Distribution-Dependent SDEs for Landau Type Equations

By Feng-Yu Wang. 2016

333 citations recorded by the source corpus.

The distribution-dependent stochastic differential equations (DDSDEs) describe stochastic systems whose evolution is determined by both the microcosmic site and the macrocosmic distribution of the particle. The density function associated with a DDSDE solves a non-linear PDE. Due to the distribution-dependence, some standard techniques developed for SDEs do not apply. By iterating in distributions, a strong solution is constructed using SDEs with control. By proving the uniqueness, the distribution of solutions is identified with a non-linear semigroup $P_t^*$ on the space of probability measures. The exponential contraction as well as Harnack inequalities and applications are investigated for the non-linear semigroup $P_t^*$ using coupling by change of measures. The main results are illustrated by homogeneous Landau equations.

Subfields: Finance, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2da41c8d9be84d999ad860f83a40d6153376757b

How researchers see this paper

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In Finance

1. Feng-Yu Wang
2. Jianhai Bao
3. Michael Röckner
4. Xing Huang
5. Panpan Ren
6. Xicheng Zhang
7. Chenggui Yuan
8. Wei Liu

In Mathematical Physics

1. Feng-Yu Wang
2. Jianhai Bao
3. Panpan Ren
4. Michael Röckner
5. Xicheng Zhang
6. Xing Huang
7. Wei Liu
8. Tusheng Zhang

In Statistical and Nonlinear Physics

1. Xing Huang
2. Panpan Ren
3. Jianhai Bao
4. Michael Röckner
5. Xicheng Zhang
6. Chenggui Yuan
7. Feng-Yu Wang

Citations in the corpus

• Well-posedness of distribution dependent SDEs with singular drifts (2018, background): “[8] Huang X. and Wang F.Y.: Distribution dependent SDEs with singular coefficients. arXiv:1805.01682v1.”
• Strong convergence order for slow–fast McKean–Vlasov stochastic differential equations (2019, background): “1), which can be obtained by using the result due to Wang in [34] and whose detailed proof will be presented in the Appendix.”
• Distribution dependent SDEs driven by fractional Brownian motions (2021, background): “mention, for examples, the works in [10, 11, 18] for large scale social interactions within the memory of mean-field games, [8, 13, 23] for value functions and related PDEs, [19, 20, 32, 36] for (shift) Harnack type inequalities and gradient estimates, and the references therein.”
• Distribution dependent SDEs with singular coefficients (2018, unknown): “1) with singular coefficients, gradient estimates and Harnack type inequalities can be proved as in the regular situation considered in [18].”
• Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance (2020, background): “This type of SDEs are also called McKean-Vlasov SDEs and mean field SDEs, and have been intensively investigated due to its wide applications, see for instance [1, 2, 5, 8, 10, 11, 12, 21, 23] and references within.”
• Bismut formula for Lions derivative of distribution dependent SDEs and applications (2018, background): “,φi) = Z Rd E f(Xx T)hM x T,φ(x)i µ(dx) = E f(Xµ T)hM Xµ 0 T ,φ(X µ 0 )i . Therefore, (1.10) holds for Mµ,φ T = hM Xµ 0 T ,φ(X µ 0 )i. However, when the SDE is distribution dependent, as explained in [30] that in general (1.12) does not hold, so it is non-trivial to establish the Bismut type formula”
• McKean–Vlasov SDEs under measure dependent Lyapunov conditions (2018, unknown): “g [6, 15, 26, 37].”
• Bismut formula for Lions derivative of distribution-path dependent SDEs (2020, methodology): “Below we construct the solution by using an iterating argument as in [37].”