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Semigroups on generalized Sobolev spaces associated with Laplacians with applications Stochastic PDEs with singular boundary conditions

Sergio Albeverio, Zdzisław Brzeźniak, Szymon Peszat

TL;DR

The paper develops a functional-analytic framework for Laplacians with singular boundary conditions at the origin by introducing generalized Sobolev spaces $W^{\Delta,p}$ on $\mathbb{R}^n\setminus\{0\}$ and deriving precise representation theorems in terms of the Green function $G_n$ and its derivatives. It establishes a Green-type identity on the punctured space and classifies symmetric extensions of the Laplacian, including the Friedrichs extension and boundary-parameterized families $A_{\beta}$, clarifying how boundary data at the origin governs operator domains and resolvents. The authors then apply these results to heat evolution and stochastic boundary problems, constructing analytic semigroups $S_\beta$ and a generalized Dirichlet map to handle stochastic forcing at the boundary, proving well-posedness and Gaussian Markov properties in various $L^p$ spaces, with a detailed treatment of the 2D and 3D cases. Overall, the work links boundary-singular operator theory with SPDEs and stochastic boundary problems, providing explicit decompositions, Green formulas, and stochastic solution representations that facilitate analysis in both deterministic and probabilistic settings.

Abstract

Laplacians associated with domains with singular boundary conditions and are considered together with semigroups on generalized Sobolev spaces, they generate. Applications are given to stochastic PDEs with singular boundary conditions.

Semigroups on generalized Sobolev spaces associated with Laplacians with applications Stochastic PDEs with singular boundary conditions

TL;DR

The paper develops a functional-analytic framework for Laplacians with singular boundary conditions at the origin by introducing generalized Sobolev spaces on and deriving precise representation theorems in terms of the Green function and its derivatives. It establishes a Green-type identity on the punctured space and classifies symmetric extensions of the Laplacian, including the Friedrichs extension and boundary-parameterized families , clarifying how boundary data at the origin governs operator domains and resolvents. The authors then apply these results to heat evolution and stochastic boundary problems, constructing analytic semigroups and a generalized Dirichlet map to handle stochastic forcing at the boundary, proving well-posedness and Gaussian Markov properties in various spaces, with a detailed treatment of the 2D and 3D cases. Overall, the work links boundary-singular operator theory with SPDEs and stochastic boundary problems, providing explicit decompositions, Green formulas, and stochastic solution representations that facilitate analysis in both deterministic and probabilistic settings.

Abstract

Laplacians associated with domains with singular boundary conditions and are considered together with semigroups on generalized Sobolev spaces, they generate. Applications are given to stochastic PDEs with singular boundary conditions.
Paper Structure (17 sections, 45 theorems, 350 equations)

This paper contains 17 sections, 45 theorems, 350 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. The Sobolev space $W^{\Delta,p}(\mathbb{R}^n\setminus\{0\})$
  5. Representation Theorem
  6. A Green formula (or Stokes Theorem) on $\mathbb{R}^n\setminus\{0\}$
  7. Non-uniqueness of extensions of a certain class
  8. Some density results.
  9. Symmetric extensions of the Laplace operator on $\mathbb{R}^n\setminus \{0\}$
  10. Case of $n=2,3$
  11. Heat evolutions and boundary problems
  12. Stochastic boundary problem
  13. Case $\beta =0$ for $n=2,3$
  14. The case of $\beta\not =0$, the first approach
  15. Stochastic boundary problem on $\mathbb{R}^3 \setminus \{0\}$; case $\beta \not =0$
  16. ...and 2 more sections

Key Result

Proposition 3.1

Let $\tilde{\mathscr{O}}$ be an open subset of $\mathbb{R}^n$ such that its closure $\overline{\tilde{\mathscr{O}}}\subset \mathbb{R}^n\setminus\{0\}$, and let $u\in W^{\Delta,p}(\mathbb{R}^n\setminus\{0\})$. Then

Theorems & Definitions (115)

  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 105 more