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.

翻译：设$(X_t)$为$\mathbb R^d$中有界凸域上的反射扩散过程，满足随机微分方程$$dX_t = \nabla f(X_t) dt + \sqrt{2f (X_t)} dW_t, ~t \ge 0,$$其中$W_t$为$d$维布朗运动。数据$X_0, X_D, \dots, X_{ND}$由离散测量组成，且连续观测间的时间间隔$D$固定，因此无法对过程观测路径进行"放大"分析。目标在于推断扩散系数$f$及其关联的转移算子$P_{t,f}$。我们证明了映射$f \mapsto P_{t,f} \mapsto P_{D,f}, t<D$的单射定理与稳定性不等式。利用这些估计，我们建立了基于高斯过程先验的贝叶斯算法族对无穷维参数$f$的统计一致性，并证明了部分收敛速率的最优性。文中还讨论了该反问题不适定程度与谱几何中"热点"猜想之间的内在关联。