Abstract
Under suitable conditions on a family (I(t))t≥ 0 of Lipschitz mappings on a complete metric space, we show that, up to a subsequence, the strong limit \(S(t):=\lim _{n\to \infty }(I(t 2^{-n}))^{2^{n}}\) exists for all dyadic time points t, and extends to a strongly continuous semigroup (S(t))t≥ 0. The common idea in the present approach is to find conditions on the generating family (I(t))t≥ 0, which can be transferred to the semigroup. The construction relies on the Lipschitz set, which is invariant under iterations and allows to preserve Lipschitz continuity to the limit. Moreover, we provide a verifiable condition which ensures that the infinitesimal generator of the semigroup is given by \(\lim _{h\downarrow 0}\tfrac {I(h)x-x}{h}\), whenever this limit exists. The results are illustrated with several examples of nonlinear semigroups such as robustifications and perturbations of linear semigroups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces Springer Monographs in Mathematics. Springer, New York (2010)
Bénilan, P., Crandall, M.G.: Completely Accretive Operators. In: Semigroup Theory and Evolution Equations (Delft, 1989), Volume 135 of Lecture Notes in Pure and Appl. Math., pp 41–75. Dekker, New York (1991)
Blessing, J., Kupper, M.: Viscous hamilton–Jacobi equations in exponential Orlicz hearts. arXiv:2104.06433 (2021)
Brézis, H.: Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. In: Contributions to Nonlinear Functional Analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), pp 101–156. Academic Press, New York (1971)
Butko, Y.A.: The method of Chernoff approximation. In: Conference on Semigroups of Operators: Theory and Applications, pp 19–46. Springer (2018)
Butko, Y.A., Smolyanov, O.G., Shilling, R.L.: Feynman formulas for Feller semigroups. Dokl. Akad. Nauk 434(1), 7–11 (2010)
Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funct Anal. 2, 238–242 (1968)
Chernoff, P.R.: Product formulas, nonlinear semigroups, and addition of unbounded operators, volume 140 American Mathematical Soc. (1974)
Coquet, F., Hu, Y., Mémin, J., Peng, S.: Filtration-consistent nonlinear expectations and related g-expectations. Probab. Theory Related Fields 123(1), 1–27 (2002)
Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math Soc. (N.S.) 27(1), 1–67 (1992)
Delbaen, F., Hu, Y., Bao, X.: Backward SDEs with superquadratic growth. Probab. Theory Relat. Fields 150(1-2), 145–192 (2011)
Denk, R., Kupper, M., Nendel, M.: A semigroup approach to nonlinear Lévy processes. Stochastic Process Appl. 130, 1616–1642 (2020)
Denk, R., Kupper, M., Nendel, M.: Convex semigroups on lattices of continuous functions. Forthcoming Publ. Res. Inst. Math. Sci (2021)
Denk, R., Kupper, M., Nendel, M.: Convex semigroups on Lp-like spaces. J. Evol. Equ. 21(2), 2491–2521 (2021)
El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Financ. 7(1), 1–71 (1997)
Evans, L.C.: Nonlinear Semigroup Theory and Viscosity Solutions of Hamilton-Jacobi PDE. In: Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, D.C., 1985), volume 1248 of Lecture Notes in Math, pp 63–77. Springer, Berlin (1987)
Gomilko, A., Kosowicz, S., Tomilov, Y.: A general approach to approximation theory of operator semigroups. J. Math. Pures Appl. 127(9), 216–267 (2019)
Hollender, J.: Lévy-Type Processes under Uncertainty and Related Nonlocal Equations, Phd thesis TU Dresden (2016)
Hu, M., Peng, S.: g-Lévy processes under sublinear expectations. Preprint (2009)
Kato, T.: Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19, 508–520 (1967)
Kazi-Tani, N., Possamaï D., Zhou, C., et al.: Second-order bsdes with jumps: formulation and uniqueness. Ann. Appl. Probab. 25(5), 2867–2908 (2015)
Kühn, F.: Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear Lévy(-type) processes. ALEA Lat. Am. J. Probab. Math. Stat., (16):531–559 (2019)
Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems. Modern Birkhäuser Classics. Birkhäuser/Springer, Basel (1995). [2013 reprint of the 1995 original] [MR1329547]
Nendel, M., Röckner, M.: Upper envelopes of families of Feller semigroups and viscosity solutions to a class of nonlinear Cauchy problems. Forthcoming in SIAM J. Control Optim (2021)
Neufeld, A., Nutz, M.: Nonlinear lévy processes and their characteristics. Trans. Amer. Math. Soc. 369(1), 69–95 (2017)
Nisio, M.: On a non-linear semi-group attached to stochastic optimal control. Publ. Res. Inst. Math. Sci. 12(2), 513–537 (1976/77)
Orlov, Y.N., Sakbaev, V.Z., Smolyanov, O.G.: Feynman formulas for nonlinear evolution equations. Dokl. Akad Nauk 477(3), 271–275 (2017)
Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Volume 44 of Applied Mathematical Sciences. Springer, New York (1983)
Peng, S.: G-expectation, G-Brownian Motion and Related Stochastic Calculus of Itô Type. In: Stochastic Analysis and Applications, Volume 2 of Abel Symp, pp 541–567. Springer, Berlin (2007)
Peng, S.: Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process Appl. 118(12), 2223–2253 (2008)
Smolyanov, O.G., Tokarev, A.G., Truman, A.: Hamiltonian Feynman path integrals via the Chernoff formula. J. Math Phys. 43(10), 5161–5171 (2002)
Soner, H.M., Touzi, N., Zhang, J.: Wellposedness of second order backward SDEs. Probab. Theory Relat. Fields 153(1-2), 149–190 (2012)
Trotter, H.F.: Approximation of semi-groups of operators. Pacific J. Math. 8, 887–919 (1958)
Trotter, H.F.: On the product of semi-groups of operators. Proc. Amer. Math. Soc. 10, 545–551 (1959)
Wnuk, W.: Banach Lattices with Order Continuous Norms. Advanced topics in mathematics Polish Scientific Publishers PWN (1999)
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Robert Denk, Stephan Eckstein, Karsten Herth, Markus Kunze, Max Nendel, Reinhard Racke and Liming Yin for helpful comments and discussions. Furthermore, we thank an anonymous referee for valuable comments and feedback on an earlier version of the paper.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Blessing, J., Kupper, M. Nonlinear Semigroups Built on Generating Families and their Lipschitz Sets. Potential Anal 59, 857–895 (2023). https://doi.org/10.1007/s11118-022-09985-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-022-09985-w