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

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 $\Delta_{W,V} u = \lambda u$ on the torus $\mathbb{T}$, where $\Delta_{W,V} = \frac{d^{+}}{dV}\frac{d^{-}}{dW}$ 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 $\Delta_{W,V}$ 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.