Space-Time Fractional Cauchy Problems and Trace Estimates for Relativistic
Stable Processes
by
Jebessa Bulti Mijena
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 03, 2013
Keywords: Distributed-order Cauchy problems, Caputo fractional derivative,
Reimann-Liouville fractional derivative, Strongly analytic solution
Copyright 2013 by Jebessa Bulti Mijena
Approved by
Erkan Nane, Chair, Associate Professor of Mathematics
Olav Kallenberg, Professor of Mathematics
Jerzy Szulga, Professor of Mathematics
Ming Liao, Professor of Mathematics
George Flowers, Dean of the Graduate school
Abstract
Fractional derivatives can be used to model time delays in a di usion process. When
the order of the fractional derivative is distributed over the unit interval, it is useful for
modeling a mixture of delay sources. In some special cases distributed order derivative can
be used to model ultra-slow di usion. In the st part of the thesis, we extend the results
of Baeumer and Meerschaert [3] in the single order fractional derivative case to distributed
order fractional derivative case. In particular, we develop the strong analytic solutions of
distributed order fractional Cauchy problems.
In this thesis, we also study the asymptotic behavior of the trace of the semigroup of
a killed relativistic -stable process in any bounded R smooth boundary open set. More
precisely, we establish two-term estimates of the trace with an error bound of e2mtt(2 d)= .
When m = 0; our result reduces to the result established by Ba~nuelos and Kulczycki for
stable processes given in [7].
ii
Acknowledgments
As my graduate education nears its end it is a pleasure to think back at these past
four years and remember all those people who have made this part of my life so enjoyable at
times and just bearable at others. I have been fortunate enough to have had the opportunity
of higher education, a right so often denied to people because of political, socio-cultural, or
economic circumstances. This, as well as many other things, I owe to the foresight of my
mother and father.
I have had the good fortune of having Dr. Erkan Nane as my thesis advisor. He has
not only supported me but also given me complete freedom in my research without ignoring
me. He has respected all my decisions and he is always available whenever I needed him. I
am grateful for his trust, his encouragement, and his good natured disposition. I also have
to thank the members of my PhD committee, Professors Olav Kallenberg, Jerzy Szulga, and
Ming Liao for their helpful advice and suggestions in general. I am also grateful to all of you
for making me speak in our probability seminars during my third and fourth years in which
I learned a great deal of probability theory. I would also like to thank Professor Rodrigo
Ba~nuelos for his collaboration.
I especially thank my mom, dad, brothers and sisters. My hard-working parents have
sacri ced their lives for my siblings and myself and provided unconditional love and care.
I love them so much, and I would not have made it this far without them. Dad, your are
the best of best father that any one could ever ask for. I admire the way you teach us
everything in life. I remember when I was a little boy you told me I will be a very successful
man in my education. Thank you for showing big con dence in me throughout my school
years. Mom, thank you for keeping our home so warm and full of love. Thank you for your
proud smile whenever I achieve anything it keeps me to go for more. My brothers (Tagel
iii
and Abdi) and sisters (Mulu, Demie and Misir) have been my best friends all my life and I
love them dearly and thank you for your unconditional love and support. I know I always
have my family to count on when times are rough. Special thanks to my beautiful wife,
Mek, as well as her wonderful family, Ermule, Mommy, Sofu , and Esete, who all have been
supportive and caring. Thank you Ermule for all advice and support you have given me and
Mek. Mommy(Etenu) thank you for your kindness and caring nature.
One of the best decision I made in my life is nding my best friend, soul-mate, and
wife. I married the best person out there for me. There are no words to convey how much
I love her. Mek has been a true and great supporter and has unconditionally loved me
during my good and bad times. She has been non-judgmental of me. She has faith in me
and my intellect. These past several years have not been an easy ride, both academically
and personally. I truly thank Mek for providing all the help I needed over all these years.
I feel that what we both learned a lot about life and strengthened our commitment and
determination to each other and to live life to the fullest.
Last but not least, I would like to thank my friends (too many to list here but you know
who you are!) for providing support and friendship that I needed.
iv
I dedicate this thesis to
my wife, my mother, my father, my brothers and sisters
for their constant support and unconditional love.
I love you all dearly.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Cauchy problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Relativistic stable processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Di usion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Generator Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Distributed Order Fractional Derivatives . . . . . . . . . . . . . . . . 17
2.3 Time-fractional Di usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Basic Facts of Relativistic Stable Process . . . . . . . . . . . . . . . . . . . . 23
3 Strong Analytic Solution to Distributed Order Time Fractional Cauchy problems 31
4 Two-term trace estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Mixed Stable Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6 Open Problems and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vi
Chapter 1
Introduction
In this chapter, we give an introduction to Cauchy problems and relativistic stable
processes.
1.1 Cauchy problems
Cauchy problems @u@t = Lu model di usion processes and have appeared as an essential
tool for the study of dynamics of various complex stochastic processes arising in anomalous
di usion in physics [39, 50], nance [21], hydrology [11], and cell biology [46]. Complexity
includes phenomena such as the presence of weak or strong correlations, di erent sub-or
super-di usive modes, and jump e ects. For example, experimental studies of the motion
of macromolecules in a cell membrane show apparent subdi usive motion with several si-
multaneous di usive modes (see [46]). When L = = Pj@2u=@x2j, Cauchy problem is a
tradition di usion equation.
Traditional di usion represents the long-time limit of a random walk, where nite vari-
ance jumps occur at regularly spaced intervals. Eventually, after each particle makes a series
of random steps, a histogram of particle locations follows a bell-shaped normal density. The
central limit theorem of probability ensures that this same bell-shaped curve will eventually
emerge from any random walk with nite variance jumps, so that this di usion model can
be considered universal. The random walk limit is a Brownian motion, whose probability
densities solve the di usion equation.
The fractional Cauchy problem @ u=@t = Lu with 0 < < 1 models anomalous
sub-di usion, in which a cloud of particles spreads slower than the square root of time.
The "particles" might be pollutants in ground water, stock prices, sound waves, proteins
1
crossing a cell boundary, or animals invading a new ecosystem. When L = , the solution
u(t;x) is the density of a time-changed Brownian motion B(E(t)), where the non-Markovian
time change E(t) = inff > 0;D( ) > tg is the inverse, or rst passage time of a stable
subordinator D(t) with index .
The process B(E(t)) is the long-time scaling limit of a random walk [31, 32], when the
random waiting times between jumps belong to the -stable domain of attraction. Roughly
speaking, a power-law distribution of waiting times leads to a fractional time derivative in
the governing equation. Recently, Barlow and Cern y [9] obtained B(E(t)) as the scaling limit
of a random walk in a random environment. More generally, for a uniformly elliptic operator
L on a bounded domain D Rd, under suitable technical conditions and assuming Dirichlet
boundary conditions, the di usion equation @u=@t = Lu governs a Markov process Y(t)
killed at the boundary, and the corresponding fractional di usion equation @ u=@t = Lu
governs the time-changed process Y(E(t)) [36].
In some applications, waiting times between particle jumps evolve according to a more
complicated process, which cannot be adequately described by a single power law. A mixture
of power laws leads to a distributed-order fractional derivative in time [16, 28, 29, 30, 34, 41].
An important application of distributed-order di usions is to model ultraslow di usion
where a plume of particles spreads at a logarithmic rate [34, 47]. This thesis considers
the distributed-order time-fractional di usion equations with the generator L of a uniformly
bounded and strongly continuous semigroup in a Banach space. Hahn et al. [22] discussed
the solutions of such equations on Rd, and the connections with certain subordinated pro-
cesses. Kochubei [25] proved strong solutions on Rd for the case L = . Luchko [27] proved
the uniqueness and continuous dependence on initial conditions on bounded domains. Meer-
schaert et al. [37] established the strong solutions of distributed order fractional Cauchy
problems in bounded domains with Dirichlet boundary conditions.
2
When L is the generator of a uniformly bounded and continuous semigroup on a Banach
spaces, Baeumer and Meerschaert [3] showed that the solution of
@ u=@t = Lu
is analytic in a sectorial region. A similar problem has been considered in the literature on
a purely analytic level, without a probabilistic interpretation of the subordination represen-
tation: see, for example, Pr uss [44, Corollary 4.5] . The case of a single fractional order was
also considered by Bazhlekova [10]. In this thesis, we extend the results of Baeumer and
Meerschaert [3] to distributed order fractional di usion case. Our proofs work for operators
L that are generators of uniformly bounded and continuous semigroups on Banach spaces.
Our result for this case is given in chapter 3.
In the next few paragraphs we introduce basic facts about relativistic stable processes.
1.2 Relativistic stable processes
In Ryznar [45] Green function estimates of the Sch odinger operator with the free Hamil-
tonian of the form
( +m2= ) =2 m;
were investigated, where m > 0 and is the Laplace operator acting on L2(Rd). Some
of these estimates (see Lemma 2.13 below) and (essentially the same) proof in Ba~nuelos
and Kulczycki (2008) can be used to provide an extension of the asymptotics in [7] to the
relativistic stable processes for any 0 < < 2.
An Rd-valued process with independent, stationary increments having the following
characteristic function
Eei X ;mt = e tf(m2= +j j2) =2 mg; 2Rd
3
is called relativistic -stable process with mass m. We assume that sample paths of X ;mt
are right continuous and have left-hand limits a.s. If we put m = 0 we obtain the symmetric
rotation invariant stable process with the characteristic function e tj j ; 2Rd: We refer
to this process as standard stable process. The in nitesimal generator of X ;mt is m
(m2= ) =2; which is a non-local operator. Note that when m = 1; this in nitesimal
generator reduces to 1 (1 ) =2: Thus the 1 resolvent kernel of the relativistic stable
process X ;1t on Rd is just the Bessel potential kernel. When = 1; the in nitesimal
generator reduces to the so-called free relativistic Hamiltonian m p +m2: The operator
m p +m2 is very important in mathematical physics due to its application to relativistic
quantum mechanics. For the rest of the thesis we keep ;m and d 2 xed and drop ;m
in the notation, when it does not lead to confusion. Hence from now on the relativistic -
stable process is denoted by Xt and its standard stable counterpart by ~Xt . We keep this
notational convention consistently throughout the paper, e.g., if pt(x y) is the transition
density of Xt then ~pt(x y) is the transition density of ~Xt.
Brownian motion has characteristic function
E0ei Bt = e tj j2; 2Rd
Let = 2 . Ryznar showed that Xt is subordinated to Brownian motion. Let T (t); t> 0,
denote the strictly -stable subordinator with the following Laplace transform
E0e T (t) = e t ; > 0: (1.1)
Let (t;u); u > 0, denote the density function of T (t). Then the process BT (t) is the
standard symmetric -stable process.
Ryznar [45, Lemma 1] showed that we can obtain Xt = BT (t;m), where T (t;m) is
a positive in nitely divisible process with stationary increments with probability density
4
function
(t;u;m) = e m1= u+mt (t;u); u> 0:
Transition density of T (t;m) is given by (t;u v;m). Hence the transition density
of Xt is given by
p(t;x) = emt
Z 1
0
1
(4 u)d=2e
jxj2
4u e m
1= u
(t;u)du (1.2)
Then p(t;x;y) = p(t;x y): Since the transition density is obtained from the characteristic
function by inverse Fourier transform, it follows that p(t;x) is a radially symmetric decreasing
function and that
p(t;x) p(t;0) emt
Z 1
0
1
(4 u)d=2 (t;u)du = e
mtt d= !d (d= )
(2 )d (1.3)
where !d = 2 d=2 (d=2) is the surface area of the unit sphere in Rd. For A Rd we de ne the rst
exit time from A by A = infft 0 : Xt =2Ag.
Let D Rd be a domain.
We set
rD(t;x;y) = Ex[p(t D;X D;y); D 0. For a nonnegative Borel function f and t> 0, let
PDt f(x) = Ex[f(Xt) : t< D] =
Z
D
pD(t;x;y)f(y)dy
be the semigroup of the killed process acting on L2(D), see, Ryznar [45, Theorem 1].
Let D be a bounded domain (or of nite volume). Then the operator PDt is a Hilbert-
Schmidt operator mapping L2(D) into L1(D) for every t> 0. This follows from (1.3), (1.4),
and the general theory of heat semigroups as described in [18]. It follows that there exists
5
an orthonormal basis of eigenfunctionsf?n : n = 1; 2; 3; gfor L2(D) and corresponding
eigenvalues f n : n = 1; 2; 3; g of the generator of the semigroup PDt satisfying
1 < 2 3
with n!1 as n!1. By de nition, the pair f?n; ng satis es
PDt ?n(x) = e nt?n(x); x2D; t> 0:
Under such assumptions we also have
pD(t;x;y) =
1X
n=1
e nt?n(x)?n(y) (1.6)
In this thesis we are interested in the behavior of the trace of this semigroup de ned by
ZD(t) =
Z
D
pD(t;x;x)dx: (1.7)
Because of (1.6) we can write (5.9) as
ZD(t) =
1X
n=1
e nt
Z
D
?2n(x)dx =
1X
n=1
e nt: (1.8)
We denote d-dimensional volume of D by jDj. It is shown in [7] that for any open set
D with nite volume, it holds that
limt!0td= ~ZD(t) = C1jDj; C1 = !d (d= )(2 )d (1.9)
This is closely related to the growth of the eigenvalues of ~PDt : Let N( ) be the number
of eigenvalues f jg of ~PDt which do not exceed , it follows from the classical Tauberian
6
theorem (see for example [20], p.445 Theorem 2) that
lim
!1
d= N( ) = C1jDj (1 +d= ) (1.10)
This is the analogue for killed stable processes of the celebrated Weyl?s asymptotic
formula for the eigenvalues of the Laplacian. We will prove later in (2.38) that similar
formula is true for relativistic stable processes.
De nition 1.1. The boundary, @D, of an open set D in Rd is said to be R smooth if for
each point x0 2@D there are two open balls B1 and B2 with radii R such that B1 D;B2
Rdn(D[@D) and @B1\@B2 = x0:
The asymptotic for the trace of the heat kernel when = 2 (the case of the Laplacian
with Dirichlet boundary condition in a domain of Rd), have been extensively studied by
many authors. The van den Berg [49] result states that under the R smoothness condition
when = 2;
ZD(t) (4 t) d=2
jDj
p t
2 j@Dj
CdjDjt1 d=2
R2 ;t> 0: (1.11)
For domains with C1 boundaries the result
ZD(t) = (4 t) d=2
jDj
p t
2 j@Dj+o(t
1=2)
(1.12)
was proved by Brossard and Carmona [13]. The asymptotic behavior for the trace of killed
symmetric stable processes, 2(0;2); for an open bounded set with R smooth boundary
was given in [7]
~ZD(t)
C1jDj
td= +
C2j@Djt1=
td=
C3jDjt2=
R2td= (1.13)
7
where C1;C2; and C3 are some constants depending on d; ;X: In [8], the authors proved
that, for any bounded Lipschitz domain D; ~ZD(t) satis es
td= ~ZD(t) = C1jDj C2Hd 1(@D)t1= +o(t1= ) (1.14)
where C1;C2 are some constants depending on d; andHd 1(@D) denote the (d 1) dimen-
sional Hausdor measure of @D.
In the second part of the thesis we obtained the second term in the asymptotics of ZD(t)
for bounded open set with R smooth boundary. Our result is inspired by result for Trace
estimates for stable processes by Ba~nuelos and Kulczycki [7].
1.3 Outline of Dissertation
This dissertation is divided into two more or less independent parts. The rst part is
dedicated to nding strong analytic solution of distributed order fractional Cauchy prob-
lems. Whereas the second part is more concerned with nding two-term trace estimate of
relativistic stable processes on bounded R smooth open set.
Chapter 2 consists of preliminary mathematical material, which serves the purpose of
setting up the vocabulary and the framework for the rest of the dissertation. In Section
2.1 we discuss the basic facts of traditional di usion model and shows that the solution
of a di usion equation is a density of a Brownian motion. Section 2.2 and Section 2.2.2
discusses fractional calculus. We give a generator form, Caputo, and Riemann-Liouville
form of fractional derivative with some simple examples. In Section 2.4 we state a de ning
property of a semigroup. There we give all of the results on a semigroup we shall use in
the proof our main results in chapter 3. Finally in Section 2.5 we present the basic facts
about relativistic -stable process. There we give all of the results on a relativistic stable
process we shall use in the proof of our main result for such process in chapter 4. We also
8
give our rst result in proposition 2.9, which is the analogue for relativistic stable process of
the celebrated Weyl?s asymptotic formula for the eigenvalues of the Laplacian.
In chapter 3 we state and prove our two main results about distributed order time
fractional Cauchy problems. Our proofs work for operators L that are generators of uni-
formly bounded and continuous semigroups on Banach spaces. This chapter describes work
contained in paper [40].
Chapter 4 discusses a trace estimates for relativistic stable process on open bounded
domain with R smooth boundary. There we state and proof our main result. Similar result
is obtained for stable process in [7].
Chapter 5 is the last chapter and discusses a basic facts about sum of two independent
stable processes. So far we are only able to nd the rst term asymptotic expansion of trace
of such process. I am still looking in nding the best estimate for the trace on R smooth
boundary domains and then extend to Lipschitz domains if possible.
9
Chapter 2
Preliminaries
In this chapter, we summarize some results from fractional calculus and relativistic
stable processes. We mainly focus on trandional di usion model, fractional di usion
models and relativistic stable processes. Results in the following sections are mainly
adapted from [33], [42] and [43].
2.1 Di usion Model
The traditional model for di usion combines elements of probability, di erential equa-
tions, and physics. A random walk provides the basic physical model of particle motion.
The central limit theorem gives convergence to a Brownian motion, whose probability den-
sities solves the di usion equation. We start with a sequence of independent and identically
distributed (iid) random variables Y1;Y2; that represent the jumps of a randomly selected
particle. The random walk
Sn = Y1 +Y2 + +Yn
gives the location of that particle after n jumps. Next we recall the well-known central limit
theorem, which shows that the probability distribution of Sn converges to a normal limit.
Here we sketch the argument in the simplest case, using Fourier transforms. For complete
proof of the central theorem and to see the same normal limit governs a somewhat broader
class of random walk (see [33], Theorem 3.5 and 4.5).
10
Let F(x) = P[Y x] denote the cumulative distribution function (cdf) of the jumps,
and assume that the probability density function (pdf) f(x) = F0(x) exists. Then we have
P[a Y b] =
Z b
a
f(x)dx
for any real numbers a 0 at any time scale c > 0: Increasing the time scale
c makes time to go faster. The long-time limit of the rescaled random walk is a Brownian
motion: As c!1 we have
E
h
e ikc 1=2S[ct]
i
=
1 k
2
c +o(c
1)
[ct]
=
1 k
2
c +o(c
1)
c [ct]c
!e tk2 (2.4)
where the limit
e tk2 = ^p(k;t) =
Z 1
1
e ikxp(x;t)dx
12
is the FT of a normal density
p(x;t) = 1p4 te x2=4t
with mean zero and variance 2t. Then the continuity theorem for FT implies that
c 1=2S[ct] )Zt
where the Brownian motion Zt is normal with mean zero and variance 2t:
Clearly the FT ^p(k;t) = e tk2 solves a di erential equation
d^p
dt = k
2 ^p = (ik)2 ^p: (2.5)
If f0 exists and if f;f0 are integrable, then the FT of f0(x) is (ik) ^f(k) [33, p. 8]. Using this
fact, we can invert the FT on both sides of (2.5) to get
@p
@t =
@2p
@x2 (2.6)
This shows that the pdf of Zt solves the di usion equation (2.6). The di usion equation
models the spreading of a cloud of particles. The random walk Sn gives the location of a
randomly selected particle, and the long-time limit density p(x;t) gives the relative concen-
tration of particles at location x at time t> 0:
More generally, suppose that 1 = E[Yn] = 0 and 2 = E[Y 2n] = 2 > 0: Then
^f(k) = 1 1
2
2k2 +o(k2)
leads to
E
h
e ikn 1=2Sn
i
=
1
2k2
2n +o(n
1)
n
!exp( 12 2k2)
13
and
E
h
e ikc 1=2S[ct]
i
=
1
2k2
2c +o(c
1)
[ct]
!exp( 12t 2k2) = ^p(k;t): (2.7)
This FT inverts to a normal density
p(x;t) = 1p2 2te x2=(2 2t)
with mean zero and variance 2t: The FT solves
d^p
dt =
2
2 k
2 ^p =
2
2 (ik)
2 ^p
which inverts to
dp
dt =
2
2
@2p
@x2 (2.8)
This form of the di usion equation shows the relation between the dispersivity D = 2=2
and the particle jump variance. Apply the continuity theorem for FT to (2.7) to get random
walk convergence:
c 1=2Sn)Zt
where Zt is a Brownian motion, normal with mean zero and variance 2t:
In many applications, it is useful to add a drift: vt+Zt has FT
E e ik(vt+Zt) = exp( ikvt 12t 2k2) = ^p(k;t);
which solves
d^p
dt =
ikv +
2
2 (ik)
2
^p
Invert the FT to obtain the di usion equation with drift:
@p
@t = v
@p
@x +
2
2
@2p
@x2 (2.9)
14
This represents the long-time limit of a random walk whose jumps have a non-zero mean
v = 1 [33, p. 9].
2.2 Fractional Derivatives
The concept of di erentiation operator D = d=dx is familiar to all who have stud-
ied elementary calculus. And for suitable function f, the nth derivative of f; denoted by
Dnf(x) = dnf(x)=dxn is well de ned-provided n is a positive integer. In 1695; L? H^opital
inquired of Leibniz what meaning could be ascribed to Dnf(x) is n were a fraction. Since
that time the fractional calculus has drawn the attention of many famous mathematicians,
such as Euler, Laplace, Fourier, Abel, Liouville, and Laurent. But it was not until 1884 that
the theory of generalized operators achieved a level in its development suitable as a point of
departure for the modern mathematician. By then the theory had been extended to include
operators Dv, where v could be rational or irrational, positive or negative, real or complex.
In the past few years fractional calculus appeared as an important tool to deal with anoma-
lous di usion processes. An anomalous di usion process can be visualized as an ant in a
labyrinth where the average square of the distance covered by the ant ishx2(t)i/t2 where
is a phenomenological constant; for = 1=2 we have the ordinary di usion processes. A
more physical approach of anomalous di usion processes has several applications in many
eld such as di usion in porous media or long range correlation of DNA sequence.
2.2.1 Generator Form
The generator form of the fractional derivative for 0 < < 1 is given by
d f(x)
dx =
Z 1
0
[f(x) f(x y)] (1 )y 1dy: (2.10)
[33, see p. 30], where ( ) =
Z 1
0
e xx 1dx is a gamma function.
15
For continuously di erentiable and bounded function integrate by parts with u = f(x)
f(x y) to get the caputo form
d f(x)
dx =
1
(1 )
Z 1
0
f0(x y)y dy = 1 (1 )
Z 1
0
d
dxf(x y)y
dy (2.11)
Take the derivative outside the integral to get the Riemann-Liouville form
d
dt
f(x) = 1 (1 ) ddx
Z 1
0
f(x y)y dy (2.12)
For 1 < < 2 we can write the generator form
d f(x)
dx =
( 1)
(2 )
Z 1
0
[f(x) f(x y) +yf0(x)]y 1dy: (2.13)
Integrate by parts twice to get the Caputo form for 1 < < 2 :
d f(x)
dx =
1
(2 )
Z 1
0
d2
dx2f(x y)y
1 dy (2.14)
Move the derivative outside to get the Riemann-Liouville form for 1 < < 2 :
d
dt
f(x) = 1 (2 ) d
2
dx2
Z 1
0
f(x y)y1 dy (2.15)
In general, Caputo?s de nition can be written as
d f(x)
dx =
1
(n )
Z 1
0
dn
dxnf(x y)y
n 1 dy; (n 1 < 0, so that f0(x) = e x: Using the Caputo
form for 0 < < 1, a substitution u = y, and the de nition of gamma function ( ) =
R1
0 e
xx 1dx, we get
d
dx [e
x] = e x
which agrees with the integer order case. Using the Riemann-Liouville form we get
d
dt
[e x] = e x
which agrees with the Caputo. In this case, both forms lead to the same result.
2.2.2 Distributed Order Fractional Derivatives
In this section, we give an equivalent form of Caputo and Riemann-Liouville form for a
function de ned on a nonnegative real line.
For a function u(t;x); the Caputo fractional derivative [15] for t 0; is de ned as
@ u(t;x)
@t =
1
(1 )
Z t
0
@u(r;x)
@r
dr
(t r) for 0 < < 1: (2.18)
For 0 < < 1, its Laplace transform is given by
Z 1
0
e st@
u(t;x)
@t ds = s
~u(s;x) s 1u(0;x) (2.19)
where ~u(s;x) = R10 e stu(t;x)dt. For 1 < < 2; @ u(t;x)@t has Laplace transform s ~u(s;x)
s 1u(0;x) @@tu(0;x) and incorporates the initial condition in the usual way as the regular
derivative. The distributed order fractional derivative is
D( )u(t;x) :=
Z 1
0
@ u(t;x)
@t (d ); (2.20)
where is a nite Borel measure with (0;1) > 0.
17
For a function u(t;x) continuous in t 0, the Riemann-Liouville fractional derivative
of order 0 < < 1 is de ned by
@
@t
u(t;x) = 1 (1 ) @@t
Z t
0
u(r;x)
(t r) dr: (2.21)
Its Laplace transform Z
1
0
e st
@
@t
u(t;x)ds = s ~u(s;x): (2.22)
The main advantage of Caputo?s approach is that the initial conditions for fractional
di erential equations with Caputo derivates take on the same form as for integer-order
di erential equations, i.e. contain the limit values of integer-order derivatives of unknown
functions at the lower terminal t = 0:
Example 2.2. Let f(t) = 1 for t 0 and f(t) = 0 for t< 0: Then f0(t) = 0 for t6= 0; so the
Caputo fractional derivative is zero. In fact, the Caputo fractional derivative of a constant
function is always zero, just like the integer order derivative. But the Riemann-Liouville
derivative is not. For t> 0 and 0 < < 1; use (2.21) to get
d
dt
f(t) = 1 (1 ) ddt
Z t
0
1(t y) dy
= 1 (1 ) ddt
Z t
0
u du
= x
(1 ) 6= 0:
If u( ;x) is absolutely continuous on bounded intervals (e.g., if the derivative exists
everywhere and is integrable) then the Riemann-Liouville and Caputo derivatives are related
by
@ u(t;x)
@t =
@
@t
u(t;x) t
u(0;x)
(1 ) : (2.23)
The Riemann-Liouville fractional derivative is more general, as it does not require the rst
derivative to exist. It is also possible to adopt the right-hand side of (2.23) as the de nition of
18
the Caputo derivative, see for example Kochubei [25]. Hence we adopt this as our de nition
of Caputo derivative in this paper. Then the (extended) distributed order derivative is
D( )1 u(t;x) :=
Z 1
0
"
@
@t
u(t;x) t
u(0;x)
(1 )
#
(d ); (2.24)
which exists for u(t;x) continuous, and agrees with the usual de nition (2.20) when u(t;x)
is absolutely continuous.
2.3 Time-fractional Di usion
In this section we will outline the stochastic model for time-fractional di usion. For
additional details and precise mathematical proofs see [33, chapter 4].
Distributed order fractional derivatives are connected with random walk limits. For
each c > 0, take a sequence of i.i.d. waiting times (Jcn) and i.i.d. jumps (Ycn). Let Xc(n) =
Yc1 + + Ycn be the particle location after n jumps, and Tc(n) = Jc1 + + Jcn the time
of the nth jump. Suppose that Xc(ct) )A(t) and Tc(ct) )D (t) as c!1, where the
limits A(t) and D (t) are independent L evy processes. The number of jumps by time t 0
is Nct = maxfn 0 : Tc(n) tg, and [35, Theorem 2.1] shows that the continuous time
random walk (CTRW) Xc(Nct ))A(E (t)), where
E (t) = inff : D ( ) >tg: (2.25)
A speci c mixture model from [34] gives rise to distributed order fractional derivatives:
Let (Bi), 0 ujBi = g= c 1u , for
u c 1= . Then Tc(ct))D (t), a subordinator with E[e sD (t)] = e t (s), where
(s) =
Z 1
0
(e sx 1) (dx): (2.26)
19
Then the associated L evy measure is
(t;1) =
Z 1
0
t (d ); (2.27)
where is the distribution of Bi. An easy computation gives
(s) =
Z 1
0
s (1 ) (d ) =
Z 1
0
s (d ): (2.28)
Here we de ne (d ) = (1 ) (d ). Then, Theorem 3.10 in [34] shows that c 1Nct )
E (t), where E (t) is given by (2.25). The L evy process A(t) de nes a strongly continuous
convolution semigroup with generator L, and A(E (t)) is the stochastic solution to the
distributed order-fractional di usion equation
D( )1 u(t;x) = Lu(t;x); (2.29)
where D( )1 is given by (2.24) with (d ) = (1 ) (d ). The condition
Z 1
0
1
1 (d ) <1 (2.30)
is imposed to ensure that (0;1) <1. Since (0;1) = 1 in (2.26), Theorem 3.1 in [35]
implies that E (t) has a Lebesgue density
gE (t)(x) =
Z t
0
(t y;1)PD (x)(dy): (2.31)
Note that E (t) is almost surely continuous and nondecreasing.
The CTRW model provides a physical explanation for fractional di usion. A power law
jump distribution with P[Ycn > x] = Cx leads to a fractional derivative in space @ =@x
of the same order. A power law waiting time distribution P[Jcn > t] = Bt leads to a
fractional time derivative @ =@t of the same order. Long power-law jumps re ect a heavy
20
tailed velocity distribution, which allows particles to make occasional long jumps, leading to
anomalous super-di usion. Long waiting times model particle sticking and trapping, leading
to anomalous sub-di usion:
B(Ect)?B(c Et)?c =2B(Et):
where B is a Brownian motion. Since < 1, the density of this process spreads slower that
a Brownian motion. For detailed discussion see for example [33].
2.4 Semigroup
This section will serve as a basic introduction to semigroups of linear operators. In
general, semigroups can be used to solve a large class of problems commonly known as
evolution equations. These types of equations appear in many disciplines including physics,
chemistry, biology, engineering, and economics. A semigroup is a family of linear operator
on a Banach space. A Banach space X is a complete normed vector space. That is, if fn2X
is a Cauchy sequence in this vector space, such thatjjfn fmjj!0 as m;n!1; then there
exists some f2X such that jjfn fjj!0 as n!1 in the Banach space norm.
De nition 2.3. Let X be a Banach space. A family of linear operators fTt : t 0g from
X into X is called a semigroup if
(i) T(0) = I; (I is the identity operator on X).
(ii) T(t+s) = T(t)T(s) for all t;s 0 (the semigroup property).
We say that T(t) is uniformly bounded ifjjT(t)fjj Mjjfjjfor all f2X and all t 0.
If T(tn)f ! T(t)f in X for all f 2 X whenever tn ! t then the operator T is strongly
continuous. It is easy to check that fT(t);t 0g is strongly continuous if T(t)f !f in
X for all f 2X as t! 0. A strongly continuous, bounded semigroup is also called a C0
semigroup.
21
For any strongly continuous semigroup fT(t);t > 0g on a Banach space X we de ne
the in nitesimal generator as
Lf = lim
t!0+
T(t)f f
t in X (2.32)
meaning that jjt 1(T(t)f f) Lfjj!0 in the Banach space norm. The domain D(L) of
this linear operator is the set of all f2X for which the limit in (2.32) exists. The domain
D(L) is dense in X, and L is closed, meaning that if fn ! f and Lfn ! g in X then
f2D(L) and Lf = g (see, for example Corollary I.2.5 in [42]).
Theorem 2.4. Let T(t) be a C0 semigroup. There exists a constant a 0 and M 1 such
that
jjT(t)jj Meat for 0 t<1:
Proof. See, for example, Pazy [42, Theorem I.2.2].
Corollary 2.5. If T(t) is a C0 semigroup then for every f2X; t!T(t)f is a continuous
function from R+ (the nonnegative real line) into X:
Proof. See, for example, Pazy [42, Theorem I.2.3].
Theorem 2.6. Let T(t) be a C0 semigroup and let L be its in nitesimal generator. Then
(i) For f2X;Rt0 T(s)fds2D(L) and
L
Z t
0
T(s)fds
= T(t)f f
(ii) For f2D(L); T(t)f2D(L) and
d
dtT(t)f = LT(t)f = T(t)Lf
Proof. See, for example, Pazy [42, Theorem I.2.4].
22
2.5 Basic Facts of Relativistic Stable Process
In this section we assemble the basic notation and facts of relativistic stable process that
will be used in the sequel. We also give a proof to some simple lemmas and propositions.
Next we introduce some notations. For x2Rd; let D(x) denote the Euclidean distance
between x and @D and the ball in Rd center at x and radius r;fy : jy xj < rg will be
denoted by B(x;r): De ne
( ) =
Z 1
0
e vvp 1=2( +v=2)p 1=2dv; 0;
wherep = (d+ )=2:We putR( ;d) =A( ;d)= (0);whereA(v;d) = ( ((d v)=2))=( d=22vj (v=2)j):
Let (x);~ (x) be the densities of the L evy measures of the relativistic stable process and
the standard stable process, respectively. These densities, are given by
(x) = R( ;d)jxjd+ e m1= jxj (m1= jxj) (2.33)
~v(x) = A( ;d)jxjd+ (2.34)
We need the following estimate of the transition probabilities of the process Xt which
is given in ([26], Lemma 2.2): For any x;y2Rd and t> 0 there exist constants c1 > 0 and
c2 > 0;
p(t;x;y) c1emt min
t
jx yjd+ e
c2jx yj;t d=
(2.35)
We will use the fact([14], Lemma 6) that if D Rd is an open bounded set satisfying a
uniform outer cone condition, then Px(X( D) 2 @D) = 0 for all x 2 D: For the open
bounded set D we will be denoted by GD(x;y) the Green function for the set D equal to
GD(x;y) =
Z 1
0
pD(t;x;y)dt;x;y2Rd
23
and for any such D the expectation of the exit time of the processes Xt from D is given by
the integral of the Green function over the domain. That is,
Ex( D) =
Z
D
GD(x;y)dy:
Now we state a simple lemma about the upper bound of rD(t;x;y); which is an analogue
of [7, Lemma 2.1.] for stable processes.
Lemma 2.7. Let D Rd be an open set. For any x;y2D we have
rD(t;x;y) c1emt
t
d+ D (x)e
c2 D(x)^t d=
Proof. Using (1.4) and (2.35) we have
rD(t;x;y) = Ey(p(t D;X( D);x); D 0: Then
Px(X( D)2A; D <1) =
Z
D
GD(x;y)
Z
A
v(y z)dzdy;x2D (2.44)
Here we need the following generalization already stated and used in [7].
Proposition 2.11. [26, Proposition 2.5] Assume that D is an open, nonempty, bounded
subset of Rd; and A is a Borel set such that A Dcn@D and 0 t1 0 the
following estimates hold;
pD(t;x;y) emt~pD(t;x;y)
rD(t;x;y) e2mt~rD(t;x;y)
(2.45)
We need the following lemma given by Van den Berg in [49].
Lemma 2.14. [49, Lemma 5] Let D be an open bounded set in Rd with R-smooth boundary
@D and de ne for 0 qqg
and denote the area of its boundary @Dq by j@Dqj. Then
R q
R
d 1
j@Dj j@Dqj
R
R q
d 1
j@Dj;0 q 0 such that t1= R=2 we have
jrD(t;x;x) rH(x)(t;x;x)j ce
2mtt1=
Rtd=
t1=
D(x)
d+ =2 1
^1
(2.47)
Proof. Exactly as in [7], let x 2@D be a unique point such thatjx x j= dist(x;@D) and
B1 and B2 be balls with radius R such that B1 D;B2 Rdn(D[@D);@B1\@B2 = x :
Let us also assume that x = 0 and choose an orthonormal coordinate system (x1;x2;:::;xd)
so that the positive axis 0x1 is in the direction of !0p where p is the center of the ball B1:
Note that x lies on the interval 0p so x = (jxj;0;0;:::;0): Note also that B1 D (B2)c
and B1 H(x) (B2)c: For any open sets A1;A2 such that A1 A2 we have rA1(t;x;y)
rA2(t;x;y) so
jrD(t;x;x) rH(x)(t;x;x)j rB1(t;x;x) r(B2)c(t;x;x):
So in order to prove the proposition it su ces to show that
rB1(t;x;x) r(B2)c(t;x;x) ce
2mtt1=
Rtd=
t1=
D(x)
d+ =2 1
^1
for any x = (jxj;0;:::;0);jxj2(0;R=2]: Such an estimate was proved for the case m = 0 in
[7]. In order to complete the proof it is enough to prove that
rB1(t;x;x) r(B2)c(t;x;x) ce2mt
n
~rB1(t;x;x) ~r(B2)c(t;x;x)
o
:
But this follows from Propositions 2.11, 2.12 and 2.13.
29
Given the ballB2, we setU = (B2)c:Now using the generalized Ikeda-Watanabe formula,
Proposition (2.12) and Lemma 2.4 in [26] we have
rB1(t;x;x) rU(t;x;x)
= Ex [t> B1;X( B1)2UnB1;pU(t B1;X( B1);x)]
=
Z
B1
Z t
0
pB1(s;x;y)ds
Z
UnB1
v(y z)pU(t s;z;x)dzdy
e2mt
Z
B1
Z t
0
~pB1(s;x;y)ds
Z
UnB1
~v(y z)~pU(t s;z;x)dzdy
ce2mtEx
h
t> ~ B1; ~X(~ B1)2UnB1; ~pU(t ~ B1; ~X(~ B1);x)
i
= ce2mt (~rB1(t;x;x) ~rU(t;x;x))
ce
2mtt1=
Rtd=
t1=
D(x)
d+ =2 1
^1
The last inequality follows by Proposition 3.1 in [7].
30
Chapter 3
Strong Analytic Solution to Distributed Order Time Fractional Cauchy problems
In this chapter we give strong analytic solution to distributed order time fractional
Cauchy problems. Our proofs work for operators L that are generators of uniformly bounded
and continuous semigroups on Banach spaces. In our rst main result the L evy subordinator
is written as the sum of n independent stable subordinators of index 0 < 1 < 2 < <
n < 1 and Theorem 3.4 provides an extension with subordinator D (t) as the weighted
average of an arbitrary number of independent stable subordinators.
Let D (t) be a strictly increasing L evy process (subordinator) with E[e sD (t)] = e t (s),
where the Laplace exponent
(s) = bs+
Z 1
0
(e sx 1) (dx); (3.1)
b 0, and is the L evy measure of D . Then we must have either
(0;1) =1; (3.2)
or b> 0, or both, see [35]. Let
E (t) = inff 0 : D ( ) >tg (3.3)
be the inverse subordinator.
Let T be a uniformly bounded, strongly continuous semigroup on a Banach space. Let
S(t)f =
Z 1
0
(T(l)f)gE (t)(l)dl (3.4)
31
where gE (t)(l) is a Lebesgue density of E (t).
Using (2.31), it is easy to show that
Z 1
0
e stgE (t)(l)dt = 1s (s)e l (s):
Using Fubini?s Theorem, we get
Z 1
0
(s)e l (s)T(l)fdl = s
Z 1
0
e stS(t)fdt: (3.5)
We de ne a sectorial region of the complex plane C( ) = frei 2C : r > 0;j j< g:
Note that C( =2) = C+ = fRe(Z) > 0g: We call a family of linear operators on a Banach
space X strongly analytic in a sectorial region if for some > 0 the mapping t!T(t)f has
an analytic extension to the sectorial region C( ) for all f 2X (see, for example, section
3.12 in [23]).
Next we state two theorem that is very important in proving our main results. The rst
theorem is about Bochner intergal and the next gives analytic representation of an operators.
Theorem 3.1. (Bochner). A function f : I!X is Bochner integrable if and only if f is
measurable and jfj is integrable. If f is Bochner integrable, then
ww
ww
Z
I
f(t)dt
ww
ww
Z
I
kf(t)kdt
Proof. See, for example, [1, Theorem 1.1.4].
Theorem 3.2. (Analytic Representation). Let 0 < 2;!2R and q : (!;1)!X:
The following are equivalent:
i) There exists a holomorphic function f : C( )!X such that supz2C( )jje !zf(z)jj<1
for all 0 < < and q( ) = ~f( ) for all >!:
32
ii) The function q has a holomorphic extension q : ! + C( + =2) ! X such that
sup 2!+C( + =2)jj( !) q( )jj<1 for all 0 < < :
Proof. See, for example, [1, Theorem 2.6.1].
Now we give our main result for distributed order fractional Cauchy problems. Our
solution works for operators L that are generators of uniformly bounded and continuous
semigroups on Banach spaces.
Let 0 < 1 < 2 < < n < 1. In the next theorem we consider the case where
(s) = c1s 1 +c2s 2 + +cns n:
In this case the L evy subordinator can be written as
D (t) = (c1)1= 1D1(t) + (c2)1= 2D2(t) + + (cn)1= nDn(t)
where D1(t);D2(t); ;Dn(t) are independent stable subordinators of index 0 < 1 < 2 <
< n < 1.
Theorem 3.3. Let (X;jj:jj) be a Banach space and L be the generator of a uniformly
bounded, strongly continuous semigroup fT(t) : t 0g. Then the family fS(t) : t 0g of
linear operators from X into X given by (3.4) is uniformly bounded and strongly analytic in
a sectorial region. Furthermore, fS(t) : t 0g is strongly continuous and h(x;t) = S(t)f(x)
is a solution of
nX
i=1
ci@
ih(x;t)
@t i = Lh(x;t); h(x;0) = f(x):
for 1 < 2 < < n2(0;1)
Proof. We adapt the methods of Baeumer and Meerschaert [3, Theorem 3.1] with some
very crucial changes in the following. For the purpose of completeness of the arguments we
included some parts verbatim from Baeumer and Meerschaert [3].
33
Since fT(t) : t 0g is uniformly bounded we have jjT(t)fjj Mjjfjj for all f 2X.
Theorem 3.1 implies that a functionF : R1 !X is integrable if and only ifF(s) is measurable
and jjF(s)jj is integrable, in which case
ww
ww
Z
F(l)dl
ww
ww
Z
jjF(l)jjdl:
For xed f2X and applying Bochner?s Theorem with F(l) = (T(l)f)gE (t)(l) we have that
jjS(t)fjj =
ww
ww
Z 1
0
(T(l)f)gE (t)(l)dl
ww
ww
Z 1
0
jj(T(l)f)gE (t)(l)jjdl
=
Z 1
0
jjT(l)fjjgE (t)(l)dl
Z 1
0
MjjfjjgE (t)(l)dl = Mjjfjj
since gE (t)(l) is the Lebesgue density for E (t). This shows that fS(t) : t 0g is well
de ned and uniformly bounded family of linear operators on X.
The de nition of T(t) and dominated convergence theorem implies
jjS(t)f fjj =
ww
ww
Z 1
0
(T(l)f f)gE (t)(l)dl
ww
ww
Z 1
0
jjT(l)f fjjgE (t)(l)dl
! jjT(0)f fjj= 0
as t!0+. This shows lim
t!0+
S(t)f = f. Now if t;h> 0 then we have
jjS(t+h)f S(t)fjj
Z 1
0
jjT(l)fjjjgE (t+h)(l) gE (t)(l)jdl!0
as h!0+ since E (t+h) =) E (t) as h!0.
34
This shows that fS(t) : t> 0g is strongly continuous.
Let q(s) = R10 e stT(t)fdt and r(s) = R10 e stS(t)fdt for any s > 0; so that we can write
(3.5) in the form
(s)q( (s)) = sr(s) (3.6)
for any s> 0. Now we want to show that this relation holds for certain complex numbers.
Fix s 2 C+ = fz 2 C : R(z) > 0g, and let F(t) = e stT(t)f. Since F is continuous, it
is measurable, and we have jjF(t)jj je stjMjjfjj = e tR(s)Mjjfjj since jjT(t)fjj Mjjfjj;
so that the function jjF(t)jj is integrable. Then Bochner?s Theorem implies that q(s) =
R1
0 F(t)dt exists for all s2C+; with
jjq(s)jj=
ww
ww
Z 1
0
F(t)dt
ww
ww
Z 1
0
jjF(t)jjdt
Z 1
0
e tR(s)Mjjfjjdt = MjjfjjR(s) : (3.7)
Since q(s) is the Laplace transform of the bounded continuous function t7!T(t)f, Theorem
1.5.1 of [1] shows that q(s) is an analytic function on s2C+.
Now we carry out the details of the proof for only in the case n = 2. We want to show that
r(s) is the Laplace transform of an analytic function de ned on a sectorial region. Theorem
3.2 implies that if for some real x and some 2 (0; =2] the function r(s) has an analytic
extension to the region x+C( + =2) and if supfjj(s x)r(s)jj: s2x+C( 0+ =2)g<1
for all 0 < 0 < , then there exists an analytic function r(t) on t 2 C( ) such that
r(s) is the Laplace transform of r(t). We will apply the theorem with x = 0. It follows
from (3.6) that r(s) = 1s (s)q( (s)) for all s > 0, but the right hand side here is well
de ned and analytic on the set of complex s that are not on the branch cut and are such
that R( (s)) = R(c1s 1 + c2s 2) > 0, since 1 < 2, it su ces to consider R(s 2) > 0,
so if 1=2 < 2 < 1, then r(s) has a unique analytic extension to the sectorial region
C( =2 2) = fs 2 C : Re(s 2) > 0g (e.g., [23, 3.11.5] ), and note that =2 2 = =2 +
35
for some > 0. If 2 < 1=2 then r(s) has an analytic extension to the sectorial region
s2C( =2+ ) for any < =2 andR(s 2) > 0 for all such s. Now for any complex s = rei
such that s2C( =2 + 0) for any 0 < 0< , we have in view of (3.6) and (3.7) that
jjsr(s)jj = jj (s)q( (s))jj
= jc1s 1 +c2s 2jjjq(c1s 1 +c2s 2)jj
=
c1r 1ei 1 +c2r 2ei 2
c1r 1 cos( 1 ) +c2r 2 cos( 2 )
jjR(c1s 1 +c2s 2)q(c1s 1 +c2s 2)jj
c1r 1ei 1
c1r 1 cos( 1 ) +c2r 2 cos( 2 )
Mjjfjj
+
c2r 2ei 2
c1r 1 cos( 1 ) +c2r 2 cos( 2 )
Mjjfjj
1
cos( 1 ) +
1
cos( 2 )
Mjjfjj (3.8)
which is nite since j 1 j 0. Since r(s) is the Laplace transform of t 7! S(t)f,
it follows from the convolution property of the Laplace transform (e.g. property 1.6.4 [1])
that the function (3.9) has Laplace transform s i 1r(s) for all s > 0. Since r(s) has an
36
analytic extension to the sectorial region s2C( =2 + ), so does s i 1r(s). For any x> 0,
if s = x + rei for some r > 0 and j j< =2 + 0 for any 0 < 0 < then in view of (3.8)
we have
jj(s x)s i 1r(s)jj = jj(s x)s i 2sr(s)jj
rjjs i 2jj
1
cos( 1 ) +
1
cos( 2 )
Mjjfjj
wherejjsjjis bounded away from zero,jjsjj r+x and i 2 < 1, so thatjj(s x)s i 1r(s)jj
is bounded on the region x+ C( 0 + =2) for all 0 < 0< . Then it follows as before that
the function (3.9) has an analytic extension to the sectorial region t2C( ).
Since fT(t) : t 0g is a strongly continuous semigroup with generator L, Theorem 2.6
implies that Rt0 T(s)fds is in the domain of the operator L and
T(t)f = L
Z t
0
T(s)fds+f:
Since the Laplace transform q(s) of t7!T(t)f exists, Corollary 1.6.5 of [1] show that the
Laplace transform of t7!Rt0 T(s)fds exists and equals s 1q(s). Corollary 1.2.5 [42] shows
that L is closed. Fix s and let g = q(s) = R10 e stT(t)fdt and let gn be a nite Riemann
sum approximating this integral, so that gn!g in X. Let hn = s 1gn and h = s 1g. Then
gn;g are in the domain of L;gn ! g and hn ! h. Since hn is a nite sum we also have
L(hn) = s 1L(gn) !s 1L(g). Since L is closed, this implies that h is in the domain of L
and that L(h) = s 1L(g). In other words, the Laplace transform of t7!LRt0 T(s)fds exists
and equals s 1Lq(s). Then we have by taking the Laplace transform of each term
Z 1
0
e slT(l)fdt = s 1L
Z 1
0
e slT(l)fdl +s 1f
37
for all s> 0. Multiply through by s to obtain
s
Z 1
0
e slT(l)fdl = L
Z 1
0
e slT(l)fdl +f
and substitute c1s 1 +c2s 2 for s to get
(c1s 1 +c2s 2)R10 e (c1s 1+c2s 2)lT(l)fdl = LR10 e (c1s 1+c2s 2)lT(l)fdl +f
for all s> 0. Now use (3.5) twice to get
s
Z 1
0
e slS(l)fdl = L
s
c1s 1 +c2s 2
Z 1
0
e slS(l)fdl
+f
and multiplying both sides by c1s 1 2 +c2s 2 2 we get
(c1s 1 1 +c2s 2 1)
Z 1
0
e slS(l)fdl = Ls 1
Z 1
0
e stS(l)fdl +c1s 1 2f +c2s 2 2f: (3.11)
where we have again used the fact that L is closed. The term on the left hand side of (3.11)
is c1s 1 1r(s) + c2s 2 1r(s) which was already shown to be the Laplace transform of the
function c1Rt0 (t u) 1 (1 1) S(u)fdu+c2Rt0 (t u) 2 (1 2) S(u)fdu, which is analytic in a sectorial region.
Equation (3.10) also shows that s i 2 is the Laplace transform of t7! t1 i (2 ). Now take the
term c1s 1 2f +c2s 2 2f to the other side and invert the Laplace transforms. Using the fact
that fS(t) : t 0g is uniformly bounded, we can apply the Phragmen-Mikusinski Inversion
formula for the Laplace transform (see [2, Corollary 1.4]) to obtain
c1
Rt
0
(t u) 1
(1 1) S(u)fdu
t1 1
(2 1)f
+c2
Rt
0
(t u) 2
(1 2) S(u)fdu
t1 2
(2 2)f
= limn!1L
NnX
j=1
n;je
cnjl
cnj
Z 1
0
e cnjlS(l)fdl
38
where the constants Nn; n;j, and cnj are given by the inversion formula and the limit is
uniform on compact sets. Using again the fact that L is closed we get
c1
Z t
0
(t u) 1
(1 1)S(u)fdu
t1 1
(2 1)f
+c2
Z t
0
(t u) 2
(1 2)S(u)fdu
t1 2
(2 2)f
= L
Z t
0
S(l)fdl (3.12)
and since the function (3.9) is analytic in a sectorial region, the left hand side of (3.12) is
di erentiable for t> 0 Corollary 1.6.6 of [1] shows that
d
dt
Z t
0
(t u) i
(1 i)S(u)fdu (3.13)
has Laplace transform s ir(s) and hence (3.13) equals d iS(t)fdt i . Now take the derivative with
respect to t on both sides of (3.12) to obtain
c1
d
1
dt 1S(t)f
t 1
(1 1)f
+c2
d
2
dt 2S(t)f
t 2
(1 2)f
= LS(t)f
for all t > 0, where we use the fact that L is closed to justify taking the derivative inside.
Using the relation (2.23) between the Rieman-Liuoville and Caputo fractional derivatives we
proved the theorem
The next theorem provides an extension with subordinatorD (t) as the weighted average
of an arbitrary number of independent stable subordinators. Let E (t) be the inverse of the
subordinator D (t) with Laplace exponent (s) = R10 s d ( ) where supp (0;1).
39
Theorem 3.4. Let (X;jj jj) be a Banach space and be a positive nite measure with
supp (0;1). Then the family fS(t) : t 0g of linear operators from X into X given
by S(t)f =
Z 1
0
(T(l)f)gE (t)(l)dl, is uniformly bounded and strongly analytic in a sectorial
region. Furthermore, fS(t) : t 0g is strongly continuous and h(x;t) = S(t)f(x) is a
solution of
D( )1 h(x;t) =
Z 1
0
@ th(x;t) (d ) = Lh(x;t); h(x;0) = f(x): (3.14)
Proof. Since supp (0;1), the density gE (t)(l);l 0, exists and since jjT(l)fjj Mjjfjj,
then S(t)f exists and jjS(t)fjj Mjjfjj. Also, S(t)f is strongly continuous as in Theorem
3.3.
Let q(s) = R10 e stT(t)fdt and r(s) = R10 e stS(t)fdt for any s > 0, then by (3.5) we
have
(s)q( (s)) = sr(s) where (s) =
Z 1
0
s (d ) (3.15)
for anys> 0. Now we want to show that this relation holds for certain complex numberss. In
Theorem 3.3, we have shown that q(s) is an analytic function on s2C+ andjjq(s)jj MjjfjjR(s) .
Now we want to show that r(s) is the Laplace transform of an analytic function de ned
on a sectorial region. It follows from equation (3.15) that
r(s) =
Z 1
0
s 1 (d )
q
Z 1
0
s (d )
for all s> 0, but the right hand side here is well de ned and analytic on the set of complex s
such thatR
R1
0 s
(d )
> 0. Let 1 = supfsupp gand x > 0 small such that =2 1 >
=2. So if 1=2 < 1 < 1, then r(s) has a unique analytic extension to the sectorial region
C( =2 1 ) fs2C :R(R10 s (d )) > 0g and note that =2 1 = =2 + for some > 0. If
0 < 1 < 1=2 then r(s) has an analytic extension to the sectorial region s2C( =2 + ) for
any < =2, andR(R10 s (d )) > 0 for all such s. Now for any complex s = rei such that
40
s2C( =2 + 0) for any 0 < 0 < we have that
jjsr(s)jj =
Z 1
0
s (d )
q
Z 1
0
s (d )
Z 1
0
s (d )
R
Z 1
0
s (d
Mjjfjj
=
R1
0 r
cos( ) (d ) +iR1
0 r
sin( ) (d )
R1
0 r
cos( ) (d )
Mjjfjj
0
BB
@1 +
Z 1
0
r sin( ) (d )
Z 1
0
r cos( ) (d )
1
CC
AMjjfjj
0
BB
@1 +
Z 1
0
r (d )
cos( =2 )
Z 1
0
r (d )
1
CC
AMjjfjj
=
1 + 1cos( =2 )
Mjjfjj<1:
Hence Theorem 2.6.1 of [1] implies that there exists an analytic function r(t) on t2C( )
with Laplace transform r(s). Using the uniqueness of the Laplace transform it follows that
t7!S(t)f has an analytic extension r(t) to the sectorial region t2C( ).
As in Theorem 3.3 for any 2supp the function
t7!
Z t
0
(t u)
(1 )S(u)fdu (3.16)
has analytic extension to the sectorial region t2C( ).
Next we wish to apply Theorem 2.6.1 of [1] again to show that for any 0 < < 1 the
function
t7!
Z t
0
Z 1
0
(t u)
(1 ) (d )
S(u)fdu (3.17)
41
has analytic extension to the sectorial region t2C( ).
Since
Z 1
0
Z 1
0
t
(1 ) (d )
e stdt =
Z 1
0
s 1 (d )
for any 0 < < 1 and any s > 0 and r(s) is the Laplace transform of t 7! S(t)f it
follows from convolution property of the Laplace transform that the function (3.17) has
Laplace transform s 1 (s)r(s) for all s > 0. Since r(s) has an analytic extension to the
sectorial region s2C( =2 + ), so does s 1 (s)r(s). For any x> 0, if s = x+rei for some
r> 0 and j j< =2 + 0 for any 0 < 0 < then we have
ww
ww(s x)
Z 1
0
s 1 (d )
r(s)
ww
ww =
ww
ww(s x)
Z 1
0
s 2s:r(s) (d )
ww
ww
r
ww
ww
Z 1
0
s 2 (d )
ww
ww(1 + 1
cos( =2 ))Mjjfjj
r
Z 1
0
jjs 2jj (d )
1 + 1cos( =2 )
Mjjfjj
r
Z 1
0
x 2 (d )
1 + 1cos( =2 )
Mjjfjj
= rx2
Z 1
0
x (d )
1 + 1cos( =2 )
Mjjfjj:
Since positive nite measure and x> 0, so that jj(s x)s 1 (s)r(s)jj is bounded on the
region x+C( 0 + =2) for all 0 < 0 < . Then it follows as before that the function (3.17)
has an analytic extension to the sectorial region C( ).
SincefT(t) : t 0gis a strongly continuous semigroup with generator L, Theorem 1.2.4
(b) in [42] implies that Rt0 T(l)fdl is in the domain of the operator L and
T(t)f = L
Z t
0
T(l)fdl +f:
42
Then by taking Laplace transform of both sides we have
Z 1
0
e stT(t)fdt = s 1L
Z 1
0
e stT(t)fdt+s 1f
for all s> 0. Multiply both sides by s to obtain
s
Z 1
0
e stT(t)fdt = L
Z 1
0
e stT(t)fdt+f
and substitute (s) = R10 s (d ) for s to obtain
(s)
Z 1
0
e (s)tT(t)fdt = L
Z 1
0
e (s)tT(t)fdt+f
for all s> 0. Now use (3.15) twice to get
s
Z 1
0
e stS(t)fdt = L s (s)
Z 1
0
e stS(t)fdt+f
and multiplying through by s 2 (s) to get
s 1 (s)
Z 1
0
e stS(t)fdt = Ls 1
Z 1
0
e stS(t)fdt+ (s)s 2f
since L is closed. Invert the Laplace transform to get
Z t
0
Z 1
0
(t u)
(1 ) (d )
S(u)fdu
Z 1
0
t1
(2 )f (d )
= limn!1L
NnX
j=1
n;je
cnjt
cnj
Z 1
0
e CnjtS(t)fdt
(3.18)
where the constant Nn; n;j, and cn are given by the inversion formula.
43
Next using Fubini?s theorem we show thatR10 Rt0 (t u) (1 ) S(u)fdu (d ) have same Laplace
transform s 1 (s)r(s): This is true because
Z 1
0
Z 1
0
ww
wwe st
Z t
0
(t u)
(1 )S(u)f
ww
wwdudt (d )
Mjjfjj
Z 1
0
Z 1
0
e st
Z t
0
(t u)
(1 )dudt (d )
= Mjjfjj
Z 1
0
Z 1
0
e stt1
(2 )dt (d )
Mjjfjj
Z 1
0
s 2 (d ) <1:
(3.19)
Since is positive nite measure and S(t)f is uniformly bounded then using Fubini?s theorem
and the uniqueness of the Laplace transform for functions in L1loc(Rd) (Theorem 1.7.3 in [1])
we have
Z 1
0
Z t
0
(t u)
(1 )S(u)fdu
t1
(2 )f
(d )
= limn!1L
NnX
j=1
n;je
cnjt
cnj
Z 1
0
e CnjtS(t)fdt:
(3.20)
Using again the fact that L is closed we get
Z 1
0
Z t
0
(t u)
(1 )S(u)fdu
t1
(2 )f
(d ) = L
Z t
0
S(u)fdu
and now take the derivative with respect to t on both sides to obtain
Z 1
0
d
dt S(t)f
t
(1 )f
(d ) = LS(t)f
for all t> 0, where we use the fact that L is closed to justify taking the derivative inside.
Corollary 3.5. Let 0 < 2. Let ( ) =2 be fractional Laplacian on L1(Rd) correspond-
ing to the semigroup T(t) on L1(Rd). Let Y(t) be the corresponding symmetric stable process
(i.e. T(t)f(x) = Ex(f(Y(t))) ). Then the family fS(t) : t 0g of linear operators from X
44
into X given by S(t)f =
Z 1
0
(T(l)f)gE (t)(l)dl = E(f(Y(E (t)))), is uniformly bounded and
strongly analytic in a sectorial region. Furthermore, fS(t) : t 0g is strongly continuous
and h(x;t) = S(t)f(x) is a solution of
Z 1
0
@ th(x;t) (d ) = ( ) =2h(x;t); h(x;0) = f(x): (3.21)
45
Chapter 4
Two-term trace estimates
In this chapter we state and proof our main result about trace of relativistic stable
processes for R smooth boundary domains. The asymptotic behavior for the trace of killed
symmetric stable processes, 2(0;2); for an open bounded set with R smooth boundary
was given in [7]. When m = 0 our result reduces to the result for stable processes as
given in [7].
Theorem 4.1. Let D Rd;d 2; be an open bounded set with R smooth boundary. Let
jDj denote the volume (d dimensional Lebesgue measure) of D and @D denote its surface
area ((d 1) dimensional Lebesgue measure) of its boundary. Suppose 2(0;2): Then
ZD(t) C1(t)emtjDj
td= +C2(t)j@Dj
C3e2mtjDjt2=
R2td= ; t> 0; (4.1)
where
C1(t) = 1(4 )d=2
Z 1
0
z d=2e (mt)1= z (1;z)dz!C1 = !d (d= )(2 )d as t!0;
C2(t) =
Z 1
0
rH(t;(x1;0; ;0);(x1;0; ;0))dx1 C4e
2mtt1=
td= ; t> 0
C4 =
Z 1
0
~rH(1;(x1;0; ;0);(x1;0; ;0))dx1
C3 = C3(d; );H = (x1; ;xd)2Rd : x1 > 0 and rH is given by (1.4)
When m = 0, 0 < 2 our result becomes for bounded domains with R smooth
boundary
ZD(t)
C5jDj
td= +
C4j@Djt1=
td=
C7jDjt2=
R2td= (4.2)
46
where c5 = !d (d= )(2 )d ;C4 as in Theorem 4.1. This was established by Ba~nuelos and Kulczycki
[7] for stable processes.
Proof of Theorem 4.1. For the case t1= >R=2 the theorem holds trivially. This is true
because for such t0s we have by Equation (1.3)
ZD(t)
Z
D
p(t;x;x)dx c1e
mtjDj
td=
c1emtjDjt2=
R2td=
By Corollary 2.15 and Lemma 2.13 we also have
C2(t)j@Dj C4e
2mtj@Djt1=
td=
2dC4e2mtjDjt1=
Rtd=
2d+1C4e2mtjDjt2=
R2td=
C1(t)emtjDj
td=
C1emtjDjt2=
R2td=
Therefore for t1= > R=2 (4.1) holds. Here and in sequel we consider the case t1= R=2.
From (1.5) and the fact that p(t;x;x) = C1(t)emttd= ; we have that
ZD(t) C1(t)e
mtjDj
td= =
Z
D
pD(t;x;x)dx
Z
D
p(t;x;x)dx
=
Z
D
rD(t;x;x)dx; (4.3)
where C1(t) is as stated in the theorem. Therefore we must estimate (4.3). We break our
domain into two pieces, DR=2 and its complement. We will rst consider the contribution of
DR=2:
Claim 1: Z
DR=2
rD(t;x;x)dx ce
2mtjDjt2=
R2td= (4.4)
for t1= R=2:
47
Proof of Claim 1: By Lemma 2.13 we have
Z
DR=2
rD(t;x;x)dx e2mt
Z
DR=2
~rD(t;x;x)dx; (4.5)
and by scaling of the stable density the right hand side of (4.5) equals
e2mt
td=
Z
DR=2
~rD=t1= (1; xt1= ; xt1= )dx: (4.6)
For x2DR=2 we have D=t1= (x=t1= ) R=(2t1= ) 1: By [7, Lemma 2.1], we get
~rD=t1=
1; xt1= ; xt1=
c d+
D=t1= (x=t
1= )
c
2D=t1= (x=t1= )
ct2=
R2 :
Using the above inequality, we get
Z
DR=2
rD(t;x;x)dx e
2mt
td=
Z
DR=2
ct2=
R2 dx
ce2mtjDjt2=
R2td= ;
which proves (4.4).
Now using proposition 2.16 we estimate the contribution from DnDR=2 to the integral
of rD(t;x;x) in (4.3).
Claim 2:
Z
DnDR=2
rD(t;x;x)dx
Z
DnDR=2
rH(x)(t;x;x)dx
ce2mtjDjt2=
R2td= (4.7)
for t1= R=2:
Proof of Claim 2: By Proposition 2.16 the left hand side of (4.7) is bounded above by
ce2mtt1=
Rtd=
Z R=2
0
j@Dqj
t1=
q
d+ =2 1
^1
dq:
48
By Corollary 2.15, (i), the last quantity is smaller than or equal to
ce2mt(emtt1= j@Dj
Rtd=
Z R=2
0
t1=
q
d+ =2 1
^1
dq:
The integral in last quantity is bounded above by ct1= : To see this observe that since
t1= R=2 the above integral is equal to
ce2mtt1= j@Dj
Rtd=
Z t1=
0
t1=
q
d+ =2 1
^1
dq +
Z R=2
t1=
t1=
q
d+ =2 1
^1
dq
= ce
2mtt1= j@Dj
Rtd=
Z t1=
0
1dq +
Z R=2
t1=
t1=
q
d+ =2 1
dq
ce
2mtt2= j@Dj
Rtd= :
Using this and Corollary (2.15), (ii), we get (4.7).
Recall that H =f(x1; ;xd)2Rd : x1 > 0g. For abbreviation let us denote
fH(t;q) = rH(t;(q;0; ;0);(q;0; ;0)); t;q> 0:
Of course we have rH(x)(t;x;x) = fH(t; H(x)): In the next step we will show that
Z
DnDR=2
rH(x)(t;x;x)dx j@Dj
Z R=2
0
fH(t;q)dq
ce2mtjDjt2=
R2td= (4.8)
We have Z
DnDR=2
rH(x)(t;x;x)dx =
Z R=2
0
j@DqjfH(t;q)dq
Hence the left hand side of (4.8) is bounded above by
Z R=2
0
jj@Dqj j@DjjfH(t;q)dq
49
By Corollary 2.15, (iii), this is smaller than
cjDj
R2
Z R=2
0
qfH(t;q)dq
cjDje
2mt
R2
Z R=2
0
q ~fH(t;q)dq
= cjDje
2mt
R2
Z R=2
0
qt d= ~fH(1;qt 1= )dq
= cjDje
2mt
R2td=
Z R=2t1=
0
qt2= ~fH(1;q)dq
cjDje
2mtt2=
R2td=
Z 1
0
q q d ^1 dq cjDje
2mtt2=
R2td=
This shows (4.8). Finally, we have
j@Dj
Z R=2
0
fH(t;q)dq j@Dj
Z 1
0
fH(t;q)dq
j@Dj
Z 1
R=2
fH(t;q)dq
cjDjR
Z 1
R=2
fH(t;q)dq by Corollary 2.15, (ii)
cjDje
2mt
Rtd=
Z 1
R=2
~fH(1;qt 1= )dq
= cjDje
2mtt1=
Rtd=
Z 1
R=2t1=
~fH(1;q)dq
Since R=2t1= 1; so for q R=2t1= 1 we have ~fH(1;q) cq d cq 2: Therefore,
Z 1
R=2t1=
~fH(1;q)dq c
Z 1
R=2t1=
dq
q2
ct1=
R :
Hence,
j@Dj
Z R=2
0
fH(t;q)dq j@Dj
Z 1
0
fH(t;q)dq
cjDje2mtt2=
R2td= (4.9)
50
Note that the constant C2(t) which appears in the formulation of Theorem (4.1) satis es
C2(t) = R10 fH(t;q)dq: Now equations (4.3), (4.4), (4.7), (4.8), (4.9) give (4.1).
51
Chapter 5
Mixed Stable Processes
In this chapter we explore the basic general properties of the sum of two independent
stable processes and give a rst term asymptotic expansion of the trace. I?m still working on
nding a better trace estimate for such processes for a domain with R smooth boundary.
Most of the notations and results of this chapter are adapted from [17].
Let X be a L evy process that is the independent sum of an -stable process Y and a
-stable process W in bounded open subset of Rd: The in nitesimal generator of the L evy
process X is =2 + =2. Let p1D(t;x;y) and G1D(x;y) denote the transition density function
and the Green function of the subprocess XD of X killed upon exiting an open set D Rd:
Let pD(t;x;y) and GD(x;y) denote the transition density function and Green function of the
subprocess YD of Y killed upon exiting D: Intuitively, one expects the following Duhamel?s
formulas (or Trotter-Kato formula) hold:
p1D(t;x;y) = pD(t;x;y) +
Z t
0
Z
D
p1D(s;x;z) =2z pD(t s;z;y)dz (5.1)
G1D(x;y) = GD(x;y) +
Z
D
G1D(x;z) =2z GD(z;y)dz (5.2)
The L evy process X runs on two di erent scales: on the small spatial scale, the component
dominates, while on the large spatial scale the component takes over. Both components
play essential roles, and so in general this process can not be regarded as a perturbation of
the stable process or of the stable process.
Let us rst recall some basic facts about the independent sum of stable processes and
state our main result. Throughout the remainder of this paper, we assume that d 1 and
0 < < < 2: The Euclidean distance between x and y will be denoted as jx yj:
52
Suppose X is a symmetric stable process and Y is a symmetric stable process on
Rd and that X and Y are independent. For any a 0, we de ne Xa by Xat := Xt + aYt:
We will call the process Xa the independent sum of the symmetric stable process X
and the symmetric stable process Y with weight a: The in nitesimal generator of Xa
is =2 + a =2: Let pa(t;x;y) denote the transition density of Xa (or equivalently the
heat kernel of =2 + a =2) with respect to the Lebesgue measure on Rd. We will use
p(t;x;y) = p0(t;x;y) to denote the transition density of X = X0: Recently it is proven in
[17] that
p1(t;x;y) t d= ^t d= ^
t
jx yjd+ +
a t
jx yjd+
on (0;1) Rd Rd (5.3)
Here and in the sequel, for a;b2R;a^b := minfa;bg and a_b := maxfa;bg; for any two
positive functions f and g; f g means that there is a positive constant c 1 so that
c 1g f cg on their common domain of de nition.
For every open subset D Rd; we denote by Xa;D the subprocess of Xa killed upon
leaving D: The in nitesimal generator of Xa;D is =2 +a =2jD, the sum of two fractional
Laplacians in D with zero exterior condition. It is known [17] that Xa;D has a H older
continuous transition density paD(t;x;y) with respect to the Lebesgue measure.
Unlike the case of the symmetric stable process X := X0;Xa does not have the
stable scaling for a> 0: Instead, the following approximate scaling property is true and will
be used several times in the rest of this paper: IffXa;D;t 0gis the subprocess of Xa killed
upon t leaving D, thenf 1Xa;D at;t 0gis the subprocess offXa ( )= t ;t 0gkilled upon
leaving 1D :=f 1y : y2Dg: Consequently, for any > 0, we have
pa ( )= 1D (t;x;y) = dpaD( t; x; y) for t> 0 and x;y2 1D (5.4)
53
In particular, letting a = 1; = a =( ) and D = Rd; we get
pa(t;x;y) = ad =( )p1(a =( )t;a =( )x;a =( )y) for t> 0 and x;y2Rd: (5.5)
So we deduce from (5.3) that there exists a constants C > 1 depending only on d; and
such that for every a> 0 and (t;x;y)2(0;1) Rd Rd
C 1fa(t;x;y) pa(t;x;y) Cfa(t;x;y); (5.6)
where
fa(t;x;y) = t d= ^(a t) d= ^
t
jx yjd+ +
a t
jx yjd+
:
For a domain D Rd, we de ne the rst exit time from D by aD = infft 0 : Xat =2Dg.
We set
raD(t;x;y) = Ex[pa(t D;Xa a
D
;y); aD 0.
We are interested in the behavior of the trace of this semigroup de ned by
ZaD(t) =
Z
D
paD(t;x;x)dx: (5.9)
Lemma 5.1. Let D Rd be an open set. For any x;y2D we have
raD(t;x;y) c
t d= ^(a t) d= ^
t
( aD(x))d+ +
a t
( aD(x))d+
!!
54
Proof.
raD(t;x;y) = Ey [ D 0 [34, Theorem 3.9]. Currently,
I?m working on nding strong analytic solution for Tempered fractional Cauchy problems
and then extending all this time fractional result to more general time operator.
Laplace symbol: (s) inverse subordinator time operator
R1
0 (1 e
sy) (dy) E (t) (@t) (0) (t;1)
s E(t) @ t , Caputo
R1
0 s
(1 ) (d ) E (t) R1
0 @
t (1 ) (d )
(s+ ) E (t) @ ; t in (6.1)
@ ; t g(t) = (@t)g(t) g(0) (t;1)
= e t 1 (1 )dt
Z t
0
e sg(s)ds
(t s)
g(t)
g(0) (1 )
Z 1
t
e r r 1dr:
(6.1)
Subdi usion: 0 < < 1, Ex(W(E(t)))2 = E(E(t)) t .
57
Ultraslow di usion: For special 2RV0( 1) for some > 0: Ex(W(E (t)))2 =
E(E (t)) (logt) [34, Theorem 3.9].
Intermediate between subdi usion and di usion: Tempered fractional di usion
Ex(W(E (t)))2
8
><
>:
t = (1 + ); t<< 1
t= ; t>> 1:
W(E (t)) occupies an intermediate place between subdi usion and di usion (Stanislavsky
et al., 2008)
I am also working on estimating the trace of general processes like sum of two inde-
pendent stable process over a bounded domains with R smooth boundary and Lipschitz
domains. So far I am able nd the rst term asymptotic for the trace of such processes over
a bounded domains with R smooth boundary.
58
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