About semilinear low dimension Bessel PDEs

Alberto Ohashi; Francesco Russo; Alan Teixeira

doi:10.1007/s40072-025-00386-9

About semilinear low dimension Bessel PDEs

Alberto Ohashi
, Francesco Russo
, Alan Teixeira

Departamento de Matemática
Universidade de Brasília
Unité de Mathématiques Appliquées

Research output: Contribution to journal › Article › peer-review

Abstract

We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator Lδ with low dimension 0<δ<1. Lδ is a local operator, whose drift component is the derivative of x↦log(|x|): in particular it is a Schwartz distribution, which is not the derivative of a continuous real function. The solutions are intended in a duality (weak) sense with respect to state space L2(R+,dμ),μ being an invariant measure for the Bessel semigroup.

Original language	English
Pages (from-to)	1144-1178
Number of pages	35
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Volume	14
Issue number	3
DOIs	https://doi.org/10.1007/s40072-025-00386-9
Publication status	Published - 1 Aug 2026

Keywords

Bessel processes
Friedrichs extension
Kolmogorov equation
Mild and weak solutions
SDEs with distributional drift
Self-adjoint operators

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10.1007/s40072-025-00386-9

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abstract = "We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator Lδ with low dimension 0<δ<1. Lδ is a local operator, whose drift component is the derivative of x↦log(|x|): in particular it is a Schwartz distribution, which is not the derivative of a continuous real function. The solutions are intended in a duality (weak) sense with respect to state space L2(R+,dμ),μ being an invariant measure for the Bessel semigroup.",

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AU - Ohashi, Alberto

AU - Russo, Francesco

AU - Teixeira, Alan

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.

PY - 2026/8/1

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N2 - We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator Lδ with low dimension 0<δ<1. Lδ is a local operator, whose drift component is the derivative of x↦log(|x|): in particular it is a Schwartz distribution, which is not the derivative of a continuous real function. The solutions are intended in a duality (weak) sense with respect to state space L2(R+,dμ),μ being an invariant measure for the Bessel semigroup.

AB - We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator Lδ with low dimension 0<δ<1. Lδ is a local operator, whose drift component is the derivative of x↦log(|x|): in particular it is a Schwartz distribution, which is not the derivative of a continuous real function. The solutions are intended in a duality (weak) sense with respect to state space L2(R+,dμ),μ being an invariant measure for the Bessel semigroup.

KW - Bessel processes

KW - Friedrichs extension

KW - Kolmogorov equation

KW - Mild and weak solutions

KW - SDEs with distributional drift

KW - Self-adjoint operators

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SN - 2194-0401

VL - 14

SP - 1144

EP - 1178

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

IS - 3

ER -

About semilinear low dimension Bessel PDEs

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