abstract:
A well-known method to sample from a given target distribution on a
finite dimensional space is to simulate the solution of a stochastic
differential equation (SDE) which has the target distribution as its
invariant measure. Assuming ergodicity of the SDE, the solution of
the SDE at a ``large\'\' time can be used as an approximation to a
sample from the target distribution.
If the target distribution lives on an infinite dimensional space, the
situation is less clear, but it transpires that often one can still
employ the same idea. We show how one can use SDEs with values in the
space of continuous functions to sample from certain target
distributions on this space. In the cases we consider, the sampling
equation turn out to be stochastic partical differential equations
(SPDEs).
In order to derive an implementable MCMC method from these results,
one needs to discretise the resulting SPDEs. We give some preliminary
results comparing different discretisation schemes for these
equations. |