%% file LaTeX  fm60_pinazza.tex
%% Bologna 20 December 2002 : FM60:
%% Contribution by C. Manzoni and O. Pinazza
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\begin{document}
\cen{INTERDISCIPLINARY WORKSHOP}
%\vsn
\cen{{\bf \lq\lq FROM  WAVES TO DIFFUSION AND BEYOND\rq\rq} }
%\vsn
\cen{{\bf BOLOGNA, Italy : December 20, 2002}}
\vs \hrule \vvs
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\cen{{\bf SELF-SIMILAR STOCHASTIC PROCESSES}}
\vs
\cen {\bf Claudia MANZONI$^1$ and Ombretta PINAZZA$^2$}
\vsh
\cen{$^1$ Mathematics Institute,
University of Warwick}
\cen{Coventry CV4 7AL
England,  UK}
\cen{{\tt  manzoni@maths.warwick.ac.uk}}
\vsh
\cen{
$^2$ INFN -  Istituto Nazionale di Fisica Nucleare,
Sezione di Bologna}
\cen{Via Irnerio 46,  I-40126 Bologna}
\cen{{\tt Ombretta.Pinazza@bo.infn.it}}
\vsp
This work is concerned with the analysis of self-similar stochastic processes, 
where statistical self-similarity refers to invariance in distribution 
under a suitable change of scale.
In particular, emphasis is placed upon those processes which, 
being non-local in time, can  display long memory and/or 
long-range correlation effects, and 
are connected to anomalous diffusion.
%%  a phenomenon met in a number of physical, chemical and biological systems.
Whereas normal diffusion is characterized by a mean-square displacement
which is asymptotically linear in time according to Einstein law, 
in anomalous diffusion,
a phenomenon met
in a number of physical, chemical and biological systems,
  we find appreciable deviations from this law. 
%% mean-square displacement is not proportional to $t$. 
%% As a consequence, memory effects are observed.
\\
Fractional Brownian motion (fBm) and time fractional diffusion are both 
self-similar stochastic processes, that display 
deviations from Einstein law. 
%% {\em long memory} or {\em long-range dependence}. 
The first
one is Gaussian, non-stationary, with stationary increments; 
the second is non-Gaussian,
non-stationary with increments that are also non-stationary.
%% \\
The class of self-similar stochastic processes is related, via a simple
one-to-one connection, called Lamperti
transformation, to another important class of stochastic processes, 
the stationary processes (statistically invariant under time shifts).
The Lamperti transformation, which essentially consists of a proper 
warping of the time axis, is a bijective invertible map which 
connects any self-similar process with a stationary counterpart.
Although introduced by Lamperti in 1962, 
this fundamental result has further
received little attention until a recent past when it 
has been rediscovered and
appreciated as an important tool allowing researchers to 
apply the extensive body
of knowledge about stationary processes to examine 
the structure of the associated
self-similar processes.



\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{fm60_cmfig1.eps}
\includegraphics[width=.48\textwidth]{fm60_cmfig2.eps}
\caption[]{(Left) 
Two paths of fBm with $H=0.25$ (top) and $H=0.75$ (bottom);
(Right) Lampertization of Brownian motion}
 \end{figure}



\begin{thebibliography} {99}


\bibitem{Flandrin INDIA03}
P. Flandrin and P.O. Amblard,
    From stationarity to self-similarity and back: 
 Variations on the Lamperti transformations,
    in
    G. Rangarajan and M. Ding (Editors),
    {\it Long Range Dependent Processes: Theory and Applications},
   Springer Verlag, to appear (2003).


\bibitem{GorMai CHEMPHYS02}
  R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini and P. Paradisi,
  Discrete random walk  models for space-time fractional  diffusion,
  {\it Chemical Physics} {\bf 284} (2002) 521-544.
%% Chemical Physics, Vol. 284 No 1/2, pp. 521-544 (2002).
%% Special issue on "Strange Kinetics",
%% Guest Editors,  R. Hilfer, R. Metzler, A. Blumen, J. Klafter.


\bibitem{Lamperti 62}
J.W. Lamperti,
Semi-stable stochastic processes,
 {\it Trans. Amer. Math. Soc.} {\bf 104} (1962) 62-78.


\bibitem{MandelbrotNess 68}
 B.B. Mandelbrot and J.W. Van Ness,
 Fractional Brownian motions, fractional noises and applications,
 {\it SIAM Review} {\bf 10} (1968) 422-433.



%% \bibitem{Moretti THESIS00}
%% D. Moretti,
%% {\it Stochastic Processes for Anomalous Diffusion:
%% Analysis and Simulations},
%% Thesis for Degree in Physics, University
%% of Bologna, December 2000, in Italian.
%% [Supervisors: Prof. F. Mainardi and Dr. P. Paradisi]

\bibitem{Manzoni THESIS02}
C. Manzoni,
{\it Analysis of Self-Similar Stochastic Processes},
 Thesis for Degree in Physics, University
of Bologna, October 2002, in Italian.
[Supervisors: Prof. F. Mainardi and Dr. O. Pinazza]

\bibitem{Pinazza THESIS94}
O. Pinazza,
{\it Fractal Dimension and Fractional Calculus: Theory and Applications},
 Thesis for Degree in Physics, University
of Bologna, July 1993, in Italian.
[Supervisors: Prof. F. Mainardi and Prof. S. Rambaldi]

\bibitem{RambaldiPinazza 94}
S. Rambaldi and O. Pinazza,
  An accurate fractional Brownian motion generator,
   {\it Physica A} {\bf 208} (1994) 21-30.

%%\bibitem{Taqqu REV02}
%%   M.S. Taqqu,
%% Fractional Brownian Motion and Long-Range Dependence,
%%  to appear in: P. Doukhan, G. Oppenheim and M.S. Taqqu (Editors),
%% {\it Long-range Dependence: Theory and Applications},<
%% Birkh\"auser, Basel and Boston (2002). %%%  pp. ?-??.


\end{thebibliography}%% {99}

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