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\begin{document}
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{\noindent\small Fourth Mississippi State Conference on
Differential Equations and \newline Computational Simulations,
{\em Electronic Journal of Differential Equations}, \newline
Conference 03, 1999, pp. 39--50.\newline
URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login:
ftp)}
\thanks{\copyright 2000 Southwest Texas State University and
University of North Texas.}
\vspace{1cm}
\title[Kac walks and asymptotic analyticity]
{ The mathematics of suspensions: Kac walks and asymptotic analyticity }
\author[E.C. Eckstein, J.A. Goldstein, M. Leggas]
{ Eugene C. Eckstein, Jerome A. Goldstein, \& Mark Leggas }
\address{Eugene C. Eckstein \hfill\break\indent
School of Biomedical Engineering\\
University of Tennessee \hfill\break\indent
899 Madison Avenue, Suite 801\\
Memphis, TN 38163, USA}
\email{eeckstein@utmem.edu}
\address{Jerome A. Goldstein \hfill\break\indent
Department of Mathematical Sciences\\
University of Memphis \hfill\break\indent
Memphis, TN 38152, USA}
\email{goldstej@msci.memphis.edu}
\address{Mark Leggas \hfill\break\indent
UM/UT Joint Graduate Program in Biomedical Engineering\\
SBME, UT Memphis\hfill\break\indent
899 Madison Avenue, Suite 801\\
Memphis, TN 38163, USA}
\email{mleggas@bme.utmem.edu}
\thanks{Published July 10, 2000.}
\subjclass{76T20, 76A99, 76D99, 76M22, 76M35, 76R50, 76Z99}
\keywords{Suspensions; Telegraph equation; Kac random walk;
Semigroups of operators; \hfill\break\indent
asymptotic analyticity; Taylor dispersion; F\"urth-Ornstein-Taylor formula}
\begin{abstract}
Of concern are suspension flows. These combine directed and random motions
and are typically modelled by parabolic partial differential equations.
Sometimes they can be better modelled (in terms of fitting the data generated
by certain blood flow experiments) by hyperbolic equations, such as the
telegraph equation, which have parabolic (or analytic) asymptotics.
\end{abstract}
\maketitle
\newtheorem{thm}{Theorem}[section]
\numberwithin{equation}{section}
\section{Motivation}
Engineering models for practical suspension flows depict the
erratic motions
of the individual particles with the mathematics of Brownian motion
and
diffusion. The particles in such flows include blood cells that
move through
the body's vessels or the passages of an artificial kidney or lung,
the
pulpy material in fruit juice that is being processed or sipped
through a
straw, and bits of coal in a coal-oil slurry being fed to a burner
in an
electrical power generation plant. These particles are far from
small,
usually are not present as a dilute species, and often exhibit
large-scale
erratic motions due to interaction with one another in the shearing
flow.
Despite these differences from true Brownian motion, the
engineering models
are functional, as Acrivos \cite{A} noted in his recent review. But
he also noted
that aspects remain to be described well. In particular, the
question of
scaling for the strength of the erratic motion is open - there is
no direct
equivalent of the temperature / thermal energy that is linked to
the motion
of Brownian particles.
Such models depend on experimental measurements of effective
properties to
represent the diffusion coefficient and the viscosity. The methods
and
representations used to compile and reduce data for such
measurements
originate in basic mathematics and physics. Typically, experimental
methods
quantify the erratic motion by timing displacements of known
extents i.e.,
by observing the jumps in the walk or related measures. Two
particular
random walks are of interest. One is the simple drunkard's walk
that well is
linked to the diffusion equation; the other is the persistent
random walk,
which is linked to the telegraphers' equation. The first expects
that the
particle is essentially stationary at a position and will go in
either
direction; the next position depends only on the current position.
The
latter walk assumes that the particle is moving with a velocity and
the
probabilistic aspect is whether the sign of the velocity changes;
it depends
on the current and previous position. Other aspects of the walks
and the
models are described in two other papers related to our work
\cite{EGL,LE}.
One paper explores the physical rationale for describing the random
motion in
particular ways, and the other describes the initial experimental
results
and an alternative means to obtain effective coefficients. In this
paper
linkages between the two random walk models and their related
differential
equations and semigroups are explored. To motivate the treatments
below, a
few aspects of suspension flows are reviewed below; more material
appears in
the other papers and in a thesis by Leggas \cite{L}.
The small particles are coupled to the general flow by viscous
tractions
acting on their surfaces. The Reynolds number that provides a
fractional
measure of the inertia and viscous traction on the particle is
typically of
the order 0.001 or less, which reflects small sizes and the small
relative
velocities between particles and the local average motion of the
suspensions. The axial speed then provides a reasonable, but not
perfect,
indication of lateral position. Reasons for the imperfections
include that
particles can tumble temporarily in groups as faster and slower
bodies pass
one another. Also, many small inertial events either individually
or
cumulatively act and lead to differences between the motion of
points in an
ideal continuum fluid and the actual motion of suspended, bodies
among many
molecules that are much closer to the point approximation.
The experimental method involves tracking identifiable bodies among
equivalent unmarked bodies. An identifiable body is found at a time
when it
has a velocity similar to that of the reference frame in which the
observations are being made. In the initial studies, this reference
frame
speed is approximately the average speed of the flow and the
suspension flow
occurs in a rectangular channel. The time increments needed for the
body to
move selected net distances in either axial direction are measured.
These
first passage times are used to estimate the stochastic process
that is
associated with the suspension flow. As noted above, the change of
apparent
axial speed occurs because the particle changes its lateral
location in a
related fashion.
The form of the experiments is similar to a model of Taylor
dispersion
devised by Van den Broeck [18]. He assumed that the identifiable body
would jump
among parallel tracks of known velocities and that the body would
jump at
times that followed a Poisson distribution. The experiments collect
observations of particle motions while moving along the flow axis
as if one
of the Van den Broeck tracks were being followed. The measured time
increments are a means of approximating the first passages to other
tracks
by a continuous time random walk. The particle waits on the initial
track
until it jumps to some other track. The jump occurs instantly with
a change
of velocity that is sufficient to make it match the speed of the
new track
and provide an overall continuity of flow. A kind of dilute
condition is
implicit in this analogy because a landing space always exists in
the
receiving track.
One basic question is whether the velocity-based walk is a more
fundamental
representation of the events than the more commonly used drunkard's
walk.
The drunkard's walk implies a kind of local equilibrium; the
history of the
past motion is not applicable. Use of the velocity-based requires
extra
initial knowledge, which is reflected in its initial conditions.
Improved
engineering models of the complex events in suspension flows may
require
that such detailed considerations be a part of the measurements of
effective
properties that are entered into the model. This paper explores the
connections among the two random walks and especially focuses on
the
incorporation of initial and boundary conditions in them and their
duals.
\section{The Kac Walk}
We begin with the random walk model leading to the telegraph
equation.
The idea of this model originated with G. I. Taylor \cite{T}. It
was
developed by S. Goldstein \cite{Go}. The connection with the
Poisson process
was noted by M. Kac \cite{K}. Kac was the main pioneer in using
stochastic
processes to help in understanding hyperbolic partial differential
equations, so we like to refer to this model as a ``Kac walk".
A particle starts at the origin $0 \in {\mathbb R}$ on the real
line. After each
time interval of length $\Delta t > 0$ it will move, either to the
left
or right, a distance $\Delta x > 0$. The speed is $c=\Delta x/
\Delta
t$. The first step is determined by the flip of a fair coin; one
moves
either to the left or to the right with probability $1/2$. On each
subsequent step, the direction of the move is determined by the
flip of
a weighted coin. Let $a$ be a positive constant so that $a \Delta
t < 1$.
The probability of reversing direction is $a \Delta t;$ the
probability
of continuing in the same direction is $1-a\Delta t$. Let $S_n$ be
the
position of the random walk after $n$ steps, or, at time $n\Delta
t$.
We want to compute $E(f(S_n))$ for arbitrary functions $f$.
We shall reproduce Kac's calculation. We do this to persuade the
reader
that the surprising appearance of the telegraph equation and the
Poisson
process in the description of the solutions is really elementary.
We now define the coin flips probabilistically. Let $\xi$ be $+1$
or $-
1$, with probability $1-a \Delta t$ or $a\Delta t$, respectively.
Let
$\xi_1, \xi_2, \ldots$ be independent identically distributed
random
variables, distributed as $\xi$. Suppose for definiteness that the
first step is to the {\it right}. Then
$$S_n = S_n^+ = c\Delta t(1+\xi_1 + \xi_1 \xi_2 + \ldots + \xi_1
\ldots
\xi_{n-1}).$$
If the first step were to the left, we would have
$$S_n = S_n^- = -S_n^+.$$
Let us now write $S_n$ in place of $S_n^+$ and consider
$$u_n^{\pm} (x) = E (f(x \pm S_n))$$
for $x \in {\mathbb R}$. Conditioning on $\xi_1$, we have
\begin{eqnarray}
u_n^+ (x) &=& E\{f(x+c\Delta t + c \Delta t \xi_1 (1 + \xi_2 +
\ldots + \xi_2 \ldots \xi_{n-1}))\}\nonumber\\
&=& E\{f (x + \ldots)| \xi_1 = -1\} P\{\xi_1 = -1\}\nonumber\\
&&+ E\{f(x + \ldots ) |\xi_1=1\} P\{\xi_1 = 1\}\nonumber\\
&=& a \Delta t\;\; u_{n-1}^- (x + c\Delta t) + (1-a\Delta t)
u_{n-1}^+ (x-
c\Delta t).
\end{eqnarray}
Similarly
\begin{equation}
u_n^- (x) = a\Delta t \;\; u_{n-1}^+ (x-c\Delta t) + (1-a\Delta
t)u_{n-1}^-
(x-c\Delta t).
\end{equation}
Equation (2.1) leads to
\begin{eqnarray*}
\frac{u_n^+ (x) - u_{n-1}^+ (x)}{\Delta t} &=& \frac{u_{n-1}^+ (x +
c\Delta t) - u_{n-1}^+ (x)}{\Delta t}\\
&&- a u_{n-1}^+ (x + c\Delta t) + a u_{n-1}^- (x + c \Delta t).
\end{eqnarray*}
Taking the limit as $\Delta t, \Delta x \to 0$ with $a > 0$ and
$c=\Delta x/ \Delta t$ fixed and with $n\Delta t \to t$, we obtain
functions $u^{\pm} (x,t)$ satisfying
$$\frac{\partial u^+}{\partial t} = c \frac{\partial u^+}{\partial
x} +
au^+ + au^-.$$
Similarly (2.2) leads to
$$\frac{\partial u^-}{\partial t}=-c\frac{\partial u^-}{\partial x}
+
au^+ - au^-.$$
Letting $u = \frac{u^+ + u^-}{2}, \;\; v = \frac{u^+ - u^-}{2}$, we
can
add and subtract the above differential equations to obtain
\begin{equation}
\frac{\partial u}{\partial t}= c\frac{\partial v}{\partial x},
\quad
\frac{\partial v}{\partial t}=-c \frac{\partial u}{\partial x} -
2av.
\end{equation}
If we take $\partial/\partial t$ of the first equation in (2.3),
$\partial/ \partial x$ of the second, and eliminate
$\partial^2 v/ \partial t\partial x$ we finally obtain the
telegraph
equation
$$\frac{1}{c} \frac{\partial^2 u}{\partial t^2} =c \frac{\partial^2
u}{\partial x^2}
-\frac{2a}{c}\frac{\partial u}{\partial t},$$
or
\begin{equation}
\frac{\partial^2 u}{\partial t^2}+2a \frac{\partial u}{\partial
t}=
c^2\frac{\partial^2 u}{\partial x^2}.
\end{equation}
The initial conditions are
\begin{equation}
u(x,0)=f(x), \quad \frac{\partial u}{\partial t}(x,0)=0.
\end{equation}
The first is clear; the second is perhaps not, but it is a result
of the
assumption that the first step determined the direction to move by
flipping a fair coin.
When $a=0$, each $\xi_i$ is 1 and so
$$u^{\pm} (x) = f(x \pm n c\Delta t),$$
whence
$$u(x,t) = \frac{f(x+ct) + f (x-ct)}{2}.$$
This is the unique solution of the wave equation
\begin{equation}
\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2
u}{\partial
x^2}
\end{equation}
(which is (2.4) with $a=0$) with initial conditions (2.5). This
helps
explain the second initial condition in (2.5).
If one lets $\Delta t, \Delta x \to 0$ in such a way that $a \to
\infty,
c \to \infty, n\Delta t \to t$ and $\frac{c^2}{a} \to D > 0$, then
the
heat equation
$$\frac{\partial u}{\partial t}=\frac{D}{2} \frac{\partial^2
u}{\partial
x^2}$$
would emerge, as it did in the classical (independent) work of
Einstein,
Bachelier, and Smoluchowski. This gives the first formal
connection between
the telegraph equation and the heat equation; more on this later.
Kac's next calculation is especially interesting. Redo the random
walk
analysis in continuous time. Thus we study one dimensional
continuous
motion with constant speed $c$, which changes direction in a time
interval of length $dt$ with probability $adt$ (and maintains the
same
direction in this time interval with probability $1-adt)$. This
leads to a Poisson process $\{N(t) =N_a(t) : t \geq 0\}$ with
intensity
$a$. That is, $N(t)$ takes values in $\mathbb{N}_0 =\{0,1,2,
\ldots\},
N(0)=0, P\{N(t) = k\}= e^{-at} (at)^k /k!$ for $k \in
\mathbb{N}_0$, and
for $0 \leq t_1 < t_2 < \ldots$,
$$N(t_2)- N(t_1), N(t_3) - N(t_2), N(t_4) - N(t_3), \ldots$$
are independent. Then the number of sign changes in the velocity
process up to time $t$ is $(-1)^{N(t)}$. Hence the position (or
displacement) is
$$S(t) = \int_0^t v(\tau) d\tau = c\int_0^t (-1)^{N(\tau)} d\tau,$$
which is the continuous analogue of $S_n$. This led Kac to derive
(and
prove) that the solution of the telegraph equation problem (2.4),
(2.5) is
\begin{eqnarray*}
u(x) &=& \frac{1}{2} \{E[f(x+c \int_0^t (-1)^{N(\tau)} d\tau)] +
E[f(x-c
\int_0^t (-1)^{N(\tau)} d\tau)]\}\\
&=& E[v(x, \int_0^t (-1)^{N(\tau)} d\tau)]
\end{eqnarray*}
where $v$ is the unique solution of the wave equation (2.6), (2.5).
Thus to
get the solution $u(x,t)$ of the telegraph equation, take the
corresponding solution $v(x,\sigma)$ of the wave equation, but
evalutate
it at a random time $\sigma = \int_0^t (-1)^{N(\tau)} d\tau$
determined by
a Poisson process (with intensity $a$) and then average.
For more on random walk models, see \cite{V,We}.
We view the Kac calculations as a precise version of some heuristic
reasoning
based on some old (1930) but very interesting discussion of
Uhlenbeck and
Ornstein \cite{UO}. They were interested in the kinetic theory of
gases and
Brownian motion. Consider the formula \cite[p.826]{UO}
\begin{equation}%2.7
\beta s^2 = \alpha_1 t-1-e^{-\alpha_2 t}
\end{equation}
where $\beta =f_1^2 /2mkT, \alpha_j = f_j/m_j$, represents distance
and
$t$ time, $f$ is for friction and $m$ for mass.
This equation is due to F\"urth \cite{F} and Ornstein \cite{O}.
For large $t$, this is approximately $\beta s^2 =\alpha t$, which
is
Einstein's (1905) equation
\begin{equation}%2.8
\beta s^2 = 2Dk = \frac{2kT}{f_1} t.
\end{equation}
This corresponds to a free particle of mass $m_1$, where the
friction coefficient $f_1$ only depends on the surrounding medium.
(Here $T$ is the absolute temperature and $k$ is Boltzmann's
constant.) Suppose we consider a tagged particle in a suspension,
and look at its associated parameter $\alpha_2=f_2/m_2$. Thus $f_2
= f_1$ (since the surrounding medium is unchanged) but $m_2 > m_1$
since the particle carries some of the surrounding fluid with it
and thus has an effective mass exceeding its free mass. Thus we
take $\alpha_1 > \alpha_2$, while the earlier authors took
$\alpha_1 = \alpha_2$. For small $t$, we replace $e^{-\alpha_2 t}$
by it second order Taylor expansion $1-\alpha_2 t + \alpha_2^2
t^2/2$, and (2.7) becomes
\begin{equation}%2.9
\beta s^2 = (\alpha_1-\alpha_2)t + \frac{\alpha_2^2}{2}t^2.
\end{equation}
Now think of $s$ [resp. $t$]
as representing $\frac{\partial}{\partial x}$ [resp.
$\frac{\partial}{\partial t}$]. Then (2.8), (2.9) are formally the
first and second order (in time) equations
\begin{equation}
\beta \frac{\partial^2 u}{\partial x^2} = \alpha \frac{\partial
u}{\partial t},
\end{equation}
\begin{equation}
\beta \frac{\partial^2 u}{\partial x^2} = (\alpha_1-\alpha_2)
\frac{\partial u}{\partial t}+\frac{\alpha_2^2}{2} \frac{\partial^2
u}{\partial t^2}.
\end{equation}
Velocities play no role in (2.10) (Brownian particles have infinite
speed), but they play a crucial role in the telegraph equation
(11). (In the Ornstein-Uhlenbeck stochastic process, particles
have finite velocity but infinite acceleration.)
\section{The Cosine Function Version}
%\setcounter{equation}{0}
The initial value problem for
\begin{equation}
v''(t)=Av(t) \;\; (t \geq 0),
\end{equation}
\begin{equation}
v(0) = f, \;\; v' (0)=0
\end{equation}
(for $v''=d^2v/dt^2, v: (0,\infty) \to X$, and $A$ a linear
operator on a
Banach space $X$) is well posed if and only if $A$ generates a
cosine
function $C=\{C(t) : t \geq 0\}$ on $X$, in which case the unique
solution of (3.1), (3.2) is given by
$$v(t) = C(t)f;$$
it is a strong solution if $f \in Dom (A)$ and a mild solution
otherwise (Cf. \cite{G1}.) If we replace (3.2) by
\begin{equation}
v (0) = f, \quad v'(0)=g,
\end{equation}
and if $A$ is injective, then the corresponding solution of (3.1)
is
\begin{equation}
v(t) = C(t) f + \frac{d}{dt} C(t) A^{-1} g = C(t) f + \int_0^t
C(s)g ds.
\end{equation}
It is not difficult to show that the unique solution to the
abstract
telegraph equation
\begin{equation}
u'' (t) + 2a u(t) = Au(t)
\end{equation}
with (3.2) is
\begin{equation}
u(t)= E\left[v\left(\int_0^t (-1)^{N(\tau)} d\tau\right)\right],
\end{equation}
where $v(t) = C(t)f$ is the solution of the corresponding abstract
wave
equation. As before $\{N(t) : t \geq 0\}$ is a Poission process of
intensity
$a >0$.
Continue to suppose that $A$ is injective. The unique solution of
(3.5), (3.3)
is given by
\begin{eqnarray}
u(t) &=& E\{C\left(\int_0^t (-1)^{N(\tau)} d\tau\right) (f-2a
A^{-1} g)\nonumber\\
&&+ \int_0^t C \left(\int_0^s (-1)^{N(\tau)} d\tau \right) g ds+ 2a
A^{-1} g\}.
\end{eqnarray}
To see this, let $v$ satisfy
$$v''+2av'=Av, \;\;v(0) = h, \;\; v'(0)=g.$$
Then $w=v'$ satisfies
$$w''+2aw' = Aw, \;\; w(0) = g, \;\; w'(0) = Ah-2ag.$$
If $h=2a A^{-1}g$, then
$$w(t) = \left[C\left( \int_0^t
(-1)^{N(\tau)}d\tau\right)g\right].$$
Then, $z=u - v$ satisfies $z"+ 2az'=Az,$
$$z(0)=f-2aA^{-1}g, \;\; z'(0)=0;$$
and so $u=z+v$ and (3.7) follows. \hfill$\diamondsuit$\smallskip
Note that this reduces to our previous result (3.4) for the wave
equation
where $a=0$.
Kac \cite{K} gave a rigorous proof of the representation formula
(3.6) for
the solution of (3.5) with $u(0)=f$, $u'(0) = 0$ and $A=c^2
d^2/dx^2$ on
$L^2({\mathbb R})$ using Laplace transforms. The proof uses power
series
constructions. The same proof works when $A$ has enough analytic
vectors. The power series for $cos (ta)$ is $\sum_{n=0}^\infty (-
1)^n (ta)^{2n}/(2n)!$, and this function satisfies $u''=-a^2u$ (as
a function of $t$). Thus we expect $C(t)f$, the solution of
$u''=Au,\;\; u(0)=f,\;\; u'(0)=0$ to be given by
$$C(t)f = \sum_{n=0}^\infty (-1)^n \frac{t^{2n} A^nf }{(2n)!},$$
at least for all $f$ for which the above series converges nicely.
We
define $f$ to be an {\it entire vector} for $A$ if the series
$\sum_{n=0}^\infty t^n A^n f/ n!$ has an infinite radius of
convergence (in the complex $t$-plane). Let $E(A)$ be the set of
such
vectors. Let
$\widetilde{E} (A)=\{f \in X: \sum_{n=0}^\infty t^{2n}A^nf/(2n)!$
has an infinite radius of convergence\}. Here we clearly are
assuming that $X$ is a complex Banach space. Clearly $E(A)
\subset
\widetilde{E} (A)$. If $\widetilde{E} (A)$ is dense in $X$, then
Kac's arguments \cite{K} establish the validity of (3.6).
Now let $A$ be a normal operator on a Hilbert space $\mathcal{H}$.
By the spectral theorem,
$$A=UM_\alpha U^{-1}$$
where $U : L^2 (\Omega, \mu) \to \mathcal{H}$ is unitary and
$(\Omega, \mu)$
represents some measure space; and $\alpha : \Omega \to \mathbb{C}$
is
measurable, and $M_\alpha u (x) = \alpha (x)u(x)$ for $x \in
\Omega,$
$Dom (M_\alpha)=\{u:u, \alpha u \in L^2 (\Omega, \mu)\}$. The case
when
$A$ generates a cosine function corresponds to the range of
$\alpha$
being (essentially) contained in a parabolic region of the form
$$\mathop{\rm conv}\{x+iy \in \mathbb{C}: x=-c_1(y-c_2)^2 + c_3\}$$
(conv = convex hull) for any positive constants $c_1, c_2, c_3$.
Let
$B_n = \{z\in \mathbb{C}:|z| \leq n\}, \chi_n =$ the characteristic
function of
$B_n$,
$$E_n= U(\mathop{\rm Range} (\chi_n)) U^{-1} \subset X;$$
then $E=\bigcup_{n=1}^\infty E_n$ is a dense set of entire vectors
for $A$.
This completes the proof of (3.6) (and also of (3.7)) for the only
cases we
shall consider. For more general information see
\cite{GH,H1,H2,P,G2}.
\section{Remarks on Higher Dimensions}
%\setcounter{equation}{0}
In the previous section we showed how the Kac ideas involving
the Poisson process to represent solutions of the abstract
telegraph
equation work in great generality. But the random walk model is
usually
given in just one dimension. We make a few remarks here about
extensions.
Griego and Hersh \cite{GH} and Pinsky \cite{P} considered the first
order hyperbolic
system
\begin{eqnarray}
\frac{\partial u_i}{\partial t} &=& c_i \frac{\partial
u_i}{\partial x}+ \sum_{j=1}^n
q_{ij} u_j,\nonumber\\
u_i (x,0) &=& f_i(x),
\end{eqnarray}
where $u_i : {\mathbb R} \times [0, \infty) \to {\mathbb R} \;\; (i
= 1, \ldots, n),\;\; Q=(q_{ij})$
generates an $n$-state Markov chain (i.e. $P(t)=e^{tQ}$ is an $n
\times
n$ matrix of nonnegative numbers and $P(t) \mathbf{1} =
\mathbf{1}$ where $\mathbf{1}=(1,1, \ldots 1)^{tr};$ equivalently,
$q_{ij} \geq 0$ for $i \neq j$ and the row sums $\sum_{i=1}^n
q_{ij}$ are all zero), and $c_i \in {\mathbb R}.$\; Let $p_{ij}
(t)$ be the $ij$th entry of $P(t) = e^{tQ}$, and let $\xi (t)$ be
the
position at time $t$ of a particle whose velocity $v(t)$ at time
$t$
switches from $c_i$ to $c_j$ with probability $p_{ij} (t)$. Then
we
have the representation
\begin{eqnarray}
u_i(x,t) &=& E_{x,i} [f_{v(t)} (\xi (t))]\nonumber\\
&=& E\{f_{v(t)} (\xi (t)): \xi(0)=x, \xi'(0)=i\}.
\end{eqnarray}
If $n=2, \;\; Q=\binom{\,-a\quad a\,}{\;\;a\;\; -a\,}$, and
$c_2=-c_1 = -c$, then the system (4.1) reduces to the single
equation
\begin{equation}
u_{tt} + 2au_t = c^2 u_{xx}.
\end{equation}
If we replace (4.1) by
$$\frac{du_i}{dt} = A_i u_i + \sum_{i=1}^n q_{ij} u_j$$
with $A_1 = -A_2 = A$, a $(C_0)$ group generator, then (4.3)
becomes
$$u_{tt} + 2a u_t = A^2 u,$$
and the representation formula (4.2) can be shown to imply the Kac
formula
(3.6) for the solution of (3.5).
Now let $x=(x_1, \ldots, x_n)$ vary over ${\mathbb R}^n$, and let
$e_i$ be the
unit vector in the positive direction of the $i$th coordinate axis
$L_i,
\; 1 \leq i \leq n$. Perform the Kac construction on each line
$L_i.$
Then the unique solution of
$$\partial^2 u_i / \partial t^2 + 2a_i \; \partial u_i/\partial t =
c_i^2
\partial^2u_i/\partial x_i^2,$$
$$u_i(x_i, 0 )=f_i(x_i), \quad \partial u_i (x_i, 0) /\partial t =
0$$
is
$$u_i (x_i, t) = E[v_i (x, \int_0^t (-1)^{N_i(s)} ds)]$$
where $v_i$ satisfies the same problem but with $a_i$ replaced by
zero.
If $N_1 = \ldots = N_n=N$, so that there is only one Poisson
process of
intensity $a=a_1=\ldots= a_n$, then
$$U(x,t) = E[V(x,\int_0^t (-1)^{N(s)} ds)]$$
satisfies
$$U_{tt} + 2a U_t = \sum_{i=1}^n c_i^2 U_{x_i x_i},$$
$$U(x,0) = \sum_{i=1}^n f_i (x_i), \quad U_t(x,0)=0,$$
where $V$ satisfies the same problem but with $u$ replaced by zero.
Thus
a very special case of an $n$-dimensional telegraph equation is
governed
directly by the Kac random walk, but the initial function
$F(x)=\sum_{i=1}^n f_i(x_i)$
must be very special. Nevertheless, the
conclusion of Section 3 shows that this representation formula
holds
even for general $F$ in $L^2({\mathbb R}^n).$
\section{Asymptotic Analyticity}
Of concern is the problem
\begin{equation}%5.1
u_{tt} + 2au_t + A^2u=0,
\end{equation}
where $a$ is a positive constant and $A=A^* \geq 0$ is a
nonnegative
selfadjoint operator on a Hilbert space $\mathcal{H}$. By the
spectral
theorem \cite{G1}, the general solution of (5.1) is given by
$$u(t)= \sum_{j=1}^2 T_j (t) f_j$$
where
$$T_j(t)= e^{tA_j},$$
$$A_{1,2} = -a I \pm (a^2I-A^2)^{\frac{1}{2}};$$
here the subscript 1 [resp. 2] goes with the + [resp. -] sign.
Recall that
by the spectral theorem,
$$A=UM_m U^{-1}$$
where $U : \mathcal{H} \to L^2(\Omega,\mu)$ is a unitary operator
from
$\mathcal{H}$ to an $L^2$ space on some measure space, and $M_m$ is
the
operation of multiplication by the $\mu$-measurable function
$$m : \Omega \to [0, \infty).$$
Hence for any Borel function $g$ on $(0, \infty),$
$$g(A) = UM_{g(m)} U^{-1}.$$
Thus $A_{1,2}$ and $T_j(t)$ are all well defined (for $t \geq 0$).
Let
$$F(x)= -1 + (1-x)^{\frac{1}{2}}$$
for $0 < x < 1$. Then $F'(x) = -\frac{1}{2} (1-x)^{-\frac{1}{2}},
\;\; F''(x) = -\frac{1}{4} (1-x)^{-\frac{3}{2}}$, and so (since
$F(0) =
0,\;\; F'(0)= -\frac{1}{2},\;\; F''(0) = -\frac{1}{4})$
\begin{eqnarray*}
F(x) &=& -\frac{x}{2} + \bigcirc (x^2)\\
&=& -\frac{x}{2} -\frac{x^2}{8} + \bigcirc (x^3)
\end{eqnarray*}
as $x \to 0$, by Taylor's theorem. Using the spectral theorem,
$U^{-1}
T_1 (t) U$ can be approximated by the multiplication operator
\begin{eqnarray*}
e^{t(-a+(a^2-m^2)^{\frac{1}{2}})} &=& \exp
\{-ta(1-(1-\frac{m^2}{a^2})^{\frac{1}{2}})\}\\
\cong e^{-ta \frac{m^2}{2a^2}} &=& e^{-tm^2/2a},
\end{eqnarray*}
which implies that $T_{\mathbf{1}} (t)$ is approximately
$e^{-tA^2/2a}.$
Write $A=\int_0^\infty \lambda d P(\lambda)$, so that $P(B)=\int_B
dP(\lambda)$ orthogonally projects onto the maximal invariant
closed
subspace of $\mathcal{H}$ for $A$ in which $A$ has spectrum
contained in $B$,
for $B$ any Borel set in $[0,\infty).$
Let $0<\epsilon < a$. Then any solution $u$ of (5.1) is equal to
$$e^{t(-a+(a^2-A^2)^{\frac{1}{2}})} f = \bigcirc (e^{-t(a^2-
\epsilon^2)^{\frac{1}{2}}})$$
(as $t \to \infty)$ for some $f$ in $P([0, \epsilon])$. In that
sense,
the solution is given by an analytic semigroup plus a small error
in the
asymptotic limit, $t \to \infty$. By the analysis given above,
this
analytic semigroup solution is approximately
$$e^{-tA^2/2a} f,$$
which is a solution of the variant of equation (5.1) with the $u''$
term missing.
This analysis becomes most transparent when $A$ has an orthonormal
basis
$\{\varphi_n\}$ of eigenvalues. Thus $A \varphi_n = \lambda_n
\varphi_n, \;\; 0 \leq \lambda_1 \leq \lambda_2 \leq \ldots \quad
\leq
\lambda_n \to \infty$. Suppose $M$ is such that $\lambda_M < a
\leq
\lambda_{M+1}$. Then any solution $u$ of (5.1) satisfies, for
suitable constant $c_j,$
$$u(t) = \sum_{j=1}^M \exp \{t(-a +
(a^2-\lambda_j^2)^{\frac{1}{2}})\}
c_j \varphi_j + \bigcirc (e^{-ta}),$$
which approximately equals
$$\sum_{j=1}^M e^{-t\lambda_1^2/2a} c_j \varphi_j +
\bigcirc(e^{-ta}).$$
The above calculations are correct but not ``canonical", and so
they are
nonoptimal.
If our original solution of (5.1) is
$$u(t) = e^{tA_1} h_1 + e^{tA_2} h_2,$$
then we approximate it by
$$e^{t(-a+(a^2 - A^2)^{\frac{1}{2}})} P([0, \epsilon])h,$$
with an error of $\bigcirc (\exp
[t(-a+(a^2-\epsilon^2)^{\frac{1}{2}}])$, and
this in turn can be approximated by
$$e^{-tA^2/2a} P([0,\epsilon])h_1.$$
But there is no natural way of choosing $\epsilon$. In the
orthonormal
basis case, if $\lambda_1$ is a simple eigenvalue, we can write
\begin{eqnarray*}
u(t) &=& \exp \{t(-a+(a^2 -\lambda_1^2)^{\frac{1}{2}})\} c_1
\varphi_1\\
&\cong& e^{-t\lambda_1^2/2a} c_1 \varphi_1
\end{eqnarray*}
with an error of $\bigcirc (\exp [-t(a-(a^2 -
\lambda_2^2)^{\frac{1}{2}}])$. This gives all relevant ergodic
information, but says very little in the case when $c_1=0$.
We summarize the above results.
\begin{thm}
Let $A=A^*$ be a nonnegative selfadjoint operator on a Hilbert
space $\mathcal{H}$ and let $a$ be a positive constant. Then the
unique solution $u$ of
$$u^{\prime\prime} + 2au^\prime + A^2 u=0, \quad u(0) = f \quad
u^\prime (0) = g$$satisfies, for any given $\epsilon > 0,$
$$u(t)= \exp \{t (-a + (a^2 -A^2)^{\frac{1}{2}} )\} \ell + \delta
(t)$$
where
$$\delta (t) = \bigcirc (e^{-t(a^2-\epsilon^2)^{\frac{1}{2}}})$$
as $t \to \infty$. This solution is approximately equal to
$$e^{-tA^2/2a} \ell.$$
\end{thm}
This choice of $\ell$ depends on $\epsilon$ and so is not made in a
canonical fashion. We expand on this point.
Clearly solutions of the abstract telegraph equation (5.1) are
asymptotically given by an analytic semigroup, but there is not a
unique way to make this association. If the initial conditions for
(5.1)
are $u(0) = h, u'(0)=k$, then
$$u(t) = e^{tA_1} f_1 + e^{tA_2} f_2$$
where $f_1, f_2$ case be explicitly computed in terms of $h$ and
$k$.
We focus on $f_1$ and throw $f_2$ into the error term. In
projecting $f_1$ onto $P([0,\epsilon])(\mathcal{H})$, we demand $0
\leq
\epsilon < a$, and we want this projected vector to be nonzero; but
we
have no {\it natural} way of choosing $\epsilon$, especially in the
case
of continuous spectrum, which is the case in the Kac model of $A^2
= -
d^2/dx^2$ on $L^2({\mathbb R})$.
\section{Fractional Derivatives}
The model telegraph equation
\begin{equation}
D^2u(t) + 2aDu(t)+ A^2 u(t)=0
\end{equation}
(with $D=d/dt)$ can alternatively be replaced by a fractional
differential equation of the form
\begin{equation}
(D^\gamma)^2 u(t) + 2aD^\gamma u(t) + A^2 u(t)=0,
\end{equation}
where $\frac{1}{2} \leq \gamma \leq 1, \;\; \gamma = 1$
corresponding to
(6.1). The motivation comes from self similarity and experimental
considerations. This will be studied in a separate article
\cite{CFG}.
\section{Concluding Remarks}
Compared to the heat equation, the telegraph equation seems to be a
superior model for describing certain fluid flow problems involving
suspensions. For the telegraph equation (or for its fractional
derivative
analogue), one needs to specify two pieces of initial data; the
heat
equation only requires one. If the telegraph equation is to
describe an
experiment, the experiment must be able to give enough data to
produce
two initial conditions. One of these conditions may need to be
linked strongly to the experimental method and its limitations,
which
accordingly would not be natural in a mathematical sense. If or
when
such an arrangement to set a value of $\epsilon$ cannot be
provided, one
may be forced to use a first order (in time) equation as a model.
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\end{document}