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Howitt, Christopher John (2007) Stochastic flows and sticky Brownian motion. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2244193~S1
Abstract
Sticky Brownian
motion
is
a one-dimensional
diffusion
with the
property that
the
amount of
time the process spends at zero
is
of positive
Lebesgue
measure
and yet
the
process
does
not stay at zero
for
any positive interval
of time. Sticky
Brownian
motion can
be
considered as qualitatively
between
standard
Brownian
motion and
Brownian
motion absorbed at zero.
A
system of coalescing
Brownian
motions
is
a collection of paths, where
each path
behaves as a
Brownian
motion
independent
of all other paths until
the
first
time two paths meet, at which point the two
paths that have just
met
behave is
a single
Brownian
motion
independent
of all remaining paths.
Thus the
difference between
any two paths of a system of coalescing
Brownian
motion
behaves
as a
Brownian
motion absorbed at zero.
In
this thesis
we
consider systems of
Brownian
paths, where the difference between
any two
paths
behaves as a sticky
Brownian
motion rather than a coalescing Brownian
motion.
We
consider systems of sticky
Brownian
motions starting
from
points
in
continuous
time and space.
The
evolution of systems of this type
may
be
described by
means of a stochastic
flow
of
kernels. A
stochastic
flow
of
kernels is
characterised
by its N-point
motions which
form
a consistent
family
of
Brownian
motions.
We
characterise such a consistent
family
such that the difference
between
any pair of coordinates
behaves as a sticky
Brownian
motion.
The Brownian
web
is
a way of
describing
a system of coalescing Brownian
motions starting
in
any point
in
space and time. We describe
a coupling of
Brownian
webs such
that the difference between one path
in
each web
behaves
as a sticky
Brownian motion.
Then by
conditioning one
Brownian
web on the
other we can construct a stochastic
flow
of
kernels.
Finally
we
discuss the
concept of
duality in
relation to flows
and we prove
some minor results relating
to these
dualities.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Stochastic analysis , Stochastic processes, Stochastic models , Brownian motion processes, Mathematical models | ||||
Official Date: | August 2007 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Department of Statistics | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Sponsors: | Engineering and Physical Sciences Research Council (EPSRC) | ||||
Extent: | xiii, [233] leaves : illustrations | ||||
Language: | eng |
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