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Spectral Theory of Krein-Feller Type Operators and Applications in Stochastic Fractional Elliptic and Parabolic Equations

Kelvin J. R. Almeida-Sousa, Alexandre B. Simas

TL;DR

The paper develops a spectral theory for Krein–Feller type operators $\Delta_{W,V}$ on the torus with highly irregular measures, introducing the test space $C^{\infty}_{W,V}(\mathbb{T})$ and a generalized Maclaurin expansion based on recursively defined kernels $F_k$. It furnishes a full eigenstructure in terms of generalized trig functions, proves sharp eigenvalue growth bounds, and establishes the nuclearity of $C^{\infty}_{W,V}(\mathbb{T})$ alongside traceability results for fractional powers. These analytical foundations enable regularity results and well-posedness for fractional elliptic and parabolic equations, including SPDEs driven by $V$-Gaussian white noise, within a nuclear-space setting. The work culminates in rigorous existence/uniqueness results for SPDEs valued in $D^{\prime}_{W,V}(\mathbb{T})$ and demonstrates practical applicability to stochastic fractional PDEs on irregular geometries.

Abstract

It has been shown that the space $C^{\infty}_{W,V}(\mathbb{T})$, introduced in Simas and Sousa (Potential Analysis, 2025), is the natural regularity space for solutions of the eigenvalue problem $Δ_{W,V} u = λu$ on the torus $\mathbb{T}$, where $Δ_{W,V} = \frac{d^{+}}{dV}\frac{d^{-}}{dW}$ is the Krein Feller operator in the case where $W$ and $V$ are strictly increasing and right continuous (respectively left continuous), possibly with dense sets of discontinuities. In this work we provide conditions ensuring that every function in $C^{\infty}_{W,V}(\mathbb{T})$, which may be highly discontinuous, admits a series expansion that generalizes the classical Taylor expansion. A central feature of our approach is that all proofs are nonstandard, since classical analytical and spectral arguments cannot be adapted to this singular setting. Using these methods we characterize the eigenvectors of $Δ_{W,V}$ in terms of generalized trigonometric functions and obtain an asymptotic lower bound for the associated eigenvalues. We also derive a sharp upper bound for the convergence exponent of these eigenvalues, and as a consequence we prove that $C^{\infty}_{W,V}(\mathbb{T})$ is a nuclear space. Further consequences include results on the asymptotic behavior of eigenvalues of compact operators and improvements in traceability. As a final application we establish existence results for generalized fractional stochastic and deterministic differential equations, as well as for parabolic stochastic partial differential equations acting on nuclear spaces.

Spectral Theory of Krein-Feller Type Operators and Applications in Stochastic Fractional Elliptic and Parabolic Equations

TL;DR

The paper develops a spectral theory for Krein–Feller type operators on the torus with highly irregular measures, introducing the test space and a generalized Maclaurin expansion based on recursively defined kernels . It furnishes a full eigenstructure in terms of generalized trig functions, proves sharp eigenvalue growth bounds, and establishes the nuclearity of alongside traceability results for fractional powers. These analytical foundations enable regularity results and well-posedness for fractional elliptic and parabolic equations, including SPDEs driven by -Gaussian white noise, within a nuclear-space setting. The work culminates in rigorous existence/uniqueness results for SPDEs valued in and demonstrates practical applicability to stochastic fractional PDEs on irregular geometries.

Abstract

It has been shown that the space , introduced in Simas and Sousa (Potential Analysis, 2025), is the natural regularity space for solutions of the eigenvalue problem on the torus , where is the Krein Feller operator in the case where and are strictly increasing and right continuous (respectively left continuous), possibly with dense sets of discontinuities. In this work we provide conditions ensuring that every function in , which may be highly discontinuous, admits a series expansion that generalizes the classical Taylor expansion. A central feature of our approach is that all proofs are nonstandard, since classical analytical and spectral arguments cannot be adapted to this singular setting. Using these methods we characterize the eigenvectors of in terms of generalized trigonometric functions and obtain an asymptotic lower bound for the associated eigenvalues. We also derive a sharp upper bound for the convergence exponent of these eigenvalues, and as a consequence we prove that is a nuclear space. Further consequences include results on the asymptotic behavior of eigenvalues of compact operators and improvements in traceability. As a final application we establish existence results for generalized fractional stochastic and deterministic differential equations, as well as for parabolic stochastic partial differential equations acting on nuclear spaces.

Paper Structure

This paper contains 16 sections, 42 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. A Basic Convention and the Space $C^{\infty}_{W,V}(\mathbb{T})$
  3. Generalized Taylor Expansion in $C^{\infty}_{W,V}(\mathbb{T})$
  4. Spectral asymptotics of $\Delta_{W,V}$ and nuclearity of $C^{\infty}_{W,V}(\mathbb{T})$
  5. Applications
  6. Organization of the Manuscript
  7. Review of $W$-$V$-Sobolev Spaces and the Generalized Laplacian
  8. Generalized Maclaurin Expansion for Functions in $C^{\infty}_{W,V}(\mathbb{R})$
  9. Characterization of the eigenvectors of $-\Delta_{W,V}$
  10. Traceability of $\Delta_{W,V}$ under Dirichlet Boundary Conditions
  11. Asymptotic behavior of the eigenvalues of $\Delta_{W,V}$
  12. Applications
  13. Nuclearity of $C^{\infty}_{W,V}(\mathbb{T})$
  14. Fractional Stochastic Differential Equations Driven by $V$-Gaussian White Noise
  15. Stochastic Partial Differential Equations
  16. ...and 1 more sections

Key Result

Proposition 1

The space $C^{\infty}_{V,W}(\mathbb{T})$ is dense in $L^2_V(\mathbb{T})$. Furthermore, $\left\{D^{+}_{V}g : g \in C^{\infty}_{V,W}(\mathbb{T})\right\}$ is dense in $L^2_{V,0}(\mathbb{T})$.

Theorems & Definitions (94)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Theorem 2
  • Definition 4
  • Proposition 3
  • Definition 5
  • Remark 1
  • Lemma 4
  • ...and 84 more