  In this study, we introduce a novel method for generating new synthetic
samples that are independent and identically distributed (i.i.d.) from
high-dimensional real-valued probability distributions, as defined implicitly
by a set of Ground Truth (GT) samples. Central to our method is the integration
of space-time mixing strategies that extend across temporal and spatial
dimensions. Our methodology is underpinned by three interrelated stochastic
processes designed to enable optimal transport from an easily tractable initial
probability distribution to the target distribution represented by the GT
samples: (a) linear processes incorporating space-time mixing that yield
Gaussian conditional probability densities, (b) their diffusion bridge analogs
that are conditioned to the initial and final state vectors, and (c) nonlinear
stochastic processes refined through score-matching techniques. The crux of our
training regime involves fine-tuning the nonlinear model, and potentially the
linear models -- to align closely with the GT data. We validate the efficacy of
our space-time diffusion approach with numerical experiments, laying the
groundwork for more extensive future theory and experiments to fully
authenticate the method, particularly providing a more efficient (possibly
simulation-free) inference.
