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Parameter estimation from local measurements for a class of stochastic Burgers equations

Josef Janák, Enrico Priola

Abstract

We deal with a class of semilinear SPDEs driven by space-time white noise that includes the one dimensional stochastic Burgers equation. Such equations can have nonlocal and quadratic nonlinearities. We consider the problem of estimation of the diffusivity parameter in front of the second-order spatial derivative. Based on local observations in space, we study the estimator derived in [Altmeyer, Reiß, Ann. Appl. Probab.(2021)] for linear stochastic heat equation that has also been used in [Altmeyer, Cialenco, Pasemann, Bernoulli (2023)] to cover certain class of semilinear SPDEs including stochastic Burgers equations driven by trace class noise. The space-time white noise case we consider has also relevant physical motivations. After we establish new regularity results for the solution, we are able to show that our proposed estimator is strongly consistent and asymptotically normal.

Parameter estimation from local measurements for a class of stochastic Burgers equations

Abstract

We deal with a class of semilinear SPDEs driven by space-time white noise that includes the one dimensional stochastic Burgers equation. Such equations can have nonlocal and quadratic nonlinearities. We consider the problem of estimation of the diffusivity parameter in front of the second-order spatial derivative. Based on local observations in space, we study the estimator derived in [Altmeyer, Reiß, Ann. Appl. Probab.(2021)] for linear stochastic heat equation that has also been used in [Altmeyer, Cialenco, Pasemann, Bernoulli (2023)] to cover certain class of semilinear SPDEs including stochastic Burgers equations driven by trace class noise. The space-time white noise case we consider has also relevant physical motivations. After we establish new regularity results for the solution, we are able to show that our proposed estimator is strongly consistent and asymptotically normal.

Paper Structure

This paper contains 19 sections, 30 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Notation
  4. The generalized stochastic Burgers equation
  5. The observation scheme
  6. Estimation method and main results
  7. The estimator
  8. Proofs
  9. Moment estimates of the solution
  10. Regularity of the linear part
  11. Regularity of the nonlinear part
  12. Integrability of the nonlinear part
  13. Proofs of the main theorems
  14. Appendix
  15. Some results from ACP
  16. ...and 4 more sections

Key Result

Proposition 2.7

Let $X_0 \in H^{s} \cap C(\bar{\Lambda})$, for any $s \in [0, 1/2)$. Then there exists a pathwise unique mild solution $X$ to the equation eq:Burgers that satisfies $X \in C([0,T]; H^s) \cap C([0,T]; \bar{C}(\Lambda))$, ${\mathbb P}$-a.s., for any $s \in [0, 1/2)$. The nonlinear part $\widetilde{X} for any $s \in [1/2, 3/2)$ and some (random) constant $C_{\vartheta, s, \varepsilon, X_0} > 0$.

Theorems & Definitions (58)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proposition 2.9
  • proof
  • ...and 48 more