Brake orbits and heteroclinic connections for first order **Mean** **Field**
Games

7/10 relevant

arXiv

Existence and Regularity of Solutions to Multi-Dimensional **Mean**-Field
Stochastic Differential Equations with Irregular Drift

**mean**-

**field**stochastic differential equations, i.e. a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition. Expand abstract.

**mean**-

**field**stochastic differential equations with irregular drift coefficients. Furthermore, we establish Malliavin differentiability of the solution and show regularity properties such as Sobolev differentiability in the initial data as well as H\"older continuity in time and the initial data. Using the Malliavin and Sobolev differentiability we formulate a Bismut-Elworthy-Li type formula for

**mean**-

**field**stochastic differential equations, i.e. a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.

10/10 relevant

arXiv

Zero-Sum Differential Games on Probability Spaces: Hamilton-Jacobi-Isaacs Equations and Viscosity Solutions

**mean**-

**field**games,

**mean**-

**field**type control, empirical risk optimization in statistical learning, and pursuit-evasion games. Unlike ZSDGs studied in the existing literature, where the value function is a function on a finite-dimensional state space, the class of ZSDGs of this paper introduces a new technical challenge, since the corresponding (lower and upper) value functions are defined on $\mathcal{P}_2$ (the set of probability measures with finite second moments) or $\mathcal{L}_2$ (the set of random variables with finite second moments), both of which are infinite-dimensional spaces. We show that the (lower and upper) value functions on $\mathcal{P}_2$ and $\mathcal{L}_2$ are equivalent to each other and continuous, where the former implies that they are law invariant. Then we prove that these value functions satisfy dynamic programming principles. We use the notion of derivative of a function of probability measures in $\mathcal{P}_2$ and its lifted version in $\mathcal{L}_2$ to show that the (lower and upper) value functions are viscosity solutions to the associated (lower and upper) Hamilton-Jacobi-Isaacs (HJI) equations that are (infinite-dimensional) first-order partial differential equations on $\mathcal{P}_2$ and $\mathcal{L}_2$. When the dynamics and running cost are independent of time, the value function is a unique viscosity solution of the corresponding HJI equation and under the Isaacs condition, the ZSDG has a value. Numerical examples are provided to illustrate the theoretical results of the paper.

4/10 relevant

arXiv

Mean-**Field** Neural ODEs via Relaxed Optimal Control

**Mean**-

**Field**(overdamped) Langevin dynamics. Expand abstract.

**mean**-

**field**(over-damped) Langevin algorithm for solving relaxed data-driven control problems. A key step in the analysis is to derive Pontryagin's optimality principle for data-driven relaxed control problems. Subsequently, we study uniform-in-time propagation of chaos of time-discretised

**Mean**-

**Field**(overdamped) Langevin dynamics. We derive explicit convergence rate in terms of the learning rate, the number of particles/model parameters and the number of iterations of the gradient algorithm. In addition, we study the error arising when using a finite training data set and thus provide quantitive bounds on the generalisation error. Crucially, the obtained rates are dimension-independent. This is possible by exploiting the regularity of the model with respect to the measure over the parameter space (relaxed control).

10/10 relevant

arXiv

Modulated Free Energy and **Mean** **Field** Limit

**mean**-

**field**theory in BrJaWa [4]. This physical object may be seen as a combination of the modulated potential energy introduced by S. Serfaty [See Proc. Int. Cong. Math. (2018)] and of the relative entropy introduced in

**mean**

**field**limit theory by P.-E. Jabin, Z. Wang [See Inventiones 2018]. It allows to obtain, for the first time, a convergence rate in the

**mean**

**field**limit for Riesz and Coulomb repulsive kernels in the presence of viscosity using the estimates in Du [8] and Se1 [20]. The main objective in this paper is to explain how it is possible to cover more general repulsive kernels through a Fourier transform approach as announced in BrJaWa [4] first in the case $\sigma$N $\rightarrow$ 0 when N $\rightarrow$ +$\infty$ and then if $\sigma$ > 0 is fixed. Then we end the paper with comments on the particle approximation of the Patlak-Keller-Segel system which is associated to an attractive kernel and refer to [C.R.

10/10 relevant

arXiv

**Mean** **field** interaction on random graphs with dynamically changing
multi-color edges

7/10 relevant

arXiv

Probabilistic Approach to **Mean** **Field** Games and Mean Field Type Control
Problems with Multiple Populations

**mean**

**field**equilibrium for each of these cases. Expand abstract.

**mean**

**field**games and

**mean**

**field**type control problems with multiple populations using a coupled system of forward-backward stochastic differential equations of McKean-Vlasov type stemming from Pontryagin's stochastic maximum principle. Although the same cost functions as well as the coefficient functions of the state dynamics are shared among the agents within each population, they can be different population by population. We study the

**mean**

**field**limits of the three different situations; (i) every agent is non-cooperative; (ii) the agents within each population are cooperative; and (iii) only for some populations, the agents are cooperative within each population. We provide several sets of sufficient conditions for the existence of

**mean**

**field**equilibrium for each of these cases.

10/10 relevant

arXiv

Extinction scenarios in evolutionary processes: A Multinomial Wright-Fisher approach

**mean**-

**field**equilibrium. Expand abstract.

**mean**-

**field**dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the

**mean**-

**field**dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the

**mean**-

**field**equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition, to limit theorems, we propose a variation of Fisher's maximization principle, fundamental theorem of natural selection, for a completely general deterministic replicator dynamics and study implications of the deterministic maximization principle for the stochastic model.

7/10 relevant

arXiv

Proximity effect in a heterostructure of a high $T_c$ superconductor
with a topological insulator from Dynamical **mean** **field** theory

**mean**

**field**theory (CDMFT), the TI layers being included via the CDMFT self-consistency loop. Expand abstract.

**mean**

**field**theory (CDMFT), the TI layers being included via the CDMFT self-consistency loop. The penetration of superconductivity into the TI depends on the position of the Fermi level with respect to the TI gap. We illustrate the back action of the TI layer on the HTSC layer, in particular the gradual disappearance of Mott physics with increasing tunneling amplitude.

10/10 relevant

arXiv

Topology and Magnetism in the Kondo Insulator Phase Diagram

**fields**around their

**mean**-

**field**value are included in order to establish the stability of the mean-field solution through computation of the full dynamical susceptibility. Expand abstract.

**mean**-

**field**approximation, we map out the magnetic phase diagram and characterize both antiferromagnetic and paramagnetic phases by their topological properties. Among others, we identify an antiferromagnetic insulator that shows, for suitable crystal terminations, topologically protected hinge modes. Furthermore, Gaussian fluctuations of the slave boson

**fields**around their

**mean**-

**field**value are included in order to establish the stability of the

**mean**-

**field**solution through computation of the full dynamical susceptibility.

7/10 relevant

arXiv