  The Metropolis-Adjusted Langevin Algorithm (MALA), originally introduced to
sample exactly the invariant measure of certain stochastic differential
equations (SDE) on infinitely long time intervals, can also be used to
approximate pathwise the solution of these SDEs on finite time intervals.
However, when applied to an SDE with a nonglobally Lipschitz drift coefficient,
the algorithm may not have a spectral gap even when the SDE does. This paper
reconciles MALA's lack of a spectral gap with its ergodicity to the invariant
measure of the SDE and finite time accuracy. In particular, the paper shows
that its convergence to equilibrium happens at exponential rate up to terms
exponentially small in time-stepsize. This quantification relies on MALA's
ability to exactly preserve the SDE's invariant measure and accurately
represent the SDE's transition probability on finite time intervals.
