On Some Properties of a Class of Fractional Stochastic Heat Equations
On Some Properties of a Class of Fractional Stochastic Heat Equations
Wei Liu 0 1
Kuanhou Tian 0 1
Mohammud Foondun 0 1
Introduction 0 1
Main Results 0 1
B Mohammud Foondun 0 1
0 Loughborough University , Loughborough, England
1 Shanghai Normal University , Shanghai , China
We consider nonlinear parabolic stochastic equations of the form ∂t u = Lu + λσ (u)ξ˙ on the ball B(0, R), where ξ˙ denotes some Gaussian noise and σ is Lipschitz continuous. Here L corresponds to a symmetric αstable process killed upon exiting B(0, R). We will consider two types of noises: spacetime white noise and spatially correlated noise. Under a linear growth condition on σ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014). Consider the following stochastic heat equation on the interval (0, 1) with Dirichlet boundary condition: Research supported in part by EPSRC.
Stochastic partial differential equations; Fractional Laplacian; Stochastic heat equation; Heat kernel

1
∂t ut (x ) = 2 ∂x x ut (x ) + λut (x )w˙ (t, x ) for 0 < x < 1 and t > 0
ut (0) = ut (1) = 0 for t > 0.
Here w˙ denotes white noise, λ is a positive parameter and u0(x ) is the initial condition.
Set
0
1
The study of Et (λ) as λ gets large was initiated in [
13,14
]. In Ref. [9], it was shown
that Et (λ) grows like const × exp(λ4) as λ gets large. The main aim of this paper
is to extend similar results to a much wider class of stochastic equations. Existence
and uniqueness of solutions to these equations is a direct consequence of the methods
[
6,18
]. We will provide a proof in the appendix. We will first look at equations driven
by white noise. Fix R > 0 and consider the following:
∂t ut (x ) = Lut (x ) + λσ (ut (x ))w˙ (t, x ),
ut (x ) = 0, for all x ∈ B(0, R)c,
(1.1)
where w˙ denotes white noise (0, ∞) × B(0, R). Here and throughout this paper, we
will make the following assumptions on the function σ and the initial condition u0.
Assumption 1.1 The function σ : Rd → R is a Lipschitz continuous function with
lσ x  ≤ σ (x ) ≤ Lσ x  for all x ∈ Rd ,
where lσ and Lσ are some positive constants.
The above assumptions on σ are quite natural and have been used in various works;
see [
10,11
]. The lower bound is essentially a growth condition which is needed for our
results. These inequalities also imply that σ (0) = 0. This is needed for nonnegativity
of solutions to stochastic heat equations. Even though we do not need nonnegativity
of the solution in this paper, the upper bound makes our computations easier to follow.
Assumption 1.2 The initial function u0 is a nonnegative, nonrandom bounded
function which is strictly positive in a set of positive measure in B(0 , R). More precisely,
we will assume that if 0 < R, then
B(0, R− )
u0(y)dy
is strictly positive. Throughout this paper, whenever we fix
assume that it is much less than R so that the above is satisfied.
> 0, we will always
L is the generator of a symmetric αstable process killed upon exiting B(0, R) so
that (1.1) can be thought of as the Dirichlet problem for fractional Laplacian of order
α.
where
B(0, R)
Following Walsh [
18
], we say that u is a mild solution to (1.1) if it satisfies the
following evolution equation,
Here pD(t, x , y) denotes the fractional Dirichlet heat kernel. It is also well known
that this unique mild solution satisfies the following integrality condition
sup sup Eut (x )k < ∞
x∈B(0, R) t∈[0, T ]
for all T > 0 and k ∈ [2, ∞],
(1.3)
which imposes the restriction that d = 1 and 1 < α < 2, which will be in force
whenever we deal with (1.1).
Here is our first result.
Theorem 1.3 Fix
> 0 and let x ∈ B(0, R − ), then for any t > 0,
,
where ut is the unique solution to (1.1).
Set
e(t ) := λl→im∞
We then have the following corollary.
Corollary 1.5 The excitation index of the solution to (1.1) is α2−α1 .
It can be seen that when α = 2 this gives the result in [
9
]. Our second main result
concerns coloured noise driven equations. Consider
∂t ut (x ) = Lut (x ) + λσ (ut (x ))F˙ (t, x ),
ut (x ) = 0, for all x ∈ B(0, R)c.
(1.5)
This equation is exactly the same as (1.1) except for the noise which is now given by
F˙ and can be described as follows.
E[F˙ (t, x )F˙ (s, y)] = δ0(t − s) f (x , y),
where f is given by the socalled Riesz kernel:
f (x , y) :=
1
x − yβ
Here β is some positive parameter satisfying β < d. Other than the noise term, we
will work under the exact conditions as those for Eq. (1.1). The mild solution will thus
satisfy the following integral equation.
Existence–uniqueness considerations will force us to impose β < α ∧ d; see for
instance [
8
] or the appendix of this current paper. We point out that the stochastic
integral in the above display is well defined for an even larger class of coloured noises.
This is thanks to [
7,18
]. The same applies for existence and uniqueness. One can
prove wellposedness of equations which are driven by noises which are more general.
For other papers studying coloured noise driven equations on bounded domains; see
[
16,17
]. Our first result concerning (1.5) is the following.
Theorem 1.6 Fix
> 0 and let x ∈ B(0, R − ), then for any fixed t > 0,
,
where ut is the unique solution to (1.5).
Corollary 1.7 The excitation index of the solution to (1.5) is α2−αβ .
It is clear that our results are significant extensions of those in [
9,13
]. The techniques
are also considerably harder and required some new highly nontrivial ideas which we
now mention.
• We need to compare the heat kernel estimates for killed stable process with that
of “unkilled” one. To do that, we will need sharp estimates of the Dirichlet heat
kernel.
• We will also need to study some renewaltype inequalities, and by doing so, we
come across the MittagLeffler function whose asymptotic properties become
crucial.
• While the above two ideas are enough for the proof of Theorem 1.3, we will also
need to significantly modify the localisation techniques of [
13
] to complete the
proof of Theorem 1.6.
Our method seems suited for the study of a much wider class of equations. To
illustrate this, we devote a section to various extensions.
Here is a plan of the article. In Sect. 2, we collect some information about the
heat kernel and the renewaltype inequalities. In Sect. 3, we prove the main results
concerning (1.1). Section 4 contains analogous proofs for (1.5). In Sect. 5, we extend
our study to a much wider class of equations.
Finally, throughout this paper, the letter c with or without subscripts will denote
constants whose exact values are not important to us and can vary from line to line.
2 Preliminaries
Let Xt denote the αstable process on Rd with p(t, x , y) being its transition density.
It is well known that
c1 t −d/α
t
∧ x − yd+α
≤ p(t, x , y) ≤ c2 t −d/α
t
∧ x − yd+α
,
where c1 and c2 are positive constants. We define the first exit of time Xt from the ball
B(0, R) by
τB(0, R) := inf{t > 0, Xt ∈/ B(0, R)}.
We then have the following representation for pD(t, x , y)
pD(t, x , y) = p(t, x , y) − Ex [ p(t − τB(0, R), XτB(0, R) , y); τB(0, R) < t ].
From the above, it is immediate that
pD(t, x , y) ≤ p(t, x , y) for all x , y ∈ Rd .
This in turn implies that
c1
pD(t, x , y) ≤ t d/α
for all x , y ∈ Rd .
(2.1)
We now provide some sort of converse to the above inequality. Not surprisingly, this
inequality will hold for small times only.
Proposition 2.1 Fix > 0. Then for all x , y ∈ B(0, R − ), there exists a t0 > 0
and a constant c1, such that
whenever t ≤ t0. And if we further impose that x − y ≤ t 1/α, we obtain the following
pD(t, x , y) ≥ c1 p(t, x , y),
pD(t, x , y) ≥ t dc/2α ,
where c2 is some positive constant.
(2.2)
Proof Set δB(0, R)(x ) := dist(x , B(0, R)c). It is known that
pD(t, x , y) ≥ c1 1 ∧
for some constant c1. See for instance [
2
] and references therein. Since x ∈ B(0, R −
), we have δB(0, R)(x ) ≥ . Now choosing t0 = α, we have δαB/(02, R)(x ) ≥ t 1/2 for all
t ≤ t0. Similarly, we have δαB/(02, R)(y) ≥ t 1/2 which together with the above display
yield
pD(t, x , y) ≥ c2 p(t, x , y) for all x , y ∈ B(0, R − ),
whenever t ≤ t0. We now use the fact that
p(t, x , y) ≥ c3
t
x − yd+α ∧ t −d/α .
to end up with (2.2).
We now make a simple remark which will be important in the sequel.
Remark 2.2 Recall that for any t˜ > 0 and x ∈ B(0, R).
(GDu)s+t˜(x ) :=
B(0, R)
pD(s + t˜, x , y)u0(y)dy.
Fix
> 0 and note that for x ∈ B(0, R − ),
(GDu)s+t˜(x ) ≥ x, y∈Bin(0f, R− ) pD(s + t˜, x , y)
Let t > 0. Choose small enough if necessary. Then, for any 0 ≤ s ≤ t , the right
hand side is strictly positive. For “small times,” that is, for t + t˜ ≤ t0, we can use the
argument of the above result to write pD(s + t˜, x , y) ≥ c1 p(s + t˜, x , y). While for
“large times,” we have pD(s +t˜, x , y) ≥ c2e−λ(s+t˜) for some positive constant λ. This
follows from general spectral theory and can be found in [
2
] and references therein.
For x ∈ B(0, R − ) and 0 ≤ s ≤ t , we have therefore found a strictly positive lower
bound on (GDu)s+t˜(x ). We denote this bound by gt to indicate its possible dependence
on t . In a sense, this fact is analogous to the wellknown “infinite propagation of heat”
for the Laplacian.
We now give a definition of the MittagLeffler function which is denoted by Eβ
where β is some positive parameter. Define
Eβ (t ) :=
∞
n=0
t n
(nβ + 1)
This function is well studied and crops up in a variety of settings including the
study of fractional Eq. [
15
]. In our context, we encounter it in the study of the renewal
inequalities mentioned in the introduction. Even though a lot is known about this
function, we will need the following simple fact whose statement is motivated by the
use we make of it later. We will need the upper and lower bounds separately.
Proposition 2.3 For any fixed t > 0, we have
Lemma 2.4 For any fixed t > 0, we have
See for instance [
12
] and references therein for more details. Thus, for any positive
constant > 0 there exists a Z > 0 such that for all z > Z
Choosing
< 1/β, it is easy to see that
log log
1
β −
+ z1/β
≤ log log Eβ (z) ≤ log log
+ z1/β .
1
β +
Letting z = θ t , the above yield the assertions of the proposition.
What follows is a consequence of Lemma 14.1 of [
14
]. But for the sake of
completeness, we give a quick proof based on the asymptotic behaviour of the MittagLeffler
function which we used in the above proof. Fix ρ > 0 and consider the following:
and
In other words, we have
Proof By using Laplace transforms techniques, one can show that for large z,
lim sup
θ→∞
lim inf
θ→∞
lim
θ→∞
log log Eβ (θ t )
log θ
1
≤ β ,
log log Eβ (θ t )
log θ
.
An application of Proposition 2.3 proves the result.
We now present the renewal inequalities.
Proposition 2.5 Let T < ∞ and β > 0. Suppose that f (t ) is a nonnegative locally
integrable function satisfying
f (t ) ≤ c1 + κ
where c1 is some positive number. Then for any t ∈ (0, T ], we have the following
lim sup
κ→∞
log log f (t )
log κ
t
Proof We begin by setting (Aψ )(t ) := κ 0 (t − s)β−1ψ (s)ds where ψ can be any
locally integrable function. And for any fixed integer k > 1, we have (Ak ψ )(t ) :=
t
κ 0 (t − s)β−1(Ak−1ψ )(s)ds. We further set 1(s) := 1 for all 0 ≤ s ≤ T . With these
notations, (2.4) can be succinctly written as f (t ) ≤ c1 + (A f )(t ) which upon iterating
becomes
and therefore we also have
(An1)(t ) =
(κ (β))nt nβ
.
0
t
n−1
k=0
As n → ∞, we have (An f )(t ) → 0. We thus end up with
Keeping in mind that we are interested in the behaviour as κ tends to infinity while t
is fixed, we can apply Proposition 2.3 to obtain the result.
We have the “converse” of the above result.
Proposition 2.6 Let T < ∞ and β > 0. Suppose that f (t ) is a nonnegative locally
integrable function satisfying
f (t ) ≥ c2 + κ
where c2 is some positive number. Then for any t ∈ (0, T ], we have the following
lim inf
κ→∞
log log f (t )
log κ
Proof With the notations introduced in the proof of Proposition 2.5, (2.6) yields
f (t ) ≥ c2
Now similar arguments as in Proposition 2.5 prove the result. We leave it to the reader
to fill in the details.
The above inequalities are well studied; see for instance [
12
]. But the novelty here
is that, as opposed to what is usually done, instead of t , we take κ to be large.
3 Proofs of Theorem 1.3 and Corollary 1.5
We will begin with the proof of Theorem 1.3. We will prove it in two steps. Set
Proposition 3.1 Fix t > 0, then
.
Proof We start off with the representation (1.2) and take the second moment to obtain
Putting these estimates together, we have
Now an application of Proposition 2.5 proves the result.
For any fixed
> 0, set
Proposition 3.2 For any fixed
> 0, there exists a t0 > 0 such that for all 0 < t ≤ t0,
I ,t (λ) := x∈B(0, R− ) Eut (x )2.
inf
lim inf
λ→∞
log log I ,t (λ)
log λ
.
Proof As in the proof of the previous proposition, we start off with (3.2) and seek to
find lower bound on each of the terms. We fix > 0 and choose t0 as in Proposition 2.1.
Using Remark 2.2, we have that for 0 < t ≤ t0, we have inf x∈B(0, R− ) GD(t, x ) :=
g˜t0 . Hence, I1 ≥ g˜t20 . We now turn our attention to I2.
I2 ≥ (λlσ )2
≥ (λlσ )2
t
0
t
0
We now apply Proposition 2.6 to obtain the result.
Proof of Theorem 1.3 The proof of the result when t ≤ t0 follows easily from the
above two propositions. To prove the theorem for all t > 0, we only need to prove the
above proposition for all t > 0. For any fixed T , t > 0, by changing the variable we
have
EuT +t (x )2
= (GDu)T +t (x )2 + λ2
= (GDu)T +t (x )2 + λ2
+ λ2
0
t
EuT +t (x )2 ≥ (GDu)T +t (x )2 + λ2lσ2
pD2(t − s, x , y)EuT +s (y)2dyds.
0
t
Since by Remark 2.2, (GDu)T +t (x )2 is strictly positive, we can use the proof of the
above proposition with an obvious modification to conclude that
for x ∈ B(0, R − ) and small t .
Proof of Corollary 1.5 Note that
4 Proofs of Theorem 1.6 and Corollary 1.7
where here and throughout the rest of this section, ut will denote the solution to (1.5).
The following lemma will be crucial later. In what follows, f denotes the spatial
correlation of the noise F˙ .
Lemma 4.1 For all x , y ∈ B(0, R),
pD(t, x , w) pD(t, y, z) f (w, z)dwdz ≤ t βc/1α ,
(4.1)
B(0, R)×B(0, R)
for some positive constant c1.
Proof We begin by noting that
Now the scaling property of the heat kernel and a proper change of variable proves
the result.
Proposition 4.2 Fix t > 0, then
lim sup
λ→∞
Proof We start with the mild formulation to the solution to (1.5) which after taking
the second moment gives us
Eu(t, x )2 = (GDu)t (x )2
t
We obviously have I1 ≤ c1. Note that the Lipschitz assumption on σ together with
Hölder’s inequality give
We can use the above inequality and Lemma 4.1 to bound I2 as follows.
Combining the above estimates, we obtain
which immediately yields the result upon an application of Proposition 2.5.
We have the following lower bound on the second of the solution. Inspired by the
localisation arguments of [
13
], we have the following.
Proposition 4.3 Fix
> 0 and t˜ > 0. Then for all x ∈ B(0, R − 2 ) and 0 ≤ t ≤ t0,
E ut+t˜(x ) 2 ≥ gt2 + gt2
∞
k=1
where c1 is some positive constant depending on α and β.
Proof Fix > 0 and for convenience, set B := B(0, R) and B := B(0, R − ). We
will also use the following notation; B2 := B × B and B2 := B × B .
After taking the second moment, the mild formulation of the solution together with
the growth condition on σ gives us
t
E u
t t
+˜
(x)
2
≥  G
E u
×  t s
˜+ 2
(z )u
2
t s
˜+ 2
2 
(z ) f (z , z )dz dz ds .
2 2 2
2 2
p (s
s , z , z ) p (s
The above two inequalities thus give us
E u
t t
≥  G
We now replace the above by t t and use a substitution to reduce the above to
≥ G
D
( u)
+
×
+
×
+
×
×
λ l
u)
t s
˜+ 1
(z )dz dz ds
1
1 1
1
−
p(t s , x, z ) p t s , x, z f z , z
1 1 1 1
1
1
−
p (s
s , z , z ) p s
D 1 − 2 1 2 D 1 − 2
s , z , z
(4.2)
Therefore,
We set z
z
t and continue the recursion as above to obtain
2
0 B 0
2
B
 G
(z )
k 
p (s
s , z
t s
˜+ k
z
k
p (s
s , z
D i 1 − i i 1 i D
− −
, z ) p
−
i 1 − i
−
s , z
i 1 i
, z ) f z , z dz dz ds .
i i i
i i
Using the fact that for z , z
k
k ∈
B ,
( u)
D
G
t s
˜+ k
(z )(
k
D
G
u)
t s
˜+ k
z
k ≥
inf
inf (
D
G
u)
t s
˜+
(x)(
D
G
u)
t s
˜+
(y)
x,y B 0 s t
∈ ≤ ≤
we obtain
E u
(x)
2
g
2
≥ t + t
g
2
∞
k 1
=
≥ t
2
g ,
i 1
=
p (s
s , z
D i 1 − i i 1 i D
− −
, z ) p
−
i 1 − i
−
s , z
i 1 i
, z ) f z , z dz dz ds .
i i i
i i
We reduce the temporal region of integration as follows.
E ut˜+t (x )
2
Now we make a change the temporal variable, si−1 − si → si , in the following way
such that for all integers i ∈ [1, k], we have
pD(si−1 − si , zi−1, zi ) pD si−1 − si , zi−1, zi f zi , zi dsi
pD(si , zi−1, zi ) pD si , zi−1, zi ) f (zi , zi dsi .
si−1
si−1−t/k
t/k
We now focus our attention on the multiple integral appearing in the above inequality.
We will further restrict its spatial domain of integration so that we have the required
lower bound on each component of the following product,
Recall that x ∈ B(0, R − 2 ). For each i = 1, . . . , k, choose zi and zi satisfying
pD (si , zi−1, zi ) pD si , zi−1, zi f zi , zi .
(4.4)
E ut˜+t (x )
2
t/k
t/k
t/k
B2 · · · 0
B2
pD(si , zi−1, zi ) pD si , zi−1, zi f zi , zi dzi dzi dsi .
zi ∈ B z0, s11/α/2 ∩ B zi−1, si1/α
zi ∈ B z0, s11/α/2 ∩ B zi−1, si1/α ,
so that we have zi − zi  ≤ si1/α together with zi − zi−1 ≤ si1/α and zi − zi−1 ≤
si1/α. Now using Proposition 2.1, we can conclude that pD(si , zi−1, zi ) ≥ si−d/α
and pD(si , zi−1, zi ) ≥ si−d/α. Moreover, we have zi − zi  ≤ s11/α, which gives us
0
t/k
≥
f (zi , zi ) ≥ s1−β/α. In other words, we are looking at the points {si , zi , zi }ik=0 such
that the following holds
pD(si , zi−1, zi ) pD si , zi−1, zi f zi , zi ≥
k
1
× pD si , zi−1, zi f zi , zi dzi dzi dsi
t/k
A1 A1 0
A2 A2
0
t/k
· · ·
k
Ak Ak i=1
pD(si , zi−1, zi )
0
t/k
k
1
Ak Ak i=1 si2d/αsβ/α dzi dzi dsi .
1
We now use the lower bounds on the area of Ai s and Ai s to estimate the above
integrals. We note that for si ≤ s1/2, the area of Ai and Ai is bounded below by
c1sid/α. After some computations and using the fact that si ≤ s1/2, we see that the
above integral is bounded below by
c2k
2
0
t/k
k
s1/2
i=2 0
αc32k
dsi ds1 = k(α − β) k
Putting the above estimates together we obtain
E ut˜+t (x ) 2 ≥ gt2 + gt2
∞
k=1
∞
k=1
for some constant c4.
Recall that
where here ut is the solution to (1.5). We now have
Proposition 4.4 For any fixed
> 0, then for any fixed t > 0, we have
I ,t (λ) := x∈B(0, R− ) Eut (x )2,
inf
lim inf log log I ,t (λ)
λ→∞ log λ
Proof We begin by noting that any fixed t > 0 can be written as t = t˜ + t , where t˜
is strictly positive and t is small as in the previous proposition.
∞
k=1
.
Lemma 2.4 with ρ := (α − β)/α and θ := λ2 together with the above result finishes
the proof.
Proof of Theorem 1.6 The above two propositions prove the theorem for all t ≤ t0.
We now extend the result to all t > 0. As in the proof of Theorem 1.3, we only need
to extend the above proposition to any fixed t > 0. For any T , t > 0,
E uT +t (x )2 ≥ (GDu)t+T (x )2 + λ2lσ2
pD(T + t − s1, x , z1)
0 B2
× pD T + t − s1, x , z1 E us1 (z1)us1 z1
f z1, z1 dz1dz1ds1.
This leads to
E uT +t (x )2 ≥ (GDu)t+T (x )2 + λ2lσ2
Similar ideas to those used in the rest of the proof of Proposition 4.3 together with the
proof of the above proposition show that for all t ≤ t0, we have
lim inf
λ→∞
for all T > 0 and whenever x ∈ B(0, R − ).
Proof of Corollary 1.7 The proof is exactly the same as that of Corollary 1.5 and is
omitted.
× pD t − s1, x , z1 E uT +s1 (z1)uT +s1 z1
f z1, z1 dz1dz1ds1.
A similar argument to that used in the proof of Proposition 4.3 shows that
E uT +t (x )2 ≥ (GDu)T +t (x )2
∞
· · ·
×
i=1
t
5 Some Extensions
the introduction. We set Et (λ) :=
follows,
Theorem 5.1 Fix t > 0, we then have
We begin this section by showing that the methods developed in this paper can be used
to study the stochastic wave equation as well. More precisely, we give an alternative
proof of a very interesting result proved in [
13
]. Consider the following equation
∂tt ut (x ) = ∂x x ut (x ) + λσ (ut (x ))w˙ (t, x ) for x ∈ R t > 0,
(5.1)
with initial condition u0(x ) = 0 and nonrandom initial velocity v0 satisfying v0 ∈
L1(R) ∩ L2(R) and v0 L2(R) > 0. As before σ satisfies the conditions mentioned in
∞ Eut (x )2 dx and restate the result of [
13
] as
−∞
Proof We again use the theory of Walsh [
18
] to make sense of (5.1) as the solution to
the following integral equation
t
We now use Walsh’s isometry to obtain
R 1[0,t−s](x − y)σ (us (y))W (dsdy).
1
Eut (x )2 = 4
v0(x − y) dy
2
t
Recall that from the assumption on the initial velocity, we have
R
t
v0(x − y) dy
dx ≤ 4t 2 v0 L2(R).
2
This and the assumption on σ yields
Using similar ideas, we can obtain the following lower bound,
Et2(λ) ≤ t 2 v0 2L2(R) + 41 λ2 L2σ
(t − s)Es2(λ) ds.
Et2(λ) ≥ t 2 v0 2L2(R) + 41 λ2 L2σ
(t − s)Es2(λ) ds.
0
0
t
t
(5.2)
(5.3)
We now use Propositions 2.5 and 2.6 together with the above two inequalities to obtain
the result.
The method developed so far can be adapted to the study of a much wider class
of stochastic heat equations, once we have the “right” heat kernel estimates. Indeed,
(2.1) and (2.2) were two crucial elements of our method. So by considering operators
whose heat kernels behave in a nice way, we can generate examples of stochastic
heat equations for which we can apply our method. Recall that we are considering
equations of the type,
∂t ut (x ) = Lut (x ) + λσ (ut (x ))F˙ (t, x ).
(5.4)
In what follows, we will choose different Ls while keeping all the other conditions as
before. And again, the choice of these operators Ls will make the boundary conditions
clear. Some of the equations below appear to be new. We again do not prove existence–
uniqueness results as these are fairly standard once we have a grip on the heat kernel.
See [
6,18
].
Example 5.2 We choose L to be the generator of a Brownian motion defined on the
interval (0, 1) which is reflected at the point 1 and killed at the other end of the interval.
So, we are in fact looking at
∂t ut (x ) = 21 ∂x x ut (x ) + λσ (ut (x ))F˙ (t, x ) for 0 < x < 1 and t > 0
ut (0) = 0, ∂x ut (1) = 0 for t > 0.
It can be shown that for any > 0, there exists a t0 > 0, such that for all x ∈ [ , 1)
and t ≤ t0, the heat kernel of this Brownian motion satisfies
whenever x − y ≤ t 1/2. We use the method developed in this paper to conclude that
lim
λ→∞
whenever x ∈ [ , 1).
Example 5.3 Let Xt be censored stable process as introduced in [
1
]. These have been
studied in [
4
]. Roughly speaking, the censored stable process in the ball B(0, R) can
be obtained by suppressing the jump from B(0, R) to the complement of B(0, R)c.
The process is thus forced to stay inside B(0, R). We denote the generator of this
process by −(− )α/2B(0, R) and consider the following equation
∂t ut (x ) = −(− )α/2B(0, R)ut (x ) + λσ (ut (x ))F˙ (t, x ),
(5.5)
In a sense, the above equation can be regarded as fractional equation with Neumann
boundary condition. In Ref. [
4
], it was shown that the probability density function of
Xt , which we denote by p¯(t, x , y) satisfies
p¯(t, x , y)
1 ∧
So we can proceed as in the proof of Theorem 1.6 to see that we have
where the conditions on α and β are the same as those stated in Sect. 1.
Example 5.4 In this example, we choose L be the generator of the relativistic stable
process killed upon exiting the ball B(0, R). We are therefore looking at the following
equation
∂t ut (x ) = mut (x ) − (m2/α −
ut (x ) = 0, for all x ∈ B(0, R)c.
)α/2ut (x ) + λσ (ut (x ))F˙ (t, x ),
Here m is some fixed positive number. One can show that for any
a t0 > 0, such that for all x , y ∈ B(0, R − ) and t ≤ t0, we have
> 0, there exists
whenever x − y ≤ t 1/α. See for instance [
5
]. The constants involved in the above
inequality depends on m. We therefore have the same conclusion as that of Theorem
1.6. In other words, we have
lim
λ→∞
,
whenever x ∈ B(0, R − ) and the conditions on α and β are the same as those stated
in Sect. 1.
Example 5.5 Let 0 < α¯ ≤ α with 1 < α < 2 and consider the following
∂t ut (x ) = −(− )α/2ut (x ) − (−
ut (x ) = 0, for all x ∈ B(0, R)c.
)α¯ /2ut (x ) + λσ (ut (x ))F˙ (t, x ),
The Dirichlet heat kernel for the operator L := −(− )α/2 −(− )α¯ /2 has been studied
in [
3
]. Since α¯ ≤ α, it is known that for small times, the behaviour of the heat kernel
estimates is dominated by the fractional Laplacian −(− )α/2. More precisely, for any
> 0, there exists a t0 > 0, such that for all x , y ∈ B(0, R − ) and t ≤ t0, we have
whenever x − y ≤ t 1/α. Therefore, in this case also, we have (5.7).
(5.6)
(5.7)
Acknowledgments The authors thank a referee and an associate editor for insightful comments which
improved the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
6 Appendix
In this section, we will indicate how to get existence and uniqueness of random field
solutions for the equations studied in this paper. We will focus on the coloured driven
equation. We will following a standard iteration argument. Recall that from Lemma
4.1, we have that for all x , y ∈ B(0, R),
B(0, R)×B(0, R)
c1
pD(t , x , w) pD(t , y, z) f (w, z)dwdz ≤ t β/α ,
(6.1)
for some positive constant c1. Set ut(0)(x ) := (GDu)t (x ) and for n ≥ 1, set
ut(n)(x ) = (GDu)t (x ) + λ
pD(t − s, x , y)σ
u(sn−1)(y) F (ds, d y).
The stochastic integral is well defined even when the correlation function is restricted
on B(0, R). This essentially follows from [
18
]. We have by Burkholder’s inequality,
E ut(n)(x ) − ut(n−1)(x )
p
≤ cpλ
p
B(0, R)×B(0, R)
pD(t − s, x , y) pD(t − s, x , z)
× E σ (u(sn−1)(y)) − σ (u(sn−2)(y)) σ (u(sn−1)(z)) − σ (u(sn−2)(z)) f (y, z) d y dz ds
p/2
,
where cp is some positive constant. Using
and the assumption on σ , we obtain
Edn (t , x ) p ≤ cpλ p L σp
dn (t , x ) := ut(n)(x ) − ut(n−1)(x ),
0
t
0
t
B(0, R)
0
t
B(0, R)×B(0, R)
pD(t − s, x , y) pD(t − s, x , z)
p/2
We set
F (t ) :=
sup
x∈B(0, R)
Edn−1(s, x )2 f (y, z) d y dz ds
B(0, R)×B(0, R)
pD(t − s, x , y) pD(t − s, x , z) f (y, z) d y dz ds.
We now use Hölder’s inequality to obtain
Edn (t , x ) p ≤ cpλ p L σp F (t ) p/2−1
E dn−1(s, x ) p ds
sup
0 x∈B(0, R)
Since F (t ) is bounded whenever β < α, we obtain
sup
0 x∈B(0, R)
Edn−1(s, x ) pds,
for some constant C > 0. Now Gronwall’s lemma gives convergence of the sequence
ut(n)(x ) in the pth moment, thus showing existence of a solution. Uniqueness follows
from a wellknown argument. See [
6
] for details.
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