The Master equation in mean field theory
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Such motifs are known to be highly overrepresented in transcriptional, signaling, and neural networks compared with random networks.
“Such motifs are known to be highly overrepresented in transcriptional, signaling, and neural networks compared with random networks [48-50].”
From Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies (2026) · cites this paper as [48]
The Introduction discusses probability measures and mean field theory, while §4.4 discusses approaches to the Master equation; neither addresses biological network motifs. Searches for overrepresented motifs and for transcriptional, signaling, neural, and random networks found no passage supporting the claim in our copy. (LLM, conf=0.78)
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Cites
Lasry, J., & Lions, P. (2007) · Japanese journal of mathematics · doi:10.1007/s11537-007-0657-8 · Cited by 2,639 · see citing works
Mean Field Games and Mean Field Type Control Theory
Bensoussan, A., Frehse, J., & Yam, S. C. P. (2013) · SpringerBriefs in mathematics · doi:10.1007/978-1-4614-8508-7 · Cited by 789 · see citing works
Self-citationStochastic Hamilton–Jacobi–Bellman Equations
Péng, S. (1992) · SIAM Journal on Control and Optimization · doi:10.1137/0330018 · Cited by 337 · see citing works
Mean Field Games and systemic risk
Carmona, R. A., Fouque, J., & Sun, L. (2015) · Communications in Mathematical Sciences · doi:10.4310/cms.2015.v13.n4.a4 · Cited by 256 · see citing works
Linear-Quadratic Mean Field Games
Bensoussan, A., Sung, K. C. J., Yam, S. C. P., et al. (2015) · Journal of Optimization Theory and Applications · doi:10.1007/s10957-015-0819-4 · Cited by 247 · see citing works
Self-citationThe Master Equation for Large Population Equilibriums
Carmona, R. A., & Delarue, F. (2014) · Springer proceedings in mathematics & statistics · doi:10.1007/978-3-319-11292-3_4 · Cited by 118 · see citing works
Dynamic programming for mean-field type control
Laurière, M., & Pironneau, O. (2014) · Comptes Rendus Mathématique · doi:10.1016/j.crma.2014.07.008 · Cited by 66 · see citing works
Linear-Quadratic Mean Field GamesPreprint
Bensoussan, A., Sung, J. J. �., Yam, P., et al. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1404.5741 · Cited by 32 · see citing works
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On the interpretation of the Master Equation
Bensoussan, A., Frehse, J., & Yam, S. C. P. (2017) · Stochastic Processes and their Applications · doi:10.1016/j.spa.2016.10.004 · Cited by 76 · see citing works
Self-citationMemory-Limited Partially Observable Stochastic Control and Its Mean-Field Control Approach
Tottori, T., & Kobayashi, T. J. (2022) · Entropy · doi:10.3390/e24111599 · Cited by 11 · see citing works
Tottori, T., & Kobayashi, T. J. (2023) · Entropy · doi:10.3390/e25020208 · Cited by 11 · see citing works
Decentralized Stochastic Control with Finite-Dimensional Memories: A Memory Limitation Approach
Tottori, T., & Kobayashi, T. J. (2023) · Entropy · doi:10.3390/e25050791 · Cited by 7 · see citing works
Tottori, T., & Kobayashi, T. J. (2025) · Physical Review Research · doi:10.1103/gvl6-cvby · Cited by 6 · see citing works
Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies
Tottori, T., & Kobayashi, T. J. (2026) · Physical Review Letters · doi:10.1103/5ynb-7k4v
- Authors
- Bensoussan, A., Frehse, J., & Yam, S. C. P.
- Affiliations
- The University of Texas at Dallas (US)
- University of Bonn (DE)
- Chinese University of Hong Kong (HK)
- Corresponding author
- Bensoussan, Alain
- Venue
- Journal de Mathématiques Pures et Appliquées
- Publisher
- Elsevier BV
- Published
- 2015-06
- Volume
- 103
- Issue
- 6
- Pages
- 1441-1474
- ISSN
- 0021-7824 (ISSN-L)
- OpenAlex
- W2138405426
- Cited by
- 138
- References
- 8
- Citations by year
- 2026: 11 · 2025: 12 · 2024: 15 · 2023: 9 · 2022: 14 · 2021: 11 · 2020: 15 · 2019: 11 · 2018: 21 · 2017: 8 · 2016: 4 · 2015: 5 · 2014: 2
- FWCI
- 11.59
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Journal de Mathématiques Pures et Appliquées (publishedVersion)
- Funders
- National Science Foundation (DMS-1303775)
- CityU (500113)
- GRF (404012)
- Chinese University of Hong Kong (2060444)
- Topics
- Social Sciences › Economics, Econometrics and Finance › Finance › Stochastic processes and financial applications
- Physical Sciences › Mathematics › Applied Mathematics › Nonlinear Partial Differential Equations
- Social Sciences › Decision Sciences › Management Science and Operations Research › Game Theory and Applications
- Keywords
- master equation, mean field games, mean-field control, Hamilton-Jacobi-Bellman equation, Fokker-Planck equation, Nash equilibrium, large population games, representative agent, stochastic differential games, backward stochastic differential equations, stochastic optimal control, stochastic partial differential equations
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