Global stability in a nonlocal
reaction-diusion equation
Dmitri Finkelshtein
1
2
Yuri Kondratiev
3
Stanislav Molchanov
Pasha Tkachov
4
March 19, 2017
Abstract
We study stability of stationary solutions for a class of nonlocal semilinear parabolic equations. To this end, we prove the FeynmanKac formula for a Lévy processes with time-dependent potentials and arbitrary
initial condition. We propose sucient conditions for asymptotic stability
of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we nd conditions
which imply that its positive stationary solution is asymptotically stable.
We consider also the case when the initial condition is given by a random
eld.
Keywords: nonlocal diusion, FeynmanKac formula, Lévy processes,
reaction-diusion equation, semilinear parabolic equation, monostable equation, nonlocal nonlinearity
2010 Mathematics Subject Classication: 35B40, 35B35, 60J75,
60K37
1
Introduction
1.1
Overview of results
The aim of this paper is to study stability of stationary solutions to a class of
non-local semilinear parabolic equations applying the FeynmanKac formula.
Namely, we wish to investigate bounded solutions to the following equation
∂
u(x, t) = (LJ u)(x, t) + V (u(x, t))u(x, t),
∂t
where
x ∈ Rd , t > 0,
LJ = J ∗ u − kJkL1 u, cf. (1.3), is a generator of a pure jump
V : Cb (Rd ) → Cb (Rd ) is a bounded locally Lipschitz mapping,
process,
(1.1)
Markov
and the
1 Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.
([email protected]).
2 Fakultät für Mathematik, Universität Bielefeld, Postfach 110 131, 33501 Bielefeld, Ger-
many ([email protected]).
3 Department of Mathematics and Statistics, University of North Carolina Charlotte, NC
28223, USA ([email protected]);
National Research University Higher School of Eco-
nomics, Russia.
4 Fakultät für Mathematik, Universität Bielefeld, Postfach 110 131, 33501 Bielefeld, Ger-
many ([email protected]).
1
u(·, 0) ∈ Cb (Rd ) belongs to a neighborhood of u ≡ 0.
initial condition
Note that
the FeynmanKac formula for diusion processes with time-dependent potentials is known (see [8, Theorem 5.7.6]). However, the corresponding result for
general Lévy processes seems to be proved only recently in [12], where compactly
supported smooth initial conditions where assumed. We relax assumptions on
the initial conditions, considering bounded continuous functions
prove the FeynmanKac formula for the generator
LJ
Cb (Rd )
and
(see Propositions 2.1
and 2.2). We propose also sucient conditions for the asymptotic stability of
the zero solution to (1.1) uniformly in space (see Theorem 2.3), and apply this
to a particular equation
∂
u(x, t) = κ + (La+ u)(x, t) + κ − θ − (a− ∗ u)(x, t) u(x, t),
∂t
(1.2)
κ + −m
> 0, and a± are probability kernels (see [1, 4, 6,
κ−
9, 10, 13]). The equation (1.2) may be considered as a non-local version of the
where
κ ± , m > 0, θ :=
classical logistic equation (see (3.2) below).
to (1.2),
u≡0
u ≡ θ.
and
There are two constant solution
Dierent properties and the long-time behavior of
solutions to (1.2), were considered in [5].
We are interested to nd sucient conditions which ensure that a solution
to (1.2) converges to the constant non-zero solution
u≡θ
uniformly in space.
Applying Theorem 2.3, we prove (see Theorem 3.6) that a bounded initial con-
θ > 0 exponentially fast if
x ∈ Rd . The condition on Jθ
dition, which is separated from zero, tends to
only
Jθ (x) = κ + a+ (x) − θκ − a− (x) ≥ 0,
may
for all
be relaxed under more restrictive assumptions on the initial condition. Namely,
introducing a parameter in the initial condition and considering the analytical
decomposition of the corresponding solution with respect to the parameter, one
can show that if
at
θ,
kJθ kL1 < κ +
and if the initial condition lies in a ball centered
then the solution tends to
θ
exponentially fast (see Theorem 4.1).
An
example of a parameter constructed by a stationary random eld provides an
enhanced asymptotic for the convergence (see Theorem 4.4).
1.2
Basic notations
B(Rd ) be the Borel σ -algebra on the d-dimensional Euclidean space Rd ,
d ≥ 1. Let Cb (Rd ) and Bb (Rd ) denote the spaces of all bounded continuous,
d
respectively, bounded Borel measurable functions on R . The functional spaces
Let
become Banach ones being equipped with the norm
kvk∞ := sup |v(x)|.
x∈Rd
For any
J ∈ L1 (Rd ) := L1 (Rd , dx)
and
v ∈ Bb (Rd ),
one can dene the
classical convolution
Z
(J ∗ v)(x) :=
J(x − y)v(y) dy.
Rd
Let
J ∈ L1 (Rd )
be non-negative. Consider the following bounded operator
(in any of the Banach spaces above)
Z
(LJ v)(x) :=
J(x − y) v(y) − v(x) dy = (J ∗ v)(x) − µv(x),
Rd
2
(1.3)
R
µ := Rd J(y) dy > 0.
d
d
Let Xt be a jump-process with the state space (R , B(R )) and the natural
ltration Ft = σ(Xs | s ≤ t) whose generator is LJ (for details see [3]). It is
d
well known that Xt is a Markov process and, for all s, t ≥ 0, f ∈ Bb (R ), the
where
following equation holds,
E f (Xt+s )|Xt = E f (Xt+s )|Ft
Z
= f (y)p(Xt − y, s)dy = (p ∗ f )(Xt ).
(1.4)
Rd
where
p(x, t)
is the transition density of
Xt .
Namely,
p(x, 0) = δ(x)
and
p(x, t)
satises the following equation
∂p
(x, t) = (LJ p)(x, t),
∂t
x ∈ Rd , t > 0.
I ⊂ R+ := [0, ∞), consider the
E -valued functions on I , where E is a
For an interval
continuous
Banach spaces
Cb (I → E)
of
space above, with the norm
kukI := supku(·, t)k∞ .
t∈I
For the simplicity of notations, we set, for any
XT1 ,T2 := Cb [T1 , T2 ], Cb (Rd ) ,
with the corresponding norms
2
Let
T2 > T1 ≥ 0, T > 0,
XT := X0,T ,
X∞ := Cb R+ , Cb (Rd ) ,
k · kT1 ,T2 , k · kT , k · k.
The FeynmanKac Formula and Stability
u = u(x, t) describe the local density of a system at the point x ∈ Rd , d ≥ 1,
t ∈ R+ . Prove now a version of the FeynmanKac formula
the time-dependent potential and operator LJ , cf. e.g. [2, Theorem 2.5], [8,
at the moment of time
for
Theorem 5.7.6]. Consider a perturbed equation

∂
u(x, t) = (LJ u)(x, t) + W (x, t)u(x, t),
∂t
u(x, 0) = u (x) ∈ C (Rd ),
0
b
where
W ∈ XT .
t ∈ [0, T ],
Then, clearly, (2.1) has a unique solution in
XT .
(2.1)
The following
theorem states that the solution will satisfy the FeynmanKac formula.
(2.1) for t ∈ [0, T ]. Then
Z t
x
u(x, t) = E u0 (Xt ) exp
W (Xt−s , s) ds , x ∈ Rd , t ∈ [0, T ].
Proposition 2.1.
Let u solves
(2.2)
0
Proof.
For
f ∈ XT ,
we denote
Zt Z
p(x − y, t − s)W (y, s)f (y, s)dyds.
(Qf )(x, t) =
0 Rd
3
(2.3)
By Duhamel's formula,
u(x, t) = (p ∗ u0 )(x, t) + Qu (x, t) = (p ∗ u0 )(x, t) + Q(p ∗ u0 + Qu) (x, t)
= (p ∗ u0 )(x, t) + Q(p ∗ u0 ) (x, t) + Q2 (p ∗ u0 ) + Qu (x, t)
= . . . ( by induction ) . . .
n
X
=
Qj (p ∗ u0 ) (x, t) + Qn+1 u (x, t).
(2.4)
j=0
By (1.4) and (2.4),
(p ∗ u0 )(Xt−s , s) = E u0 (Xt )|Ft−s ,
Zs Z
Q(p ∗ u0 ) (Xt−s , s) =
p(Xt−s − y, s − τ )W (y, τ )(p ∗ u0 )(y, τ )dydτ
0 Rd
Zs
=
E W (Xt−τ , τ )(p ∗ u0 )(Xt−τ , τ )|Ft−s dτ
0
Zs h
i
= E W (Xt−τ , τ ) E u0 (Xt )|Ft−τ Ft−s dτ
0
Zs
= E u0 (Xt ) W Xt−τ , τ dτ
Ft−s ,
0
where the last equality holds by the tower rule and Fubini's theorem. One can
continue then
(Q2 (p ∗ u0 ))(Xt−s , s)
Zs
= E W (Xt−τ , τ )(Q(p ∗ u0 ))(Xt−τ , τ )Ft−s dτ
0
Zs Zτ
h
i = E W (Xt−τ , τ ) E u0 (Xt ) W (Xt−σ , σ)dσ Ft−τ Ft−s dτ
0
0
Zs Zτ
= E u0 (Xt )
W (Xt−τ , τ )W (Xt−σ , σ)dσdτ
0
Ft−s
0
Zs
1
= E u0 (Xt ) W (Xs−τ , τ )dτ
2
Ft−s .
0
In the same manner, we can prove, by the induction, the following equality
Zs
n
1
W Xs−τ , τ dτ
(Q (p ∗ u0 ))(Xt−s , s) = E u0 (Xt )
n!
n
0
4
Ft−s .
(2.5)
By (2.4) and (2.5) (with
u(x, t) =
n
X
s = t),
Ex Qj (p ∗ u0 ) X0 , t + Qn+1 u (x, t)
j=0
Zs
n
X
n
1
W Xs−τ , τ dτ
+ Qn+1 u (x, t).
= E u0 (Xt )
n!
j=1
x
(2.6)
0
Q̃
We write
for the operator dened by (2.3) with
W
substituted by
follows easily that the equation similar to (2.5) holds for
u0 ≡ 1
and
Q̃.
|W |.
It
In particular, for
s = t,
Zt
n
n
1 x W (Xt−τ , τ ) dτ
.
(Q̃ 1)(x, t) = E (Q̃ 1)(X0 , t) = E
n!
n
x
0
Hence
kQn ukT ≤ kQ̃n 1kT kukT ≤
As a result, for
n → ∞,
Tn
kW knT kukT .
n!
(2.6) yields (2.2), that completes the proof.
Consider now a general semi-linear evolution equation with the generator

∂
u(x, t) = (LJ u)(x, t) + V (u(x, t))u(x, t),
∂t
u(x, 0) = u (x) ∈ C (Rd ),
0
b
where
V : Cb (Rd ) → Cb (Rd )
t>0
LJ :
(2.7)
is a bounded locally Lipschitz mapping, i.e.
kV (u)k∞ ≤ Mc kuk∞ ,
kV (u) − V (v)k∞ ≤ Mc ku − vk∞ ,
(2.8)
provided that kuk∞ ≤ c, kvk∞ ≤ c. Then, evidently, kV (u)u − V (v)vk∞ ≤
(1 + c)Mc ku − vk∞ , and hence, by e.g. [11, Theorem 1.4], there exists a Tmax ≤
∞, such that the initial-value problem (2.7) has a unique mild solution u on
[0, Tmax ), i.e. u that solves the integral equation
u(x, t) = etLJ u0 (x) +
Z
t
e(t−s)LJ V (u(x, s))u(x, s) ds.
0
Tmax < ∞ implies that lim ku(·, t)k∞ = ∞.
Note also that since LJ
t↑Tmax
is a bounded operator, then the mild solution will be classical one, i.e. u ∈ XT ,
d
for any T < Tmax , and u(x, t) is dierentiable in t w.r.t. the norm in Cb (R ).
Moreover,
By Proposition 2.1, the following FeynmanKac-type expression holds for
the solution to (2.7).
Proposition 2.2. Let (2.8) hold and u be the unique classical solution to
on [0, T ], T < Tmax . Then
Z t
u(x, t) = E u0 (Xt ) exp
V (u(Xt−s , s)) ds ,
x
0
5
x ∈ Rd , t ∈ [0, T ].
(2.7)
(2.9)
Denote
XTk,l = {f ∈ X∞ |k ≤ f (x, t) ≤ l,
x ∈ Rd , t ∈ [0, T ]}, k, l ∈ R.
The following theorem provides sucient conditions for the stability of the stationary solution
u≡0
Theorem 2.3.
Let there exist p : R2 → R+ such that, for any k ≤ 0, l ≥ 0,
to (2.7),
V f (x, t) ≤ − p(k, l),
p(k, l) ≤ p(λk, λl),
x ∈ Rd , t ∈ [0, T ], f ∈ XTk,l ,
λ ∈ [0, 1].
(2.10)
Suppose that u0 ∈ E is such that, for some c ≤ 0 and d ≥ 0,
c ≤ u0 (x) ≤ d,
x ∈ Rd .
Then, for any T > 0, there exists a unique u ∈ XT , which satises the Feynmanc,d
Kac formula (2.9). Moreover, u ∈ X∞
, ku(·, t)kE does not increase in time,
and if p(c, d) > 0, then ku(·, t)kE converges to zero exponentially fast, namely,
lim sup
t→∞
Proof.
ln kut k
≤ −p(0, 0).
t
(2.11)
Let us introduce the following operator: we set, for a
Zt
[Ψwt ](x) = Ex u0 η(t) exp
[V wt−s ] η(s) ds ,
w ∈ X∞ ,
x ∈ Rd , t ∈ I.
0
Then, for any
w ∈ XTc,d ,
ce−tp(c,d) ≤ [Ψwt ](x) ≤ de−tp(c,d) ,
x ∈ Rd , t ∈ [0, T ].
(2.12)
p is non-negative, one gets Ψ(XTc,d ) ⊂ XTc,d . Since |e−x − e−y | ≤ |x − y|
c,d
all x, y ≥ 0, then, for all v, w ∈ XT , t ∈ [0, T ], the following estimate holds
[Ψwt ](x) − [Ψvt ](x) ≤ dT M kv − wk∞ ,
Since
for
M = Mmax{−c,d} is dened by (2.8). Hence Ψ is a contraction map on
1
c,d
T =
. Therefore, there exists a xed point u ∈ XT . By (2.12),
2dM
function u satises the following estimate
where
XTc,d for
the
ce−tp(c,d) ≤ u(x, t) ≤ de−tp(c,d) ,
x ∈ Rd , t ∈ [0, T ].
c1 ≤ u(x, T ) ≤ d1 , x ∈ Rd , where c1 = ce−T p(c,d) , d1 = de−T p(c,d) .
c,d
can repeat the proof on [T, 2T ] to extend u to X2T , so that the following
Hence,
We
estimate holds
c1 e−(t−T )p(c1 ,d1 ) ≤ u(x, t) ≤ d1 e−(t−T )p(c1 ,d1 ) ,
By induction,
u
can be extended to
c,d
XnT
,
x ∈ Rd , t ∈ [T, 2T ].
and for any
n ∈ N , x ∈ Rd , t ∈
[nT, (n + 1)T ],
cn e−(t−nT )p(cn ,dn ) ≤ u(x, t) ≤ dn e−(t−nT )p(cn ,dn ) ,
6
(2.13)
c0 = c, d0 = d, cn = cn−1 e−T p(cn−1 ,dn−1 ) and dn = dn−1 e−T p(cn−1 ,dn−1 ) .
c,d
u ∈ X∞
, such that (2.9) and (2.13) hold. Since p
non-negative, {cn } is increasing and {dn } is decreasing. Moreover,
where
Hence, there exists a unique
is
cn = λcn−1 ,
together with (2.10) yield that
ku(·, t)kE
λ = e−T p(cn−1 ,dn−1 ) ∈ [0, 1]
dn = λdn−1 ,
p(ck , dk ) ≤ p(cn , dn ),
does not increase in time.
for
k ≤ n.
Therefore,
Let us prove by induction the following
inequalities
ck e−T (n−k)p(ck ,dk ) ≤ cn ,
−T (n−k)p(ck ,dk )
dk e
The case
them for
≥ dn ,
n = 1 is obvious. Let (2.14) and (2.15)
0 ≤ k ≤ N + 1. Since cN ≤ 0, we have
0 ≤ k ≤ n,
(2.14)
0 ≤ k ≤ n.
(2.15)
hold for
cN +1 = cN e−T p(cN ,dN ) ≥ cN e−T p(ck ,dk ) ≥ ck e−κ
−
0 ≤ k ≤ N.
We prove
T (N +1−k)p(ck ,dk )
.
Hence (2.14) is proved. Similarly, the following estimate yields (2.15)
dN +1 = dN e−T p(cN ,dN ) ≤ dN e−T p(ck ,dk ) ≥ dk e−T (N +1−k)p(ck ,dk ) ,
where
0 ≤ k ≤ N.
k = 0, both {cn } and {dn } converge
p(c, d) > 0. Therefore, for t ∈ [nT, (n + 1)T ],
By (2.14) and (2.15) with
to zero exponentially fast if only
ln max{dn , −cn }
ln kut k
≤
,
t
Tn
and, by (2.14), (2.15), we have, for
lim sup
t→∞
k ≥ 0,
ln kut k
≤ −p(ck , dk ).
t
p(ck , dk )
As a result, (2.11) holds, because
is increasing in
k.
This proves the
theorem.
3
Spatial logistic equation
We will consider the following equation for a bounded function
u(x, t),
which
describes the (approximate) value of the local density of a system of particles
distributed in
Rd
according to the so-called spatial logistic model. More detailed
explanation and historical remarks can be found in [5, Subsection 6.1]. Namely,
let
ut (x) := u(x, t), x ∈ Rd , t ∈ I ,
solves the equation
∂ut
= κ + [La+ ut ](x) + (κ + − m)ut (x) − κ − ut (x)(a− ∗ ut )(x).
∂t
In particular,
u(x, 0) = u0 (x), x ∈ Rd .
Here
κ + La+
(3.1)
is a generator of the
underlying random walk, cf. (1.3):
[La+ h](x) = (a+ ∗ h)(x) − h(x),
x ∈ Rd ,
which spends exponentially distributed random time
sition
x, P {τ > ρ} = e−κ
+
ρ
, and it makes a jump
7
τ in each particular pox → x + X, thereafter,
where the random variable
β = κ+ − m > 0
X
has the distribution density
a+ (x). The constant
β of the birth of
is the dierence between the biological rate
a new particle and the mortality rate
m.
The last term in (3.1) describes the
κ − a− (x − y) presents the interacx, y ∈ Rd . Equation (3.1) is
competition between particles, the potential
tion between two particles located at the points
similar to the well-known logistic ordinary dierential equation:
∂w
(t) = βw(t) − κ − w(t)2 ,
∂t
whose partial solution is the constant
exponentially fast to
x ∈ Rd , t ≥ 0
θ.
θ :=
β
.
κ−
(3.2)
All other positive solutions tend
The equation (3.1) has the same solution
(we suppose
R
a± (y)dy = 1).
u(x, t) ≡ θ,
This important particular solution
R
is the exponentially stable attractor for (3.1). We will study the neighborhood
of the attractor, using variations of
u0 .
Let us denote, for any
[F h](x) = (κ + − m)h(x) − κ − h(x)(a− ∗ h)(x),
h ∈ E,
x ∈ Rd .
Then (3.1) has the following form

 ∂ut
(x) = κ + [La+ ut ](x) + [F ut ](x) x ∈ Rd , t ∈ I,
∂t
u(x, 0) = u (x)
x ∈ Rd .
0
(3.3)
The analysis of the non-linear parabolic equation (3.1) will be based on
integral equations. The rst of them is given through the standard Duhamel's
formula.
Lemma 3.1.
Function u solves
ut = e
κ + tLa+
(3.3)
Zt
u0 +
i it satises the following equation
e−(t−s)κ
+
La+
[F us ]ds, t ∈ I.
(3.4)
0
This equation has the Volterra form and can be used for the existenceuniqueness theory (see [5]).
Theorem 3.2. Let u0 ∈ E be non-negative. Then, for each T > 0, there exists
a unique non-negative solution to (3.3) in XT .
u0
be a constant
κ ∈ [0, θ].
We make the
x ∈ Rd .
(A1)
Now we will estimate solution to (3.3) from below.
Let
function
u0 ≡ q0 ∈ (0, θ).
Then the corresponding solution to (3.3) is the function
qt =
θ
1+
e−βt ( qθ0
By Theorem 3.2, this solution is unique.
− 1)
.
Let us x
following assumption
Jκ (x) := κ + a+ (x) − κκ − a− (x) ≥ 0,
8
Theorem 3.3.
Let (A1) hold with κ = q0 ∈ (0, θ). Suppose that
u0 (x) ≥ q0 ,
x ∈ Rd ,
where u0 ∈ E . Then the corresponding to u0 solution ut to
following inequality
Let us x
T > 0.
Dene
dened later. The function
vt
vt = eKt (ut − qt ), t ∈ [0, T ],
where
K
will be
satises the following linear equation
∂vt
(x) = [Gt vt ](x)
∂t
v0 (x) = u0 (x) − q0
where, for all
satises the
x ∈ Rd , t > 0.
ut (x) ≥ qt ,
Proof.
(3.3)
x ∈ Rd , t ∈ [0, T ],
x ∈ Rd ,
w ∈ E,
[Gt w] := κ + (a+ ∗ w) − κ − qt (a− ∗ w) − κ − w(a− ∗ ut ) − mw + Kw.
By Theorem 3.2, there exists
M > 0,
such that
x ∈ Rd , t ∈ [0, T ].
ut (x) ≤ M,
K := κ − M + m. Since qt ≤ q0 for t ≥ 0, we have, by (A1) with κ = q0 ,
[Gt w](x) is non-negative for all t ∈ [0, T ] and for all non-negative w ∈ E .
Dene
that
Therefore,
Z t
vt (x) = exp
Gs ds v0 (x) ≥ 0,
x ∈ Rd , t > 0,
0
since
v0
is non-negative.
Hence,
arbitrary, the same holds for any
Remark
ut (x) ≥ qt , x ∈ Rd , t ∈ [0, T ].
t > 0.
3.4 (cf. [5, Proposition 3.4])
(A1) holds with
κ = θ,
and
u0 ∈ E
.
Since
T
is
In a similar way, it can be shown that if
is such that
0 ≤ u0 (x) ≤ θ, x ∈ Rd ,
then
the corresponding solution satises the following inequality
x ∈ Rd , t > 0.
0 ≤ ut (x) ≤ θ,
The following theorem shows, based on the FeynmanKac formula, that
ut
satises another integral equation.
Let (A1) holds with κ = θ. Suppose that u ∈ XT , T ∈ (0, ∞],
is the solution to (3.1) with an initial condition u0 ∈ E . Then u satises the
following formula, for all x ∈ Rd , t ∈ [0, T ],
Theorem 3.5.
x
u(x, t) = θ + E
Zt
−
−
u0 (Xt ) − θ exp −κ
a ∗ ut−s (Xs )ds .
0
where X0 = x.
9
Proof.
Let us denote
gt := ut −θ.
If
ut
solves (3.1), then
gt
satises the following
equation

 ∂gt
(x) = [LJθ gt ](x) − κ − gt a− ∗ gt − βgt ,
∂t
g(x, 0) = g (x) = u (x) − θ,
0
0
where
LJθ
is dened by (1.3), for
V
LJ = LJθ
and the generator
x ∈ Rd ,
J = Jθ = κ + a+ −κ − θa− .
[V h](x) = −κ − (a− ∗ h)(x) − β,
For such
x ∈ Rd , t ∈ I,
(3.5)
We set
x ∈ Rd , h ∈ E.
of the jump-process
Xt ,
we apply
Proposition (2.2) to the solution of (3.5)
Z t
g(x, t) = Ex g0 (Xt ) exp
[V gt−s ](Xs )ds ,
x ∈ Rd , t ∈ [0, T ].
(3.6)
0
Substituting
gt = ut −θ into the previous representation completes the proof.
The following theorem shows the asymptotic stability of the positive stationary solution.
Theorem 3.6.
condition to
Let (A1) holds with κ = θ. Suppose that u0 ∈ E is an initial
, such that
(3.1)
c1 ≤ u0 (x) ≤ c2 ,
x ∈ Rd ,
where 0 ≤ c1 ≤ θ and c2 ≥ θ. Then there exists a unique solution u ∈ X∞ to
c ,c
(3.1). Moreover, u ∈ X∞1 2 , kut − θkE does not increase in time, and if c1 > 0,
then kut − θkE converges to zero exponentially fast, namely
lim sup
t→∞
Proof.
We consider
solution
g ∈ X∞
gt = ut − θ.
By Theorem 3.2 and 3.5, there exists a unique
to (3.5), and this solution satises (3.6). The rest of the proof
follows from Theorem 2.3 with
4
ln kut − θk
≤ −β.
t
p(c, d) = p(c) = κ − (θ + c).
Stability on the initial condition
We will be interested in initial conditions of the following form
u0 (x, λ) = θeλξ(x) ,
(4.1)
ξ : Rd → R. Since the operator La+ is linear and bounded on E , and
F is analytic on E , then the solution u to (3.3) depends analytically on the
initial condition u0 (see e.g. [7, Theorem 3.4.4, Corollary 3.4.5, 3.4.6]). Hence,
by (4.1), the E -valued function λ 7→ u(·, t, λ) is analytic on R for each t ≥ 0.
Therefore, for all λ ∈ R, it is given by the following series
where
u(·, t, λ) =
X λn
kn,t (·),
n!
n≥0
10
(4.2)
where
kn,t (·) :=
∂nu
(·, t, 0) ∈ E,
∂λn
n ≥ 0.
We substitute (4.2) in (3.4).
X λn
X θλn ξ n
+
kn,t = eκ tLa+
n!
n!
n≥0
n≥0
Zt
+
e−(t−s)κ
+
La+
[F
n≥0
0
Hence, the
n-th
kn,t = θ[e
X λn
kn,s ]ds.
n!
Taylor coecient satises the following equation
Zt
κ + tLa+ n
ξ ]+
e−(t−s)κ
+
L a+
(κ + − m)kn,s
0
−κ
−
n X
n
l=0
l
kl,s (a− ∗ kn−l,s ) ds,
n ≥ 0.
Therefore,
∂kn,t
(x) = κ + [La+ kn,t ](x) + (κ + − m)kn,t (x)
∂t
n X
n
−
−κ
kl,t (x)(a− ∗ kn−l,t )(x),
l
x ∈ Rd , t ∈ I,
(4.3)
l=0
where
kn,0 (x) = θξ n (x).
Theorem 4.1.
estimate holds
Let ξ ∈ E and γ = κ + − kJθ kL1 > 0. Then the following
−γt
kut (·, λ) − θkE ≤ θe
if only |λ| <
Proof.
γ
−
2β
s
γ
γ2
− e|λ| kξkE − 1
2
4β
β
!
,
γ
1
+1 .
ln
kξkE
4β
We will estimate
kn , n ≥ 0 .
By (4.3),
k0 ≡ θ .
The function
k1
satises
the following equation
∂k1,t
(x) = Jθ ∗ k1,t (x) − κ + k1,t (x),
∂t
where
k1,0 (x) = θξ(x).
Since
γ = κ + − kJkL1 ,
x ∈ Rd , t ∈ I,
we have
k1,t (x) ≤ e−γt kk1,0 kE = θe−γt kξkE ,
x ∈ Rd , t ≥ 0.
Suppose that
kkl,t kE ≤ Cl θe−γt kξklE ,
11
t ∈ I, 1 ≤ l ≤ n − 1,
(4.4)
where
Cl
C1 = 1).
is a positive constant. (Note that, by (4.4),
Estimate
kn .
By the mild form of (4.3), the following inequality holds
kkn,t kE ≤ e
−γt
kkn,0 kE + κ
−
Zt
e
−γ(t−s)
n−1
X
l=1
0
≤ e−γt kkn,0 kE + κ −
Zt
e−γ(t+s)
n−1
X
l=1
0
≤
1+
β
γ
n−1
X
l=1
n
kkl,s kE kkn−l,s kE ds
l
n
Cl Cn−l
l
n
Cl Cn−l θ2 kξknE ds
l
!
θkξknE e−γt .
Therefore, by induction,
kkn,t kE ≤ θCn kξknE e−γt ,
n ≥ 1,
(4.5)
,
(4.6)
where
n−1 βX n
Cl Cn−l
1+
l
γ
Cn =
!
C1 = 1.
l=1
Put
C0 = 0.
Consider the following generating function:
H(x) :=
X Cn
xn .
n!
n≥0
By (4.6),
H
satises the following equation:
β 2
H (x).
γ
√
z → 1 − z is
H(x) = ex − 1 +
Since
H(0) = C0 = 0
and the function
analytic for
|z| < 1,
one
has
γ
H(x) =
−
2β
s
γ2
γ
− (ex − 1) ,
4β 2
β
x < ln
γ
+1 .
4β
(4.7)
Therefore, (4.2), (4.5) and (4.7), we have
kut (·, λ) − θkE ≤
X
θCn
n≥1
|λ|n kξknE −γt
e
n!
= θH(|λ|kξkE )e−γt ,
|λ|kξkE < ln(
γ
+ 1).
4β
This proves the theorem.
Remark
.
4.2
Note that the estimate
|λ|kξkE < ln
γ
+1
4β
the initial condition satises
θe−
γ
4β +1
< u0 (x) < θe
12
γ
4β +1
.
holds if and only if
Let ξ : Rd × Ω → R be a random eld. Under the assumptions
of Theorem 4.1, the following estimate holds
Corollary 4.3.
Ekut (ω, λ) − θk2E ≤ θ2 e−2γt
s
!2
γ
γ
γ2
|λ|kξ(ω)k
E
−1
− e
−E
,
2β
4β 2
β
(4.8)
γ
+1 .
4β
where sup |λ|kξ(ω)kE < ln
ω∈Ω
We apply now the general results to the specic case of the random initial
data and try to estimate the rate of convergence using
bility space. Let us denote, for any
Z
fb(λ) =
1
d
f ∈ L (R ),
e−iλx f (x)dx,
L2 -norm
over a proba-
its Fourier transform by
λ ∈ Rd .
Rd
Let
p̃t (x) be a transition probability density for the jump process with the genLJ for J = Jθ (see (1.3) and (A1)). Introduce the following assumption
erator
Jθ
is bounded.
(A2)
p̃t − δ ∈ L2 (Rd ) ∩ L∞ (Rd ),
estimate (4.8), when Jθ is non-
Assumption (A2) is a sucient condition to have
t ≥ 0.
The following theorem improves the
negative.
Theorem 4.4. Let (A1) holds with κ = θ . Let ξ(x, ω) be a homogeneous
random eld with the following correlation function
B(x − y) = Eξ(x)ξ(y),
x, y ∈ Rd .
Suppose that B ∈ L1 (Rd ) and its Fourier transform Bb satises the following
asymptotic
a
b
,
B(λ)
∼
|λ|α
λ → 0,
where α ∈ (0, d], a > 0. Suppose, moreover, that the function Jb is such that the
following estimate
b
J(λ)
= 1 − b|λ|β + o(|λ|β ),
λ → 0,
b
where β ∈ (0, 2], b > 0, and let the function x → sup J(λ)
be monotonically
|λ|≤x
decreasing in a neighborhood of 0. Then the following inequality holds
2
Ek1,t
≤ θ2 e−2βt (D1 t
α−d
β
+ D2 e−2∆t ),
where D1 , D2 , ∆ are some xed positive constants.
13
Proof.
By assumptions of the theorem,
2
Ek1,t
= θ2 e−2βt E
Z
p̃t (z)ξ(z)dz
Rd
Z
=θ e
p̃t (y)p̃t (z)B(y − z)dydz
p̃t (y)p̃t (z)Eξ(z)ξ(y)dydz = θ e
Z
Rd
2 −2βt
pb̃t (λ)(p̃\
t ∗ B)(λ)dλ = θ e
Rd
= θ2 e−2βt
Z
2 −2βt
Rd
= θ2 e−2βt
Therefore,
Z
p̃t (y)ξ(y)dy
Rd
2 −2βt
p̃t ∗ B ∈ L1 (Rd ) ∩ L2 (Rd ).
Z pb̃t (λ)
2
b
B(λ)dλ
Rd
Z
e
b
2(J(λ)−1)t
b
B(λ)dλ,
Rd
where Parseval's theorem were used.
Therefore, by assumption on
b
B
and
Jb there
exist
ε > 0, δ > 0
and
∆>0
such that
Z
b
2(J(λ)−1)t
e
Z
a(1 + ε) −2(b−ε)|λ|β t
e
dλ +
|λ|α
b
B(λ)dλ
≤
Rd
≤ D1 t
where
D1
and
D2
b
e−2∆t B(λ)dλ
Rd \Bδ (0)
Bδ (0)
α−d
β
Z
+ D2 e−2∆t ,
are some constants, that yields the statement.
Acknowledgments
Financial supports by the DFG through CRC 701, Research Group Stochastic Dynamics: Mathematical Theory and Applications, and by the European
Commission under the project STREVCOMS PIRSES-2013-612669 are gratefully acknowledged.
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