  We construct a family of non-Gaussian martingales the marginals of which are
all Gaussian. We give the predictable quadratic variation of these processes
and show they do not have continuous paths. These processes are Markovian and
inhomogeneous in time, and we give their infinitesimal generators. Within this
family we find a class of piecewise deterministic pure jump processes and
describe the laws of jumps and times between the jumps.
