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Quantitative uniqueness estimates for stochastic parabolic equations on the whole Euclidean space

Yuanhang Liu, Donghui Yang, Xingwu Zeng, Can Zhang

Abstract

In this paper, a quantitative estimate of unique continuation for the stochastic heat equation with bounded potentials on the whole Euclidean space is established. This paper generalizes the earlier results in [29] and [17] from a bounded domain to an unbounded one. The proof is based on the locally parabolic-type frequency function method. An observability estimate from measurable sets in time for the same equation is also derived.

Quantitative uniqueness estimates for stochastic parabolic equations on the whole Euclidean space

Abstract

In this paper, a quantitative estimate of unique continuation for the stochastic heat equation with bounded potentials on the whole Euclidean space is established. This paper generalizes the earlier results in [29] and [17] from a bounded domain to an unbounded one. The proof is based on the locally parabolic-type frequency function method. An observability estimate from measurable sets in time for the same equation is also derived.
Paper Structure (9 sections, 9 theorems, 114 equations)

This paper contains 9 sections, 9 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and main result
  3. Preliminary lemmas
  4. Proof of Theorem \ref{['2.3']}
  5. Proof of Corollary \ref{['Thm1']}
  6. Further comments
  7. Controllability for the backward stochastic parabolic equation
  8. Controllability for the forward stochastic parabolic equation
  9. Appendix

Key Result

Theorem 2.1

Let $0<r<R<+\infty$ and $T>0$. Assume that there is a sequence $\{x_i\}_{i\geq1}\subset\mathbb R^N$ so that Let Then there are two constants $C:= C(R)>0$ and $\theta:=\theta(r,R)\in (0,1)$ such that for any $\varphi_{0}\in L_{\mathcal{F}_0}^{2}(\Omega;L^2 (\mathbb{R}^{N}))$, the corresponding solution $\varphi$ of (1.1) satisfies

Theorems & Definitions (16)

  • Theorem 2.1
  • Corollary 2.2
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • Remark 4.3
  • ...and 6 more