The solutions of Hamiltonian equations are known to describe the underlying
phase space of the mechanical system. In Bayesian Statistics, the only place,
where the properties of solutions to the Hamiltonian equations are successfully
applied, is Hamiltonian Monte Carlo. In this article, we propose a novel
spatio-temporal model using a strategic modification of the Hamiltonian
equations, incorporating appropriate stochasticity via Gaussian processes. The
resultant sptaio-temporal process, continuously varying with time, turns out to
be nonparametric, nonstationary, nonseparable and no-Gaussian. Besides, the
lagged correlations tend to zero as the spatio-temporal lag tends to infinity.
We investigate the theoretical properties of the new spatio-temporal process,
along with its continuity and smoothness properties. Considering the Bayesian
paradigm, we derive methods for complete Bayesian inference using MCMC
techniques. Applications of our new model and methods to two simulation
experiments and two real data sets revealed encouraging performance