FRACTIONAL STOCHASTIC HEAT EQUATION
WITH PIECEWISE CONSTANT COEFFICIENTS
arXiv:1910.12655v1 [math.PR] 28 Oct 2019
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
Abstract. We introduce a fractional stochastic heat equation with second
order elliptic operator in divergence form, having a piecewise constant diffusion
coefficient, and driven by an infinite-dimensional fractional Brownian motion.
We characterize the fundamental solution of its deterministic part, and prove
the existence and the uniqueness of its solution.
1. Introduction
In the last years, stochastic partial differential equations (SPDEs) driven by
different types of fractional noises have attracted a great attention of the probability community. In several cases, the noise is expressed as a function of a formal
derivative of a one-dimensional fractional Brownian motion.
A one-dimensional fractional Brownian motion (fBm) (B H (t))t≥0 with Hurst
index H ∈ (0, 1) is a centered Gaussian process on some probability space (Ω, F , P)
with covariance function
1 2H
(1.1)
t + s2H − |t − s|2H .
E(B H (t)B H (s)) =
2
So, an fBm is a natural extension of a standard Brownian motion because taking
H = 12 in (1.1) we get
E(B 1/2 (t)B 1/2 (s)) = s ∧ t.
Firstly introduced by Kolmogorov in [9], since the appearance of the paper [13],
the interest in this process has increased enormously as an important ingredient of
fractal models because of such characteristics as self-similarity, Hölder continuity
and long-range dependence. Theory of stochastic calculus with respect to fBm has
been developed, leading to the consideration of several types of SPDEs driven by
a noise depending, in one way or another, on a fBm (e. g. [1–4, 8, 17, 18, 20] and
references therein).
In this paper, we consider the following SPDE:
(
du(t, x) = Lu(t, x) dt + h u(t, x) W H (dt, x), t ∈ (0, T ], x ∈ R,
(1.2)
u(0, x) = 0,
x ∈ R,
where h is an affine function,
h(z) = h1 z + h2 ,
(1.3)
h1 , h2 ∈ R and the operator L is defined by
1 d
d
L=
ρ(x)A(x)
.
2ρ(x) dx
dx
Here the coefficients A and ρ have the following form
A(x) = a1 1{x≤0} + a2 1{x>0}
and ρ(x) = ρ1 1{x≤0} + ρ2 1{x>0} ,
2010 Mathematics Subject Classification. 60G22, 60H15, 35R60.
Key words and phrases. Stochastic partial differential equation; discontinuity of coefficients;
fundamental solution; mild solution; infinite-dimensional fractional Brownian motion.
1
2
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
df
ai , ρi (i = 1, 2) are positive constants, and dx
denotes the derivative of f in the
distributional sense. The term W H refers to an L2 (R)-valued fractional Brownian
motion with Hurst index H ∈ ( 12 , 1), defined by
W H (·, t) :=
∞
X
λj ej (·) BjH (t),
(1.4)
j=1
where BjH = {BjH (t), t ≥ 0}, j ∈ N is a sequence of one-dimensional fractional
Brownian motions with the Hurst index H ∈ (1/2, 1) starting at the origin, {λj , j ∈
N} is a sequence of positive real numbers and {ej , j ∈ N} is an orthonormal basis
of L2 (R), such that:
sup kej k∞ < ∞, and
j
∞
X
j=1
λj < ∞,
(1.5)
where k · k∞ denotes the norm in L∞ (R). The series (1.4) converges a. s. in L2 (R)
because of the assumptions (1.5), see [15].
Equation (1.2) can be considered as a stochastic counterpart of the parabolic
equation
∂u(t, x)
= Lu(t, x),
(1.6)
∂t
which arises in mathematical modelling of diffusion phenomena in many areas,
such as ecology [5], biology [14] etc. The non-smoothness of the coefficients reflects
the heterogeneity of the medium in which the process under study propagates.
An explicit expression of the fundamendal solution of equation (1.6) was given
in [6, 22–24].
SPDEs with the operator L have been introduced and investigated in [25, 26],
with an additive noise W defined as a centered Gaussian field W = {W (t, C); t ∈
[0, T ], C ∈ Bb (R)} with covariance
E(W (t, C)W (s, D)) = (t ∧ s)λ(C ∩ D),
where λ denotes the Lebesgue measure. Comparing to those two articles, the equation (1.2) contains more complicated noise, which is multiplicative and fractional.
Its study requires more sophisticated stochastic and Hilbert analysis tools.
Other forms of SPDE (1.2), with other different operators and similar or different
Gaussian noises, have been recently studied by many authors (e. g. [17, 19–21] and
references therein).
We make here a first step in the study of SPDEs of the form (1.2). More precisely,
we prove existence and uniqueness of a mild solution to equation (1.2). In addition
to the introduction of a Besov-type Banach space and the use of the Riemann–
Liouville fractional derivatives, a part of the theory of generalized Lebesgue–Stieljes
integration with respect to fBm with Hurst index H > 1/2 is employed, and the
proofs require many integration techniques, calculation and analysis tools; they are
particularly based on a deep characterization of the explicit form of the fundamental
solution of (1.6).
The paper is organized as follows. In the second section we present some useful
characterizations and upper bounds of the fundamental solution of equation (1.6),
and the third one is devoted to the proof of the existence and uniqueness of a mild
solution to equation (1.2). Some technical lemmata are proved in the appendix.
2. Some properties of the fundamental solution
The fundamental solution of the deterministic partial differential equation (1.6)
is given by
FRACTIONAL STOCHASTIC HEAT EQUATION
"
3
1{y>0}
1{y≤0}
(f (x) − f (y))2
+ √
exp −
√
a1
a2
2(t − s)
#
(| f (x) | + | f (y) |)2
+ β sign(y) exp −
1{s<t} ,
2(t − s)
1
G(t − s, x, y) = p
2π(t − s)
with
y
y
f (y) = √ 1{y≤0} + √ 1{y>0} ,
a1
a2
α = 1−
√
√
a1 + a2 (α − 1)
and β = √
.
√
a1 − a2 (α − 1)
ρ1 a 1
,
ρ2 a 2
For the proof see, e. g., [22, 23] and [6].
The fundamental solution G satisfies the following properties.
Lemma 2.1. For every 0 ≤ s < t ≤ T and x, y ∈ R, we have
1
(f (x) − f (y))2
,
exp −
|G(t − s, x, y)| ≤ Ca1 ,a2 √
2(t − s)
t−s
√
√1 + √1
with Ca1 ,a2 := 1+|β|
.
a
a
1
2
2π
Proof. We have for every x, y ∈ R and 0 ≤ s < t ≤ T ,
1{y>0}
1{y≤0}
(f (x) − f (y))2
1
exp −
+ √
|G(t − s, x, y)| = p
√
a1
a2
2(t − s)
2π(t − s)
(|f (x)| + |f (y)|)2
+ β sign(y) exp −
2(t − s)
1
(f (x) − f (y))2
1
1
≤p
exp −
√ +√
a1
a2
2(t − s)
2π(t − s)
2
(|f (x)| + |f (y)|)
.
+ |β| exp −
2(t − s)
Since the function x 7→ exp(−x2 ) is decreasing on the interval [0, +∞), we have
!
(|f (x)| + |f (y)|)2
(f (x) − f (y))2
exp −
≤ exp −
,
2(t − s)
2(t − s)
and consequently
(1 + |β|)
|G(t − s, x, y)| ≤ p
2π(t − s)
1
1
√ +√
a1
a2
(f (x) − f (y))2
.
exp −
2(t − s)
Corollary 2.1. For every 0 ≤ s < t ≤ T x ∈ R and y ∈ R, we have
Z
Z
max
|G(t − s, z, y)| dz, |G(t − s, x, z)| dz ≤ C1 (a1 , a2 ),
R
with C1 (a1 , a2 ) =
R
√1
a1
+
√1
a2
√ √
(1 + |β|) max( a1 , a2 ).
Proof. : By the virtue of Lemma 2.1, we get:
Z
Z
(1 + |β|)
1
(f (z) − f (y))2
1
|G(t − s, z, y)| dz ≤ p
dz.
exp −
√ +√
a1
a2
2(t − s)
2π(t − s)
R
R
Further,
!
Z 0
Z
( √za1 − f (y))2
(f (z) − f (y))2
dz =
exp −
dz
exp −
2(t − s)
2(t − s)
−∞
R
4
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
+
Z
0
√z
a
By the change of variables U = √i
−f (y)
2(t−s)
∞
exp −
( √za2 − f (y))2
2(t − s)
dz
,
2(t−s) ai
=⇒ dU = √
!
dz.
with i = 1 in the
first integral and i = 2 in the second one, we obtain:
Z
Z
p
√ √
(f (z) − f (y))2
dz ≤ max( a1 , a2 ) 2(t − s) exp(−U 2 ) dU
exp −
2(t
−
s)
R
R
p
√ √
= max( a1 , a2 ) 2(t − s) π.
(2.1)
Thus,
Z
R
|G(t − s, z, y)| dz ≤
1
1
√ +√
a1
a2
√ √
(1 + |β|) max( a1 , a2 ).
Concerning the second integral, also by the virtue of Lemma 2.1 and (2.1), we get:
Z
Z
1
(f (x) − f (z))2
1
(1 + |β|)
dz
+
exp
−
|G(t − s, x, z)| dz ≤ p
√
√
a1
a2
2(t − s)
2π(t − s)
R
R
√ √
1
1
≤ √ +√
(1 + |β|) max( a1 , a2 ).
a1
a2
Lemma 2.2. (i) For every η > 0, there exists a strictly positive constant C such
that, for every 0 < t ≤ T and x, y ∈ R,
Z
Z
1
3
η
η
max
|∂t G(t, z, y)| dz, |∂t G(t, x, z)| dz ≤ Ct− 2 η+ 2 .
R
R
(ii) For every η > 0, there exists a strictly positive constant C such that, for every
0 ≤ s < t ≤ T and y ∈ R,
Z
η
1
5
∂2
G(t − s, z, y) dz ≤ C(t − s)− 2 η+ 2 .
∂t∂s
R
Proof. (i) For every x, y ∈ R, we have
"
1
1
(f (x) − f (y))2
1
1 −3/2
|∂t G(t, x, y)| ≤ √
exp
−
+
t
√
√
a1
a2
2
2t
2π
(|f (x)| + |f (y)|)2
+ |β| exp −
2t
n
1
(f (x) − f (y))2
(f (x) − f (y))2
+ t−1/2
exp
−
2
t2
2t
#
2
(|f (x)| + |f (y)|)2 o
(|f (x)| + |f (y)|)
exp −
+ |β|
t2
2t
"
1
(1 + |β|) −3/2
1
1
(f (x) − f (y))2
√
≤
t
exp −
√ +√
a1
a2
2
2t
2π
n (f (x) − f (y))2
(f (x) − f (y))2
1
exp −
+ t−3/2
2
t
2t
#
(|f (x)| + |f (y)|)2 o
(|f (x)| + |f (y)|)2
exp −
+ |β|
t
2t
FRACTIONAL STOCHASTIC HEAT EQUATION
5
Then, by using the fact that for every a, x, y, z > 0, we have (x + y + z)a ≤
C(xa + y a + z a ), we get
η "
1
(1 + |β|)η − 3 η
(f (x) − f (y))2
1
2
exp −η
t
√ +√
a1
a2
2η
2t
η
(f (x) − f (y))2
1 3 n (f (x) − f (y))2
exp −η
+ η t− 2 η
2
t
2t
#
2η
(|f (x)| + |f (y)|)2 o
η (|f (x)| + |f (y)|)
+ |β|
.
exp −η
tη
2t
C
|∂t G(t, x, y)| ≤ √
( 2π)η
η
Thus,
η
3
1
t− 2 η
1
η
|∂t G(t, z, y)| dz ≤ √
+√
√
a1
a2
(2 2π)η
R
"
Z
(f (z) − f (y))2
dz
× (1 + |β|)η
exp −η
2t
R
η
Z
(f (z) − f (y))2
(f (z) − f (y))2
+
dz
exp −η
t
2t
R
η
#
Z
(|f (z)| + |f (y)|)2
(|f (z)| + |f (y)|)2
η
dz .
+ |β|
exp −η
t
2t
R
Z
On the one hand, by (2.1) we have
(f (z)−f (y))2
exp
−η
dz
R
2t
R
≤
√1
η
√ √ √
max( a1 , a2 ) 2πt.
On the other hand, we have
Z
R
η
(f (z) − f (y))2
(f (z) − f (y))2
dz
exp −η
t
2t
!
Z +∞ ( √z − f (y))2 !η
( √za2 − f (y))2
a2
=
dz
exp −η
t
2t
0
!η
!
Z 0
( √za1 − f (y))2
( √za1 − f (y))2
dz
exp −η
+
t
2t
−∞
!η
!
Z
( √za2 − f (y))2
( √za2 − f (y))2
exp −η
dz
≤
t
2t
R
!η
!
Z
( √za1 − f (y))2
( √za1 − f (y))2
+
dz
exp −η
t
2t
R
η+ 21
Z
√
2
√ √
2
≤
max( a1 , a2 ) t z 2η e−z dz.
η
R
It is clear that the last integral converges for every η > 0. Therefore, we get
Z
R
(f (z) − f (y))2
t
η
√
(f (z) − f (y))2
dz ≤ C(η, a1 , a2 ) t.
exp −η
2t
6
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
Concerning the last integral, we have
η
Z
(|f (z)| + |f (y)|)2
(|f (z)| + |f (y)|)2
dz
exp −η
t
2t
R
η
Z
(f (z) − f (y))2
(f (z) − f (y))2
=
dz
exp −η
t
2t
{zy≤0}
η
Z
(f (z) + f (y))2
(f (z) + f (y))2
+
dz
exp −η
t
2t
{zy≥0}
η
Z
(f (z) − f (y))2
(f (z) − f (y))2
≤
exp −η
dz
t
2t
R
η
Z
(f (z) + f (y))2
(f (z) + f (y))2
dz.
exp −η
+
t
2t
R
By the same technique as above, we get
η
Z
(|f (z)| + |f (y)|)2
(|f (z)| + |f (y)|)2
dz
exp −η
t
2t
R
r
η+ 21
√ √
2
πt
≤
max( a1 , a2 )
.
η
2
All this implies that
Z
R
η
3
1
|∂t G(t, z, y)| dz ≤ C(η, a1 , a2 , β)t− 2 η+ 2 ,
where C(η, a1 , a2 , β) denotes a strictly positive constant depending only on η, a1 , a2 , β.
Following the same steps, we can prove that
Z
3
1
|∂t G(t, x, z)|η dz ≤ C(T, η, a1 , a2 , β)t− 2 η+ 2
R
with the same constant C(η, a1 , a2 , β) as above.
(ii) The mixed partial derivative of G equals
2
1{y≤0}
1{y>0}
1
3
∂
(f (z) − f (y))2
− 25
+ √
G(t − s, z, y) = √
(t − s) exp −
√
∂t∂s
a1
a2
4
2(t − s)
2π
2
2
7
(f (z) − f (y))
(f (z) − f (y))
− 3(t − s)− 2 exp −
2(t − s)
2
2
9
(f
(z)
−
f
(y))
(f
(z)
−
f (y))4
+ (t − s)− 2 exp −
2(t − s)
4
2
5
(|f
(z)|
+
|f
(y)|)
3
+ β sign(y)(t − s)− 2 exp −
4
2(t − s)
(|f
(z)|
+ |f (y)|)2 (|f (z)| + |f (y)|)2
7
− 3β sign(y)(t − s)− 2 exp −
2(t − s)
2
2
(|f
(z)|
+
|f
(y)|)
9
(|f
(z)|
+
|f (y)|)4
+ β sign(y)(t − s)− 2 exp −
.
2(t − s)
4
Therefore
(f (z) − f (y))2
∂2
− 25
exp −
G(t − s, z, y) ≤ C(t − s)
∂t∂s
2(t − s)
2
2
(f (z) − f (y))
(f (z) − f (y))
+ exp −
2(t − s)
2(t − s)
2
(f (z) − f (y))4
(f (z) − f (y))
+ exp −
2(t − s)
4(t − s)2
FRACTIONAL STOCHASTIC HEAT EQUATION
7
(|f (z)| + |f (y)|)2
(|f (z)| + |f (y)|)2 (|f (z)| + |f (y)|)2
+ exp −
+ exp −
2(t − s)
2(t − s)
2(t − s)
2
4
(|f (z)| + |f (y)|)
(|f (z)| + |f (y)|)
+ exp −
.
2(t − s)
4(t − s)2
The rest of proof can be done similarly to that of (i).
Corollary 2.2. For all δ ∈ ( 13 , 1) there exists a strictly positive constant C > 0,
such that, for every x, y ∈ R, for every s, t ∈ (0, T ] with s < t,
Z
Z
max
G(t, z, y) − G(s, z, y) dz,
G(t, x, z) − G(s, x, z) dz ≤ C s−δ (t − s)δ .
R
R
Proof. For every fixed z, y ∈ R, 0 < s < t ≤ T and δ ∈ [0, 1] we have, by the
triangular inequality,
G(t, z, y) − G(s, z, y) = G(t, z, y) − G(s, z, y)
≤
1−δ
G(t, z, y) + G(s, z, y)
G(t, z, y) − G(s, z, y)
1−δ
δ
δ
G(t, z, y) − G(s, z, y) .
By mean-value theorem, there exists t∗ ∈ (s, t) such that
G(t, z, y) − G(s, z, y)
δ
≤ (t − s)δ |∂t G(t∗ , z, y)|δ .
2
Therefore, by Lemma 2.1 and since for every x ∈ R, e−x ≤ 1, we have
1−δ
1
1
(t − s)δ |∂t G(t∗ , z, y)|δ .
G(t, z, y) − G(s, z, y) ≤ C t− 2 + s− 2
Since for every s < t, we have t−1/2 < s−1/2 , by Lemma 2.2, we get:
Z
Z
δ
− 21 (1−δ)
(t − s)
|∂t G(t∗ , z, y)|δ dz
G(t, z, y) − G(s, z, y) dz ≤ Cs
R
R
1
3
1
≤ Cs− 2 (1−δ) (t − s)δ (t∗ )− 2 δ+ 2 .
Note that for δ >
1
3
3
1
3
1
(t∗ )− 2 δ+ 2 ≤ s− 2 δ+ 2 .
Consequently, we obtain:
Z
3
1
1
G(t, z, y) − G(s, z, y) dz ≤ Cs− 2 (1−δ) s− 2 δ+ 2 (t − s)δ ≤ Cs−δ (t − s)δ .
R
Following the same technique we also get
Z
G(t, x, z) − G(s, x, z) dz ≤ C s−δ (t − s)δ .
R
Lemma 2.3. For every δ ∈ ( 15 , 1) there exists a strictly positive constant C such
that, for all 0 < σ < τ < s < t < T,
Z
|G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| dz
R
≤ C (t − s)δ (s − τ )−2δ (τ − σ)δ . (2.2)
Proof. On the one hand, by Lemma 2.1,
|G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)|
1
1
1
1
1
≤ C (t − τ )− 2 + (s − τ )− 2 + (t − σ)− 2 + (s − σ)− 2 ≤ C(s − τ )− 2 .
8
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
On the other hand, by the mean value theorem, there exist θ∗ ∈ (s, t) and ρ∗ ∈ (σ, τ )
such that
|G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)|
Z tZ τ
∂2
∂2
=
G(θ − ρ, z, y) dρ dθ =
G(θ∗ − ρ∗ , z, y) (t − s)(τ − σ).
∂θ∂ρ
s σ ∂θ∂ρ
Hence, for every δ ∈ (0, 1), we have
Z
|G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| dz
R
Z
1−δ
=
|G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)|
R
δ
× |G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| dz
Z
δ
∂2
1
G(θ∗ − ρ∗ , z, y) dz.
≤ C(s − τ )− 2 (1−δ) (t − s)δ (τ − σ)δ
R ∂θ∂ρ
Taking into account Lemma 2.2 (ii), we obtain
Z
|G(t − τ, z, y) − G(s − τ, z, y) − G(t − σ, z, y) + G(s − σ, z, y)| dz
R
1
5
1
≤ C(s − τ )− 2 (1−δ) (t − s)δ (τ − σ)δ (θ∗ − ρ∗ )− 2 δ+ 2 .
5
1
5
1
If δ > 51 , then (θ∗ − ρ∗ )− 2 δ+ 2 ≤ (s − τ )− 2 δ+ 2 and we arrive at (2.2).
3. Mild solution
3.1. Definitions and notation.
3.1.1. Norms and spaces. Throughout the paper, the symbol C will denote a generic
constant, the precise value of which is not important and may vary between different
equations and inequalities. Let k · k2 be the norm in L2 (R). Let 0 < σ < 1. For
every measurable function u : [0, T ] × R → R and t ∈ [0, T ] denote
2
Z t Z s
ku(s, ·) − u(v, ·)k2
2
dv ds,
kukσ,1,t :=
(s − v)σ+1
0
0
2
2
2
kukσ,2,t := sup ku(s, ·)k2 + kukσ,1,t .
(3.1)
s∈[0,t]
We define also the following seminorm for f : [0, T ] → R and t ∈ [0, T ]:
Z v
|f (u) − f (v)|
|f (u) − f (z)|
kf kσ,0,t := sup
+
dz
(v − u)1−σ
(z − u)2−σ
0≤u<v≤t
u
For σ ∈ (0, 1) we denote by B σ,2 [0, T ]; L2(R) the following Banach space:
n
B σ,2 [0, T ]; L2(R) := u : [0, T ] × R → R Lebesgue measurable mapping
o
2
such that kukσ,2,T < ∞ .
3.1.2. Integration with respect to B H for every H ∈ ( 12 , 1). Let a, b ∈ R, a < b. Let
ϕ ∈ L1 ([a, b]) and σ ∈ (0, 1). The Riemann–Liouville left- and right-sided fractional
integrals of ϕ of order σ are defined for almost all x ∈ [a, b] by
Z x
1
σ
Ia+ ϕ(x) =
(x − y)σ−1 ϕ(y) dy
Γ(σ) a
Z b
1
σ
(x − y)σ−1 ϕ(y) dy.
Ib− ϕ(x) =
Γ(σ) x
FRACTIONAL STOCHASTIC HEAT EQUATION
9
Let Iaσ+ (Lp )(resp. Ibσ− (Lp )) denote the image of Lp ([a, b]) by the operator
(resp. Ibσ− ).
If ϕ ∈ Iaσ+ (Lp )(resp. ϕ ∈ Ibσ− (Lp )) then the Riemann–Liouville left- and right-sided
fractional derivatives are defined by
Z x
1
ϕ(x)
ϕ(x) − ϕ(y)
σ
Da+ ϕ(x) =
+σ
dy ,
Γ(1 − σ) (x − a)σ
(x − y)1+σ
a
!
Z b
ϕ(x)
ϕ(x)
−
ϕ(y)
1
+σ
dy .
Dbσ− ϕ(x) =
1+σ
Γ(1 − σ) (b − x)σ
x (y − x)
Iaσ+
For every two functions ϕ, ψ : [a, b] → R such that Daσ+ ϕ ∈ L1 ([a, b]) and
∈ L∞ ([a, b]) where ψb− (x) = ψ(b− ) − ψ(x), we define the generalized
Lebesgue–Stieljes integral by
Z b
Z b
Daσ+ ϕ(x) Db1−σ
(3.2)
ϕ(x) dψ(x) :=
− ψb− (x) dx.
Db1−σ
− ψb−
a
a
Furthermore, this integral admits the following bound:
Z x
Z b
Z b
|ϕ(x)|
|ϕ(x) − ϕ(y)|
+
dy
dx. (3.3)
ϕ(x) dψ(x) ≤ Cσ kψkσ,0,b
(x − a)σ
(x − y)1+σ
a
a
a
For more information on generalized Lebesgue–Stieljes integration with respect
to fractional Brownian motion with Hurst index H > 1/2, the reader can see,
e. g., [16] and references therein.
Now we suppose that σ ∈ (1 − H, 1/2), and denote by
L L2 (R) the space of
linear operators on L2 (R). Let F : [0, T ] × Ω → L L2 (R) be an operator function
such that
Z t
Z T
k(F (s, ω) − F (v, ω))ej k2
kF (s, ω)ej k2
+
dv ds < ∞
a. s. (3.4)
sup
sσ
(s − v)σ+1
j∈N 0
0
Following [15], we define the integral with respect to the L2 (R)-valued fractional
Brownian motion by
Z b
Z b
∞
X
λj
F (s, ω)ej dBjH (s),
(3.5)
F (s, ω) dB H (s) =
a
a
j=1
where the integral with respect to BjH is the path-wise generalized Lebesgue–
Stieltjes integral defined in (3.2) and the convergence of the series should be understood as P-a. s. convergence in L2 (R).
Applying assumptions (1.5) and [15, Proposition 2.1], we deduce that the integral
(3.5) is well defined.
From (3.3), one can obtain the following inequality for 0 ≤ a < b ≤ T :
Z
a
b
F (s, ω) dB H (s)
2
≤ Cξσ,H,T sup
i∈N
+
Z b
a
Z
s
a
kF (s, ω)ei k2
(s − a)σ
kF (s, ω)ei − F (v, ω)ei k2
dv
ds, (3.6)
(s − v)1+σ
where C = C(σ) is a constant, and the random variable
ξσ,H,T :=
∞
X
λj BjH
σ,0,T
j=1
is finite a. s., see [15]. For more details on the integration with respect to the
L2 (R)-valued fractional Brownian motion, see e. g. [15] and references therein.
10
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
3.1.3. Mild solution.
Definition 3.1. An L2 (R)-valued random process {u(t, ·), t ∈ [0, T ]} is called a
mild solution to the problem (1.2) if it satisfies the following assumptions:
(1) u ∈ B σ,2 [0, T ]; L2(R) a.s. for some σ ∈ (1 − H, 1/2).
(2) For any ∀t ∈ [0, T ] it holds that
Z tZ
∞
X
λj
G(t − s, x, y) h(u(s, y)) ej (y) dy dBjH (s)
a.s.
(3.7)
u(t, x) =
0
j=1
R
Here, the integrals with respect to BjH , j ∈ N are the generalized Lebesgue–Stieltjes
integrals defined in (3.2).
Let us introduce some notations for u ∈ B σ,2 [0, T ]; L2(R) . Namely, let
Z
:=
ςj,t (u)(s, x)
G(t − s, x, y) h(u(s, y)) ej (y) dy,
(3.8)
R
and
∗
ςj,t,s
(u)(v, x) := ςj,t (u)(v, x) − ςj,s (u)(v, x).
(3.9)
Also let us denote the right-hand side of (3.7) by
Z tZ
∞
X
λj
G(t − s, x, y) h(u(s, y)) ej (y) dy dBjH (s)
(Au)(t, x) :=
=
0
j=1
∞
X
λj
Z
0
j=1
t
R
ςj,t (u)(s, x) dBjH (s).
Note that Au is an integral with respect to an L2 (R)-valued fractional Brownian
motion, defined by (3.5). The condition (3.4) for it has the form
Z T
Z s
kςj,t (u)(s, ·)k2
kςj,t (u)(s, ·) − ςj,t (u)(v, ·)k2
sup
+
dv ds < ∞
a. s.
sσ
(s − v)σ+1
j∈N 0
0
In fact, this condition holds for any u ∈ B σ,2 [0, T ]; L2(R) . It will be checked
in the proof of Proposition 3.1 below, in which we also establish that Au actually
determines an L2 (R)-valued stochastic process from the class B σ,2 [0, T ]; L2(R) .
3.2. A priori estimates. Let us fix H ∈ 21 , 1 and σ ∈ 1 − H, 21 . Abbreviate
ξ = ξσ,H,T .
Proposition 3.1. Let u ∈ B σ,2 [0, T ]; L2(R) . Then Au ∈ B σ,2 [0, T ]; L2(R)
a. s. Moreover, for any t ∈ (0, T ],
2
2
kAukσ,2,t ≤ Cξ 2 kukσ,2,t + 1 .
Proof. Let δ ∈ (max{2σ, 31 }, 1) be fixed throughout the proof. It follows from (3.7)
and Lemma A.1 that for any t ∈ (0, T ],
k(Au)(t, ·)k2 =
∞
X
j=1
λj
Z
0
t
ςj,t (u)(s, ·)dBjH (s)
Z t
2
Z s
kςj,t (u)(s, ·)k2
kςj,t (u)(s, ·) − ςj,t (u)(v, ·)k2
≤ C ξ sup
+
dv
ds
sσ
(s − v)σ+1
j∈N 0
0
Z tZ s
Z t
ku(s, ·) − u(v, ·)k2
ku(s, ·)k2 + 1
ds
+
dv ds
≤Cξ
σ
s
(s − v)σ+1
0 0
0
FRACTIONAL STOCHASTIC HEAT EQUATION
Z
+
t
0
− 2δ
(t − s)
s
Z
(s − v)
0
δ
2 −σ−1
11
!
(ku(v, ·)k2 + 1) dv ds .
By the Cauchy–Schwarz inequality, we have
Z
0
t
Z
ku(s, ·)k2 + 1
ds ≤
sσ
Since σ < 1/2, we have
t
Z
0
Rt
1
0 s2σ
t
0
1
ds
s2σ
1/2 Z
t
2
(ku(s, ·)k2 + 1) ds
0
ku(s, ·)k2 + 1
ds ≤ C
sσ
Z t
0
2
ku(s, ·)k2
+1
ds
1/2
.
s−v
t−v
(3.10)
in the last
v
0
=
.
ds < ∞. Hence,
By Fubini’s Theorem and applying the change of variables x =
integral, we obtain
Z t
Z s
δ
− δ2
(t − s)
(s − v) 2 −σ−1 (ku(v, ·)k2 + 1) dv ds
0
0
Z t Z t
δ
− 2δ
−σ−1
2
(t − s) (s − v)
=
ds (ku(v, ·)k2 + 1) dv
Z
1/2
t
0
=B
≤C
≤ Ct
(t − v)−σ
δ
2
1
Z
0
− σ, 1 −
δ
2
δ
δ
(1 − x)− 2 x 2 −σ−1 dx (ku(v, ·)k2 + 1) dv
Z
0
t
(t − v)−σ (ku(v, ·)k2 + 1) dv
!Z
t
sup ku(v, ·)k2 + 1
v∈[0,t]
1−σ
0
(3.11)
(t − v)−σ dv
!
sup ku(v, ·)k2 + 1 ,
v∈[0,t]
where B denotes the beta function defined, for every p, q > 0, by B(p, q) :=
R 1 p−1
t
(1 − t)q−1 dt, and in the last inequality we used the fact that σ < 1/2.
0
Therefore, using (3.10) and (3.11), we obtain
"
! Z
t
1/2
2
2
2
1−σ
k(Au)(t, ·)k2 ≤ C ξ t
sup ku(v, ·)k2 + 1 +
ku(s, ·)k2 + 1 ds
v∈[0,t]
+
Z tZ
0
≤Cξ
+
s
0
2
t
sup
v∈[0,t]
0
≤ C ξ2
+t
ku(s, ·) − u(v, ·)k2
dv ds
(s − v)σ+1
2−2σ
Z t Z
0
s
0
2
ku(v, ·)k2
sup
v∈[0,t]
0
0
!
ku(s, ·) − u(v, ·)k2
dv ds
(s − v)σ+1
(t2−2σ + t)
Z t Z
+1
#2
s
2
ku(v, ·)k2
ku(s, ·) − u(v, ·)k2
dv
(s − v)σ+1
Z t
2
ku(s, ·)k2 + 1 ds
+
0
2 !
+1
2
!
!
ds .
12
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
Since t2−2σ = t1−2σ t ≤ T 1−2σ t = Ct for σ < 1/2, we arrive at
2 !
Z t Z s
ku(s,
·)
−
u(v,
·)k
2
2
2
k(Au)(t, ·)k2 ≤ C ξ 2 t sup ku(v, ·)k2 + 1 +
dv
ds .
(s − v)σ+1
v∈[0,t]
0
0
(3.12)
By the definition of the norm k.kσ,2,t , we get
2
2
k(Au)(t, ·)k2 ≤ C ξ 2 t kukσ,2,t + 1 .
Consequently,
2
2
sup k(Au)(s, ·)k2 ≤ C ξ 2 kukσ,2,t + 1 ,
(3.13)
s∈[0,t]
because kuk2σ,2,s ≤ kuk2σ,2,t for s ≤ t.
Futhermore,
k(Au)(s, ·) − (Au)(v, ·)k2
Z s
Z
∞
X
=
λj
ςj,s (u)(z, ·)dBjH (z) +
v
j=1
∞
X
≤
λj
∞
X
0
j=1
2
2
v
Z
λj
[ςj,s (u)(z, ·) −
0
ςj,v (u)(z, ·)] dBjH (z)
ςj,s (u)(z, ·)dBjH (z)
v
j=1
+
s
Z
v
[ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] dBjH (z)
.
(3.14)
2
For the first term in the right-hand side of (3.14) we have that
∞
X
s
Z
λj
v
j=1
ςj,s (u)(z, ·)dBjH (z)
2
≤ C ξ sup
j∈N
Z
s
v
kςj,s (u)(z, ·)k2
+
(z − v)σ
Z
z
v
kςj,s (u)(z, ·) − ςj,s (u)(r, ·)k2
dr dz.
(z − r)σ+1
Then, using Lemma A.1 and the same technique as used to prove (3.12), we get
∞
X
Z
λj
s
v
j=1
2
ςj,s (u)(z, ·) dBjH (z)
2
2
≤ C ξ (s − v)
sup
r∈[v,s]
2
ku(r, ·)k2
+1+
Z
v
s
Z
v
z
ku(z, ·) − u(r, ·)k2
dr
(z − r)σ+1
2
!
dz .
(3.15)
For the second term in the right-hand side of (3.14) we have that
Z v
∞
X
λj
[ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] dBjH (z)
j=1
0
2
≤ C ξ sup
j∈N
+ sup
j∈N
Z
0
Z vZ
0
v
0
z
kςj,s (u)(z, ·) − ςj,v (u)(z, ·)k2
dz
zσ
[ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] − [ςj,s (u)(r, ·) − ςj,v (u)(r, ·)]
(z − r)σ+1
2
drdz
!
FRACTIONAL STOCHASTIC HEAT EQUATION
13
=: C ξ L1 + L2 .
Using Lemma A.2, we obtain
δ
δ
kςj,s (u)(z, ·) − ςj,v (u)(z, ·)k2 ≤ C (v − z)− 2 (s − v) 2 (ku(z, ·)k2 + 1) .
Thus,
L1 ≤ C(s − v)
δ
2
δ
≤ C(s − v) 2
δ
= C(s − v) 2
≤ C(s − v)
δ
v
(v − z)− 2
(ku(z, ·)k2 + 1) dz
zσ
0
!Z
δ
v
(v − z)− 2
sup ku(z, ·)k2 + 1
dz
zσ
z∈[0,v]
0
!
Z
sup ku(z, ·)k2 + 1 v −δ/2+1−σ B 1 − 2δ , 1 − σ
z∈[0,v]
δ
2
sup ku(z, ·)k2 + 1
z∈[0,v]
!
because σ < 1/2 and δ < 1.
Now by Lemma A.2, we have
kςj,s (u)(z, ·) − ςj,v (u)(z, ·) − ςj,s (u)(r, ·) + ςj,v (u)(r, ·)k2
δ
δ
≤ C (s − v) 2 (v − z)− 2 ku(z, ·) − u(r, ·)k2
δ
δ
+ (s − v) 2 (v − z)−δ (z − r) 2 (ku(r, ·)k2 + 1) .
Thus,
L2 = sup
j∈N
≤C
Z vZ
0
(z − r)σ+1
0
Z vZ
0
+C
[ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] − [ςj,s (u)(r, ·) − ςj,v (u)(r, ·)]
z
δ
z
0
Z vZ
0
δ
z
0
δ
δ
δ
(s − v) 2 (v − z)−δ (z − r) 2 (ku(r, ·)k2 + 1)
drdz
(z − r)σ+1
By Fubini’s theorem, we have
Z vZ z
δ
L2,2 =
(z − r) 2 −σ−1 (v − z)−δ (ku(r, ·)k2 + 1) dr dz
Z v
Z0 v 0
δ
=
(ku(r, ·)k2 + 1)
(z − r) 2 −σ−1 (v − z)−δ dz dr.
0
r
Since 2σ < δ < 1, we see that
Z v
δ
δ
(z − r) 2 −σ−1 (v − z)−δ dz = B 1 − δ, δ2 − σ (v − r)− 2 −σ .
r
Therefore,
≤C
Z
v
0
drdz
(s − v) 2 (v − z)− 2 ku(z, ·) − u(r, ·)k2
drdz
(z − r)σ+1
=: C(s − v) 2 (L2,1 + L2,2 ) .
L2,2 ≤ C
2
δ
(ku(r, ·)k2 + 1) (v − r)− 2 −σ dr
!Z
v
sup ku(r, ·)k2 + 1
r∈[0,v]
0
δ
(v − r)− 2 −σ dr
(3.16)
14
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
≤ Cv
1− 2δ −σ
sup ku(r, ·)k2 + 1
r∈[0,v]
!
≤C
!
sup ku(r, ·)k2 + 1 .
r∈[0,v]
(3.17)
It follows from (3.16) and (3.17) that
!
Z vZ z
δ
(v − z)− 2 ku(z, ·) − u(r, ·)k2
δ
L2 ≤ C(s−v) 2
drdz + sup ku(r, ·)k2 + 1 ,
(z − r)σ+1
r∈[0,v]
0
0
and consequently,
∞
X
λj
j=1
Z
v
0
≤ C ξ(s−v)
≤ C ξ L1 + L2
[ςj,s (u)(z, ·) − ςj,v (u)(z, ·)] dBjH (z)
δ
2
sup ku(r, ·)k2 +1+
r∈[0,v]
Z
v
z
Z
0
0
2
!
δ
(v − z)− 2 ku(z, ·) − u(r, ·)k2
drdz .
(z − r)σ+1
(3.18)
Hence, combining (3.14)–(3.18), we obtain
"
k(Au)(s, ·) − (Au)(v, ·)k2 ≤ C ξ (s − v)1/2
Z
1/2
+ (s − v)
v
δ
v
r∈[0,v]
Z
δ
2
v
0
1
2
z
Z
ku(z, ·) − u(r, ·)k2 2
dr dz
(z − r)σ+1
!
!1/2
sup ku(r, ·)k2 + 1
+ (s − v) 2
+ (s − v)
s
sup ku(r, ·)k2 + 1
r∈[v,s]
!
Since (s − v) = (s − v)
1−δ
2
− 2δ
(v − z)
z
Z
0
δ
2
(s − v) ≤ T
#
ku(z, ·) − u(r, ·)k2
dr dz .
(z − r)σ+1
1−δ
2
δ
(s − v) 2 , we see that
"
δ
k(Au)(s, ·) − (Au)(v, ·)k2 ≤ C ξ(s − v) 2
sup ku(r, ·)k2 + 1
r∈[0,s]
!1/2
ku(z, ·) − u(r, ·)k2 2
+
dr dz
(z − r)σ+1
v
v
#
Z v
Z z
ku(z,
·)
−
u(r,
·)k
δ
2
(v − z)− 2
+
dr dz .
(z − r)σ+1
0
0
Z
s
Z
z
Therefore,
2
kAukσ,1,t =
Z t Z
0
s
0
k(Au)(s, ·) − (Au)(v, ·)k2
dv
(s − v)σ+1
2
ds ≤ C ξ 2
Z tX
3
0 i=1
Hi2 (s) ds,
where
H1 (s) =
sup ku(r, ·)k2 + 1
r∈[0,s]
!Z
s
0
δ
(s − v) 2 −σ−1 dv,
!1/2
ku(z, ·) − u(r, ·)k2 2
dr dz
dv,
H2 (s) =
(s − v)
(z − r)σ+1
0
v
v
Z z
Z v
Z s
ku(z, ·) − u(r, ·)k2
δ
−σ−1
− δ2
2
(v − z)
H3 (s) =
(s − v)
dr dz dv.
(z − r)σ+1
0
0
0
Z
s
δ
2 −σ−1
Z
s
Z
z
(3.19)
FRACTIONAL STOCHASTIC HEAT EQUATION
The first term can be bounded by
!
H1 (s) =
sup ku(r, ·)k2 + 1 s
δ
2 −σ
r∈[0,s]
≤C
15
!
sup ku(r, ·)k2 + 1 ,
r∈[0,s]
(3.20)
because 2δ > σ.
Concerning H2 , we have
H2 (s) ≤ kukσ,1,s
Z
s
0
δ
(s − v) 2 −σ−1 dv ≤ C kukσ,1,s .
(3.21)
For the last one, applying Fubini’s theorem, we get
Z z
Z s Z s
2
ku(z, ·) − u(r, ·)k2
δ
δ
H32 (s) =
dr
dz
(s − v) 2 −σ−1 (v − z)− 2 dv
(z − r)σ+1
0
0
z
2
Z s
Z z
2
ku(z, ·) − u(r, ·)k2
dr
dz
.
≤ B 1 − 2δ , δ2 − σ
(s − z)−σ
(z − r)σ+1
0
0
Since σ < 1/2, we can write, applying the Cauchy–Schwartz inequality,
2
Z s Z z
Z s
ku(z, ·) − u(r, ·)k2
2
(s − z)−2σ dz
H32 (s) ≤ C
dr
dz ≤ C kukσ,1,s .
σ+1
(z
−
r)
0
0
0
(3.22)
Finally, combining (3.19)–(3.22), we get
Z t
2
2
2
2
(3.23)
kAukσ,1,t ≤ C ξ
kukσ,2,s ds + 1 ≤ C ξ 2 kukσ,2,t + 1 ,
0
kuk2σ,2,s
kuk2σ,2,t
because
≤
for s ≤ t.
Hence, from (3.1), (3.13) and (3.23) we get the result.
Proposition 3.2. Let u, ũ ∈ B σ,2 [0, T ]; L2(R) . Then for all t ∈ [0, T ],
Z t
2
2
2
kAu − Aũkσ,2,t ≤ Cξ
ku − ũkσ,2,s ds.
0
Proof. Recall that σ ∈ 1 − H, . Fix δ ∈ max{σ, 31 }, 1 .
By the definition of A, we can write
Z t
∞
X
λj
[ςj,t (u)(s, ·) − ςj,t (ũ)(s, ·)] dBjH (s)
k(Au)(t, ·) − (Aũ)(t, ·)k2 =
1
2
j=1
≤ C ξ sup
j∈N
+
Z
0
s
Z
t
0
2
kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·)k2
sσ
0
!
kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·) − ςj,t (u)(r, ·) + ςj,t (ũ)(r, ·)k2
dr ds.
(s − r)σ+1
It follows from Lemma A.3 that
k(Au)(t, ·) − (Aũ)(t, ·)k2 ≤ C ξ
+
Z
s
−δ
(t − s)
0
(Z
0
t
ku(s, ·) − ũ(s, ·)k2
sσ
!
ku(r, ·) − ũ(r, ·)k2
dr ds
(s − r)σ+1−δ
)
ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2
dr ds ,
+
(s − r)σ+1
0
0
where the right-hand side is finite, because u, ũ ∈ B σ,2 [0, T ]; L2(R) .
Z tZ
s
16
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
First, the Cauchy–Schwarz inequality implies that for every σ < 12
Z t
1/2 Z t
1/2
Z t
ku(s, ·) − ũ(s, ·)k2
2
−2σ
ds
≤
s
ds
ku(s,
·)
−
ũ(s,
·)k
ds
2
sσ
0
0
0
!
1/2
Z t
2
≤C
sup ku(r, ·) − ũ(r, ·)k2 ds
0 r∈(0,s)
Second, it follows from Fubini’s theorem and the Cauchy–Schwarz inequality that
for every σ < δ < 1
Z tZ s
(t − s)−δ ku(r, ·) − ũ(r, ·)k2
dr ds
(s − r)σ+1−δ
0 0
Z t
Z t
=
ku(r, ·) − ũ(r, ·)k2
(t − s)−δ (s − r)−σ−1+δ ds dr
0
r
Z t
= B(1 − δ, δ − σ)
(t − r)−σ ku(r, ·) − ũ(r, ·)k2 dr
0
t
≤C
Z
≤C
Z
0
(t − r)−2σ dr
21
Z
0
t
0
t
2
sup ku(r, ·) − ũ(r, ·)k2 ds
r∈(0,s)
2
ũ(r, ·)k2
sup ku(r, ·) −
r∈(0,s)
ds
! 21
! 12
.
Hence,
k(Au)(t, ·) − (Aũ)(t, ·)k2 ≤ C ξ
+t
Z t Z
0
s
0
( Z
0
t
sup ku(r, ·) −
r∈(0,s)
2
ũ(r, ·)k2
ds
!1/2
ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2
dr
(s − r)σ+1
2
)
ds .
Therefore, we obtain
2
sup k(Au)(s, ·) − (Aũ)(s, ·)k2 ≤ C ξ 2
s∈(0,t)
Z
0
t
2
ku − ũkσ,2,s ds,
On the other hand, we have that for r < s
k(Au)(s, ·) − (Aũ)(s, ·) − (Au)(r, ·) + (Aũ)(r, ·)k2
Z s
∞
X
λj
[ςj,s (u)(v, ·) − ςj,s (ũ)(v, ·)] dBjH (v)
≤
r
j=1
+
∞
X
λj
Z
r
[ςj,s (u)(v, ·) − ςj,s (ũ)(v, ·) − ςj,r (u)(v, ·) + ςj,r (ũ)(v, ·)] dBjH (v)
0
j=1
2
2
=: J1 + J2 .
The first term can be bounded as follows
Z s
∞
X
λj
[ςj,s (u)(v, ·) − ςj,s (ũ)(v, ·)] dBjH (v)
J1 =
r
j=1
≤ C ξ sup
j∈N
Z
r
s
kςj,s (u)(v, ·) − ςj,s (ũ)(v, ·)k2
(v − r)σ
2
FRACTIONAL STOCHASTIC HEAT EQUATION
+
!
kςj,s (u)(v, ·) − ςj,s (ũ)(v, ·) − ςj,s (u)(z, ·) + ςj,s (ũ)(z, ·)k2
dz dv.
(v − z)σ+1
v
Z
17
r
By Lemma A.3,
Z
J1 ≤ Cξ
s
Z v
ku(v, ·) − ũ(v, ·)k2
ku(z, ·) − ũ(z, ·)k2
+
(s − v)−δ
dz
σ
(v − r)
(v − z)σ+1−δ
r
!
Z v
ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2
+
dz dv.
(v − z)σ+1
r
r
The term J2 can be written and bounded as follows
J2 =
=
∞
X
λj
λj
≤ C ξ sup
j∈N
+
Z
r
Z
r
[ςj,s (u)(v, ·) − ςj,s (ũ)(v, ·) − ςj,r (u)(v, ·) + ςj,r (ũ)(v, ·)] dBjH (v)
2
0
j=1
Z
r
0
j=1
∞
X
Z
∗
∗
ςj,s,r
(u)(v, ·) − ςj,s,r
(ũ)(v, ·) dBjH (v)
∗
∗
ςj,s,r
(u)(v, ·) − ςj,s,r
(ũ)(v, ·)
2
2
vσ
0
∗
∗
∗
∗
ςj,s,r
(u)(v, ·) − ςj,s,r
(ũ)(v, ·) − ςj,s,r
(u)(z, ·) + ςj,s,r
(ũ)(z, ·)
v
(v − z)σ+1
0
2
!
dz dv.
By Lemma A.4, we get
Z r
δ ku(v, ·) − ũ(v, ·)k
δ
2
J2 ≤ C ξ
(s − r) 2 (r − v)− 2
vσ
0
Z v
δ ku(z, ·) − ũ(z, ·)k
δ
2
+
(s − r) 2 (r − v)− 2
dz
δ
(v − z)σ+1− 2
0
Z v
δ ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k
δ
2
+
(s − r) 2 (r − v)− 2
dz
dv.
(v − z)σ+1
0
Therefore
Z t Z s
0
0
k(Au)(s, ·) − (Aũ)(s, ·) − (Au)(r, ·) + (Aũ)(r, ·)k2
dr
(s − r)σ+1
=Cξ
2
2
ds
Z tX
6
0 i=1
where
s
s
G2i (s) ds,
ku(v, ·) − ũ(v, ·)k2
dv dr,
(v − r)σ
0
r
Z s
Z sZ v
ku(z, ·) − ũ(z, ·)k2
(s − r)−σ−1
G2 (s) :=
(s − v)−δ
dz dv dr,
(v − z)σ+1−δ
Z0 s
Zr sZr v
ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2
G3 (s) :=
(s − r)−σ−1
dz dv dr,
(v − z)σ+1
0
r r
Z s
Z r
δ
δ ku(v, ·) − ũ(v, ·)k
2
G4 (s) :=
(s − r) 2 −σ−1
(r − v)− 2
dv dr,
vσ
0
0
Z rZ v
Z s
δ ku(z, ·) − ũ(z, ·)k
δ
2
dz dv dr,
(r − v)− 2
G5 (s) :=
(s − r) 2 −σ−1
δ
σ+1−
2
(v − z)
0 0
0
G1 (s) :=
Z
−σ−1
(s − r)
Z
18
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
Z
s
δ
(s − r) 2 −σ−1
0
Z rZ v
δ ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k
2
dz dv dr.
×
(r − v)− 2
(v − z)σ+1
0 0
G6 (s) :=
Let us bound each of terms Gi , i = 1, . . . , 6. For every σ ∈ (0, 1/2)
Z s
Z s
−σ−1
−σ
G1 (s) ≤ sup ku(z, ·) − ũ(z, ·)k2
(s − r)
(v − r) dv dr
z∈[0,s]
0
r
= C sup ku(z, ·) − ũ(z, ·)k2 .
z∈[0,s]
and by the same technique we get for every δ > σ
G2 (s) ≤ C sup ku(z, ·) − ũ(z, ·)k2 .
z∈[0,s]
By the Cauchy–Schwartz inequality, we obtain
G3 (s) ≤
Z
s
Z s Z
−σ− 21
(s−r)
r
0
r
v
ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2
dz
(v − z)σ+1
2
dv
! 12
dr.
Similarly to (3.21), we get the bound
"Z Z
2 #1/2
v
s
ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2
dz
G3 (s) ≤ C
dv
.
(v − z)σ+1
0
0
It is not hard to see that
G4 (s) ≤ sup ku(z, ·) − ũ(z, ·)k2
z∈[0,s]
Z
s
0
δ
(s − r) 2 −σ−1
≤ C sup ku(z, ·) − ũ(z, ·)k2 .
Z
r
0
δ
(r − v)− 2 v −σ dv dr
z∈[0,s]
In the same way,
G5 (s) ≤ C sup ku(z, ·) − ũ(z, ·)k2 .
z∈[0,s]
Finally, the term G6 can be bounded similarly to (3.22). We obtain
"Z Z
2 #1/2
v
s
ku(v, ·) − ũ(v, ·) − u(z, ·) + ũ(z, ·)k2
dz
G6 (s) ≤ C
dv
.
(v − z)σ+1
0
0
Then we have
k(Au)(s, ·) −
2
(Aũ)(s, ·)kσ,2,t
≤ Cξ
2
Z
0
t
2
ku − ũkσ,2,s ds.
3.3. Existence and uniqueness of mild solution.
Theorem 3.1. For every H ∈ ( 21 , 1), there exists a unique mild solution to the
problem (1.2).
Proof. Existence. We define the following sequence of random processes up : [0, T ] →
L2 (R):
u0 ≡ 0, up+1 = Aup , p ≥ 1.
Reasoning by induction and using Proposition 3.1, we easily get that up ∈
B σ,2 [0, T ]; L2(R) a. s. for every p ≥ 0. By Proposition 3.2, for any p ≥ 1,
Z T
kup+1 − up k2σ,2,T = kAup − Aup−1 k2σ,2,T ≤ Cξ 2
kup − up−1 k2σ,2,s ds.
0
FRACTIONAL STOCHASTIC HEAT EQUATION
19
By induction, we get
2
kup+1 − up kσ,2,t ≤ C p ξ 2p
T
Z
Z
s1
···
0
0
Z
sp−1
2
ku1 − u0 kσ,2,sp dsp . . . ds2 ds1 .
0
Since u0 ≡ 0, we see that by Proposition 3.1,
2
2
2
2
ku1 − u0 kσ,2,sp = ku1 kσ,2,sp = kAu0 kσ,2,sp ≤ Cξ 2 ku0 kσ,2,sp + 1 = Cξ 2 .
Hence,
2
kup+1 − up kσ,2,T ≤ Cξ 2
p+1
Then for m > n ≥ 0 we get
kum − un kσ,2,t =
≤
Z
T
0
Z
s1
0
···
Z
m−1
X
p=n
(up+1 − up )
Cξ
2 p+1
p!
dsp . . . ds2 ds1 = Cξ 2
0
m−1
X
p=n
sp−1
T
σ,2,t
p
≤
!1/2
m−1
X
p=n
p+1 T p
.
p!
kup+1 − up kσ,2,t
→0
a. s. as m, n → ∞.
σ,2
2
Therefore, {up , p ≥ 0} is a Cauchy sequence
in
B
[0,
T
];
L
(R)
a. s. Then
σ,2
2
[0, T ]; L (R) such that
there exists a process u∞ ∈ B
kup − u∞ kσ,2,T → 0 a. s., as p → ∞.
(3.24)
Now we prove that u∞ is a mild solution. For any p ≥ 0,
ku∞ − Au∞ k2σ,2,T ≤ 2 ku∞ − up+1 k2σ,2,T + 2 kup+1 − Au∞ k2σ,2,T
2
2
= 2 ku∞ − up+1 kσ,2,T + 2 kAup − Au∞ kσ,2,T
Z T
2
2
≤ 2 ku∞ − up+1 kσ,2,T + Cξ
kup − u∞ kσ,2,s ds
0
2
≤ 2 ku∞ − up+1 kσ,2,T + Cξ 2 T kup − u∞ kσ,2,T ,
a. s., by Proposition 3.2. Letting p → ∞ and taking into account (3.24), we get
that u∞ = Au∞ a. s. Hence, u∞ is a mild solution.
Uniqueness. Let u and ũ be two mild solutions. Then u = Au and ũ = Aũ a. s.,
and for all t ∈ [0, T ],
Z t
2
2
2
ku − ũkσ,2,t = kAu − Aũkσ,2,t ≤ Cξ 2
ku − ũkσ,2,s ds
a. s.,
0
2
by Proposition 3.2. Then, by Gronwall’s lemma, ku − ũkσ,2,T = 0 a. s., which means
the uniqueness of the mild solution.
Appendix A.
In this appendix we establish upper bounds for L2 -norms of the functions ςj,t (u)
∗
(u) and their differences (see (3.8) and (3.9) for the definitions of that
and ςj,t,s
functions). The results of the appendix are used in the proofs of Propositions 3.1
and 3.2.
Lemma A.1. Let {u(t, ·), t ∈ [0, T ]} be an L2 (R)-valued random process. Then
(i) for all 0 < s < t < T ,
sup kςj,t (u)(s, ·)k2 ≤ C (ku(s, ·)k2 + 1) ;
j∈N
20
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
(ii) for all 0 < r < s < t < T , and for any δ ∈ ( 13 , 1),
sup kςj,t (u)(s, ·) − ςj,t (u)(r, ·)k2
j∈N
o
n
δ
δ
≤ C ku(s, ·) − u(r, ·)k2 + (t − s)− 2 (s − r) 2 (ku(r, ·)k2 + 1) .
Proof. Let us start with the assertion (i) and produce the following transformations:
Z Z
Z
2
2
2
|ςj,t (u)(s, x)| dx =
kςj,t (u)(s, ·)k2 =
G(t − s, x, y) h(u(s, y)) ej (y) dy dx.
R
R
R
It follows from the Hölder inequality that
Z
Z Z
2
2
|G(t − s, x, y)| dy |G(t − s, x, y)| |h(u(s, y))| |ej (y)|2 dy dx.
kςj,t (s, ·)k2 ≤
R
R
R
Corollary 2.1 implies that
Z Z
2
2
|G(t − s, x, y)| |h(u(s, y))| |ej (y)|2 dy dx.
kςj,t (u)(s, ·)k2 ≤ C
R
By (1.3), for all z ∈ R,
R
2
|h(z)| ≤ C z 2 + 1 .
Therefore, applying Fubini’s Theorem, we get
Z
2
2
|G(t − s, x, y)| |u(s, y)| + 1 |ej (y)|2 dx dy
kςj,t (u)(s, ·)k2 ≤ C
2
Z
ZR
2
2
|u(s, y)| + 1 |ej (y)|
|G(t − s, x, y)| dx dy.
=C
R
R
Using again Corollary 2.1, we get
kςj,t (u)(s, ·)k22 ≤ C
Z
R
|u(s, y)|2 + 1 |ej (y)|2 dy.
However,
Z
|u(s, y)|2 + 1 |ej (y)|2 dy ≤ C sup kej k2∞ ku(s, ·)k22 + kej k22
j∈N
R
2
2
≤ C ku(s, ·)k2 + 1 ,
(A.1)
since supj∈N kej k∞ is bounded by (1.5), and kej k2 = 1 due to orthonormality.
Consequently, the assertion (i) follows.
Now, let us prove assertion (ii). On the one hand, obviously,
Z
1/2
2
kςj,t (u)(s, ·) − ςj,t (u)(r, ·)k2 :=
|ςj,t (u)(s, x) − ςj,t (u)(r, x)| dx
.
R
On the other hand,
|ςj,t (u)(s, x) − ςj,t (u)(r, x)|
Z
=
G(t − s, x, y) h(u(s, y)) ej (y) − G(t − r, x, y) h(u(r, y)) ej (y) dy
R
Z
h
i
=
G(t − s, x, y) h(u(s, y)) − h(u(r, y)) ej (y) dy
R
Z h
i
G(t − s, x, y) − G(t − r, x, y) h(u(r, y)) ej (y) dy
+
R
FRACTIONAL STOCHASTIC HEAT EQUATION
21
Z
≤ sup kej k∞ |G(t − s, x, y)| |h(u(s, y)) − h(u(r, y))| dy
j∈N
R
Z
+ |G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y))| |ej (y)| dy.
(A.2)
R
By Holder’s inequality, we obtain from (A.2) that
2
|ςj,t (u)(s, x) − ςj,t (u)(r, x)| ≤ C
Z
R
|G(t − s, x, y)| |h(u(s, y)) − h(u(r, y))| dy
2
Z
2
+C
|G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y))| |ej (y)| dy
Z
Z R
|G(t − s, x, y)| dy |G(t − s, x, y)| |h(u(s, y)) − h(u(r, y))|2 dy
≤ C
R
R
Z
|G(t − s, x, y) − G(t − r, x, y)| dy
+C
ZR
2
2
(A.3)
× |G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y))| |ej (y)| dy.
R
Applying Corollaries 2.1 and 2.2, we get from (A.3) the following bounds
Z
2
2
|G(t − s, x, y)| |h(u(s, y)) − h(u(r, y))| dy
|ςj,t (u)(s, x) − ςj,t (u)(r, x)| ≤ C
R
Z
2
2
−δ
δ
+ C(t − s) (s − r)
|G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y))| |ej (y)| dy
R
Z
2
≤C
|G(t − s, x, y)| |u(s, y) − u(r, y)| dy + C(t − s)−δ (s − r)δ
ZR
2
2
× |G(t − s, x, y) − G(t − r, x, y)| |u(r, y)| + 1 |ej (y)| dy.
R
Applying Fubini’s theorem, we get
Z
|ςj,t (u)(s, x) − ςj,t (u)(r, x)|2 dx
R
Z
Z
2
|u(s, y) − u(r, y)|
≤C
|G(t − s, x, y)| dx dy
R
Z R
2
2
|u(r, y)| + 1 |ej (y)|
+ C(t − s)−δ (s − r)δ
R
Z
×
|G(t − s, x, y) − G(t − r, x, y)| dx dy.
R
2
≤ C ku(s, ·) − u(r, ·)k2 + C(t − s)−δ (s − r)δ
Z
Z
Z
|u(r, y)|2 + 1 |ej (y)|2
|G(t − s, x, y)| dx + |G(t − r, x, y)| dx dy.
×
R
R
R
Using again Corollary 2.1 and the bound (A.1), we get
Z
2
2
|ςj,t (u)(s, x) − ςj,t (u)(r, x)| dx ≤ C ku(s, ·) − u(r, ·)k2
R
Z
|u(r, y)|2 + 1 |ej (y)|2 dy
+ C(t − s)−δ (s − r)δ
R
i
h
2
2
≤ C ku(s, ·) − u(r, ·)k2 + (t − s)−δ (s − r)δ ku(r, ·)k2 + 1 .
22
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
Consequently,
kςj,t (u)(s, ·) − ςj,t (u)(r, ·)k2
h
i
δ
δ
≤ C ku(s, ·) − u(r, ·)k2 + (t − s)− 2 (s − r) 2 (ku(r, ·)k2 + 1) .
Hence, the statement (ii) is obtained.
Lemma A.2. Let {u(t, ·), t ∈ [0, T ]} be an L2 (R)-valued random process. Then
(i) for all 0 < v < s < t < T and δ ∈ ( 13 , 1),
δ
δ
sup kςj,t (u)(v, ·) − ςj,s (u)(v, ·)k2 ≤ C(s − v)− 2 (t − s) 2 (ku(v, ·)k2 + 1) ,
j∈N
(ii) for all 0 < r < v < s < t < T and δ ∈ ( 31 , 1), δ ′ ∈ ( 51 , 1),
sup kςj,t (u)(v, ·) − ςj,s (u)(v, ·) − ςj,t (u)(r, ·) + ςj,s (u)(r, ·)k2
j∈N
δ
δ
≤ C (t − s) 2 (s − v)− 2 ku(v, ·) − u(r, ·)k2
′
δ′
δ′
+ (t − s) 2 (s − v)−δ (v − r) 2 (ku(r, ·)k2 + 1) .
Proof. (i) First, we can produce the relations
|ςj,t (u)(v, x) − ςj,s (u)(v, x)|
Z
=
G(t − v, x, y) h(u(v, y)) ej (y) − G(s − v, x, y) h(u(v, y)) ej (y) dy
R
Z h
i
G(t − v, x, y) − G(s − v, x, y) h(u(v, y)) ej (y) dy .
=
R
By using Holder’s inequality, we obtain
Z
2
|ςj,t (u)(v, x) − ςj,s (u)(v, x)| ≤ C
|G(t − v, x, y) − G(s − v, x, y)| dy
R
Z
× |G(t − v, x, y) − G(s − v, x, y)| |h(u(v, y))|2 |ej (y)|2 dy
R
Then, we can deduce from Corollary 2.1
2
|ςj,t (u)(v, x) − ςj,s (u)(v, x)|
Z
2
2
|G(t − v, x, y) − G(s − v, x, y)| |h(u(v, y))| |ej (y)| dy
≤ C
R
Z
2
2
|G(t − v, x, y) − G(s − v, x, y)| |u(v, y)| + 1 |ej (y)| dy.
≤ C
R
Hence, it follows from Fubini theorem that
Z
2
|ςj,t (u)(v, x) − ςj,s (u)(v, x)| dx
R
Z
Z
|u(v, y)|2 + 1 |ej (y)|2
|G(t − v, x, y) − G(s − v, x, y)| dx dy.
≤ C
R
R
By Corollary 2.2, we obtain
Z
Z
2
2
2
−δ
δ
|ςj,t (u)(v, x) − ςj,s (u)(v, x)| dx ≤ C(s−v) (t−s)
|u(v, y)| + 1 |ej (y)| dy.
R
R
Consequently, by (A.1),
δ
δ
kςj,t (u)(v, ·) − ςj,s (u)(v, ·)k2 ≤ C (s − v)− 2 (t − s) 2 (ku(v, ·)k2 + 1) .
FRACTIONAL STOCHASTIC HEAT EQUATION
23
(ii) For the second statement, we have
2
kςj,t (u)(v, ·) − ςj,s (u)(v, ·) − ςj,t (u)(r, ·) + ςj,s (u)(r, ·)k2
Z
2
|ςj,t (u)(v, x) − ςj,s (u)(v, x) − ςj,t (u)(r, x) + ςj,s (u)(r, x)| dx
=
R
Z Z h
i
=
G(t − v, x, y) − G(s − v, x, y) h(u(v, y)) ej (y) dy
R
Z hR
i
2
−
G(t − r, x, y) − G(s − r, x, y) h(u(r, y)) ej (y) dy dx
R
Z Z h
=
R
R
i
G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)
× h(u(r, y)) ej (y) dy +
Z h
R
i
G(t − v, x, y) − G(s − v, x, y)
2
h
i
× h(u(v, y)) − h(u(r, y)) ej (y) dy dx
≤C
Z Z h
i
G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)
R
R
2
× h(u(r, y)) ej (y) dy dx + C
Z Z h
i
G(t − v, x, y) − G(s − v, x, y)
R
R
2
h
i
× h(u(v, y)) − h(u(r, y)) ej (y) dy dx = K1 + K2 .
In order to get the upper bound for K1 , we apply Holder’s inequality and produce
that
Z Z h
i
K1 =
G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)
R
R
2
× h(u(r, y)) ej (y) dy dx
Z Z
G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y) dy
≤
R
Z R
×
|G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)|
R
2
2
× |h(u(r, y))| |ej (y)| dy dx
Z Z
≤ C
|G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)| dy
R
R
Z
×
|G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)|
R
2
2
× (|u(r, y)| + 1) |ej (y)| dy dx
By Corollary 2.1,
Z
K1 ≤ C
|G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)|
R2
2
2
× (|u(r, y)| + 1) |ej (y)| dy dx
24
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
Lemma 2.3 and Fubini’s theorem allow us to get
Z
|u(r, y)|2 + 1 |ej (y)|2
K1 ≤ C
ZR
× [G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)] dx dy
R
Z
2
2
δ′
−2δ ′
δ′
|u(r, y)| + 1 |ej (y)| dy.
≤ C (t − s) (s − v)
(v − r)
R
Therefore, by (A.1),
′
′
K1 ≤ C (t − s)δ (s − v)−2δ (v − r)δ
′
ku(r, ·)k22 + 1 .
In order to get the upper bound for K2 , we apply again the Hölder’s inequality and
get that
Z Z h
ih
i
2
G(t − v, x, y) − G(s − v, x, y) h(u(v, y)) − h(u(r, y)) ej (y) dy dx
K2 =
ZR ZR
G(t − v, x, y) − G(s − v, x, y) dy
≤
R
R
Z
h
i2
2
×
G(t − v, x, y) − G(s − v, x, y) h(u(v, y)) − h(u(r, y)) |ej (y)| dy dx
R
Z Z
2
≤ C sup kej k∞
G(t − v, x, y) − G(s − v, x, y) dy
j∈N
×
Z
R
R
R
2
G(t − v, x, y) − G(s − v, x, y) u(v, y) − u(r, y) dy dx.
By Corollary 2.1,
Z
K2 ≤ C
2
R2
G(t − v, x, y) − G(s − v, x, y) u(v, y) − u(r, y) dy dx
Therefore, applying Corollary 2.2 and Fubini’s theorem we get that
Z
2
δ
−δ
K2 ≤ C (t − s) (s − v)
u(v, y) − u(r, y) dy
R
2
= C (t − s)δ (s − v)−δ ku(v, ·) − u(r, ·)k2
≤ C (t − s)δ (s − v)−δ ku(v, ·) − u(r, ·)k22
Consequently,
2
kςj,t (u)(v, ·) − ςj,s (u)(v, ·) − ςj,t (u)(r, ·) + ςj,s (u)(r, ·)k2
≤ C (t − s)δ (s − v)−δ ku(v, ·) − u(r, ·)k22
′
′
′
2
+ (t − s)δ (s − v)−2δ (v − r)δ ku(r, ·)k2 + 1 .
Lemma A.3. Let {u(t, ·), t ∈ [0, T ]} and {ũ(t, ·), t ∈ [0, T ]} be two L2 (R)-valued
random processes. Then
(i) for all 0 < s < t < T ,
sup kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·)k2 ≤ C ku(s, ·) − ũ(s, ·)k2 ,
j∈N
FRACTIONAL STOCHASTIC HEAT EQUATION
25
(ii) for all 0 < r < s < t < T and for every δ ∈ ( 13 , 1),
sup kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·) − ςj,t (u)(r, ·) + ςj,t (ũ)(r, ·)k2
j∈N
≤ C(t − s)−δ (s − r)δ ku(r, ·) − ũ(r, ·)k2 + C ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 .
Proof. In order to prove (i), we can apply the Cauchy–Schwartz inequality and
Corollary 2.1, and get the following inequalities:
Z
2
2
|ςj,t (u)(s, x) − ςj,t (ũ)(s, x)| dx
kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·)k2 :=
R
=
Z
2
Z
G(t − s, x, y) [h(u(s, y)) − h(ũ(s, y))] ej (y) dy dx
R
R
Z
Z Z
2
≤C
|G(t − s, x, y)|dy
|G(t − s, x, y)| [h(u(s, y)) − h(ũ(s, y))] dy dx
R
R
R
Z
Z
2
[h(u(s, y)) − h(ũ(s, y))]
≤C
|G(t − s, x, y)|dx dy
R
R
≤ C ku(s, ·) − ũ(s, ·)k22 ,
whence (i) follows. To proceed with (ii), we apply the Minkowski inequality in
order to get the following bounds:
2
kςj,t (u)(s, ·) − ςj,t (ũ)(s, ·) − ςj,t (u)(r, ·) + ςj,t (ũ)(r, ·)k2
Z Z
=
[G(t − s, x, y) − G(t − r, x, y)] [h(u(r, y)) − h(ũ(r, y))] ej (y)
R
R
2
+ G(t − s, x, y) [h(u(s, y)) − h(ũ(s, y)) − h(u(r, y)) + h(ũ(r, y))] ej (y) dy dx
Z Z
2
≤C
|G(t − s, x, y) − G(t − r, x, y)| |h(u(r, y)) − h(ũ(r, y))| dy dx
R
+C
R
Z Z
R
R
2
|G(t − s, x, y)| |h(u(s, y)) − h(ũ(s, y)) − h(u(r, y)) + h(ũ(r, y))| dy dx
=: H1 + H2 .
It follows from the Cauchy–Schwartz inequality that
Z Z
|G(t − s, x, y) − G(t − r, x, y)| dy
H1 ≤ C
R
Z R
2
×
|G(t − s, x, y) − G(t − r, x, y)| [h(u(r, y)) − h(ũ(r, y))] dy dx.
R
By Fubini’s theorem,
Z
H1 ≤ C (t − s)−δ (s − r)δ
R
Z
×
|G(t − s, x, y) − G(t − r, x, y)| [h(u(r, y)) − h(ũ(r, y))]2 dy dx
R
≤ C(t − s)−δ (s − r)δ
Z
Z
2
×
[h(u(r, y)) − h(ũ(r, y))]
|G(t − s, x, y) − G(t − r, x, y)| dx dy.
R
R
Then, using Corollary 2.2 we obtain
2
H1 ≤ C (t − s)−2δ (s − r)2δ ku(r, ·) − ũ(r, ·)k2 .
26
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
For the other term, we get by using Cauchy-Schwartz inequality, Lemma 2.1, and
Fubini’s theorem,
Z Z
|G(t − s, x, y)| dy
H2 ≤ C
R
R
Z
2
×
|G(t − s, x, y)| [h(u(s, y)) − h(ũ(s, y)) − h(u(r, y)) + h(ũ(r, y))] dy dx
R
Z
[h(u(s, y)) − h(ũ(s, y)) − h(u(r, y)) + h(ũ(r, y))]2
≤C
R
Z
× |G(t − s, x, y)|dx dy,
R
and Lemma 2.1 allow us to get
2
H2 ≤ C kh(u(s, ·)) − h(ũ(s, ·)) − h(u(r, ·)) + h(ũ(r, ·))k2 .
It follows from (1.3) that
2
H2 ≤ C ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 .
Hence, we get the result.
Lemma A.4. Let {u(t, ·), t ∈ [0, T ]} and {ũ(t, ·), t ∈ [0, T ]} be two L2 (R)-valued
random processes. Then
(i) for all 0 < v < s < t < T and for every δ ∈ ( 31 , 1),
∗
∗
sup ςj,t,s
(u)(v, ·) − ςj,t,s
(ũ)(v, ·)
j∈N
δ
δ
≤ C (t − s) 2 (s − v)− 2 ku(v, ·) − ũ(v, ·)k2 ,
2
(ii) for all 0 < r < v < s < t < T and for any δ ∈ ( 15 , 1), δ ′ ∈ ( 31 , 1),
∗
∗
∗
∗
sup ςj,t,s
(u)(v, ·) − ςj,t,s
(ũ)(v, ·) − ςj,t,s
(u)(r, ·) + ςj,t,s
(ũ)(r, ·)
j∈N
δ
2
δ
≤ C(t − s) 2 (s − v)−δ (v − r) 2 ku(r, ·) − ũ(r, ·)k2
δ′
δ′
+ C(t − s) 2 (s − v)− 2 ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 .
Proof. In order to prove (i), we can apply the Cauchy–Schwartz inequality, Corollaries 2.1 and 2.2, and get the following inequalities:
Z
2
2
∗
∗
∗
∗
ςj,t,s
(u)(v, ·) − ςj,t,s
(ũ)(v, ·) 2 =
ςj,t,s
(u)(v, x) − ςj,t,s
(ũ)(v, x) dx
R
=
Z
2
Z
[G(t − v, x, y) − G(s − v, x, y)] [h(u(v, y)) − h(ũ(v, y))] ej (y) dy dx
R
R
Z Z
|G(t − v, x, y) − G(s − v, x, y)| dy
≤C
R
R
Z
2
×
|G(t − v, x, y) − G(s − v, x, y)| [h(u(v, y)) − h(ũ(v, y))] dy dx
R
Z
Z
2
[h(u(s, y)) − h(ũ(s, y))]
≤C
|G(t − v, x, y) − G(s − v, x, y)|dx dy
R
R
≤ C(t − s)δ (s − v)−δ ku(s, ·) − ũ(s, ·)k22 .
In the other hand, we have by using the fact that for every a, b ∈ R, (a + b)2 ≤
2(a2 + b2 )
2
∗
∗
∗
∗
ςj,t,s
(u)(v, ·) − ςj,t,s
(ũ)(v, ·) − ςj,t,s
(u)(r, ·) + ςj,t,s
(ũ)(r, ·) 2
Z
∗
∗
∗
∗
=
ςj,t,s
(u)(v, x) − ςj,t,s
(ũ)(v, x) − ςj,t,s
(u)(r, x) + ςj,t,s
(ũ)(r, x)
R
2
dx
FRACTIONAL STOCHASTIC HEAT EQUATION
=
Z Z
R
R
27
[G(t − v, x, y) − G(s − v, x, y)]
× [h(u(v, y)) − h(ũ(v, y)) − h(u(r, y)) + h(ũ(r, y))]ej (y) dy
Z
+ [h(u(r, y)) − h(ũ(r, y))]
R
2
× [G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)]ej (y) dy dx
Z Z
≤C
[G(t − v, x, y) − G(s − v, x, y)]
R
R
2
× [h(u(v, y)) − h(ũ(v, y)) − h(u(r, y)) + h(ũ(r, y))]ej (y) dy dx
Z Z
+C
[h(u(r, y)) − h(ũ(r, y))]
R
R
2
× [G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)]ej (y) dy dx
=: D1 + D2 .
Applying the Cauchy–Schwarz inequality and Fubini’s theorem, combined with
Corollaries 2.1 and 2.2, we get
Z Z
[G(t − v, x, y) − G(s − v, x, y)]
D1 =
R
R
2
× [h(u(v, y)) − h(ũ(v, y)) − h(u(r, y)) + h(ũ(r, y))]ej (y) dy dx
Z
Z Z
|G(t − v, x, y) − G(s − v, x, y)|dy
≤
|G(t − v, x, y) − G(s − v, x, y)|
R
R
R
× |h(u(v, y)) − h(ũ(v, y)) − h(u(r, y)) + h(ũ(r, y))|2 dy dx
′
2
′
≤ C (t − s)δ (s − v)−δ ku(s, ·) − ũ(s, ·) − u(r, ·) + ũ(r, ·)k2 .
By the same technique as before and from Lemma 2.3, we obtain
Z Z
D2 =
[G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)]
R
R
2
× [h(u(r, y)) − h(ũ(r, y))]ej (y) dy dx
Z Z
|G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)|dy
≤
R
Z R
×
|G(t − v, x, y) − G(s − v, x, y) − G(t − r, x, y) + G(s − r, x, y)|
R
× |h(u(r, y)) − h(ũ(r, y))|2 dy dx
≤ C (t − s)δ (s − v)−2δ (v − r)δ ku(r, ·) − ũ(r, ·)k22 .
Hence, we get the result.
Acknowledgments
Y. Mishura and K. Ralchenko acknowledge that the present research is carried
through within the frame and support of the ToppForsk project nr. 274410 of the
28
YULIYA MISHURA, KOSTIANTYN RALCHENKO, MOUNIR ZILI, AND EYA ZOUGAR
Research Council of Norway with title STORM: Stochastics for Time-Space Risk
Models.
References
[1] R. M. Balan and C. A. Tudor, Stochastic heat equation with multiplicative fractional-colored
noise, J. Theoret. Probab. 23 (2010) 834–870.
[2] R. M. Balan and C. A. Tudor, The stochastic wave equation with fractional noise: a random
field approach, Stoch. Process. Appl. 120 (2010) 2468–2494.
[3] L. Bo, Y. Jiang and Y. Wang, Stochastic Cahn–Hilliard equation with fractional noise, Stoch.
Dyn. 8 (2008) 643–665.
[4] L. Bo, Y. Jiang and Y.Wang, On a class of stochastic Anderson models with fractional noises,
Stoch. Anal. Appl. 26 (2008) 256–273.
[5] R. Cantrell and C. Cosner, Diffusion models for population dynamics incorporating individual
behavior at boundaries: Applications to refuge design, Theor. Population Biology 55 (1999)
198–207.
[6] Z. Q. Chen and M. Zili, One-dimensional heat equation with discontinuous conductance, Sci.
China Math. 58 (2015) 97–108.
[7] W. Grecksch and V. V. Ahn, A parabolic stochastic differential equation with fractional Brownian motion input, Statist. Probab. Lett. 41 (1999) 337–346.
[8] Y. Hu, Heat equations with fractional white noise potentials, Appl. Math. Optim. 43 (2001)
221–243.
[9] A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen
Raum, C.R. (Doklady) Acad. URSS (N.S.) 26 (1940) 115–118.
[10] A. Lejay, A scheme for simulating one-dimensional diffusion processes with discontinuous
coefficients, Ann. Appl. Probab. 6 (2016) 107–139.
[11] W. Leland, M. Taqqu, W. Willinger and D. Wilson, On the self-similar nature of ethernet
traffic, IEEE/ACM Trans. Networking 2 (1994), 1–15.
[12] B. B. Mandelbrot, The variation of certain speculative prices, J. Business 36 (1963) 394–419.
[13] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and
applications, SIAM Rev. 10 (1968) 422–437.
[14] S. Nicas, Some results on spectral theory over networks, applied to nerve impulse transmission,
Orthogonal Polynomials and Applications (Bar-le-Duc, 1984), Lect. Notes in Math. 1171
532–541 (Springer, 1985).
[15] B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion,
J. Funct. Anal 202 (2003) 277–305.
[16] Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,
Lecture Notes in Math. 1929 (Springer, 2008).
[17] Y. Mishura, K. Ralchenko and G. Shevchenko, Existence and uniqueness of mild solution to
stochastic heat equation with white and fractional noises, Theory Probab. Math. Statist., 98
(2018).
[18] D. Nualart and Y. Ouknine, Regularization of quasilinear heat equations by a fractional noise,
Stoch. Dyn 4 (2004) 201–221.
[19] M. Sanz-Solé and P.-A. Vuillermot, Equivalence and Hölder–Sobolev regularity of solutions
for a class of non-autonomous stochastic partial differential equations, Ann. Inst. H. Poincaré
Probab. Statist. 39 (2003) 703–742.
[20] M. Sanz-Solé and P. A. Vuillermot, Mild solutions for a class of fractional SPDEs and their
sample paths, J. Evol. Equ. 9 (2009) 235–265.
[21] S. Tindel, C. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian
motion, Probab. Theory Related Fields 127 (2003) 186–204.
[22] M. Zili, Développement asymptotique en temps petits de la solution d’une équation aux
dérivées partielles de type parabolique généralisée au sens des distributions-mesures, C. R.
Math. Acad. Sci. Paris, 321 (1995) 1049–1052.
[23] M. Zili, Construction d’une solution fondamentale d’une équation aux dérivées partielles à
coefficients constants par morceaux, Bull. Sci. Math. 123 (1999) 115–155.
[24] M. Zili, Fundamental solution of a parabolic partial differential equation with piecewise constant coefficients and admitting a generalized drift, Int. J. Appl. Math. 2 (2000) 1073–1110.
[25] M. Zili and E. Zougar, One-dimensional stochastic heat equation with discontinuous conductance, Appl. Anal. 98 (2019), 2178–2191.
[26] M. Zili and E. Zougar, Exact variations for stochastic heat equations with piecewise constant
coefficients and application to parameter estimation, Theory Probab. Math. Statist. 100 (2019),
75–101.
FRACTIONAL STOCHASTIC HEAT EQUATION
29
(Y. Mishura) Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska St., Kyiv, 01601,
Ukraine
E-mail address: myus@univ.kiev.ua
(K. Ralchenko) Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska St., Kyiv,
01601, Ukraine
E-mail address: k.ralchenko@gmail.com
(M. Zili) University of Monastir, Department of Mathematics, Faculty of sciences
of Monastir, Avenue de l’environnement, 5019 Monastir, Tunisia
E-mail address: mounir.zili@fsm.rnu.tn
(E. Zougar) University of Monastir, Department of Mathematics, Faculty of sciences
of Monastir, Avenue de l’environnement, 5019 Monastir, Tunisia
E-mail address: zougareya@gmail.com