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 $W_t$ 是 $d$ 维布朗运动。观测数据 $X_0, X_D, \dots, X_{ND}$ 为离散测量值，相邻观测时间间隔 $D$ 固定，因此无法对过程路径进行“缩放”分析。本文旨在推断扩散系数 $f$ 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无穷维参数 $f$ 贝叶斯算法的统计一致性，并证明了部分收敛速度的最优性。我们讨论了该反问题病态程度与谱几何中“热点猜想”之间的内在联系。