Open Access
2023 On the parabolic Cauchy problem for quantum graphs with vertex noise
Mihály Kovács, Eszter Sikolya
Author Affiliations +
Electron. J. Probab. 28: 1-20 (2023). DOI: 10.1214/23-EJP962


We investigate the parabolic Cauchy problem associated with quantum graphs including Lipschitz or polynomial type nonlinearities and additive Gaussian noise perturbed vertex conditions. The vertex conditions are the standard continuity and Kirchhoff assumptions in each vertex. In the case when only Kirchhoff conditions are perturbed, we can prove existence and uniqueness of a mild solution with continuous paths in the standard state space H of square integrable functions on the edges. We also show that the solution is Markov and Feller. Furthermore, assuming that the vertex values of the normalized eigenfunctions of the self-adjoint operator governing the problem are uniformly bounded, we show that the mild solution has continuous paths in the fractional domain space associated with the Hamiltonian operator, Hα for α<14. This is the case when the Hamiltonian operator is the standard Laplacian perturbed by a potential. We also show that if noise is present in both type of vertex conditions, then the problem admits a mild solution with continuous paths in the fractional domain space Hα with α<14 only. These regularity results are the quantum graph analogues obtained by da Prato and Zabczyk [9] in case of a single interval and classical boundary Dirichlet or Neumann noise.

Funding Statement

M. Kovács acknowledges the support of the Marsden Fund of the Royal Society of New Zealand through grant no. 18-UOO-143, the Swedish Research Council (VR) through grant no. 2017-04274 and the National Research, Development, and Innovation Fund of Hungary under Grant no. TKP2021-NVA-02 and Grant no. K-131545. E. Sikolya was supported by the OTKA grant no. 135241.


The authors would like to thank the anonymous referee for the careful reading of the manuscript and for the useful comments that helped them to improve the presentation and the results of the paper significantly.


Download Citation

Mihály Kovács. Eszter Sikolya. "On the parabolic Cauchy problem for quantum graphs with vertex noise." Electron. J. Probab. 28 1 - 20, 2023.


Received: 7 July 2022; Accepted: 13 May 2023; Published: 2023
First available in Project Euclid: 2 June 2023

arXiv: 2207.01944
MathSciNet: MR4596349
MathSciNet: MR4529085
Digital Object Identifier: 10.1214/23-EJP962

Primary: 35R60 , 60H15 , 81Q35
Secondary: 35R02 , 47D06

Keywords: Hamiltonian operator , quantum graph , white-noise vertex conditions

Vol.28 • 2023
Back to Top