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Information bounds for inference in stochastic evolution equations observed under noise

Gregor Pasemann, Markus Reiß

TL;DR

The paper develops a comprehensive information-theoretic framework for estimation in stochastic evolution equations observed with noise, deriving tight minimax lower bounds via Hellinger distances between cylindrical Gaussian observation laws. It shows how the information content depends on the noise level, observation time, spatial dimension, and the way a coefficient enters the generator, and it provides parametric estimators that achieve these bounds in many commuting scenarios. The results cover key SPDEs, including Ornstein–Uhlenbeck, fractional Laplacians, transport, and source terms, and reveal an ellbow phenomenon in nonparametric settings where rates interpolate between parametric-like and nonparametric regimes depending on T and ε. The findings illuminate the intrinsic difficulty of inference in SPDEs and establish precise rate-optimal strategies and limitations for both parametric and nonparametric coefficients, with implications for spatially inhomogeneous models and high-dimensional settings.

Abstract

We consider statistics for stochastic evolution equations in Hilbert space with emphasis on stochastic partial differential equations (SPDEs). We observe a solution process under additional measurement errors and want to estimate a real or functional parameter in the drift. Main targets of estimation are the diffusivity, transport or source coefficient in a parabolic SPDE. By bounding the Hellinger distance between observation laws under different parameters we derive lower bounds on the estimation error, which reveal the underlying information structure. The estimation rates depend on the measurement noise level, the observation time, the covariance of the dynamic noise, the dimension and the order, at which the parametrised coefficient appears in the differential operator. A general estimation procedure attains these rates in many parametric cases and proves their minimax optimality. For nonparametric estimation problems, where the parameter is an unknown function, the lower bounds exhibit an even more complex information structure. The proofs are to a large extent based on functional calculus, perturbation theory and monotonicity of the semigroup generators.

Information bounds for inference in stochastic evolution equations observed under noise

TL;DR

The paper develops a comprehensive information-theoretic framework for estimation in stochastic evolution equations observed with noise, deriving tight minimax lower bounds via Hellinger distances between cylindrical Gaussian observation laws. It shows how the information content depends on the noise level, observation time, spatial dimension, and the way a coefficient enters the generator, and it provides parametric estimators that achieve these bounds in many commuting scenarios. The results cover key SPDEs, including Ornstein–Uhlenbeck, fractional Laplacians, transport, and source terms, and reveal an ellbow phenomenon in nonparametric settings where rates interpolate between parametric-like and nonparametric regimes depending on T and ε. The findings illuminate the intrinsic difficulty of inference in SPDEs and establish precise rate-optimal strategies and limitations for both parametric and nonparametric coefficients, with implications for spatially inhomogeneous models and high-dimensional settings.

Abstract

We consider statistics for stochastic evolution equations in Hilbert space with emphasis on stochastic partial differential equations (SPDEs). We observe a solution process under additional measurement errors and want to estimate a real or functional parameter in the drift. Main targets of estimation are the diffusivity, transport or source coefficient in a parabolic SPDE. By bounding the Hellinger distance between observation laws under different parameters we derive lower bounds on the estimation error, which reveal the underlying information structure. The estimation rates depend on the measurement noise level, the observation time, the covariance of the dynamic noise, the dimension and the order, at which the parametrised coefficient appears in the differential operator. A general estimation procedure attains these rates in many parametric cases and proves their minimax optimality. For nonparametric estimation problems, where the parameter is an unknown function, the lower bounds exhibit an even more complex information structure. The proofs are to a large extent based on functional calculus, perturbation theory and monotonicity of the semigroup generators.

Paper Structure

This paper contains 20 sections, 34 theorems, 215 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Hellinger bounds for cylindrical Gaussian measures
  3. Stochastic evolution equations under noise
  4. Parametric minimax rates
  5. Fundamental parametric examples
  6. Scalar Ornstein-Uhlenbeck processes
  7. Stochastic evolution equations without noise
  8. Parameter in the fractional Laplacian
  9. Parameter in the transport term
  10. Parameter in the source term
  11. Fundamental nonparametric examples
  12. Space-dependent diffusivity
  13. Space-dependent transport
  14. Space-dependent source
  15. Notation
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $\delta>0$. Assume there are $\vartheta_0,\vartheta_1\in\Theta$ with semi-distance $d(\vartheta_0,\vartheta_1)\geqslant \delta$ such that their respective laws have Hellinger distance $H(\mathop{\mathrm{\mathbb P}}\nolimits_{\vartheta_0},\mathop{\mathrm{\mathbb P}}\nolimits_{\vartheta_1})\leqsla holds where the infimum is taken over all estimators (measurable $\Theta$-valued functions) in the

Theorems & Definitions (79)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Remark 2.4
  • proof
  • Example 3.1
  • Lemma 3.2
  • Remark 3.3
  • proof
  • ...and 69 more