  We consider a parameter estimation problem for one dimensional stochastic
heat equations, when data is sampled discretely in time or spatial component.
We prove that, the real valued parameter next to the Laplacian (the drift), and
the constant parameter in front of the noise (the volatility) can be
consistently estimated under somewhat surprisingly minimal information. Namely,
it is enough to observe the solution at a fixed time and on a discrete spatial
grid, or at a fixed space point and at discrete time instances of a finite
interval, assuming that the mesh-size goes to zero. The proposed estimators
have the same form and asymptotic properties regardless of the nature of the
domain - bounded domain or whole space. The derivation of the estimators and
the proofs of their asymptotic properties are based on computations of power
variations of some relevant stochastic processes. We use elements of Malliavin
calculus to establish the asymptotic normality properties in the case of
bounded domain. We also discuss the joint estimation problem of the drift and
volatility coefficient. We conclude with some numerical experiments that
illustrate the obtained theoretical results.
