2025/02/04 by Alexandre B. Simas, Simas, Alexandre B., Kelvin J. R. Sousa +1 · 1 citation
Computer Science · Economics, Econometrics and Finance · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2502.02264
openalex publication_date 2025/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define the operator D+VD-W:=ΔW,V on the one-dimensional torus \mathbbT. Here, W and V are functions inducing (possibly atomic) positive Borel measures on \mathbbT, and the derivatives are generalized lateral derivatives. For the first time in this work, the space of test functions C∞W,V(\mathbbT) emerges as the natural regularity space for solutions of the eigenproblem associated with ΔW,V. Moreover, these spaces are essential for characterizing the energetic space HW,V(\mathbbT) as a Sobolev-type space. By observing that the Sobolev-type spaces HW,V(\mathbbT) with additional Dirichlet conditions are reproducing kernel Hilbert spaces, we introduce the so-called W-Brownian bridges as mean-zero Gaussian processes with associated Cameron-Martin spaces derived from these spaces. This framework allows us to introduce W-Brownian motion as a Feller process with a two-parameter semigroup and càdlàg sample paths, whose jumps are subordinated to the jumps of W. We establish a deep connection between W-Brownian motion and these Sobolev-type spaces through their associated Cameron-Martin spaces. Finally, as applications of the developed theory, we demonstrate the existence and uniqueness of related deterministic and stochastic differential equations.