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STOCHASTIC STABILITY
1.
Introduction
Deterministic stability is a branch of the qualitative theory of dy-
namical systems. In particular, the majority of presently available
results which are termed stability results pertain to certain qualitative
and quantitative (which do not involve the actual computation of a
solution) properties of differential equations. Consider the differential
equation
P
=f(x,
t)
with initial condition
xo
belonging to a set
R.
In
what follows,
R
may vary but will always be a nonempty bounded
open set containing the origin
x
=
(0).
Let
P
be a set containing
R.
Some typical problems which may be grouped under the title “sta-
bility problems” are
:
Let
P
be given.
Is
there an
R
such that if
xOe
R,
then
X,E
P
for
all finite
t?
In reference to
(Pl),
is there some
R
corresponding to each open
set
P
containing the origin?
In reference to
(Pl),
estimate the largest set
R.
(The estimate is
a quantitative property.)
For
a given set of initial values
R,
is the set
P,
containing the
range of the trajectory
for
all
t
<
00,
bounded?
What is the smallest set containing the asymptotic values of the
solution for all
xo
in
R?
For
fixed
P
and
xo
in
R,
estimate the first time of exit of
x,
from
P.
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