Geometric and stochastic framework for a regularized p(x)-Laplacian
Abstract
For a prescribed classical profile and a fixed positive regularization parameter, we construct an Itô diffusion whose interior generator reproduces the regularized p(x)-Laplacian when applied to that profile. We determine the exact tangent spectra of the unregularized and regularized fluxes, characterize their behavior at zero gradient, and establish quantitative ellipticity bounds on bounded gradient ranges. The full divergence expansion identifies a covariance equal to twice the gradient derivative of the regularized flux, evaluated at the profile gradient, and a logarithmic drift generated by spatial variation of the variable exponent p(x). Under explicit classical regularity assumptions, we establish boundedness and local Lipschitz continuity of the coefficients and prove strong existence and pathwise uniqueness up to domain exit. We identify the interior generator of the boundary-stopped process and verify the profile-reproduction identity. Further consequences include initial-point stability for a fixed global extension, localized martingale identities, and Dynkin's formula at bounded stopping times. An explicit nonconstant exponent example illustrates the construction at a critical point, and the coefficient selection is shown to be canonical under the flux-based prescriptions rather than determined uniquely by profile reproduction alone.
Related
- The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the p-Laplacian
- Regularized Small Area Estimation with Graph Laplacian Benchmarking priors
- Positive normalized solutions for a singular regularized p(x)-Laplacian Dirichlet problem
- A Stochastic Proximal-Gradient Method for Regularized Generally Constrained Continuous Optimization
- A Functional SVD Framework for Regularized Multivariate Functional PCA with Dual Penalization
BibTeX
@misc{avci2026geometric,
title = {{Geometric and stochastic framework for a regularized $p(x)$-Laplacian}},
author = {Mustafa Avci},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05609},
eprint = {2610.05609},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05609},
}
RIS
TY - UNPB
TI - Geometric and stochastic framework for a regularized p(x)-Laplacian
AU - Mustafa Avci
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05609
UR - https://arxiv.org/abs/2610.05609
AB - For a prescribed classical profile and a fixed positive regularization parameter, we construct an Itô diffusion whose interior generator reproduces the regularized p(x)-Laplacian when applied to that profile. We determine the exact tangent spectra of the unregularized and regularized fluxes, characterize their behavior at zero gradient, and establish quantitative ellipticity bounds on bounded gradient ranges. The full divergence expansion identifies a covariance equal to twice the gradient derivative of the regularized flux, evaluated at the profile gradient, and a logarithmic drift generated by spatial variation of the variable exponent p(x). Under explicit classical regularity assumptions, we establish boundedness and local Lipschitz continuity of the coefficients and prove strong existence and pathwise uniqueness up to domain exit. We identify the interior generator of the boundary-stopped process and verify the profile-reproduction identity. Further consequences include initial-point stability for a fixed global extension, localized martingale identities, and Dynkin's formula at bounded stopping times. An explicit nonconstant exponent example illustrates the construction at a critical point, and the coefficient selection is shown to be canonical under the flux-based prescriptions rather than determined uniquely by profile reproduction alone.
ER -