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1. Deep Learning Methods for Mean Field Control Problems with Delay

   https://doi.org/10.3389/fams.2020.00011  

   JP Fouque, Zaoyu Zhang (January 2020, Frontiers in applied mathematics and statistics)

   We consider a general class of mean field control problems described by stochastic delayed differential equations of McKean-Vlasov type. Two numerical algorithms are provided based on deep learning techniques, one is to directly parameterizing the optimal control using neural networks, the other is based on numerically solving the McKean-Vlasov forward anticipated backward stochastic differential equation (MV-FABSDE) system. In addition, we establish the necessary and sufficient stochastic maximum principle of this class of mean field control problems with delay based on the differential calculus on function of measures, and the exis- tence and uniqueness results are proved for the associated MV-FABSDE system under suitable conditions. 

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2. Systemic Robustness: A Mean‐Field Particle System Approach

   https://doi.org/10.1111/mafi.12459  

   Bayraktar, Erhan; Guo, Gaoyue; Tang, Wenpin; Zhang, Yuming Paul (February 2025, Mathematical Finance)

   ABSTRACT This paper is concerned with the problem of capital provision in a large particle system modeled by stochastic differential equations involving hitting times, which arises from considerations of systemic risk in a financial network. Motivated by Tang and Tsai, we focus on the number or proportion of surviving entities that never default to measure the systemic robustness. First we show that the mean‐field particle system and its limit McKean–Vlasov equation are both well‐posed by virtue of the notion of minimal solutions. We then establish a connection between the proportion of surviving entities in the large particle system and the probability of default in the McKean–Vlasov equation as the size of the interacting particle system tends to infinity. Finally, we study the asymptotic efficiency of capital provision for different drift , which is linked to the economy regime: The expected number of surviving entities has a uniform upper bound if ; it is of order if ; and it is of order if , where the effect of capital provision is negligible. 

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3. Rate of homogenization for fully-coupled McKean–Vlasov SDEs

   https://doi.org/10.1142/S0219493723500132  

   Bezemek, Zachary William; Spiliopoulos, Konstantinos (January 2023, Stochastics and Dynamics)

   In this paper, we consider a fully-coupled slow–fast system of McKean–Vlasov stochastic differential equations with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy problem on the Wasserstein space that are of independent interest. 

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4. Control of kinetic plasma instabilities by laser fields

   https://doi.org/10.1063/5.0315595  

   Crouseilles, Nicolas; Einkemmer, Lukas; Li, Qin; Shumlak, Uri; Yue, Yukun (July 2026, Physics of Plasmas)

   We study the possibility of controlling kinetic plasma instabilities by using lasers to apply external transverse electromagnetic fields. We derive the dispersion relation for the corresponding mathematical description, a reduced Vlasov–Maxwell system, by extending the well-known Penrose condition. It is observed that, under very mild assumptions, the dispersion relation decouples into two parts. The first part is identical to the classic Penrose condition for the Vlasov–Poisson system, while the second part describes the influence of the laser on the transverse dynamics (e.g., a Weibel instability). In particular, this means that the longitudinal dynamics (e.g., a two-stream instability) cannot be stabilized in this manner as far as linear theory is concerned. We show that, however, nonlinear effects can be used to couple the two parts and achieve an effective control. This is done by determining the control parameters (i.e., the form of the external electric and magnetic fields) by solving a partial differential equation (PDE)-constrained optimization problem. 

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5. Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows

   https://doi.org/10.1142/S0218202520500475  

   Liang, Zhilei; Wang, Dehua (November 2020, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent. 

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