%Plain TeX version of "Local Time of a Two-Dimensional Diffusion"
%
\input mssymb
\input macros
\input /users/mwb/files/math/thesis/defns
\headline={\hfil}
\sno=1
\def\roysoc{Royal Society}
\def\newsect#1{\bigbreak\centerline{\bf\the\sno\quad#1}\advance\sno by 1}
\def\newchap#1#2{\relax}
\magnification=\magstep1
%
\hsize=425truept
\hoffset=13truept
\vsize=669truept
\voffset=13truept
\parskip=\medskipamount
\parindent=25pt
\baselineskip=23pt   % directives for increased leading
\lineskip=2pt
\lineskiplimit=1pt

% Font selections

\font\sc=cmcsc10
%\font\twelvebi=cmbxsl10 at 12pt

%
\def\S{\Sigma}
\def\Pz{{\Bbb P_{\bf0}}}
\def\Ez{{\Bbb E_{\bf0}}}
\def\baxfog{Baxter (1993)} \def\kennedy{Kennedy (1976)}
\def\vervaat{Vervaat (1979)} \def\marcusrosen{Marcus and Rosen (1992)}
\def\blumgetoor{Blumenthal and Getoor (1968)} \def\rogers{Rogers (1989)}
\def\barlow{Barlow (1988)}
\def\bingham{Bingham {\it et al.} (1987)}
\def\bretagnolle{Bretagnolle (1971)}
\def\kesten{Kesten (1969)}
\def\portstone{Port and Stone (1971)}
\def\zxfone{theorem 1 of \baxfog}
%
\centerline{\bf The local time of a two-dimensional diffusion}
\medskip
\centerline{\sc Martin Baxter}
\medskip
\centerline{\it Statistical Laboratory, University of Cambridge,
16 Mill Lane, Cambridge CB2 1SB\footnote{$^1$}{\rm During 1994:
\it Department of
Mathematics, University of British Columbia, Vancouver V6T 1Z2, B.C. Canada}}
\bigskip\bigskip
%
\centerline{\bf Abstract}
\bigskip
When a Markovian random process taking values in a continuous
state-space, such as $\Bbb R$, visits a particular point repeatedly,
it is natural to seek some quantity which records how long it spends there.
Typically however, the number of visits made to the point is uncountably
infinite, and the (Lebesgue) length of time spent there is zero. One
interesting object to consider is the local time, sometimes thought of
as the occupation density of the process, which at each point is a random
Cantor function that increases only when the process visits the point.
The review article by \rogers\ contains a good introduction to the local
time of a one-dimensional Brownian motion and its relevance to the
excursions of Brownian motion from zero. In two-dimensions, a typical
diffusion, such as Brownian motion in the plane, never revisits a point,
so it does not have a local time. In this paper we shall construct the local
times of some particular two-dimensional diffusions on a one-dimensional
subspace, and show that
they are jointly continuous in both time and space.
%
\footline={\hfil}
\vfill\eject
\footline{\hfil\rm\folio\hfil}\pageno=1
%
\input /users/mwb/files/math/thesis/chap6.tex

\noindent {\bf Acknowledgement} I am glad to be able to acknowledge
Profs T.J.Lyons and D.Williams who encouraged me to consider this
problem, and also to thank D.G.Hobson and T.Chan for their advice.

\bigbreak\bigskip
\centerline{\bf References}
\nobreak
{\frenchspacing\parindent=0pt\parskip=\medskipamount
\everypar={\hangindent=25pt\hangafter=1}
Barlow, M.T. (1988) Necessary and sufficient conditions for the
continuity of local time of L\'evy processes. {\it Ann. Probab.}
{\bf 16}, 1389--1427.

Baxter, M.W. (1993) Markov processes on the boundary of the
binary tree. In {\it S\'eminaire de Probabilit\'es XXVI} (ed.
J.Az\'ema, P.A.Meyer \& M.Yor), Lecture Notes in
Mathematics 1526, pp. 210--224. Berlin and Heidelberg: Springer-Verlag.
\par\penalty-400

Bingham, N.H, Goldie, C.M., and Teugels, J.L. (1987) {\it Regular
Variation}, Encyclopedia of Mathematics and its Applications, vol.27.
Cambridge University Press.
\par\penalty-400

Blumenthal, R.M., and Getoor, R.K. (1968) {\it Markov Processes
and Potential Theory}. New York and London: Academic Press.

Bretagnolle, J. (1971) Resultats de Kesten sur les
processus \`a accroissements ind\'ep\-endants. In {\it S\'eminaire de
Probabilit\'es V} (ed. A.Dold \& B.Eckmann), Lecture Notes in Mathematics~191,
\quad pp. 21-36.\quad Berlin and Heidelberg: Springer-Verlag.

Kennedy, D.P. (1976) The distribution of the maximum
Brownian excursion. {\it J. Appl. Probab.} {\bf 13}, 371--376.

Kesten, H. (1969) Hitting probabilities of single points
for processes with stationary independent increments. {\it Mem. Amer.
Math. Soc.} {\bf 109}.

Marcus, M.B., and Rosen, J. (1992) Sample Path
Properties of the Local Times of Strongly Symmetric Markov Processes
via Gaussian Processes. {\it Ann. Probab.} {\bf 20}, 1603--1684.

Port, S.C., and Stone, C.J. (1971) Infinitely
divisible processes and their potential theory. {\it Ann. Inst. Fourier}
{\bf 21}, 2:157--275 and 4:179-267.

Rogers, L.C.G. (1989) A Guided Tour through Excursions.
{\it Bull. London Math. Soc.} {\bf 21}, 305--341.

Vervaat, W. (1979) A relation between Brownian bridge and
Brownian excursion. {\it Ann. Probab.} {\bf 7}, 143--149.
\par}
\vfill\eject
\end