New Bernstein functions and subordination representations for time-nonlocal evolution equations
Abstract
This paper establishes subordination representations from multi-term time-fractional differential equations to single-term ones, a direction complementary to the previously known subordination from single-term to multi-term equations; combining the two complementary directions also yields subordination representations between two multi-term equations. The crucial step is to prove that certain implicitly defined functions are Bernstein functions. We show that the quotient (∑j=1m ajλαj)/(∑j=1m ajαjλαj) is a Stieltjes function whenever 0≤α1−αm≤ 1, and consequently that the inverse function of λ=(∑j=1m ajφαj)1/β is a Bernstein function for every 0<β≤αm. As applications, we obtain subordination representations with a clear probabilistic meaning: from multi-term to single-term time-fractional diffusion-wave equations, between abstract nonlocal-in-time equations with kernels of type (𝒫𝒞) (including equations of tempered stable type), and between nonlocal-in-time telegraph equations. A necessary and sufficient condition for [(λ+b)β−bβ]γ to be a Bernstein function also yields a negative result: in general, no such subordination exists from an α-stable type dynamics to a tempered β-stable one when α<1.
Related
- The Lévy-Khinchine type representations and generalized functions on Bernstein spaces
- Dissipative Hamiltonian representations of linear evolution equations
- Regression Functions as Fields on Covariate Space: Evolution Equations, the Hessian, and Identification
- Bernstein and Stieltjes properties of some functions involving logarithmic functions
- Calabi-Bernstein type theorems for entire solutions of certain nonlinear elliptic equations in the plane
BibTeX
@misc{khor2026bernstein,
title = {{New Bernstein functions and subordination representations for time-nonlocal evolution equations}},
author = {Calvin Khor and Cheng-Gang Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25278},
eprint = {2610.25278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25278},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - New Bernstein functions and subordination representations for time-nonlocal evolution equations
AU - Calvin Khor
AU - Cheng-Gang Li
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25278
UR - https://arxiv.org/abs/2610.25278
AB - This paper establishes subordination representations from multi-term time-fractional differential equations to single-term ones, a direction complementary to the previously known subordination from single-term to multi-term equations; combining the two complementary directions also yields subordination representations between two multi-term equations. The crucial step is to prove that certain implicitly defined functions are Bernstein functions. We show that the quotient (∑_j=1^m a_jλ^α_j)/(∑_j=1^m a_jα_jλ^α_j) is a Stieltjes function whenever 0≤α_1−α_m≤ 1, and consequently that the inverse function of λ=(∑_j=1^m a_jφ^α_j)^1/β is a Bernstein function for every 0<β≤α_m. As applications, we obtain subordination representations with a clear probabilistic meaning: from multi-term to single-term time-fractional diffusion-wave equations, between abstract nonlocal-in-time equations with kernels of type (𝒫𝒞) (including equations of tempered stable type), and between nonlocal-in-time telegraph equations. A necessary and sufficient condition for [(λ+b)^β−b^β]^γ to be a Bernstein function also yields a negative result: in general, no such subordination exists from an α-stable type dynamics to a tempered β-stable one when α<1.
ER -