Papers Published
Abstract:
The ergodic properties of SDEs, and various time
discretizations for
SDEs, are studied. The ergodicity of SDEs is
established by
using
techniques from the theory of Markov chains
on general state
spaces.
Application of these Markov chain results
leads to
straightforward
proofs of ergodicity for a variety of SDEs,
in particular
for problems
with degenerate noise and for problems with
locally
Lipschitz vector
fields. The key points which need to be
verified are the
existence of
a Lyapunov function inducing returns to a
compact set, a
uniformly
reachable point from within that set, and
some smoothness of the
probability densities; the last two points
imply a minorization
condition. Together the minorization
condition and Lyapunov
structure
give geometric ergodicity. Applications
include the Langevin
equation, the Lorenz equation with degenerate
noise and gradient
systems. The ergodic theorems proved are
strong, yielding
exponential convergence of expectations for
classes of
measurable
functions restricted only by the condition
that they grow no
faster
than the Lyapunov function.
The same Markov chain theory is then used to
study
time-discrete approximations
of these SDEs. It is shown that the
minorization condition
is robust
under approximation. For globally Lipschitz
vector fields
this is also
true of the Lyapunov condition. However in
the locally
Lipschitz case
the Lyapunov condition fails for explicit
methods such as
Euler-Maruyama;
it is, in general, only inherited
by specially constructed implicit
discretizations.
Examples of such discretization based on
backward Euler methods
are given, and approximation of the Langevin
equation
studied in some detail.