  Let $(X_t)$ be a reflected diffusion process in a bounded convex domain in
$\mathbb R^d$, solving the stochastic differential equation $$dX_t = \nabla
f(X_t) dt + \sqrt{2f (X_t)} dW_t, ~t \ge 0,$$ with $W_t$ a $d$-dimensional
Brownian motion. The data $X_0, X_D, \dots, X_{ND}$ consist of discrete
measurements and the time interval $D$ between consecutive observations is
fixed so that one cannot `zoom' into the observed path of the process. The goal
is to infer the diffusivity $f$ and the associated transition operator
$P_{t,f}$. We prove injectivity theorems and stability inequalities for the
maps $f \mapsto P_{t,f} \mapsto P_{D,f}, t<D$. Using these estimates we
establish the statistical consistency of a class of Bayesian algorithms based
on Gaussian process priors for the infinite-dimensional parameter $f$, and show
optimality of some of the convergence rates obtained. We discuss an underlying
relationship between the degree of ill-posedness of this inverse problem and
the `hot spots' conjecture from spectral geometry.
