∣vh⟨h,v⟩H∣2<∞}
and
∀v∈D(A):Av=∑h∈Hvh⟨h,v⟩Hh,
and
let
(Hr,⟨⋅,⋅⟩Hr,∥⋅∥Hr),
r∈\mathbbmR, be a family of interpolation spaces associated to −A
(cf., e.g., [37, Section 3.7]).
Note that the assumption in Setting 1.2 above that
(Hr,⟨⋅,⋅⟩Hr,∥⋅∥Hr),
r∈\mathbbmR,
is a family of interpolation spaces associated to
−A ensures that for every
r∈[0,∞)
it holds that
(Hr,⟨⋅,⋅⟩Hr,∥⋅∥Hr)=(D((−A)r),⟨(−A)r(⋅),(−A)r(⋅)⟩H,∥(−A)r(⋅)∥H).
2 Pathwise solvability for a class of random ODEs
In this section we analyze in Corollary 2.4 the solvability of a specific class of abstract random ODEs. The considered equations can be thought of as spectral Galerkin discretizations in space of an underlying stochastic Burgers equation. Corollary 2.4 is based on an elementary and essentially well-known
pathwise existence and uniqueness result for random ODEs with non-globally Lipschitz continuous coefficient functions presented in Lemma 2.3
(cf., e.g., Liu & Röckner [29, Theorem 3.1.1]).
In addition,
we also recall
elementary results on
measurability
in
Lemma 2.1
(see, e.g.,
Aliprantis & Border [1, Lemma 4.51])
and Lemma 2.2
(cf., e.g.,
in Klenke [26, Theorem 14.16]).
For the sake of completeness we include the proof of Lemma 2.2.
Lemma 2.1**.**
Let (Ω,F) be a measurable space,
let (X,dX) be a separable metric space,
let (Y,dY) be a metric space,
let
f:X×Ω→Y
be a function,
assume for every
x∈X
that
Ω∋ω↦f(x,ω)∈Y
is
F/B(Y)-measurable,
and assume for every
ω∈Ω
that
(X∋x↦f(x,ω)∈Y)∈C(X,Y).
Then it holds that
f:X×Ω→Y
is
(B(X)⊗F)/B(Y)-measurable.
Note that for every topological space (X,τ)
it holds that B(X) is the smallest sigma-algebra
on X which contains all elements of τ.
Lemma 2.2**.**
Let
(X,∥⋅∥X)
be an \mathbbmR-Banach space,
let (Ω,F)
be a measurable space,
let
a∈\mathbbmR, b∈(a,∞),
let
f:[a,b]×Ω→X
be a
strongly
(B([a,b])⊗F)/(X,∥⋅∥X)-measurable function,
assume for every
ω∈Ω
that
∫ab∥f(s,ω)∥Xds<∞,
and let
F:Ω→X
be the function which satisfies
for every
ω∈Ω
that
F(ω)=∫abf(s,ω)ds.
Then it holds that
F is
strongly
F/(X,∥⋅∥X)-measurable.
Throughout this proof
let
λ:B(\mathbbmR)→[0,∞]
be the Lebesgue-Borel
measure on
\mathbbmR,
let
C⊆(B([a,b])⊗F)
be the set given by
[TABLE]
for every set S
let P(S)
be the power set of S, for every set
S
and every
A⊆P(S)
let
σS(A)
be the smallest
sigma-algebra on S
which contains A,
and
for every set
S
and every
A⊆P(S)
let
δS(A)
be
the smallest
Dynkin system on S
which contains A.
First, we intend to prove that
[TABLE]
For this note that for every
A∈B([a,b]),
B∈F,
ω∈Ω
it holds that
[TABLE]
This ensures that
[TABLE]
The fact that
{A×B:A∈B([a,b]),B∈F} is ∩-stable
and
Dynkin’s Lemma
therefore
prove that
[TABLE]
This
shows that
[TABLE]
Moreover, note that for every
C∈C,
ω∈Ω
it holds that
[TABLE]
This and (5) imply that
for
every C∈C
it holds that
[TABLE]
Furthermore,
note that the monotone convergence theorem
proves that for all
pairwise disjoint sets
Cn∈C,
n∈\mathbbmN,
it holds that
[TABLE]
Therefore, we obtain that
for all
pairwise disjoint sets
Cn∈C,
n∈\mathbbmN,
it holds that
∪n∈\mathbbmNCn∈C.
Combining this, (5), and (9)
implies that
C
is a Dynkin system on
[a,b]×Ω.
Combining this
and (7)
establishes (3).
Next we intend to establish the statement of Lemma 2.2.
For this observe that the fact that
f:[a,b]×Ω→X
is strongly
(B([a,b])⊗F)/(X,∥⋅∥X)-measurable and, e.g.,
Prévôt & Röckner [33, Lemma A.1.4]
imply that there exist
(B([a,b])⊗F)/B(X)-measurable functions
fn:[a,b]×Ω→X,
n∈\mathbbmN,
which satisfy that
(a)
it holds
for every n∈\mathbbmN that
fn([a,b]×Ω)
is a finite set and
2. (b)
it holds for every
ω∈Ω
that
[TABLE]
Note that item (a)
shows that
for every
n∈\mathbbmN,
s∈[a,b],
ω∈Ω
it holds that
[TABLE]
The fact that for every n∈\mathbbmN
it holds that
fn([a,b]×Ω)
is a finite set, the fact that
for every n∈\mathbbmN,
x∈X
it holds that
(fn)−1({x})∈(B([a,b])⊗F),
and (3)
hence prove that
for every n∈\mathbbmN
it holds that
Ω∋ω↦∫abfn(s,ω)ds∈X
is strongly
F/(X,∥⋅∥X)-measurable.
Combining
item (a),
item (b),
and, e.g.,
Prévôt & Röckner [33, item (i)
of Proposition A.1.3]
therefore establishes that
F
is strongly
F/(X,∥⋅∥X)-measurable.
The proof of
Lemma 2.2
is thus completed.
∎
Lemma 2.3**.**
Let
(H,∥⋅∥H,⟨⋅,⋅⟩H)
be a separable \mathbbmR-Hilbert space,
let
T∈(0,∞),
s∈[0,T),
let (Ω,F,P,(Ft)t∈[s,T])
be a filtered probability space,
let
ξ:Ω→H
be an Fs/B(H)-measurable function,
let
f:[s,T]×H×Ω→H
and
K:[s,T]×(0,∞)×Ω→[0,∞)
be functions,
assume for every
t∈[s,T],
x∈H
that
Ω∋ω↦f(t,x,ω)∈H
is Ft/B(H)-measurable,
assume for every ω∈Ω,
r∈(0,∞)
that
([s,T]×H∋(t,x)↦f(t,x,ω)∈H)∈C([s,T]×H,H),
([s,T]∋t↦Kt(r,ω)∈[0,∞))∈C([s,T],[0,∞)),
and
supt∈[s,T]supx∈H,∥x∥H≤r∥f(t,x,ω)∥H<∞,
and assume for every
t∈[s,T],
x,y∈H,
ω∈Ω,
r∈(0,∞)
with
max{∥x∥H,∥y∥H}≤r
that
2⟨x,f(t,x,ω)⟩H≤Kt(1,ω)(1+∥x∥H2)
and
[TABLE]
*Then
*
(i)
there exists a unique
function
X:[s,T]×Ω→H
which satisfies
for every
t∈[s,T], ω∈Ω
that
([s,T]∋u↦Xu(ω)∈H)∈C([s,T],H)
and
[TABLE]
and
2. (ii)
it holds that
X:[s,T]×Ω→H
is (Ft)t∈[s,T]-adapted.
Throughout this proof
let
Xn:[s,T]×Ω→H, n∈\mathbbmN,
be the functions
which satisfy for every
n∈\mathbbmN,
k∈{0,1,…,n−1},
t∈(s+nk(T−s),s+n(k+1)(T−s)],
ω∈Ω
that
Xsn(ω)=ξ(ω)
and
[TABLE]
let
L:(0,∞)×Ω→[0,∞)
be the function which satisfies for every
r∈(0,∞),
ω∈Ω
that
[TABLE]
let
κ:\mathbbmN×[s,T]→[s,T]
be the function which
satisfies for every
n∈\mathbbmN,
k∈{0,1,…,n−1},
t∈(s+nk(T−s),s+n(k+1)(T−s)]
that
κ(n,s)=s
and
[TABLE]
let
K:[s,T]×(0,∞)×Ω→[0,∞)
and
α:[s,T]×(0,∞)×Ω→[0,∞)
be the functions which satisfy for every
t∈[s,T],
r∈(0,∞),
ω∈Ω
that
[TABLE]
let
τn:(0,∞)×Ω→[0,T],
n∈\mathbbmN,
be the functions which satisfy for every
n∈\mathbbmN,
r∈(0,∞),
ω∈Ω
that
[TABLE]
and
let
pn:[s,T]×Ω→H, n∈\mathbbmN,
be the functions
which satisfy for every
n∈\mathbbmN,
t∈[s,T],
ω∈Ω
that
[TABLE]
First, we
establish
item (i).
For this note
that
for every
r∈(0,∞),
n∈\mathbbmN,
ω∈Ω,
t∈[s,τrn(ω)]
it holds that
[TABLE]
This ensures
for every
r∈(0,∞),
ω∈Ω
that
[TABLE]
The dominated convergence theorem hence
shows that for every
r∈(0,∞),
ω∈Ω it holds that
[TABLE]
In the next step we observe that
for every
t∈[s,T],
n∈\mathbbmN,
ω∈Ω it holds that
[TABLE]
Furthermore, note that the fact that
for every
ω∈Ω,
x∈H
it holds that
([s,T]∋u↦f(u,x,ω)∈H)∈C([s,T],H)
and, e.g., [19, Corollary 2.7]
(with
V=H,
W=\mathbbmR,
a=s,
b=T,
ϕ=([s,T]×H∋(t,x)↦∥x∥H2e−αt(1,ω)∈\mathbbmR),
f=([s,T]∋t↦f(t,Xκ(n,t)n(ω),ω)∈H),
F=([s,T]∋t↦Xtn(ω)∈H)
for
n∈\mathbbmN,
ω∈Ω
in the notation of [19, Corollary 2.7])
prove that for every
r∈(0,∞),
n∈\mathbbmN,
ω∈Ω,
t∈[s,τrn(ω)]
it holds that
[TABLE]
Combining this,
the assumption that
for every
t∈[s,T],
x∈H,
ω∈Ω
it holds that
2⟨x,f(t,x,ω)⟩H≤Kt(1,ω)(1+∥x∥H2),
the Cauchy-Schwarz inequality,
and (24)
implies that for every
r∈(0,∞),
n∈\mathbbmN,
ω∈Ω,
t∈[s,τrn(ω)]
it holds that
[TABLE]
The fact that
for every
r∈(0,∞),
n∈\mathbbmN,
ω∈Ω,
u∈[s,T]
it holds that
\mathbbm1[s,τrn(ω)](u)∥Xun(ω)∥H2≤r2,
Fatou’s Lemma,
and (23)
hence assure that
for every
r∈(0,∞),
ω∈Ω,
t∈[s,T]
it holds that
[TABLE]
Gronwall’s lemma
therefore demonstrates
that
for every
r∈(0,∞),
ω∈Ω,
t∈[s,T]
it holds that
[TABLE]
The change of variables formula
hence establishes
that
for every
r∈(0,∞),
ω∈Ω,
t∈[s,T]
it holds that
[TABLE]
This shows for every
r∈(0,∞),
ω∈Ω
that
[TABLE]
Therefore, we obtain that
there exist functions
N:Ω→\mathbbmN
and
M:Ω→(0,∞)
which satisfy that
for every
ω∈Ω,
n∈[N(ω),∞)∩\mathbbmN
it holds that
M(ω)=1+eαT(1,ω)∥ξ(ω)∥H2+1
and
[TABLE]
Note that (31) shows that
for every
ω∈Ω,
n∈[N(ω),∞)∩\mathbbmN
it holds that
τM(ω)n(ω)=T
and
[TABLE]
Furthermore,
note that
for every
t∈[s,T],
r∈(0,∞),
ω∈Ω,
m,n∈\mathbbmN
it holds that
[TABLE]
Combining (13),
(32),
and the Cauchy-Schwarz inequality
hence
ensures that for every
t∈[s,T],
ω∈Ω,
m,n∈[N(ω),∞)∩\mathbbmN
it holds that
[TABLE]
This implies
for every
t∈[s,T],
ω∈Ω,
m,n∈[N(ω),∞)∩\mathbbmN
that
[TABLE]
Moreover,
note that (32)
establishes for every
u∈[s,T],
ω∈Ω,
m,n∈[N(ω),∞)∩\mathbbmN
that
[TABLE]
Combining
this and (LABEL:eq:BigEstimate)
shows that for every
t∈[s,T],
ω∈Ω,
m,n∈[N(ω),∞)∩\mathbbmN
it holds that
[TABLE]
In addition, observe that
the fact that
for every ω∈Ω,
n∈[N(ω),∞)∩\mathbbmN
it holds that
τM(ω)n(ω)=T,
(23),
and (32)
assure that for every ω∈Ω it holds that
[TABLE]
This and (LABEL:eq:CauchyEstimate)
demonstrate that for every
ω∈Ω
it holds
that
([s,T]∋t↦Xtn(ω)∈H)∈C([s,T],H),
n∈\mathbbmN,
is a Cauchy sequence.
The fact that the space
C([s,T],H)
with the supremum norm
is complete
hence ensures that
there exists
a function
X:[s,T]×Ω→H
which satisfies for every
ω∈Ω
that
([s,T]∋t↦Xt(ω)∈H)∈C([s,T],H)
and
[TABLE]
Observe that
the assumption that
for every
ω∈Ω
it holds that
([s,T]×H∋(t,x)↦f(t,x,ω)∈H)∈C([s,T]×H,H),
the assumption that
for every
r∈(0,∞),
ω∈Ω
it holds that
supt∈[s,T]supx∈H,∥x∥H≤r∥f(t,x,ω)∥H<∞,
(32),
(39),
and
the
dominated convergence theorem
prove that for every
t∈[s,T],
ω∈Ω
it holds that
[TABLE]
Moreover,
observe that (39)
assures that for every
ω∈Ω
it holds that
the sequence
Xn(ω)∈C([s,T],H),
n∈\mathbbmN,
is uniformly equicontinuous.
This implies for every
ω∈Ω that
[TABLE]
The assumption that
for every
ω∈Ω
it holds that
([s,T]×H∋(t,x)↦f(t,x,ω)∈H)∈C([s,T]×H,H),
the assumption that
for every
r∈(0,∞),
ω∈Ω
it holds that
supt∈[s,T]supx∈H,∥x∥H≤r∥f(t,x,ω)∥H<∞,
(32),
and the dominated convergence theorem therefore show that
for every
t∈[s,T],
ω∈Ω
it holds that
[TABLE]
The triangle inequality and (LABEL:eq:Part1)
hence
ensure
that
for every
t∈[s,T],
ω∈Ω
it holds that
[TABLE]
Combining this, (24),
and (39)
implies
that for every
t∈[s,T],
ω∈Ω
it holds that
[TABLE]
Next note that (13)
proves that for every function
X:[s,T]×Ω→H
with
∀ω∈Ω:([s,T]∋t↦Xt(ω)∈H)∈C([s,T],H)
and
∀t∈[s,T],
ω∈Ω:Xt(ω)=ξ(ω)+∫stf(u,Xu(ω),ω)du
and every
t∈[s,T],
ω∈Ω,
r∈(supu∈[s,T]max{∥Xu(ω)∥H,∥Xu(ω)∥H},∞)
it holds that
[TABLE]
Gronwall’s lemma hence implies that
for every function
X:[s,T]×Ω→H
with
∀ω∈Ω:([s,T]∋t↦Xt(ω)∈H)∈C([s,T],H)
and
∀t∈[s,T],
ω∈Ω:Xt(ω)=ξ(ω)+∫stf(u,Xu(ω),ω)du
and every
t∈[s,T], ω∈Ω
it holds
that
[TABLE]
Combining this
and (44) establishes item (i).
In addition,
note that
the assumption that
for every
ω∈Ω
it holds that
([s,T]×H∋(t,x)↦f(t,x,ω)∈H)∈C([s,T]×H,H),
the assumption that
for every
t∈[s,T],
u∈[s,t],
x∈H
it holds that
Ω∋ω↦f(u,x,ω)∈H
is Ft/B(H)-measurable,
and
Lemma 2.1
(with
(Ω,F)=(Ω,Ft),
X=[s,T]×H,
dX=([s,T]×H×[s,T]×H∋(t1,x1,t2,x2)↦∣t1−t2∣+∥x1−x2∥H∈[0,∞)),
Y=H,
dY=(H×H∋(x1,x2)↦∥x1−x2∥H∈[0,∞)),
f=([s,t]×H×Ω∋(u,x,ω)↦f(u,x,ω)∈H)
for
t∈[s,T]
in the notation of Lemma 2.1)
show that
for every
t∈[s,T]
it holds that
[TABLE]
is (B([s,t])⊗B(H)⊗Ft)/B(H)-measurable.
The fact that
for every t∈[s,T]
and every Ft/B(H)-measurable
function ζ:Ω→H
it holds that
[s,t]×Ω∋(u,ω)↦(u,ζ(ω),ω)∈[s,t]×H×Ω
is
(B([s,t])⊗Ft)/(B([s,t])⊗B(H)⊗Ft)-measurable
hence assures that
for every
t∈[s,T]
and every Ft/B(H)-measurable
function ζ:Ω→H
it holds that
[TABLE]
is
(B([s,t])⊗Ft)/B(H)-measurable.
The assumption that
ξ:Ω→H
is Fs/B(H)-measurable,
(15),
and
Lemma 2.2
(with
X=H,
Ω=Ω,
F=Ft,
a=s+(\nicefrack(T−s)n),
b=t,
f=([s+(\nicefrack(T−s)n),t]×Ω∋(u,ω)↦f(u,Xs+(\nicefrack(T−s)n)n(ω),ω)∈H),
F=(Ω∋ω→∫s+(\nicefrack(T−s)n)tf(u,Xs+(\nicefrack(T−s)n)n(ω),ω)du∈H)
for
t∈(s+(\nicefrack(T−s)n),s+(\nicefrac(k+1)(T−s)n)],
k∈{0,1,…,n−1},
n∈\mathbbmN
in the notation of
Lemma 2.2)
therefore
imply that
for every n∈\mathbbmN it holds that
(Xtn)t∈[s,T]
is (Ft)t∈[s,T]-adapted.
Combining this and (39)
establishes item (ii).
The proof of Lemma 2.3
is thus completed.
∎
Corollary 2.4**.**
Assume Setting 1.2,
assume that
dim(H)<∞,
let
T∈(0,∞),
s∈[0,T],
C,c∈[0,∞),
δ,κ∈\mathbbmR,
F∈C(H,H),
Φ∈C(H,[0,∞)),
let
(Ω,F,P,(Ft)t∈[0,T])
be a filtered probability space,
let
ξ∈M(Fs,B(H)),
let
O:[0,T]×Ω→H
be an (Ft)t∈[0,T]-adapted
stochastic
process with continuous sample paths,
and assume
for every
x,y∈H
that
∥F(x)−F(y)∥H≤C∥x−y∥Hδ(1+∥x∥Hκc+∥y∥Hκc)
and
⟨x,Ax+F(x+y)⟩H≤Φ(y)(1+∥x∥H2).
Then
(i)
there exists a unique function
X:[s,T]×Ω→H
which satisfies for every
t∈[s,T],
ω∈Ω
that
([s,T]∋u↦Xu(ω)∈H)∈C([s,T],H)
and
[TABLE]
and
2. (ii)
it holds that
X:[s,T]×Ω→H
is (Ft)t∈[s,T]-adapted.
Throughout this proof let
K:(0,∞)×Ω→[0,∞)
be the function which satisfies for every
r∈(0,∞),
ω∈Ω
that
K(r,ω)=max{C∥(−A)δ∥L(H)max{∥(−A)κ∥L(H)c,1}(1+2(r+supu∈[0,T]∥Ou(ω)∥H)c),supu∈[0,T]Φ(Ou(ω))}.
Note that
the assumption that
for every
x,y∈H
it holds that
∥F(x)−F(y)∥H≤C∥x−y∥Hδ(1+∥x∥Hκc+∥y∥Hκc)
implies that for every
t∈[0,T],
x,y∈H,
r∈(0,∞),
ω∈Ω
with
min{∥x∥H,∥y∥H}≤r
it holds that
[TABLE]
In addition,
observe that
the assumption that for every
x,y∈H
it holds that
⟨x,Ax+F(x+y)⟩H≤Φ(y)(1+∥x∥H2)
shows that for every
t∈[0,T],
x,y∈H,
ω∈Ω
it holds that
[TABLE]
Moreover, note that the
assumption that
dim(H)<∞,
the assumption that F∈C(H,H),
and
the assumption that
O:[0,T]×Ω→H
has continuous sample paths
ensure that
for every
r∈(0,∞),
ω∈Ω
it holds that
([s,T]×H∋(u,x)↦(Ax+F(x+Ou(ω)))∈H)∈C([s,T]×H,H)
and
[TABLE]
The assumption that
(Ot)t∈[0,T] is
(Ft)t∈[0,T]-adapted,
(LABEL:eq:Monotone),
(51),
and Lemma 2.3
(with
H=H,
T=T,
s=s,
(Ω,F,P,(Fu)u∈[s,T])=(Ω,F,P,(Fu)u∈[s,T]),
ξ=ξ−Os,
f=([s,T]×H×Ω∋(u,h,ω)↦Ah+F(h+Ou(ω))∈H),
Kt(r,ω)=2K(r,ω)
for
t∈[s,T],
r∈(0,∞)
in the notation of Lemma 2.3)
therefore prove that
(a)
there exists a unique function
X:[s,T]×Ω→H
which satisfies for every
t∈[s,T],
ω∈Ω
that
([s,T]∋u↦Xu(ω)∈H)∈C([s,T],H)
and
[TABLE]
and
2. (b)
it holds that
X:[s,T]×Ω→H
is (Ft)t∈[s,T]-adapted.
Next let X:[s,T]×Ω→H
be the stochastic process with continuous sample paths
which satisfies for every
t∈[s,T],
ω∈Ω
that
[TABLE]
In addition, observe that (53)
implies for every
t∈[s,T],
ω∈Ω
that
[TABLE]
This and (54)
show that for every
t∈[s,T],
ω∈Ω
it holds that
[TABLE]
Moreover, observe that for every function
Y:[s,T]×Ω→H
with
∀ω∈Ω:([s,T]∋t↦Yt(ω))∈C([s,T],H)
and
∀t∈[s,T],
ω∈Ω:Yt(ω)=e(t−s)Aξ(ω)+∫ste(t−u)AF(Yu(ω))du+Ot(ω)−e(t−s)AOs(ω)
and every
t∈[s,T],
ω∈Ω
it holds that
Yt(ω)−Ot(ω)=ξ(ω)−Os(ω)+∫st[A(Yu(ω)−Ou(ω))+F([Yu(ω)−Ou(ω)]+Ou(ω))]du.
The fact that
for every
ω∈Ω
it holds that
([s,T]∋t↦[Xt(ω)−Ot(ω)]∈H)∈C([s,T],H),
item (a),
and (56)
therefore establish
item (i).
Furthermore,
note that
item (b),
the fact that
(Ot)t∈[0,T] is (Ft)t∈[0,T]-adapted,
and (54)
establish item (ii).
The proof of
Corollary 2.4
is thus completed.
∎
3 Strong a priori bounds based on bootstrap-type arguments
In this section we provide in
Lemmas 3.2–3.4
appropriate a priori bounds for the
approximation process (Yt)t∈[0,T] introduced in
Setting 3.1 below.
The considered equations can,
in particular, be thought of as discretizations in space and time of an underlying stochastic Burgers equation.
The proofs of Lemmas 3.2–3.4
are based on suitable bootstrap-type arguments,
which have been intensively used in the literature to establish regularity properties of solutions to
(stochastic) evolution equations
(cf., e.g.,
[21, 22]
and the references mentioned therein).
Setting 3.1**.**
Assume Setting 1.2,
let (Ω,F,P)
be a probability space,
let
T∈(0,∞),
β∈[0,1),
γ∈[0,β],
ξ∈M(F,B(Hβ)),
F∈M(B(Hγ),B(H)),
κ∈M(B([0,T]),B([0,T])),
Z∈M(B([0,T])⊗F,B(Hγ))
satisfy for every
t∈[0,T] that
κ(t)≤t
and
supu∈[0,T]∥Zu∥H+∫0t∥e(t−κ(s))AF(Zs)∥Hds<∞,
and let
O:[0,T]×Ω→Hβ
and
Y:[0,T]×Ω→H
be
stochastic processes
with continuous sample paths
which satisfy for every t∈[0,T] that
Yt=etAξ+∫0te(t−κ(s))AF(Zs)ds+Ot.
Lemma 3.2**.**
Assume Setting 3.1,
let
p∈[1,∞),
ρ∈[0,β],
α∈[0,1−ρ),
and assume that
[TABLE]
Then
(i)
it holds for every
t∈[0,T] that Yt(Ω)⊆Hρ,
2. (ii)
Throughout this proof assume w.l.o.g. that
supv∈Hγ∥F(v)∥H>0.
Note that
the assumption that
∀t∈[0,T]:κ(t)≤t
implies
that
for every
t∈(0,T]
it holds
that
[TABLE]
Hence, we obtain that for every t∈(0,T] it holds that
[TABLE]
The triangle inequality,
the assumption that
supu∈[0,T]∥Zu∥H<∞,
and (57)
therefore
prove that for every
t∈[0,T] it holds that
Yt(Ω)⊆Hρ
and
[TABLE]
This establishes items (i)
and (ii).
Next note that (LABEL:eq:SomeImport)
and
Minkowski’s integral inequality
(see, e.g., [18, Proposition 8 in A.1])
ensure that
for every t∈(0,T] it holds that
[TABLE]
The triangle inequality
therefore
establishes
item (iii).
This completes the proof of Lemma 3.2.
∎
Lemma 3.3**.**
Assume Setting 3.1,
let
p∈[1,∞),
ρ∈[0,β],
η∈[ρ,β],
α1∈[0,1−ρ),
α2∈[0,1−η),
and
assume for every
t∈[0,T] that
Zt(Ω)⊆Hρ,
supu∈[0,T]∥Zu∥Hρ≤supu∈[0,T]∥Yu∥Hρ,
supu∈[0,T]∥Zu∥L2p(P;Hρ)≤supu∈[0,T]∥Yu∥L2p(P;Hρ),
and
[TABLE]
Then
(i)
it holds for every t∈[0,T] that
Yt(Ω)⊆Hη,
2. (ii)
Throughout this proof assume w.l.o.g. that
supv∈Hγ∥F(v)∥H>0.
Note that
the assumption that
∀t∈[0,T]:κ(t)≤t
implies
that
for every
t∈(0,T]
it holds
that
[TABLE]
Hence, we obtain that for every t∈(0,T] it holds that
[TABLE]
Next observe that (LABEL:eq:more_regularity_drift0)
and
Minkowski’s integral inequality
(see, e.g., [18, Proposition 8 in A.1])
ensure that
for every t∈(0,T] it holds that
[TABLE]
Moreover,
note that (64)
and
Lemma 3.2
(with
p=2p,
ρ=ρ,
α=α1
in the notation of Lemma 3.2)
imply that
(a)
it holds for every
t∈[0,T] that Yt(Ω)⊆Hρ,
2. (b)
it holds
that
[TABLE]
and
3. (c)
it holds
that
[TABLE]
Observe that the triangle inequality,
(64),
(68)
and
item (b)
ensure that for every t∈[0,T] it holds that
Yt(Ω)⊆Hη
and
[TABLE]
This
establishes items (i)
and (ii).
Furthermore,
observe that
the triangle inequality
and (LABEL:eq:more_regularity_drift0b)
prove that for every t∈[0,T] it holds that
[TABLE]
Combining this
and
item (c)
establishes
item (iii).
The proof of Lemma 3.3
is thus completed.
∎
Lemma 3.4**.**
Assume Setting 3.1,
let
p∈[1,∞),
ρ∈[0,β],
η∈[ρ,β],
ι∈[η,β],
α1∈[0,1−ρ),
α2∈[0,1−η),
and assume for every
t∈[0,T] that
Zt(Ω)⊆Hη,
supu∈[0,T]∥Zu∥Hρ≤supu∈[0,T]∥Yu∥Hρ,
supu∈[0,T]∥Zu∥Hη≤supu∈[0,T]∥Yu∥Hη,
supu∈[0,T]∥Zu∥L4p(P;Hρ)≤supu∈[0,T]∥Yu∥L4p(P;Hρ),
supu∈[0,T]∥Zu∥L2p(P;Hη)≤supu∈[0,T]∥Yu∥L2p(P;Hη),
and
[TABLE]
Then
(i)
it holds for every
t∈[0,T] that
Yt(Ω)⊆Hι,
2. (ii)
Throughout this proof
assume w.l.o.g. that
supv∈Hγ∥F(v)∥H>0.
Observe that
the assumption that
∀t∈[0,T]:κ(t)≤t
implies
that
for every
t∈(0,T]
it holds
that
[TABLE]
Hence, we obtain that for every t∈(0,T] it holds that
[TABLE]
Moreover, note that (LABEL:eq:estimateTT0)
and
Minkowski’s integral inequality
(see, e.g., [18, Proposition 8 in A.1])
prove that for every t∈(0,T] it holds that
[TABLE]
Next observe that (74),
the assumption that
supu∈[0,T]∥Zu∥Hρ≤supu∈[0,T]∥Yu∥Hρ,
the assumption that
supu∈[0,T]∥Zu∥L4p(P;Hρ)≤supu∈[0,T]∥Yu∥L4p(P;Hρ),
and
Lemma 3.3
(with p=2p,
ρ=ρ,
η=η,
α1=α1,
α2=α2
in the notation of
Lemma 3.3)
show that
(a)
it holds for every t∈[0,T] that
Yt(Ω)⊆Hη,
2. (b)
it holds
that
[TABLE]
and
3. (c)
it holds
that
[TABLE]
Note that
the triangle inequality,
(74),
(78),
and item (b)
ensure that for every t∈[0,T] it holds that
Yt(Ω)⊆Hι
and
[TABLE]
This establishes
items (i)
and (ii).
Furthermore, observe that the triangle inequality
and (LABEL:eq:estimateTT0b)
prove that for every t∈[0,T] it holds that
[TABLE]
Combining
this
and
item (c)
establishes item (iii).
The proof of Lemma 3.4
is thus completed.
∎
4 Properties of the nonlinearity
In this section
we recall and derive
in
Subsection 4.1
and
in Subsection 4.2 some
partially well-known properties of
certain Sobolev spaces
and the nonlinearity
appearing in the stochastic Burgers equations,
respectively.
We employ these results
to establish
in
Theorem 5.10
in
Section 5
below
the main result of this article.
Setting 4.1**.**
Assume Setting 1.2,
let
λ:B((0,1))→[0,1]
be the Lebesgue-Borel
measure on
(0,1),
for every measure space
(Ω,F,μ),
every measurable space (S,S),
every set R,
and every function
f:Ω→R
let
[f]μ,S={g∈M(F,S):(∃D∈F:μ(D)=0and{ω∈Ω:f(ω)=g(ω)}⊆D)},
let
c0∈(0,∞),
c1∈\mathbbmR,
assume that
(H,⟨⋅,⋅⟩H,∥⋅∥H)=(L2(λ;\mathbbmR),⟨⋅,⋅⟩L2(λ;\mathbbmR),
∥⋅∥L2(λ;\mathbbmR)),
let
(en)n∈\mathbbmN⊆H
satisfy
for every
n∈\mathbbmN
that
en=[(2sin(nπx))x∈(0,1)]λ,B(\mathbbmR),
assume that
H={en:n∈\mathbbmN},
assume for every n∈\mathbbmN that
ven=−c0π2n2,
for every
v∈W1,2((0,1),\mathbbmR)
let
∂v∈H
satisfy for every
φ∈Ccpt∞((0,1),\mathbbmR)
that
⟨∂v,[φ]λ,B(\mathbbmR)⟩H=−⟨v,[φ′]λ,B(\mathbbmR)⟩H,
and
let
F:H\nicefrac12→H
be the function which
satisfies
for every
w∈H\nicefrac12
that
F(w)=c1w∂w.
Note that for every s∈[0,∞), p∈[1,∞) it holds that
(Ws,p((0,1),\mathbbmR),∥⋅∥Ws,p((0,1),\mathbbmR))
is the Sobolev-Slobodeckij space with smoothness parameter s and integrability parameter p
of equivalence classes
of B((0,1))/B(\mathbbmR)-measurable
functions.
4.1 Auxiliary results on Sobolev and interpolation spaces
In this subsection we recall some elementary
properties of the involved Sobolev
and interpolation spaces.
Lemmas 4.2–4.5,
Lemma 4.6
(cf., e.g., Fujiwara [13]),
Lemmas 4.7–4.10,
Lemma 4.11
(cf., e.g., Brezis [5, Exercise 8.15 and (42)
in the section
Comments on Chapter 8]
and Nirenberg [32]),
and
Lemma 4.12
(see, e.g., Sell & You [37, Theorem B.2])
below
are used for the regularity analysis of the
considered
nonlinearity
in Subsection 4.2 below.
Lemma 4.2**.**
Assume Setting 4.1.
Then it holds
for every
ρ∈[\nicefrac12,∞)
that
∑h∈H∣vh∣−2ρ≤\nicefrac∣c0∣−2ρ6,
suph∈H∥∂h∥H∣vh∣−ρ≤∣c0∣−ρ,
and
suph∈H∥h∥L∞(λ;\mathbbmR)=2.
Note that, e.g.,
Lunardi [30, Example 4.34]
ensures that
[TABLE]
This,
the fact that
for every
v∈W1,2((0,1),\mathbbmR)
it holds that
[TABLE]
and
the fact that
for every
v∈H\nicefrac12
it holds that
[TABLE]
(see, e.g., [23, Lemma 6.1])
establish item (i).
Moreover, observe
that (87)–(89)
and
Poincaré’s inequality
(see,
e.g., Brezis [5, Proposition 8.13])
show
item (ii).
Next note that
Lemma 4.2
(with ρ=\nicefrac12
in the notation of Lemma 4.2)
and,
e.g., [20, Lemma 4.3]
(with
d=1,
H=H,
ρ=\nicefrac12,
v=v
for
v∈H\nicefrac12
in the notation of [20, Lemma 4.3])
prove that for every
v∈H\nicefrac12 it holds that
[TABLE]
This
and item (i)
establish item (iii).
Moreover, note that (89)
shows item (iv).
In addition, observe that (90)
establishes item (v).
The proof of Lemma 4.3
is thus completed.
∎
Lemma 4.4**.**
Assume Setting 4.1
and let
u∈W01,2((0,1),\mathbbmR),
v∈W1,2((0,1),\mathbbmR).
Then it holds that
Throughout this proof let
(un)n∈\mathbbmN⊆Cc∞(\mathbbmR,\mathbbmR),
(vn)n∈\mathbbmN⊆Cc∞(\mathbbmR,\mathbbmR),
(un)n∈\mathbbmN⊆W01,2((0,1),\mathbbmR),
(vn)n∈\mathbbmN⊆W1,2((0,1),\mathbbmR)
satisfy for every
n∈\mathbbmN,
x∈((−∞,0]∪[1,∞))
that
un(x)=0,
un=[un∣(0,1)]λ,B(\mathbbmR),
vn=[vn∣(0,1)]λ,B(\mathbbmR),
and
limsupm→∞(∥u−um∥W1,2((0,1),\mathbbmR)+∥v−vm∥W1,2((0,1),\mathbbmR))=0.
Observe that
integration by parts
and the fact that
for every
n∈\mathbbmN it holds that
un(0)=un(1)=0
demonstrate that
(cf., e.g.,
Lunardi [30, Example 4.34]
and
Sell & You [37, Section 3.8.1])
and
the fact that
D(−A)=H1
prove that
[TABLE]
Hence, we obtain
that
for every v∈H1
it holds that
∂v∈W1,2((0,1),\mathbbmR).
The fact that
for every
n∈\mathbbmN
it holds that
en∈W01,2((0,1),\mathbbmR)
and
Lemma 4.4
(with
u=en,
v=∂v
for
n∈\mathbbmN,
v∈H1
in the notation of
Lemma 4.4)
therefore prove
that for every v∈H1
it holds that
[TABLE]
Furthermore,
note that
item (ii)
of Lemma 4.3
assures that for every
v∈H1
it holds that
v∈W01,2((0,1),\mathbbmR).
Combining (96),
the fact that
for every
n∈\mathbbmN
it holds that
∂en∈W1,2((0,1),\mathbbmR),
(95),
and
Lemma 4.4
(with
u=v,
v=∂en
for
n∈\mathbbmN,
v∈H1
in the notation of
Lemma 4.4)
hence shows that
for every v∈H1 it holds that
[TABLE]
This proves that for every
v∈H1
it holds that
∂2v∈H
and
[TABLE]
The fact that
for every
v∈W2,2((0,1),\mathbbmR)
it holds that
∥v∥W2,2((0,1),\mathbbmR)2=∥v∥H2+∥∂v∥H2+∥∂2v∥H2
and (95)
hence
ensure that for every v∈H1
it holds that
[TABLE]
Next note that
item (ii)
of Lemma 4.3
and
Poincaré’s inequality
(see,
e.g., Brezis [5, Proposition 8.13])
imply that
there exists C∈(0,∞)
such that
for every
v∈H\nicefrac12 it holds that
∥v∥W1,2((0,1),\mathbbmR)≤C∥∂v∥H.
Combining this,
(95),
and (98)
proves that
there exists C∈(0,∞)
such that
for every
v∈H1 it holds that
[TABLE]
Item (iv)
of Lemma 4.3
hence shows that
there exists C∈(0,∞)
such that
for every v∈H1 it holds that
v∈W2,2((0,1),\mathbbmR)
and
[TABLE]
This
establishes item (i).
Moreover, observe that item (i)
and (99)
imply
item (ii).
The proof of Lemma 4.5
is thus completed.
∎
Throughout this proof consider the notation in
Triebel [38, Section 1.3.2 on page 24]
(cf., e.g., Lunardi [30, Definition 1.2]).
Note that
item (i) of
Lemma 4.5
ensures that
[TABLE]
continuously.
Furthermore,
observe that,
e.g.,
Triebel [38, the theorem in Section 1.18.10 on page 142]
(cf., e.g., Lunardi [30, Theorem 4.36])
and the fact that
∀s∈[0,∞):(D((−A)s),∥(−A)s(⋅)∥H)=(Hs,∥⋅∥Hs)
prove that for every
s∈(0,1) it holds that
[TABLE]
and
[TABLE]
This,
(102),
and, e.g.,
Lunardi [30, Theorem 1.6]
imply that for every
s∈(0,1) it holds that
[TABLE]
continuously.
The fact that
for every s∈(0,1) it holds that
[TABLE]
continuously
(cf., e.g.,
Triebel [38, Definition 1
in Section 4.2.1
on page 310,
Theorem 1 in Section 4.3.1
on page 317,
item (a) in Theorem 1
in Section 4.4.2
on page 323,
and
Remark 2 in Section 4.4.2
on page 324])
hence establishes item (i).
Moreover,
note that,
e.g.,
Triebel [38, the theorem in Section 1.18.10 on page 142]
(cf., e.g.,
Lunardi [30, Theorem 4.36])
and the fact that
∀s∈[0,∞):(D((−A)s),∥(−A)s(⋅)∥H)=(Hs,∥⋅∥Hs)
prove that for every
s∈(0,1) it holds that
[TABLE]
and
[TABLE]
The fact that
for every
s∈(0,1)\{\nicefrac12}
it holds that
[TABLE]
(cf., e.g.,
Triebel [38, Definition 1 and Definition 2 in Section 4.2.1 on page 310,
the definition in Section 4.3.2 on page 317,
item (c) in Theorem 1
and
Theorem 2 in Section 4.3.2 on page 318,
item (a) in Theorem 1
in Section 4.4.2
on page 323,
and
Remark 2 in Section 4.4.2 on page 324]),
items (i) and (ii)
of Lemma 4.3,
and, e.g.,
Lunardi [30, Theorem 1.6]
therefore
assure that for every
s∈(0,1)\{\nicefrac12} it holds that
[TABLE]
and
[TABLE]
This establishes items (ii)
and (iii).
The proof of Lemma 4.6
is thus completed.
∎
Lemma 4.7**.**
*Let
s∈[0,∞),
q,r∈[s,∞)
satisfy
r+q−s>\nicefrac12.
Then
*
(i)
it holds
for every
f∈Wq,2((0,1),\mathbbmR),
g∈Wr,2((0,1),\mathbbmR) that
fg∈Ws,2((0,1),\mathbbmR)
and
2. (ii)
Throughout this proof consider the notation in
Triebel [38, Section 1.3.2 on page 24]
(cf., e.g., Lunardi [30, Definition 1.2])
and
let
∂ˉ:H→H−\nicefrac12
be the continuous linear function
which satisfies for every
v∈W1,2((0,1),\mathbbmR)
that
∂ˉv=∂v
(cf. item (i) of Lemma 4.8).
Observe that,
e.g.,
Triebel [38, the theorem in Section 1.18.10 on page 142]
(cf., e.g.,
Lunardi [30, Theorem 4.36])
and the fact that
∀s∈[0,∞):(D((−A)s),∥(−A)s(⋅)∥H)=(Hs,∥⋅∥Hs)
prove that for every
s∈(0,1)
it holds that
[TABLE]
and
[TABLE]
The fact that
for every r∈[0,∞)
it holds that
(Hr)′
and
H−r are isometrically isomorphic
and, e.g.,
Triebel [38, item (b) of
the theorem
in Section 1.3.3
on page 25
and the theorem in Section 1.11.2
on page 69]
(cf., e.g.,
Lunardi [30, Theorem 1.18])
hence
imply that for every
s∈(0,1)
it holds that
[TABLE]
and
[TABLE]
In addition, note that, e.g.,
Triebel [38, Definition 1
in Section 4.2.1
on page 310,
Theorem 1 in Section 4.3.1
on page 317,
item (a) in Theorem 1
in Section 4.4.2
on page 323,
and
Remark 2 in Section 4.4.2
on page 324]
ensures that
for every s∈(0,1)
it holds that
[TABLE]
and
[TABLE]
Furthermore,
observe that
item (ii)
of
Lemma 4.8
ensures that for every
v∈H
it holds
that
[TABLE]
Combining this,
the fact that
for every v∈W1,2((0,1),\mathbbmR)
it holds that
[TABLE]
(117)–(120),
and, e.g,
Lunardi [30, Theorem 1.6]
establishes (114).
The proof of
Lemma 4.9
is thus completed.
∎
Lemma 4.10**.**
Assume Setting 4.1
and let
α∈(\nicefrac14,∞).
Then it holds for every
v∈Hα+(\nicefrac12)
that
Note that the fact that
∀v∈H\nicefrac12:∑n=1∞∣ven∣∣⟨en,v⟩H∣2=∥(−A)\nicefrac12v∥H2=∥v∥H\nicefrac122<∞
shows that
for every
v∈H\nicefrac12
it holds that
[TABLE]
In addition, observe that
items (ii)
and (iv)
of Lemma 4.3
ensure that
(H\nicefrac12∋u↦∂u∈H)∈L(H\nicefrac12,H).
Combining (124)
and the Cauchy-Schwarz inequality
hence
implies that
for every
v∈Hα+(\nicefrac12)
it holds that
Let λ:B((0,1))→[0,1] be the
Lebesgue-Borel measure on (0,1)
and
let
q,r∈[1,∞),
α∈(0,1)
satisfy
α(q1+1−r1)=q1.
Then there exists
C∈(0,∞)
such that for every
u∈W01,r((0,1),\mathbbmR)
it holds that
Throughout this proof
let
p∈\mathbbmR
satisfy
q=p(α1−1),
for every function
f:(0,1)→\mathbbmR
let
[f]λ,B(\mathbbmR)
be the set given by
[TABLE]
let
(⋅):{[v]λ,B(\mathbbmR):(v:(0,1)→\mathbbmR is uniformly continous)}→C([0,1],\mathbbmR)
be the function which satisfies
for every
v∈C([0,1],\mathbbmR)
that
[TABLE]
for every
u∈W1,r((0,1),\mathbbmR)
let
∂u∈Lr(λ;\mathbbmR)
satisfy for every
φ∈Ccpt∞((0,1),\mathbbmR),
v∈Lr(λ;\mathbbmR)
with v∈∂u
that
∫(0,1)u(x)φ′(x)dx=−∫(0,1)v(x)φ(x)dx,
and let
G:\mathbbmR→\mathbbmR
be the function
which satisfies for every
t∈\mathbbmR
that
G(t)=∣t∣α1−1t.
Note that
(a)
it holds that
G(0)=0,
2. (b)
it holds that
G∈C1(\mathbbmR,\mathbbmR),
and
3. (c)
it holds for every
t∈[0,1] that
G′(t)=α1∣t∣α1−1.
This and, e.g.,
Brezis [5, Corollary 8.1]
show that for every
u∈W1,r((0,1),\mathbbmR)
it holds that
[TABLE]
and
[TABLE]
Combining this and, e.g.,
Brezis [5, Theorem 8.2]
ensures that
for every
u∈W1,r((0,1),\mathbbmR),
v∈Lr(λ;\mathbbmR),
x∈[0,1]
with
v∈∂u
it holds that
[TABLE]
This implies that
for every
u∈W1,r((0,1),\mathbbmR),
v∈Lr(λ;\mathbbmR)
with
v∈∂u,
u(0)=0
it holds that
[TABLE]
Next observe that the fact that
r1=q1+1−qα1
and
the fact that
p1=qα1−q1
ensure that
p1+r1=1.
Combining this with (LABEL:eq:sup_Estimate)
and
Hölder’s inequality
demonstrates that
for every
u∈W01,r((0,1),\mathbbmR)
it holds that
Let λ:B((0,1))→[0,1] be the
Lebesgue-Borel measure on (0,1)
and
let
q∈[1,∞),
p∈(q,∞),
r∈(1,∞),
α∈(0,1)
satisfy
α(q1+1−r1)=q1−p1.
Then there exists
C∈(0,∞)
such that for every
u∈W01,r((0,1),\mathbbmR) it holds that
Throughout this proof let
β=p−qαp.
Note that
Hölder’s inequality
proves that
for every
u∈W1,r((0,1),\mathbbmR)
it holds that
[TABLE]
Lemma 4.11
(with
q=q,
r=r,
α=β
in the notation of
Lemma 4.11)
hence shows that
there exists
C∈(0,∞)
such that for every
u∈W01,r((0,1),\mathbbmR)
it holds that
[TABLE]
This implies that
there exists
C∈(0,∞)
such that for every
u∈W01,r((0,1),\mathbbmR)
it holds that
In this subsection we recall
in
Lemmas 4.13–4.17,
Corollary 4.18,
Lemmas 4.19–4.21,
Corollary 4.22,
Lemma 4.23,
and
Corollary 4.24
below
a few elementary
and well-known
properties of the
nonlinearity appearing in
the stochastic Burgers equation.
Corollaries 4.22
and 4.24
are then used in Section 5
below
to establish
in Theorem 5.10 the main result of this article.
Observe that
items (ii)
and (iii)
of
Lemma 4.3
and,
e.g.,
[20, Lemma 4.5]
imply
item (i).
Furthermore, note that for every
v,w∈H\nicefrac12
it holds that
[TABLE]
Items (iv)
and (v)
of Lemma 4.3
therefore
show that for every
v,w∈H\nicefrac12
it holds that
[TABLE]
This establishes item (ii).
In addition, note that for every
v,w∈H\nicefrac12 it holds that
[TABLE]
Items (iv)
and (v)
of Lemma 4.3
hence imply
that for every
v,w∈H\nicefrac12
it holds that
[TABLE]
Therefore, we obtain that
(a)
it holds that F:H\nicefrac12→H is differentiable
and
2. (b)
it holds for every v,w∈H\nicefrac12 that
F′(v)w=c1(w∂v+v∂w).
Items (iv)
and (v)
of Lemma 4.3
hence
assure that for every
u,v∈H\nicefrac12 it holds that
[TABLE]
Combining items (a)
and (b)
therefore establishes items (iii)
and (iv).
The proof of Lemma 4.13
is thus completed.
∎
Lemma 4.14**.**
Let (X,dX)
be a metric space,
let (Y,dY) be a complete metric space,
let
S⊆X be a dense subset,
and let
F:S→Y be a locally uniformly continuous function.
Then there exists a unique
continuous function
Fˉ:X→Y
which satisfies
for every x∈S that Fˉ(x)=F(x).
Throughout this proof let
Ux⊆X, x∈S,
be non-empty open sets which satisfy
that
(a)
it holds
for every
x∈S
that
F∣Ux∩S:Ux∩S→Y
is uniformly continuous
and
2. (b)
it holds for every x∈S that
x∈Ux.
Observe that the fact that
for every x∈S it holds that
Ux∩S is a dense
subset of
Ux
and, e.g.,
Searcóid [36, Theorem 10.9.1]
show that
there exist
unique uniformly continuous functions
Fx:Ux→Y,
x∈S,
which satisfy for every
x∈S,
u∈(Ux∩S) that
[TABLE]
Note that (144)
and the fact that
for every
x,x∈S
with
(Ux∩Ux)=∅
it holds that
(Ux∩Ux)∩S
is a dense subset of (Ux∩Ux)
ensure that
for every
x,x∈S,
u∈(Ux∩Ux)
there exist
(un)n∈\mathbbmN⊆(Ux∩Ux∩S)
such that
limsupn→∞∥u−un∥X=0
and
[TABLE]
This proves that
for every
x,x∈S,
u∈(Ux∩Ux)
it holds that
[TABLE]
Moreover, observe that the
assumption that
S⊆X
is a dense subset
ensures that
X=∪x∈SUx.
Combining (144)
and (146)
hence
shows that
there exists
a unique continuous function
Fˉ:X→Y
which satisfies for every
u∈S that
Note that the fact that
ν>43−2γ
ensures that
(2γ)+(2γ)−(1−2ν)>21.
Combining this,
Lemma 4.7
(with
s=1−2ν,
q=2γ,
r=2γ
in the notation of
Lemma 4.7),
Lemma 4.9
(with
α=ν
in the notation of Lemma 4.9),
and
item (i)
of Lemma 4.13
shows that
there exists
C∈[1,∞)
such that for every
v,w∈H\nicefrac12
it holds that
(v2−w2)∈W1,2((0,1),\mathbbmR)
and
[TABLE]
Item (i) of Lemma 4.6
hence proves that
there exists C∈\mathbbmR
such that for every
v,w∈H\nicefrac12 it holds that
[TABLE]
Item (i) of Lemma 4.13
therefore establishes (LABEL:eq:LocLipSimple).
The proof of Lemma 4.15
is thus completed.
∎
Lemma 4.16**.**
Assume Setting 4.1
and let
γ∈(81,21],
ν∈([21−γ,21]∩(43−2γ,∞)).
Then
(i)
there exists
a unique continuous function
Fˉ:Hγ→H−ν
which satisfies
for every v∈H\nicefrac12
that
Fˉ(v)=F(v)
and
2. (ii)
there exists
C∈\mathbbmR
which satisfies
for every
v,w∈Hγ
that
Observe that
Lemma 4.15
(with
γ=γ,
ν=ν
in the notation of
Lemma 4.15)
ensures
that
there exists C∈\mathbbmR
such that
for every
v,w∈H\nicefrac12
it
holds that
[TABLE]
Lemma 4.14
(with
X=Hγ,
dX=((Hγ×Hγ)∋(h1,h2)↦∥h1−h2∥Hγ∈[0,∞)),
Y=H−ν,
dY=((H−ν×H−ν)∋(h1,h2)↦∥h1−h2∥H−ν∈[0,∞)),
S=H\nicefrac12,
F=F
in the notation of
Lemma 4.14)
therefore
establishes item (i).
Moreover, note that the
fact that
H\nicefrac12⊆Hγ
continuously and densely
ensures that
for every
v∈Hγ
there exist
(vn)n∈\mathbbmN⊆H\nicefrac12
such
that
limsupn→∞∥v−vn∥Hγ=0.
Item (i)
therefore
implies that
for every
v,w∈Hγ there
exist
(vn)n∈\mathbbmN⊆H\nicefrac12
and
(wn)n∈\mathbbmN⊆H\nicefrac12
such that
limsupn→∞(∥v−vn∥Hγ+∥w−wn∥Hγ)=0
and
[TABLE]
Combining this and (LABEL:eq:LocLipSimple_Application) shows that
there exists
C∈\mathbbmR
such that
for every v,w∈Hγ
there exist
(vn)n∈\mathbbmN⊆H\nicefrac12
and
(wn)n∈\mathbbmN⊆H\nicefrac12
such that
[TABLE]
This establish item (ii).
The proof of Lemma 4.16
is thus completed.
∎
Lemma 4.17**.**
Assume Setting 4.1
and let
∂ˉ:H→H−\nicefrac12
be the continuous function
which satisfies for every
v∈W1,2((0,1),\mathbbmR)
that
∂ˉv=∂v
(cf. item (i) of Lemma 4.8).
Then there exists C∈\mathbbmR
such that
for every
v,w∈H\nicefrac18
it holds that
(v2−w2)∈H
and
Note that
item (i)
of Lemma 4.6
(with
s=\nicefrac18
in the notation
of item (i)
of Lemma 4.6)
ensures that
H\nicefrac18⊆W\nicefrac14,2((0,1),\mathbbmR)
continuously.
The Sobolev embedding theorem
hence shows that
[TABLE]
continuously.
This implies
that
for every v∈H\nicefrac18
it holds that
v2∈H
and
[TABLE]
Item (ii)
of Lemma 4.8
and the Cauchy-Schwarz inequality
hence prove that
(a)
it holds for every
v,w∈H\nicefrac18
that
(v2−w2)∈H
and
2. (b)
there exists
C∈[1,∞)
such that
for every
v,w∈H\nicefrac18
it holds that
Throughout this proof let
∂ˉ:H→H−\nicefrac12
be the continuous function
which satisfies for every
v∈W1,2((0,1),\mathbbmR) that
∂ˉv=∂v
(cf. item (i) of Lemma 4.8).
Note that
item (i)
of Lemma 4.13
ensures that for every
v∈H\nicefrac12
it holds that
v2∈W1,2((0,1),\mathbbmR)
and
[TABLE]
Lemma 4.17
(with
∂ˉ=∂ˉ
in the notation of Lemma 4.17)
hence shows that
there exists
C∈\mathbbmR
such that
for every
v,w∈H\nicefrac12
it holds that
[TABLE]
Lemma 4.14
(with
X=H\nicefrac18,
dX=((H\nicefrac18×H\nicefrac18)∋(h1,h2)↦∥h1−h2∥H\nicefrac18∈[0,∞)),
Y=H−\nicefrac12,
dY=((H−\nicefrac12×H−\nicefrac12)∋(h1,h2)↦∥h1−h2∥H−\nicefrac12∈[0,∞)),
S=H\nicefrac12,
F=F
in the notation of
Lemma 4.14)
therefore
establishes
item (i).
Moreover, note that the
fact that
H\nicefrac12⊆H\nicefrac18
continuously and densely
ensures that
for every
v∈H\nicefrac18
there exist
(vn)n∈\mathbbmN⊆H\nicefrac12
such
that
limsupn→∞∥v−vn∥H\nicefrac18=0.
This and item (i) imply that
for every
v,w∈H\nicefrac18 there
exist
(vn)n∈\mathbbmN⊆H\nicefrac12
and
(wn)n∈\mathbbmN⊆H\nicefrac12
such that
limsupn→∞(∥v−vn∥H\nicefrac18+∥w−wn∥H\nicefrac18)=0
and
[TABLE]
Combining
this
and (161)
shows that
there exists C∈\mathbbmR
such that
for every v,w∈H\nicefrac18
there exist
(vn)n∈\mathbbmN⊆H\nicefrac12
and
(wn)n∈\mathbbmN⊆H\nicefrac12
such that
[TABLE]
This establishes
item (ii).
The proof of Corollary 4.18
is thus completed.
∎
Note that item (iii) of Lemma 4.13
establishes item (i). Next observe that item (ii)
of Lemma 4.3,
Lemma 4.4,
items (i) and (iv) of Lemma 4.13,
and the Cauchy-Schwarz inequality
imply that for every
v,w∈H\nicefrac12 it holds that
[TABLE]
Moreover, note that
Lemma 4.12
(with
q=2,
p=4,
r=2,
α=\nicefrac14
in the notation of
Lemma 4.12)
and
item (ii) of Lemma 4.3
prove that there exists
C∈\mathbbmR
such that for every w∈H\nicefrac12⊆W01,2((0,1),\mathbbmR) it holds that
[TABLE]
Items (ii)
and (iv) of
Lemma 4.3,
(LABEL:eq:starting_estimate),
and the fact that
for every
x1,x2,x3,x4∈\mathbbmR
it holds that
4x1x2x3x4≤∣x1∣4+∣x2∣4+∣x3∣4+∣x4∣4
hence show that
there exists
C∈(0,∞)
such that
for every
ε∈(0,∞),
v,w∈H\nicefrac12 it holds that
[TABLE]
This establishes item (ii).
The proof of Lemma 4.19
is thus completed.
∎
Lemma 4.20**.**
Assume Setting 4.1,
let
α∈[0,\nicefrac12]\{\nicefrac14},
let P(H) be the
power set of H,
let P0(H)={θ∈P(H):θ is a finite set},
and
let
(PI)I∈P(H)⊆L(H)
satisfy
for every
I∈P(H),
v∈H
that
PI(v)=∑h∈I⟨h,v⟩Hh.
Then it holds
that
Throughout this proof consider the notation in
Triebel [38, Section 1.3.2 on page 24]
(cf., e.g., Lunardi [30, Definition 1.2]).
Note that
item (iii)
of
Lemma 4.6
shows that
for every
I∈P0(H),
v∈Hα+(\nicefrac12)
it holds that
[TABLE]
Moreover,
observe that
the fact that
[TABLE]
(cf., e.g.,
Triebel [38, item (a) of Theorem 1 in Section 4.4.2
on page 323,
and Remark 2 in Section 4.4.2
on page 323]),
item (i)
of Lemma 4.6, Lemma 4.7
(with
s=2α+1,
q=2α+1,
r=2α+1
in the notation of
Lemma 4.7),
and
item (i)
of
Lemma 4.13
imply that
there exists C∈(0,∞)
such that
for every
v∈Hα+(\nicefrac12)
it holds that
v∈W2α+1,2((0,1),\mathbbmR),
v2∈W2α+1,2((0,1),\mathbbmR),
F(v)∈W2α,2((0,1),\mathbbmR),
and
[TABLE]
Moreover, note that, e.g.,
Triebel [38, Definition 1
in Section 4.2.1
on page 310,
Theorem 1 in Section 4.3.1
on page 317,
item (a) in Theorem 1
in Section 4.4.2
on page 323,
and
Remark 2 in Section 4.4.2
on page 324]
shows that for every
ι∈(0,\nicefrac12)
it holds that
[TABLE]
and
[TABLE]
In addition, observe that the fact that
H⊆W1,2((0,1),\mathbbmR)
is an orthogonal system
ensures that for every
I∈P0(H),
v∈W1,2((0,1),\mathbbmR)
it holds that
[TABLE]
The fact that
for every
I∈P0(H),
v∈H
it holds that
∥PIv∥H≤∥v∥H,
(172),
(173),
and, e.g.,
Lunardi [30, Theorem 1.6]
therefore prove that for every
I∈P0(H),
v∈W2α,2((0,1),\mathbbmR)
it holds that
[TABLE]
Combining (169),
(LABEL:eq:Est),
and
item (i) of
Lemma 4.6
hence
implies that
there exists C∈(0,∞)
such that
for every
I∈P0(H),
v∈Hα+(\nicefrac12)
it holds that
Note that items (i)
and (ii)
of
Lemma 4.3
ensures that
[TABLE]
Next observe that
item (i) of Lemma 4.13
shows that for every
v∈H\nicefrac12
it holds that
v2∈W1,2((0,1),\mathbbmR).
This,
(180),
and
Lemma 4.4
(with
u=u,
v=v2
for
u,v∈H\nicefrac12
in the notation of
Lemma 4.4)
ensure that for every
u,v∈H\nicefrac12
it holds that
⟨∂(v2),u⟩H=−⟨v2,∂u⟩H.
Item (i) of Lemma 4.13
and
Lemma 4.10
(with
α=α−21
for
α∈(43,∞)
in the notation of Lemma 4.10)
therefore
prove that
for every
v∈H\nicefrac12,
α∈(43,∞)
it holds
that
[TABLE]
This establishes
item (i).
Next note that
item (i) of Lemma 4.13,
Lemma 4.9
(with
α=α
for
α∈[0,\nicefrac12]
in the notation of Lemma 4.9),
and
Lemma 4.7
(with
s=1−2α,
q=\nicefrac2(1−α)3,
r=\nicefrac2(1−α)3
for α∈(\nicefrac14,\nicefrac12]
in the notation of Lemma 4.7)
show that for every
v∈H\nicefrac12,
α∈(\nicefrac14,\nicefrac12]
it holds that
[TABLE]
Item (i) of
Lemma 4.6
hence implies item (ii).
Furthermore, observe that
items (iv)
and (v)
of Lemma 4.3
imply that for every
v∈H\nicefrac12
it holds that
[TABLE]
This establishes item (iii).
The proof of Lemma 4.21 is thus completed.
∎
Corollary 4.22**.**
Assume Setting 4.1
and let
α1∈(\nicefrac34,∞),
α2∈(\nicefrac14,\nicefrac12].
Then
Note that
items (i)
and (ii)
of Lemma 4.3,
item (i) of
Lemma 4.13,
and
Lemma 4.4
(with
u=x,
v=x2
for
x∈H\nicefrac12
in the notation of
Lemma 4.4)
ensure that for every
x∈H\nicefrac12=W01,2((0,1),\mathbbmR)
it holds that
x2∈W1,2((0,1),\mathbbmR)
and
Throughout this proof assume
w.l.o.g. that
c1=0
and
let
C∈[0,∞]
satisfy that
[TABLE]
Note that the Sobolev embedding theorem
and
item (i)
of
Lemma 4.6
ensure that C∈(0,∞).
Next observe that
Lemma 4.23,
item (i) of Lemma 4.13,
and
Lemma 4.4
(with
u=v,
v=w2
in the notation of Lemma 4.4)
ensure that
[TABLE]
Lemma 4.4
(with
u=w,
v=v2
in the notation of Lemma 4.4)
and
item (i) of Lemma 4.13
therefore imply that
[TABLE]
Hölder’s inequality
and
item (iv)
of Lemma 4.3
hence prove that
[TABLE]
The fact that
for every
x,y∈\mathbbmR,
ε∈(0,∞)
it holds that
2xy≤εx2+εy2
therefore
shows that
5 Existence and uniqueness of mild solutions to stochastic Burgers equations
In this section
we prove
in
Theorem 5.10
below
the unique
existence of suitably regular mild solutions
to stochastic Burgers equations
with additive trace class noise.
To do so,
we first
establish
in
Lemmas 5.1–5.6
(cf., e.g., Blömker & Jentzen [4, Lemma 5.5]),
Lemma 5.7
(cf., e.g., Kloeden & Neuenkirch [27, Lemma 2.1]),
and
Lemma 5.8
(cf., e.g., Blömker & Jentzen [4, Lemma 4.3])
a
few elementary and partially well-known auxiliary results.
Only for the sake of completeness we include in this section also a proof of Lemma 5.7.
Thereafter, we combine these
auxiliary results
with
the results
from
Subsection 4.2
and
the abstract existence and uniqueness result
in Blömker & Jentzen [4, Theorem 3.1]
to establish
in
Theorem 5.10
below
the main result of this article.
Lemma 5.1**.**
Assume Setting 4.1,
let
T∈(0,∞),
ι∈(\nicefrac14,∞),
ξ∈H,
let
I⊆H
be a finite set,
let
P∈L(H)
satisfy
for every
v∈H
that
Pv=∑h∈I⟨h,v⟩Hh,
and
let O,X∈C([0,T],P(H))
satisfy for every
t∈[0,T]
that
and let
Z:[0,T]→P(H)
be the function
which satisfies for every
t∈[0,T] that
Zt=Xt−Ot.
Observe that the Sobolev embedding theorem
and
item (i)
of
Lemma 4.6
ensure that C∈[0,∞).
Next note that
for every t∈[0,T] it holds that
[TABLE]
This implies for every
t∈[0,T] that
[TABLE]
Therefore, we obtain that for every t∈[0,T]
it holds that
[TABLE]
Corollary 4.24
(with
ι=ι,
v=Zs, w=Os
for
s∈[0,T]
in the notation of
Corollary 4.24)
hence proves that for every t∈[0,T]
it holds that
[TABLE]
The fact that
O,Z∈C([0,T],P(H)) and Gronwall’s lemma
therefore establish that for every t∈[0,T] it holds
that
Furthermore, observe that
the fact that
∀v∈Hmax{α,0}:Rv∈H
ensures that
for every
v∈Hmax{α,0}
it holds that
[TABLE]
Combining
this and (203)
proves that for every
v∈Hmax{α,0}
it holds that
[TABLE]
The fact that Hmax{α,0}⊆Hα
densely
therefore
establishes
items (i) and (ii).
The proof of Lemma 5.2
is thus completed.
∎
Lemma 5.3**.**
Assume Setting 4.1,
let P(H) be the power set of H,
let
T∈(0,∞),
ι∈[0,1),
γ∈(\nicefrac14,∞),
ξ∈Hι,
P0(H)={θ∈P(H):θ is a finite set},
let
(PI)I∈P(H)⊆L(H)
satisfy
for every
I∈P(H),
v∈H
that
PI(v)=∑h∈I⟨h,v⟩Hh,
let
OI∈C([0,T],PI(H)),
I∈P0(H),
satisfy
supI∈P0(H)supu∈[0,T]∥OuI∥Hmax{γ,ι}<∞,
let
XI∈C([0,T],PI(H)),
I∈P0(H),
and assume
for every
I∈P0(H),
t∈[0,T]
that
Note that
Corollary 4.22
(with
α1=α1,
α2=α2
for
α1∈(\nicefrac34,∞),
α2∈(\nicefrac14,\nicefrac12]
in the notation of Corollary 4.22)
shows that
for every
α1∈(\nicefrac34,∞),
α2∈(\nicefrac14,\nicefrac12]
it holds that
[TABLE]
In addition, observe that
Lemma 5.2
(with
α=−α,
I=I,
R=(H∋x↦PIx∈H−α)
for
I∈P0(H),
α∈\mathbbmR
in the notation of
Lemma 5.2)
proves that
for every
x∈H,
I∈P0(H),
α∈\mathbbmR
it holds that
[TABLE]
Combining this and (208)
ensures that
for every
α1∈(\nicefrac34,∞),
α2∈(\nicefrac14,\nicefrac12]
it holds that
[TABLE]
[TABLE]
and
[TABLE]
Moreover, observe that
Lemma 5.1
(with
T=T,
ι=max{γ,ι},
ξ=ξ,
I=I,
P=PI,
O=OI,
X=XI
for
I∈P0(H)
in the notation of
Lemma 5.1)
implies that
[TABLE]
Combining (212)
and
Lemma 3.2
(with
(Ω,F,P)=({1},{∅,{1}},({∅,{1}}∋A↦\mathbbm1A(1)∈[0,1])),
T=T,
β=\nicefrac12,
γ=\nicefrac12,
ξ=({1}∋ω↦PIξ∈H\nicefrac12),
F=(H\nicefrac12∋v↦PIF(v)∈H),
κ=([0,T]∋t↦t∈[0,T]),
Z=([0,T]×{1}∋(t,ω)↦XtI∈H\nicefrac12),
O=([0,T]×{1}∋(t,ω)↦OtI∈H\nicefrac12),
Y=([0,T]×{1}∋(t,ω)↦XtI∈H),
p=1,
ρ=ρ,
α=α1
for
α1∈(\nicefrac34,1−ρ),
ρ∈[0,\nicefrac14),
I∈P0(H)
in the notation of
Lemma 3.2)
hence shows that
for every
ρ∈[0,\nicefrac14),
α1∈(\nicefrac34,1−ρ),
I∈P0(H),
t∈[0,T]
it holds that
[TABLE]
This,
(212),
(213),
and the assumption that
supI∈P0(H)supu∈[0,T]∥OuI∥Hι<∞
show
that for every
ρ∈[0,\nicefrac14)
with
ρ≤ι
it holds that
[TABLE]
Furthermore, observe that Lemma 3.3
(with
H=H,
(Ω,F,P)=({1},{∅,{1}},({∅,{1}}∋A↦\mathbbm1A(1)∈[0,1])),
T=T,
β=\nicefrac12,
γ=\nicefrac12,
ξ=({1}∋ω↦PIξ∈H\nicefrac12),
F=(H\nicefrac12∋v↦PIF(v)∈H),
κ=([0,T]∋t↦t∈[0,T]),
Z=([0,T]×{1}∋(t,ω)↦XtI∈H\nicefrac12),
O=([0,T]×{1}∋(t,ω)↦OtI∈H\nicefrac12),
Y=([0,T]×{1}∋(t,ω)↦XtI∈H),
p=1,
ρ=\nicefrac(1−α2)3,
η=η,
α1=α1,
α2=α2
for
α1∈(\nicefrac34,\nicefrac(2+α2)3),
α2∈(\nicefrac14,\nicefrac12),
η∈[\nicefrac14,\nicefrac12],
I∈P0(H)
in the notation of
Lemma 3.3),
(211),
and (212)
ensure that for every
α2∈(\nicefrac14,\nicefrac12),
α1∈(\nicefrac34,\nicefrac(2+α2)3),
η∈[\nicefrac14,\nicefrac12],
I∈P0(H),
t∈[0,T]
it holds that
[TABLE]
Combining (211)–(213)
and the assumption that
supI∈P0(H)supu∈[0,T]∥OuI∥Hι<∞
hence implies that for every
η∈[\nicefrac14,\nicefrac12]
with
η≤ι
it holds
that
[TABLE]
Moreover, note that
Lemma 3.4
(with
H=H,
(Ω,F,P)=({1},{∅,{1}},({∅,{1}}∋A↦\mathbbm1A(1)∈[0,1])),
T=T,
β=κ,
γ=\nicefrac12,
ξ=({1}∋ω↦PIξ∈Hκ),
F=(H\nicefrac12∋v↦PIF(v)∈H),
κ=([0,T]∋t↦t∈[0,T]),
Z=([0,T]×{1}∋(t,ω)↦XtI∈H\nicefrac12),
O=([0,T]×{1}∋(t,ω)↦OtI∈Hκ),
Y=([0,T]×{1}∋(t,ω)↦XtI∈H),
p=1,
ρ=\nicefrac(1−α2)3,
η=\nicefrac12,
ι=κ,
α1=α1,
α2=α2
for
α1∈[0,\nicefrac(2+α2)3),
α2∈[0,\nicefrac12),
κ∈[\nicefrac12,1),
I∈P0(H)
in the notation of
Lemma 3.4)
and (210)–(212)
prove that
for every
α2∈(\nicefrac14,\nicefrac12),
α1∈(\nicefrac34,\nicefrac(2+α2)3),
κ∈[\nicefrac12,1),
I∈P0(H),
t∈[0,T]
it holds
that
[TABLE]
Combining (210)–(213)
and the assumption that
supI∈P0(H)supu∈[0,T]∥OuI∥Hι<∞
therefore
assures that for every
κ∈[\nicefrac12,1)
with κ≤ι
it holds that
[TABLE]
This, (215),
and (217)
establish (LABEL:eq:Finite_iota).
The proof of Lemma 5.3
is thus completed.
∎
Lemma 5.4**.**
Assume Setting 1.2
and
let
T∈(0,∞),
α∈(0,1),
γ∈\mathbbmR,
Z∈C([0,T],Hγ).
Then
(i)
it holds for every t∈[0,T] that
∫0t∥(t−u)α−1e(t−u)AZu∥Hγdu<∞
and
2. (ii)
it holds that
([0,T]∋t↦∫0t(t−u)α−1e(t−u)AZudu∈Hγ)∈C([0,T],Hγ).
This establishes item (i).
Next observe that
item (i)
ensures that there exists a function
Z:[0,T]→Hγ
which satisfies for every
t∈[0,T] that
[TABLE]
Note that (221)
and the triangle inequality show
that for
every
s∈[0,T],
t∈[s,T]
it holds
that
[TABLE]
Furthermore, observe that for every
s∈[0,T],
t∈[s,T] it holds that
[TABLE]
In addition,
note that the triangle inequality assures
that for every
s∈[0,T],
t∈[s,T]
it holds that
[TABLE]
Next observe that
the fact that
for every
t∈(0,T],
s∈(0,t),
u∈[0,s)
it holds that
(t−u)α−1≤(t−s)α−1
proves that
for every
s∈[0,T],
t∈[s,T],
ρ∈(1−α,1)
it holds that
[TABLE]
Moreover, observe that the fact that
for every
x,y∈[0,T],
z∈[0,1]
it holds that
∣xz−yz∣≤∣x−y∣z
ensures that for every
t∈(0,T], s∈(0,t),
u∈[0,s)
it holds that
[TABLE]
This implies that for every
s∈[0,T],
t∈[s,T],
ε∈(0,min{\nicefracα(2(1−α)),\nicefrac12,1−α}) it holds that
[TABLE]
Combining (LABEL:eq:FirstPart)–(LABEL:eq:Est4)
therefore
demonstrates that
for every
s∈[0,T], t∈[s,T],
ρ∈(1−α,1),
ε∈(0,min{\nicefracα(2(1−α)),\nicefrac12,1−α})
it holds that
[TABLE]
This
establishes item (ii).
The proof of Lemma 5.4
is thus completed.
∎
Lemma 5.5**.**
Assume Setting 1.2,
let
T∈(0,∞),
β∈\mathbbmR,
γ∈(−∞,\nicefrac12+β),
B∈HS(H,Hβ),
let
(Ω,F,P)
be a probability space,
for every set R
and every function
f:Ω→R
let
[f]P,B(Hγ)={g∈M(F,B(Hγ)):(∃D∈F:P(D)=0and{ω∈Ω:f(ω)=g(ω)}⊆D)},
and
let
(Wt)t∈[0,T]
be an
IdH-cylindrical
Wiener process.
Then there exists an up to
indistinguishability unique
stochastic process
O:[0,T]×Ω→Hγ
with continuous sample paths
which satisfies for every
t∈[0,T]
that
[Ot]P,B(Hγ)=∫0te(t−s)ABdWs.
Note that
the fact that
γ−β<\nicefrac12
ensures that
for every
t∈[0,T]
it holds that
[TABLE]
This shows that there
exists
a stochastic process
O:[0,T]×Ω→Hγ
which satisfies for every
t∈[0,T] that
[TABLE]
Observe that (230)
and the
Burkholder-Davis-Gundy-type inequality in Da Prato & Zabczyk [9, Lemma 7.7]
prove that
for every
p∈[2,∞),
s∈[0,T],
t∈[s,T],
ρ∈(0,min{1,\nicefrac12+β−γ})
it holds that
[TABLE]
The Kolmogorov-Chentsov theorem
(cf., e.g., Kallenberg [24, Theorem 2.23])
therefore
assures that
there exists an up to indistinguishability unique stochastic
process
O:[0,T]×Ω→Hγ
with continuous sample paths
which satisfies for every
t∈[0,T]
that
[Ot]P,B(Hγ)=∫0te(t−s)ABdWs.
The proof of Lemma 5.5
is thus completed.
∎
Lemma 5.6**.**
Assume Setting 1.2,
let
T∈(0,∞),
I⊆H,
β∈\mathbbmR,
γ∈(−∞,21+β),
α∈(0,21−max{0,γ−β}),
B∈HS(H,Hβ),
let
B∈L(H),
P∈L(Hmin{0,γ})
satisfy for every
u,v∈H that
⟨Bu,v⟩H=⟨u,Bv⟩H
and
Pv=∑h∈I⟨h,v⟩Hh,
let
(Ω,F,P)
be a probability space,
for every set R
and every function
f:Ω→R
let
[f]P,B(Hγ)={g∈M(F,B(Hγ)):(∃D∈F:P(D)=0and{ω∈Ω:f(ω)=g(ω)}⊆D)},
let
(Wt)t∈[0,T]
be an
IdH-cylindrical
Wiener process,
and let O:[0,T]×Ω→Hγ be a stochastic process
with continuous sample paths
which satisfies for every
t∈[0,T]
that
[Ot]P,B(Hγ)=∫0te(t−s)ABdWs.
Then
it holds for every
p∈(\nicefrac1α,∞)
that
This ensures that
there exists a stochastic process
Z:[0,T]×Ω→Hγ
which satisfies for every
t∈[0,T]
that
[TABLE]
Note
that (234)
and the triangle inequality prove
that
for every
p∈[2,∞),
t∈(0,T], s∈[0,t)
it holds that
[TABLE]
The
Burkholder-Davis-Gundy-type inequality in Da Prato & Zabczyk [9, Lemma 7.7]
hence shows that for every
p∈[2,∞),
t∈(0,T], s∈[0,t)
it holds that
[TABLE]
Therefore, we obtain that for every
p∈[2,∞),
ε∈(0,21+β−α−γ),
t∈(0,T], s∈[0,t)
it holds
that
[TABLE]
In addition,
note
that for every
ε∈(0,21+β−α−γ),
t∈(0,T], s∈[0,t)
it holds that
[TABLE]
Next observe that the fact that
for every
x,y∈[0,T],
z∈[0,1]
it holds that
∣xz−yz∣≤∣x−y∣z
ensures that for every
t∈(0,T], s∈(0,t),
u∈(0,s)
it holds that
[TABLE]
Hölder’s inequality
hence proves that for every
ε∈(0,min{8(α+max{0,γ−β})1−41,41,α}),
t∈(0,T],
s∈[0,t)
it holds
that
[TABLE]
Combining
this,
(LABEL:eq:basic1),
and (LABEL:eq:basic2)
demonstrates that
for every
ε∈(0,min{8(α+max{0,γ−β})1−41,41,α,21+β−α−γ}),
p∈[2,∞)
it holds that
[TABLE]
The Kolmogorov-Chentsov theorem
(cf., e.g., Kallenberg [24, Theorem 2.23])
therefore
assures that there exists a stochastic process
Z:[0,T]×Ω→Hγ
with continuous sample paths
which satisfies for every
t∈[0,T] that
[TABLE]
Next note that
the fact that
0<α<21−max{0,γ−β}
ensures that
for every
t∈[0,T] it holds that
[TABLE]
Combining (234),
(242),
the fact that
Z:[0,T]×Ω→Hγ
has continuous sample paths,
item (i) of
Lemma 5.4
(with
T=T,
α=α,
γ=γ,
Z=([0,T]∋t↦PZt(ω)∈Hγ)
for
ω∈Ω
in the notation of
item (i) of
Lemma 5.4),
and, e.g.,
Da Prato & Zabczyk [11, Theorem 5.10] therefore establishes that
for every
t∈[0,T]
it holds that
[TABLE]
This,
the fact that
Z:[0,T]×Ω→Hγ
has continuous sample paths,
and
Lemma 5.4
(with
T=T,
α=α,
γ=γ,
Z=([0,T]∋t↦PZt(ω)∈Hγ)
for
ω∈Ω
in the notation of Lemma 5.4)
imply
that
for every
ω∈Ω,
p∈[1,∞)
it holds
that
([0,T]∋t↦∫0t(t−s)α−1e(t−s)APZs(ω)ds∈Hγ)∈C([0,T],Hγ)
and
[TABLE]
Hölder’s inequality and Tonelli’s theorem hence prove that for every
p∈(\nicefrac1α,∞)
it holds that
[TABLE]
In addition, observe that
the
Burkholder-Davis-Gundy-type inequality in Da Prato & Zabczyk [9, Lemma 7.7]
shows that for every
p∈(\nicefrac1α,∞),
t∈[0,T]
it holds that
[TABLE]
Tonelli’s theorem therefore
implies that for every
p∈(\nicefrac1α,∞),
t∈[0,T]
it holds that
[TABLE]
Combining this with (LABEL:eq:proj_estimate)
ensures that
Let (V,∥⋅∥V)
be an \mathbbmR-Banach space,
let (Ω,F,P)
be a probability space,
let
α∈(0,∞),
and
let
Zn:Ω→V,
n∈\mathbbmN,
be F/B(V)-measurable functions
which satisfy
for every
p∈[1,∞)
that
supn∈\mathbbmN(nα∥Zn∥Lp(P;V))<∞.
Then it holds for every
ε∈(0,∞),
p∈[1,∞)
that
Observe that
for every
ε,δ∈(0,∞),
p∈(max{\nicefrac1ε,1},∞)
it holds that
[TABLE]
Jensen’s inequality therefore
demonstrates that for every
ε∈(0,∞),
p∈[1,∞)
it holds that
[TABLE]
This
establishes (250).
The proof of Lemma 5.7
is thus completed.
∎
Lemma 5.8**.**
Assume Setting 4.1,
let T∈(0,∞),
β∈\mathbbmR,
γ∈(−∞,\nicefrac12+β),
B∈HS(H,Hβ),
let P(H) be the power set of H,
let
(PI)I∈P(H)⊆L(Hmin{0,γ})
satisfy
for every
I∈P(H),
v∈Hmin{0,γ}
that
PI(v)=∑h∈I⟨(−A)−min{0,γ}h,(−A)min{0,γ}v⟩Hh,
let
(Ω,F,P)
be a probability space,
let
(Wt)t∈[0,T]
be an
IdH-cylindrical
Wiener process,
and
let O:[0,T]×Ω→Hγ be a stochastic process
with continuous sample paths
which satisfies for every
t∈[0,T]
that
[Ot]P,B(Hγ)=∫0te(t−s)ABdWs. Then
Throughout this proof let
B∈L(H)
satisfy for every
u,v∈H that
⟨Bu,v⟩H=⟨u,Bv⟩H
and let
(In)n∈\mathbbmN⊆H
satisfy for every n∈\mathbbmN that
In={e1,…,en}.
Note that Lemma 5.6
(with
T=T,
I=H\In,
β=β,
γ=γ,
α=α,
B=B,
B=B,
P=PH\In,
(Ω,F,P)=(Ω,F,P),
(Wt)t∈[0,T]=(Wt)t∈[0,T],
O=O
for
n∈\mathbbmN,
α∈(0,21−max{0,γ−β})
in the notation of Lemma 5.6)
ensures that for every
n∈\mathbbmN,
α∈(0,21−max{0,γ−β}),
p∈(\nicefrac1α,∞)
it holds that
[TABLE]
Jensen’s inequality
hence implies that for every
α∈(0,21−max{0,γ−β}),
p∈[1,∞)
it holds
that
[TABLE]
Lemma 5.7
(with
V=\mathbbmR,
(Ω,F,P)=(Ω,F,P),
α=1+2(β−α−γ),
Zn=supt∈[0,T]∥PH\InOt∥Hγ
for
n∈\mathbbmN,
α∈(0,21−max{0,γ−β})
in the notation of
Lemma 5.7)
therefore shows that
for every
α∈(0,21−max{0,γ−β}),
η∈(0,1+2(β−α−γ))
it holds that
Assume Setting 4.1,
let
T∈(0,∞),
β∈\mathbbmR,
γ∈(−∞,\nicefrac12+β),
B∈HS(H,Hβ),
let
(Ω,F,P)
be a probability space with a normal filtration
(Ft)t∈[0,T],
let
(Wt)t∈[0,T]
be an
IdH-cylindrical
(Ft)t∈[0,T]-Wiener process,
let
ξ∈M(F0,B(H)),
let P(H) be the power set of H,
let P0(H)={θ∈P(H):θ is a finite set},
let
(PI)I∈P0(H)⊆L(Hmin{0,γ},H)
satisfy
for every
I∈P0(H),
v∈Hmin{0,γ}
that
PI(v)=∑h∈I⟨(−A)−min{0,γ}h,(−A)min{0,γ}v⟩Hh,
and
let O:[0,T]×Ω→Hγ be a stochastic process
with continuous sample paths
which satisfies for every
t∈[0,T]
that
[Ot]P,B(Hγ)=∫0te(t−s)ABdWs. Then
there exist
(Ft)t∈[0,T]-adapted stochastic
processes
XI:[0,T]×Ω→PI(H),
I∈P0(H),
with continuous sample paths
such that
for every
I∈P0(H),
t∈[0,T] it holds that
Throughout this proof
let
Φ:H\nicefrac12→[0,∞)
be the function which satisfies for every
w∈H\nicefrac12 that
[TABLE]
let
AI:PI(H)→PI(H),
I∈P0(H),
be the linear operators
which
satisfy for every
I∈P0(H),
v∈PI(H)
that
AIv=Av,
and
for every
I∈P0(H)
let
(HI,s,⟨⋅,⋅⟩HI,s,∥⋅∥HI,s),
s∈\mathbbmR,
be a family of interpolation spaces
associated
to −AI.
Note that item (ii)
of Lemma 4.13
proves that
for every
I∈P0(H),
v,w∈H\nicefrac12
it holds that
[TABLE]
Moreover, observe that
Corollary 4.24
(with ι=\nicefrac12,
v=v, w=w
for v,w∈H\nicefrac12
in the notation of
Corollary 4.24)
shows that for every
I∈P0(H),
v,w∈PI(H)⊆H\nicefrac12
it holds that
[TABLE]
Combining (259)
and
Corollary 2.4 (with
(H,⟨⋅,⋅⟩H,∥⋅∥H)=(PI(H),⟨⋅,⋅⟩H,∥⋅∥H),
H=I,
ven=−c0π2n2,
A=AI,
(Hs)s∈\mathbbmR=(HI,s)s∈\mathbbmR,
T=T,
s=0,
C=\nicefrac∣c1∣c0,
c=1,
δ=\nicefrac12,
κ=\nicefrac12,
F=(PI(H)∋x↦PIF(x)∈PI(H)),
Φ=(PI(H)∋x↦Φ(x)∈[0,∞)),
(Ω,F,P,(Ft)t∈[0,T])=(Ω,F,P,(Ft)t∈[0,T]),
ξ=(Ω∋ω↦PIξ(ω)∈PI(H)),
O=([0,T]×Ω∋(t,ω)↦PIOt(ω)∈PI(H))
for
I∈P0(H),
n∈{m∈\mathbbmN:em∈PI(H)}
in the notation of Corollary 2.4)
therefore
completes the proof of Lemma 5.9.
∎
Theorem 5.10**.**
Let
λ:B((0,1))→[0,1]
be the Lebesgue-Borel
measure on
(0,1),
for every measure space
(Ω,F,μ),
every measurable space (S,S),
every set R,
and every function
f:Ω→R
let
[f]μ,S={g:Ω→S:(∃D∈F:[μ(D)=0 and {ω∈Ω:f(ω)=g(ω)}⊆D]) and (∀D∈S:g−1(D)∈F)},
let
T,ε,c0∈(0,∞),
c1∈\mathbbmR,
β∈(−\nicefrac14,∞),
γ∈(\nicefrac14,min{1,\nicefrac12+β}),
(H,⟨⋅,⋅⟩H,∥⋅∥H)=(L2(λ;\mathbbmR),⟨⋅,⋅⟩L2(λ;\mathbbmR),∥⋅∥L2(λ;\mathbbmR)),
let
(en)n∈\mathbbmN⊆H
satisfy for every
n∈\mathbbmN
that
en=[(2sin(nπx))x∈(0,1)]λ,B(\mathbbmR),
let
A:D(A)⊆H→H
be the linear operator which satisfies
D(A)={v∈H:∑n=1∞∣n2⟨en,v⟩H∣2<∞}
and
∀v∈D(A):Av=−∑n=1∞c0π2n2⟨en,v⟩Hen,
let
(Hr,⟨⋅,⋅⟩Hr,∥⋅∥Hr),r∈\mathbbmR, be a family of interpolation spaces associated to −A
(cf., e.g., [37, Section 3.7]),
for every
v∈W1,2((0,1),\mathbbmR)
let
∂v∈H
satisfy for every
φ∈Ccpt∞((0,1),\mathbbmR)
that
⟨∂v,[φ]λ,B(\mathbbmR)⟩H=−⟨v,[φ′]λ,B(\mathbbmR)⟩H,
let
(Ω,F,P)
be a probability space with a normal filtration
(Ft)t∈[0,T],
let
(Wt)t∈[0,T]
be an
IdH-cylindrical
(Ft)t∈[0,T]-Wiener process,
let
B∈HS(H,Hβ),
and let
ξ:Ω→Hγ+ε
be an
F0/B(Hγ+ε)-measurable function.
Then
(i)
there exists a unique continuous function
F:H\nicefrac18→H−\nicefrac12
which satisfies
for every
v∈H\nicefrac12
that
F(v)=c1v∂v
and
2. (ii)
there exists an up to indistinguishability unique
(Ft)t∈[0,T]-adapted stochastic process
X:[0,T]×Ω→Hγ
with continuous sample paths which satisfies
for every
t∈[0,T]
that
Throughout this proof
let
f:H\nicefrac12→H
be the function
which satisfies for every
v∈H\nicefrac12
that
f(v)=c1v∂v,
let
H={en:n∈\mathbbmN},
let P(H) be the power set of H,
let P0(H)={θ∈P(H):θ is a finite set},
let
(PI)I∈P(H)⊆L(H)
satisfy
for every
I∈P(H),
v∈H
that
PI(v)=∑h∈I⟨h,v⟩Hh,
let
ν=\nicefrac(2−4min{γ,1/2})3,
η∈(0,min{2ε,1+2(β−γ),2(1−γ−ν)}),
and
let
(In)n∈\mathbbmN⊆P0(H)
satisfy
for every n∈\mathbbmN that
In={e1,…,en}.
Note that
item (i) of
Corollary 4.18
(with
F=f,
Fˉ=F
in the notation of
Corollary 4.18)
establishes
item (i).
Next we intend to apply Blömker & Jentzen [4, Theorem 3.1]
to prove item (ii).
For this observe that
Lemma 5.5
(with
H=H,
H=H,
ven=−c0π2n2,
A=A,
Hr=Hr,
T=T,
β=β,
γ=γ,
B=B,
(Ω,F,P)=(Ω,F,P),
(Wt)t∈[0,T]=(Wt)t∈[0,T]
for
n∈\mathbbmN,
r∈\mathbbmR
in the notation of
Lemma 5.5)
ensures
that there exists
an
(Ft)t∈[0,T]-adapted stochastic process
O:[0,T]×Ω→Hγ
with continuous sample paths
which satisfies for every
t∈[0,T] that
[TABLE]
Note that (262)
and
Lemma 5.9
(with
c0=c0,
c1=c1,
H=H,
H=H,
ven=−c0π2n2,
en=en,
A=A,
Hr=Hr,
F=f,
T=T,
β=β,
γ=γ,
B=B,
(Ω,F,P)=(Ω,F,P),
(Ft)t∈[0,T]=(Ft)t∈[0,T],
(Wt)t∈[0,T]=(Wt)t∈[0,T],
ξ=(Ω∋ω↦ξ(ω)∈H),
PI=PI,
O=O
for n∈\mathbbmN, r∈\mathbbmR, I∈P0(H)
in the notation of
Lemma 5.9)
show that
there exist
(Ft)t∈[0,T]-adapted stochastic processes
XI:[0,T]×Ω→PI(H),
I∈P0(H),
with continuous sample paths
which satisfy for every
I∈P0(H),
t∈[0,T] that
[TABLE]
Next let
Σ∈F
be the set which satisfies that
[TABLE]
let
O:[0,T]×Ω→Hγ
be the stochastic process which satisfies
for every t∈[0,T],
ω∈Ω that
[TABLE]
and let
XI:[0,T]×Ω→PI(H),
I∈P0(H),
be the stochastic processes
which satisfy for every
I∈P0(H),
t∈[0,T],
ω∈Σ
that
[TABLE]
Moreover, note that
the fact that
(γ+ν)∈(0,1)
shows that
for every
t∈(0,T] it holds that
[TABLE]
This ensures that
[TABLE]
In addition, observe that
the fact that
(γ+ν+(\nicefracη2))∈(0,1)
implies that
for every
n∈\mathbbmN,
t∈[0,T]
it holds
that
[TABLE]
This proves that
[TABLE]
Next note that
the fact that
for every x∈(\nicefrac18,\nicefrac12]
it holds that
[TABLE]
and
Lemma 4.16
(with
γ=min{γ,\nicefrac12},
ν=\nicefrac(2−4min{γ,1/2})3
in the
notation of Lemma 4.16)
ensure that
there exists
C∈[0,∞)
such that
for every
v,w∈Hγ⊆Hmin{γ,1/2}
it holds that
[TABLE]
This demonstrates that
there exists C∈\mathbbmR
such that
for every
v,w∈Hγ
it holds that
[TABLE]
Furthermore,
observe that (264),
the fact that
η∈(0,1+2(β−γ)),
and
Lemma 5.8
(with
T=T,
β=β,
γ=γ,
B=B,
PI=PI,
(Ω,F,P)=(Ω,F,P),
(Wt)t∈[0,T]=(Wt)t∈[0,T],
O=O
for
I∈P(H)
in the notation of
Lemma 5.8)
show that P(Σ)=1.
This and (265)
prove that
[TABLE]
In the next step we note that
the fact that
f(0)=0 ensures that
for every
n∈\mathbbmN,
ω∈(Ω\Σ),
t∈[0,T]
it holds that
[TABLE]
Furthermore, observe that for every
n∈\mathbbmN,
ω∈Ω,
t∈[0,T]
it holds that
[TABLE]
Combining
this,
(264),
(LABEL:eq:FirstRegularity),
and the triangle inequality
demonstrates that for every
ω∈Ω
it holds that
[TABLE]
Moreover, note that (263),
(265),
and (266)
ensure that for every
I∈P0(H),
ω∈Ω,
t∈[0,T]
it holds that
[TABLE]
In addition,
observe that
the fact that O:[0,T]×Ω→Hγ has continuous sample paths
establishes
for every ω∈Ω that
[TABLE]
The fact that γ<1,
(278),
and
Lemma 5.3
(with
F=f,
T=T,
ι=γ,
γ=γ,
ξ=ξ(ω),
OtI=PIOt(ω),
XtI=XtI(ω)
for
n∈\mathbbmN,
I∈P0(H),
t∈[0,T],
ω∈Ω
in the notation of
Lemma 5.3)
therefore prove that
for every
ω∈Ω it holds that
[TABLE]
Furthermore, note that
item (i)
and (278)
show that
for every
I∈P0(H),
ω∈Ω,
t∈[0,T]
it holds that
[TABLE]
Combining
the fact that
0<γ+ν+(\nicefracη2)<1,
(268),
(270),
(273),
(277),
and
(280)
with Blömker & Jentzen [4, Theorem 3.1]
(with
T=T,
(Ω,F,P)=(Ω,F,P),
V=Hγ,
W=H−ν,
Pn=PIn,
α=γ+ν+(\nicefracη2),
γ=η,
S=((0,T]∋s↦esA∈L(H−ν,Hγ)),
F=(Hγ∋v↦F(v)∈H−ν),
Ot=Ot+etAξ,
Xtn=XtIn
for
t∈[0,T],
n∈\mathbbmN
in the notation of Blömker & Jentzen [4, Theorem 3.1])
therefore shows that
(a)
there exists a unique
stochastic process
X:[0,T]×Ω→Hγ
with continuous sample paths
which satisfies for every
t∈[0,T] that
[TABLE]
and
2. (b)
there exists
a F/B([0,∞))-measurable
function
K:Ω→[0,∞)
such that for every
ω∈Ω,
n∈\mathbbmN
it holds that
[TABLE]
Observe that
the fact that
for every
n∈\mathbbmN
it holds that
(XtIn)t∈[0,T]
is (Ft)t∈[0,T]-adapted
and
item (b)
imply that
(Xt)t∈[0,T]
is (Ft)t∈[0,T]-adapted.
Combining (262),
(274), and
item (a)
hence
establishes item (ii).
This
completes the proof of Theorem 5.10.
∎
Acknowledgements
This project has been partially supported through the SNSF-Research project 200021_156603 ”Numerical
approximations of nonlinear stochastic ordinary and partial differential equations”.
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