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doi: 10.3934/dcdss.2022030
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## Ornstein–Uhlenbeck semigroups on star graphs

 1 Lehrgebiet Analysis, Fakultät Mathematik und Informatik, Fern-Universität in Hagen, D-58084 Hagen, Germany 2 Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (Sa), Italy

* Corresponding author: arhandi@unisa.it

Dedicated to the memory of Rosa Maria Mininni

Received  September 2021 Revised  December 2021 Early access February 2022

Fund Project: The work of D.M. was supported by the Deutsche Forschungsgemeinschaft (Grant 397230547). A.R. is member of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM). This work was started while A.R. visited the University of Hagen. He wishes to express his gratitude to the University of Hagen for the financial support. This article is based upon work from COST Action 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu

We prove first existence of a classical solution to a class of parabolic problems with unbounded coefficients on metric star graphs subject to Kirchhoff-type conditions. The result is applied to the Ornstein–Uhlenbeck and the harmonic oscillator operators on metric star graphs. We give an explicit formula for the associated Ornstein–Uhlenbeck semigroup and give the unique associated invariant measure. We show that this semigroup inherits the regularity properties of the classical Ornstein–Uhlenbeck semigroup on $\mathbb{R}$.

Citation: Delio Mugnolo, Abdelaziz Rhandi. Ornstein–Uhlenbeck semigroups on star graphs. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022030
##### References:
 [1] W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, In C. M. Dafermos and E. Feireisl, editors, Handbook of Differential Equations: Evolutionary Equations - Vol. 1. North Holland, Amsterdam, 2004. doi: 10.1016/S1874-5717(04)80003-3. [2] S. Becker, F. Gregorio and D. Mugnolo, Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and $p$-independence of spectra, J. Math. Anal. Appl., 495 (2021), 124748.  doi: 10.1016/j.jmaa.2020.124748. [3] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Math. Surveys and Monographs. Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/surv/186. [4] C. Cattaneo, The spread of the potential on a weighted graph, Rend. Semin. Mat., Torino, 57 (1999), 221-229. [5] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.  doi: 10.1017/CBO9780511566158. [6] A. Hussein and D. Mugnolo, Laplacians with point interactions-expected and unexpected spectral properties, In J. Banasiak, A. Bobrowski, M. Lachowicz, and Y. Tomilov, editors, Semigroups of Operators – Theory and Applications (Proc. Kazimierz Dolny 2018), volume 325 of Proc. Math. & Stat., pages 46–67, New York, 2020, Springer-Verlag. doi: 10.1007/978-3-030-46079-2_3. [7] V. Kostrykin, J. Potthoff and R. Schrader, Brownian motions on metric graphs, J. Math. Phys., 53 (2012), 095206, 36 pp. doi: 10.1063/1.4714661. [8] L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups. Pure and Applied Mathematics, Chapman and Hall/CRC, Boca Raton, FL, 2007. [9] L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: Generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88.  doi: 10.1007/s00028-014-0249-z. [10] L. Lorenzi and A. Rhandi, Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations, Monographs and Research Notes in Mathematics. Chapman and Hall/CRC, Boca Raton, FL, 2021. doi: 10.1201/9780429262593. [11] G. Lumer, Connecting of local operators and evolution equations on networks, In F. Hirsch, editor, Potential Theory (Proc. Copenhagen 1979), pages 230–243, Berlin, 1980. Springer-Verlag. [12] G. Lumer, Espaces ramifiés et diffusion sur les réseaux topologiques, C.R. Acad. Sc. Paris Sér. A, 291 (1980), A627–A630. [13] G. Malenovà, Spectra of Quantum Graphs, Master's thesis, Czech Technical University in Prague, 2013. [14] G. Metafune and D. Pallara, Trace formulas for some singular differential operators and applications, Math. Nachr., 211 (2000), 127-157.  doi: 10.1002/(SICI)1522-2616(200003)211:1<127::AID-MANA127>3.0.CO;2-A. [15] G. Metafune, D. Pallara and E. Priola, Spectrum of Ornstein–Uhlenbeck operators in $L^p$ spaces with respect to invariant measures, J. Funct. Anal., 196 (2002), 40-60.  doi: 10.1006/jfan.2002.3978. [16] D. Mugnolo, What is actually a metric graph?, arXiv: 1912.07549. [17] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Underst. Compl. Syst., Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-319-04621-1. [18] D. Mugnolo, D. Noja and C. Seifert, Airy-type evolution equations on star graphs, Anal. PDE, 7 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625. [19] S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission, In C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux, editors, Polynômes Orthogonaux et Applications (Proc. Bar-le-Duc 1984), volume 1171 of Lect. Notes. Math., pages 532–541, Berlin, 1985, Springer-Verlag. doi: 10.1007/BFb0076584. [20] D. Noja, Nonlinear Schrödinger equation on graphs: Recent results and open problems, Phil. Trans. Royal Soc. London A, 372 (2014), 20130002.  doi: 10.1098/rsta.2013.0002. [21] B. S. Pavlov and M. D. Faddeev, Model of free electrons and the scattering problem, Teoret. Mat. Fiz., 55 (1983), 257-268. [22] J.-P. Roth, Le spectre du laplacien sur un graphe, In G. Mokobodzki and D. Pinchon, editors, Théorie du Potentiel - Jacques Deny (Proc. Orsay 1983), volume 1096 of Lect. Notes. Math., pages 521–539, Berlin, 1984. Springer-Verlag. doi: 10.1007/BFb0100102. [23] U. Smilansky, Irreversible quantum graphs, Waves Random Media, 14 (2004), 143-153.  doi: 10.1088/0959-7174/14/1/016. [24] M. Solomyak, On a differential operator appearing in the theory of irreversible quantum graphs, Waves Random Media, 14 (2004), 173-185.  doi: 10.1088/0959-7174/14/1/018.

show all references

##### References:
 [1] W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, In C. M. Dafermos and E. Feireisl, editors, Handbook of Differential Equations: Evolutionary Equations - Vol. 1. North Holland, Amsterdam, 2004. doi: 10.1016/S1874-5717(04)80003-3. [2] S. Becker, F. Gregorio and D. Mugnolo, Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and $p$-independence of spectra, J. Math. Anal. Appl., 495 (2021), 124748.  doi: 10.1016/j.jmaa.2020.124748. [3] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, volume 186 of Math. Surveys and Monographs. Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/surv/186. [4] C. Cattaneo, The spread of the potential on a weighted graph, Rend. Semin. Mat., Torino, 57 (1999), 221-229. [5] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.  doi: 10.1017/CBO9780511566158. [6] A. Hussein and D. Mugnolo, Laplacians with point interactions-expected and unexpected spectral properties, In J. Banasiak, A. Bobrowski, M. Lachowicz, and Y. Tomilov, editors, Semigroups of Operators – Theory and Applications (Proc. Kazimierz Dolny 2018), volume 325 of Proc. Math. & Stat., pages 46–67, New York, 2020, Springer-Verlag. doi: 10.1007/978-3-030-46079-2_3. [7] V. Kostrykin, J. Potthoff and R. Schrader, Brownian motions on metric graphs, J. Math. Phys., 53 (2012), 095206, 36 pp. doi: 10.1063/1.4714661. [8] L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups. Pure and Applied Mathematics, Chapman and Hall/CRC, Boca Raton, FL, 2007. [9] L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: Generation and heat kernel estimates, J. Evol. Equ., 15 (2015), 53-88.  doi: 10.1007/s00028-014-0249-z. [10] L. Lorenzi and A. Rhandi, Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations, Monographs and Research Notes in Mathematics. Chapman and Hall/CRC, Boca Raton, FL, 2021. doi: 10.1201/9780429262593. [11] G. Lumer, Connecting of local operators and evolution equations on networks, In F. Hirsch, editor, Potential Theory (Proc. Copenhagen 1979), pages 230–243, Berlin, 1980. Springer-Verlag. [12] G. Lumer, Espaces ramifiés et diffusion sur les réseaux topologiques, C.R. Acad. Sc. Paris Sér. A, 291 (1980), A627–A630. [13] G. Malenovà, Spectra of Quantum Graphs, Master's thesis, Czech Technical University in Prague, 2013. [14] G. Metafune and D. Pallara, Trace formulas for some singular differential operators and applications, Math. Nachr., 211 (2000), 127-157.  doi: 10.1002/(SICI)1522-2616(200003)211:1<127::AID-MANA127>3.0.CO;2-A. [15] G. Metafune, D. Pallara and E. Priola, Spectrum of Ornstein–Uhlenbeck operators in $L^p$ spaces with respect to invariant measures, J. Funct. Anal., 196 (2002), 40-60.  doi: 10.1006/jfan.2002.3978. [16] D. Mugnolo, What is actually a metric graph?, arXiv: 1912.07549. [17] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Underst. Compl. Syst., Springer-Verlag, Berlin, 2014. doi: 10.1007/978-3-319-04621-1. [18] D. Mugnolo, D. Noja and C. Seifert, Airy-type evolution equations on star graphs, Anal. PDE, 7 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625. [19] S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission, In C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux, editors, Polynômes Orthogonaux et Applications (Proc. Bar-le-Duc 1984), volume 1171 of Lect. Notes. Math., pages 532–541, Berlin, 1985, Springer-Verlag. doi: 10.1007/BFb0076584. [20] D. Noja, Nonlinear Schrödinger equation on graphs: Recent results and open problems, Phil. Trans. Royal Soc. London A, 372 (2014), 20130002.  doi: 10.1098/rsta.2013.0002. [21] B. S. Pavlov and M. D. Faddeev, Model of free electrons and the scattering problem, Teoret. Mat. Fiz., 55 (1983), 257-268. [22] J.-P. Roth, Le spectre du laplacien sur un graphe, In G. Mokobodzki and D. Pinchon, editors, Théorie du Potentiel - Jacques Deny (Proc. Orsay 1983), volume 1096 of Lect. Notes. Math., pages 521–539, Berlin, 1984. Springer-Verlag. doi: 10.1007/BFb0100102. [23] U. Smilansky, Irreversible quantum graphs, Waves Random Media, 14 (2004), 143-153.  doi: 10.1088/0959-7174/14/1/016. [24] M. Solomyak, On a differential operator appearing in the theory of irreversible quantum graphs, Waves Random Media, 14 (2004), 173-185.  doi: 10.1088/0959-7174/14/1/018.
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