Stochastic parabolic equations on graphs
Abstract
A biodegradable stent is a mesh that is used to treat the narrow or closed part of the artery to open and restore normal blood flow, which is made of a biodegradable material.
{However, since the struts in the stent are thin, a simple structural model (called the one-dimensional curved rod model) can be used to mathematically model the stent.
So differential diffusion equations can describe the degradation of the stent. The movement of the blood causes microscopic trembling of the edges of the wall. Therefore, the perturbation of the corresponding differential equation by a white noise type term should correspond to the real situation.
We consider some linear parabolic equations on graphs with white noise terms. To investigate the initial-boundary value problem for these equations, we reduce it to the corresponding deterministic problem.
First, we prove the existence and uniqueness of the we ak solution to deterministic problem for parabolic equations on graphs.
Finally, same results are obtained for linear stochastic parabolic equations on graphs.
References
B. Zugec, Regularity of a parabolic differential equation on graphs, Mathematics, 11 (2023), no.21, Article ID 4453, 1–10. https://doi.org/10.3390/math11214453
M. Stoll, M. Winkler, Optimal Dirichlet control of partial differential equations on networks, Electron. Trans. Numer. Anal., 54 (2021), 392–419. https://doi.org/10.1553/etna_vol54s392
C. Chen, Y. Xiong, Z. Li, Y. Chen, Flexibility of biodegradable polymer stents with different strut geometries, Materials., 13 (2020), no.15, article ID 3332, 1–12. https://doi.org/10.3390/ma13153332
A. Colombo, E. Karvouni, Biodegradable stents: fulfilling the mission and stepping away, Circulation (J. Am. Heart Assoc.), 102 (2000), no.4, 371–73. https://doi.org/10.1161/01.CIR.102.4.371
S. Canic, J. Tambaca, Cardiovascular stents as PDE nets: 1D vs. 3D, IMA J. Appl. Math., 77 (2012), no.6, 748–770. https://doi.org/10.1093/imamat/hxs047
M.K. Fijavz, D. Mugnolo, E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), no.2, 219–240. https://doi.org/10.1007/s00245-006-0887-9
Yu. Golovaty, V. Flyud, Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions, Open Math., 15 (2017), no.1, 404–419. https://doi.org/10.1515/math-2017-0030
Yu. Golovaty, Quantum graphs: Coulomb-type potentials and exactly solvable models, Annales Henri Poincare, 24 (2023), 2557–2585. https://doi.org/10.1007/s00023-023-01270-9
Kh. Buhrii, Yu. Golovaty, SIR models on complex networks and impact of vaccination, 2023 IEEE 13th Inter. Conf. on Electronics and Information Technologies, ELIT 2023, Proceedings, 2023, 37–42.
A. Weller, Numerical methods for parabolic partial differential equations on metric graphs, PhD Thesis, Universitat zu Koln, 2024. https://kups.ub.uni-koeln.de/73182/1/weller_NumPDEMeGra_dissertation_24.pdf
A. Bensoussan, R. Temam, Equations stochastiques du type Navier-Stokes, J. Funct. Anal., 13 (1973), no.2, 195–222. https://doi.org/10.1016/0022-1236(73)90045-1
G. Vage, Stochastic differential equations and Kondratiev spaces, Dr. Ing. Thesis. Trondheim, 1995. https://www.nb.no/items/5ae9361e3c1ce3db1c4fe812813d881c?page=0
H. Liang, L. Hou, J. Ming, The velocity tracking problem for Wick-stochastic Navier–Stokes flows using Weiner chaos expansion, J. Computat. Appl. Math., 307 (2016), 25–36. https://doi.org/10.1016/j.cam.2016.04.030
D. Breit, E. Feireisl, M. Hofmanova, Stochastically forced compressible fluid flows, Berlin, Boston: Walter de Gruyter, 2018.
N. Buhrii, O. Buhrii, O. Domanska, Semilinear stochastic parabolic equation with variable exponent of nonlinearity, Visn. Lviv Univ., Ser. Mech.-Math., 93 (2022), 108–121. https://doi.org/10.1016/0022-1236(73)90045-1
O. Buhrii, M. Khoma, I.-M. Vovk, Stochastic differentiations in equations with variable exponents of nonlinearity, Visn. Lviv Univ., Ser. Mech.-Math., 96 (2024), 83–104. https://doi.org/10.30970/vmm.2024.96.071-092
M.I. Freidlin, A.D. Wentzell, Diffusion processes on graphs and the averaging principle, Ann. Probab., 21 (1993), no.4, 2215–2245. https://doi.org/10.1214/aop/1176989018
M. Freidlin, S.-J. Sheu, Diffusion processes on graphs: stochastic differential equations, large deviation principle, Probab. Theory Relat. Fields, 116 (2000), no.2, 181–220. https://doi.org/10.1007/PL00008726
M. Thorpe, T. Nguyen, H. Xia, T. Strohmer, A. Bertozzi, S. Osher, B. Wang GRAND++: Graph neural diffusion with a source term, Conference paper at ICLR-2022, 1–21.
H. Hajri, O. Raimond Stochastic flows on metric graphs, Electron. J. Probab., 19 (2014), no.12, 1–20. https://doi.org/10.1214/EJP.v19-2773
H. Hajri, O. Raimond, Stochastic flows and an interface SDE on metric graphs, Stochastic processes and their applications, 126 (2016), no.1, 33–65. https://doi.org/10.1016/j.spa.2015.07.014
R.A. Adams, Sobolev spaces, New York, San Francisco, London: Academic Press, 1975.
G. Leoni, A first course in Sobolev spaces, AMS, Providence, RI, 2010.
M. Chipot, Elements of nonlinear analysis, Basel, Boston, Berlin: Birkhauser, 2012.
O. Buhrii, N. Buhrii, Integro-differential systems with variable exponents of nonlinearity, Open Math., 15 (2017), no.1, 859–883. https://doi.org/10.1515/math-2017-0069
V. Kadets, A course in functional analysis and measure theory, Springer International Publ., 2018.
S.J. Dilworth, Some probabilistic inequalities with applications to functional analysis, Contemporary Mathematics, 144 (1993), AMS, Providence, RI, 53–67.
V. Lupulescu, C. Lungan, Random integral equations on time scales, Opuscula Math., 33 (2013), no.2, 323–335. https://doi.org/10.7494/OpMath.2013.33.2.323
O. Buhrii, N. Buhrii, V. Vlasov, On stochastic space-time Paley-Wiener-Zygmund integral, Mathematics, Informatics, Physics: Science and Education, 1 (2024), no.1, 13–26. https://doi.org/10.31652/3041-1955-2024-01-02
M. Siebenmorgen, Quadrature methods for elliptic PDEs with random diffusion, PhD Thesis. Bonn, 2016.
Q. Lu, J. Yong, X. Zhang, Representation of Ito integrals by Lebesgue-Bochner integrals, J. European Math. Soc., 14 (2012), no.6, 1795–1823. https://doi.org/10.4171/JEMS/347
Y. Wang, X. Zhu, P. Kloeden, Compactness in Lebesgue–Bochner spaces of random variables and the existence of mean-square random attractors, Stochastics and Dynamics, 19 (2019), no.4, 1950032. https://doi.org/10.1142/S0219493719500321
L.C. Evans, An introduction to stochastic differential equations, V.82, AMS, 2012.
Y. Mishura, G. Shevchenko, Theory and statistical applications of stochastic processes, London, Hoboken: ISTE Ltd and John Wiley and Sons, 2017.
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