×

McKean-Vlasov equations with positive feedback through elastic stopping times. (English) Zbl 1498.35640

Summary: We prove existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baker, G. and Shkolnikov, M.: Zero kinetic undercooling limit in the supercooled Stefan problem. arXiv preprint 2003.07239 (2020) · Zbl 1494.35197
[2] Bayraktar, E., Guo, W., Tang, W. and Zhang, Y.: McKean-Vlasov equations involving hitting times: blow-ups and global solvability. arXiv preprint 2010.14646 (2020)
[3] Cáceres, M.J., Carillo, J.A., and Perthame, B.: Analysis of nonlinear noisy integrate & fire neuron models: blow-ups and steady states. J. Math. Neurosci. 1:Art. 7, 33 (2011) · Zbl 1259.35198
[4] Carillo, J.A., González, M.D.M., Gualdani, M.P. and Schonbeck, M.E.: Classical solutions for a nonlinear Fokker-Planck equation arising in computational neuroscience. Comm. Partial Differential Equations 38(3), 385-409 (2013) · Zbl 1282.35382
[5] Cuchiero, C., Rigger, S. and Svaluto-Ferro, S.: Propagation of Minimality in the supercooled Stefan problem. arXiv preprint 2010.03580 (2020)
[6] Delarue, F., Nadtochiy, S. and Shkolnikov, M.: Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness. arXiv preprint 1902.05174 (2019) · Zbl 1489.35309
[7] Delarue, F., Inglis, D., Rubenthaler, S. and Tanré, E.: Global Solvability of a networked integrate-and-fire model of McKean-Vlasov type Ann. Appl. Prob. 25, 2096-2133 (2015) · Zbl 1322.60085
[8] Delarue, F., Inglis, D., Rubenthaler, S. and Tanré, E.: Particle Systems with a singular mean-field self-excitation. Stoch. Proc. Appl. 125, 2451-2492 (2015) · Zbl 1328.60134
[9] Hambly, B.M., Ledger, S. and Søjmark, A.: A McKean-Vlasov equation with positive feedback and blow-ups. Annals of Applied Probability 29 (4), 2338-2373 (2019) · Zbl 1423.35224
[10] Ledger, S. and Søjmark, A.: Uniqueness for the contagious McKean-Vlasov systems in the weak feedback regime. Bull. London Math. Soc. 52, 448-463 (2020) · Zbl 1439.35291
[11] Ledger, S. and Søjmark, A.: At the Mercy of the Common Noise: Blow-ups in a Conditional McKean-Vlasov Problem. Electron. J. Probab. 26, 1-39 (2021) · Zbl 1469.60161
[12] Nadtochiy, S. and Shkolnikov, M.: Particle Systems with singular interaction through hitting times: application in systemic risk. The Annals of Applied Probability 29 (1), 89-129 (2019) · Zbl 1417.35204
[13] Nadtochiy, S. and Shkolnikov, M.: Mean field systems on networks, with singular interaction through hitting times. The Annals of Probability 48 (3), 1520-1556 (2020) · Zbl 1448.82032
[14] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion Vol 3. Springer Berlin, Grundlehren der mathematischen Wissenschaften (1999) · Zbl 0917.60006
[15] Sznitman, A.: Topics in propagation of chaos. Saint-Flour XIX 1989, 165-251, Springer (1991)
[16] Whitt, W.: Stochastic-process limits: an introduction to stochastic-process limits and their application to queues. Springer Science & Business Media (2002) · Zbl 0993.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.