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Consistent inference for diffusions from low frequency measurements

Richard Nickl

TL;DR

This work addresses recovering a diffusion's spatially varying diffusivity $f$ from low-frequency observations of a reflected diffusion in a bounded convex domain, focusing on the forward transition map $P_{t,f}$ and, in particular, $P_{D,f}$. It develops a Bayesian nonparametric framework with Gaussian process priors on $f$, proves injectivity and stability of the nonlinear map from $f$ to $P_{D,f}$, and derives posterior consistency with convergence rates; it also establishes optimal recovery rates for the transition operator and links to spectral-geometry hot spots under symmetry. The methodology combines projection-based estimators for $P_{D,f}$, sharp stability results (logarithmic and Hölder), and comprehensive Bayesian analysis, including posterior contraction and mean convergence, applicable to non-linear inverse problems governed by parabolic PDEs. The results illuminate the ill-posedness of diffusion-inference from coarse measurements, show that symmetry can enable faster rates, and connect statistical identifiability with spectral properties, offering a principled framework for nonparametric inference in diffusion models with practical implications for physical and biological diffusion processes.

Abstract

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.

Consistent inference for diffusions from low frequency measurements

TL;DR

This work addresses recovering a diffusion's spatially varying diffusivity from low-frequency observations of a reflected diffusion in a bounded convex domain, focusing on the forward transition map and, in particular, . It develops a Bayesian nonparametric framework with Gaussian process priors on , proves injectivity and stability of the nonlinear map from to , and derives posterior consistency with convergence rates; it also establishes optimal recovery rates for the transition operator and links to spectral-geometry hot spots under symmetry. The methodology combines projection-based estimators for , sharp stability results (logarithmic and Hölder), and comprehensive Bayesian analysis, including posterior contraction and mean convergence, applicable to non-linear inverse problems governed by parabolic PDEs. The results illuminate the ill-posedness of diffusion-inference from coarse measurements, show that symmetry can enable faster rates, and connect statistical identifiability with spectral properties, offering a principled framework for nonparametric inference in diffusion models with practical implications for physical and biological diffusion processes.

Abstract

Let be a reflected diffusion process in a bounded convex domain in , solving the stochastic differential equation with a -dimensional Brownian motion. The data consist of discrete measurements and the time interval between consecutive observations is fixed so that one cannot `zoom' into the observed path of the process. The goal is to infer the diffusivity and the associated transition operator . We prove injectivity theorems and stability inequalities for the maps . Using these estimates we establish the statistical consistency of a class of Bayesian algorithms based on Gaussian process priors for the infinite-dimensional parameter , 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.
Paper Structure (38 sections, 28 theorems, 164 equations, 2 figures)

This paper contains 38 sections, 28 theorems, 164 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Optimal recovery of the transition operator $P_{D,f}$
  4. Injectivity of $f \mapsto P_{t,f} \mapsto P_{D,f}$, $t<D$
  5. Stability estimates
  6. Reflected diffusion and 'hot spots'
  7. Bayesian inference in the diffusion model
  8. Posterior consistency theorems
  9. Posterior reconstruction of $P_{D,f}$
  10. Consistency and convergence rates for the non-linear inverse problem
  11. Proofs
  12. Analytical background: reflected diffusions and their generators
  13. Divergence form operators
  14. Spectral resolution of the generator
  15. Heat equation, transition operator, and a perturbation identity
  16. ...and 23 more sections

Key Result

Theorem 1

Suppose positive diffusion coefficients $f_1, f_2 \in C^2(\mathcal{O})$ are bounded away from zero on $\mathcal{O}$ and such that $f_1=f_2$ near $\partial \mathcal{O}$. Then if $P_{D,f_1}= P_{D,f_2}$ coincide as bounded linear operators on $L^2(\mathcal{O})$ for some $D>0$, we must have $f_1=f_2$ on

Figures (2)

  • Figure 1: Left: a reflected diffusion path $(X_t: 0 \le t \le T)$ initialised at $X_0$ and ran until time $T=5$. Right: $N=500$ discrete observations $(X_{iD})_{i=0}^N$ at sampling frequency $D=0.05$ ($T=25$). The diffusivity $f$ is given in Fig. \ref{['fig:estimators']}.
  • Figure 2: The posterior mean estimate $f_{\bar{\theta}}$ with $\bar{\theta} = M^{-1}\sum_{m=1}^M \vartheta_m$ after $M = 10000$ pCN iterates, for sample sizes $N=2500$ (left) and $N=25000$ (center), at sampling frequency $D=0.05$; the true field $f_0$ (right).

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark 1: Stability for the 'backward heat operator'
  • Theorem 7
  • Remark 2: The one-dimensional case
  • Proposition 1
  • ...and 33 more