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Non-parametric estimation of the reaction term in semi-linear SPDEs with spatial ergodicity

Sascha Gaudlitz

Abstract

This paper discusses the non-parametric estimation of a non-linear reaction term in a semi-linear parabolic stochastic partial differential equation (SPDE). The estimator's consistency is due to the spatial ergodicity of the SPDE while the time horizon remains fixed. The analysis of the estimation error requires the concentration of spatial averages of non-linear transformations of the SPDE. The method developed in this paper combines the Clark-Ocone formula from Malliavin calculus with the Markovianity of the SPDE and density estimates. The resulting variance bound utilises the averaging effect of the conditional expectation in the Clark-Ocone formula. The method is applied to two realistic asymptotic regimes. The focus is on a coupling between the diffusivity and the noise level, where both tend to zero. Secondly, the observation of a fixed SPDE on a growing spatial observation window is considered. Furthermore, the concentration of the occupation time around the occupation measure is proved.

Non-parametric estimation of the reaction term in semi-linear SPDEs with spatial ergodicity

Abstract

This paper discusses the non-parametric estimation of a non-linear reaction term in a semi-linear parabolic stochastic partial differential equation (SPDE). The estimator's consistency is due to the spatial ergodicity of the SPDE while the time horizon remains fixed. The analysis of the estimation error requires the concentration of spatial averages of non-linear transformations of the SPDE. The method developed in this paper combines the Clark-Ocone formula from Malliavin calculus with the Markovianity of the SPDE and density estimates. The resulting variance bound utilises the averaging effect of the conditional expectation in the Clark-Ocone formula. The method is applied to two realistic asymptotic regimes. The focus is on a coupling between the diffusivity and the noise level, where both tend to zero. Secondly, the observation of a fixed SPDE on a growing spatial observation window is considered. Furthermore, the concentration of the occupation time around the occupation measure is proved.
Paper Structure (27 sections, 30 theorems, 150 equations, 2 figures)

This paper contains 27 sections, 30 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statistical methodology and contributions
  3. Probabilistic challenges and contributions
  4. Outline
  5. Setting and Assumptions
  6. Estimator and statistical main results
  7. Definition of the estimator
  8. Main results
  9. A numerical example
  10. Controlling the marginal densities and spatial ergodicity
  11. The marginal densities
  12. Concentration due to spatial ergodicity
  13. Proofs for Proposition \ref{['prop:VarianceBound']}, Lemmas \ref{['lem:BoundConditionalExpectation']} and \ref{['lem:BoundsforExpectations']}
  14. Extension: Growing observation window
  15. Results for Examples \ref{['examp:main']} (\ref{['num_Examp_a']})--(\ref{['num_Examp_c']})
  16. ...and 12 more sections

Key Result

Theorem 3.9

Fix $1\le \beta\le 2$ and $L>0$. Grant Assumptions assump:WellPosednessassump:WellPosedness, assump:keyassump:key, assump:Kernelassump:Kernel and assume $\sigma= {\mathcal{O}} (h)$, then the estimation error of $\hat{f}(x_0)_{h,\sigma}$ from eq:Estimator satisfies uniformly in $f\in \Sigma(\beta,L)$, as $\nu\to 0$ and $\sigma=\sigma(\nu)\to 0$. More precisely, we can decompose the estimation

Figures (2)

  • Figure 1: Realisations of the semi-linear stochastic heat equation with space-time white noise (Example \ref{['examp:main']} (\ref{['num_Examp_a']})) and Allen-Cahn non-linearity $f$ with stable points $\pm 3$ given by \ref{['eq:Simulation_f']} for $\nu=0.1$ (left) and $\nu=0.001$ (middle and right). Right: Those space-time points $(t,y)$, where $\abs{X_t(y)-1}\le 0.5$ are highlighted in black.
  • Figure 2: Monte Carlo simulation with $10^4$ runs of the performance of the estimator $\hat{f}(x_0)_{h,\sigma}$ at $\nu=0.001$ and $h=0.1$ with reaction function $f$ from \ref{['eq:Simulation_f']}. Left: Median, $5\%$- and $95\%$-quantiles of $\hat{f}(x_0)_{h,\sigma}$. Right: Interquartile range (IQR, difference between $75\%$- and $25\%$-quantiles) of $\hat{f}(x_0)_{h,\sigma}$.

Theorems & Definitions (91)

  • Remark 2.2: On the consequences of Assumption \ref{['assump:WellPosedness']} \ref{['assump:WellPosedness']}
  • Remark 2.4: On Assumption \ref{['assump:key']} \ref{['assump:key']}
  • Example 2.5
  • Remark 2.6: On the connection to spatial ergodicity
  • Remark 3.2: On the scaling \ref{['eq:Rescaling']}
  • Remark 3.3: On the dependence on $\nu$ and $\sigma$
  • Definition 3.4
  • Remark 3.5: On the definition of the estimator \ref{['eq:Estimator']}
  • Remark 3.6: On the decomposition of the estimation error
  • Remark 3.7: On knowing the diffusivity $\nu$
  • ...and 81 more