Related papers: Master equation for the finite state space plannin…

We study solutions to the static vacuum Einstein equations on exterior domains with prescribed metric and mean curvature on the inner boundary. It is proved that for any such boundary data near the standard round boundary data in Euclidean…

We study mean field games and corresponding $N$-player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we…

We propose a new approach to proving the uniqueness of solutions to a certain class of mean field games of controls. In this class, the equilibrium is determined by an aggregate quantity $Q(t)$, e.g. the market price or production, which…

In the framework of theory of open quantum systems, we derive quantum master equations for the ultrastrong system-bath coupling regime and, more generally, the strong-decoherence regime. In this regime, the strong decoherence is…

We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial…

We present a detailed derivation of the stochastic master equation determining the time evolution of the state of a general quantum system, which is placed inside a cavity and subjected to indirect measurements by monitoring the state of…

We study the non-linear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish…

The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence…

We consider one-dimensional stochastic differential equations with a boundary condition, driven by a Poisson process. We study existence and uniqueness of solutions and the absolute continuity of the law of the solution. In the case when…

We consider deterministic mean field games where the dynamics of a typical agent is non-linear with respect to the state variable and affine with respect to the control variable. Particular instances of the problem considered here are mean…

The determination of the stability of the long-lived consensus problem is a fundamental open problem in distributed systems. We concentrate on the memoryless binary case with geodesic paths. We offer a conjecture on the stability in this…

In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. While for Hamilton--Jacobi equations Dirichlet conditions may not be satisfied, here, we establish the existence of…

By analogy with the theory of Backward Stochastic Differential Equations, we define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. This paper considers these processes as constructions…

We study a high-dimensional stochastic optimization problem which features both control and stopping. In particular, a central planner steers a large population of particles, and can also remove particles at any time by paying a penalty. In…

In this paper, we introduce a natural learning rule for mean field games with finite state and action space, the so-called myopic adjustment process. The main motivation for these considerations are the complex computations necessary to…

We provide simple necessary and sufficient conditions under which a path constitutes a solution to an infinite-horizon, continuous-time optimal control problem. We prove transversality conditions under standard assumptions. We also present…

By extending the mean-field Hamiltonian to include nonhermitian operators, the master equations for fermions and bosons can be derived. The derived equations reduce to the Markoff master equation in the low-density limit and to the…

In this paper, we examine the stationary relaxed singular control problem within a multi-dimensional framework for a single agent, as well as its mean field game equivalent. We demonstrate that optimal relaxed controls exist for two problem…

This paper studies optimal control and stabilization problems for continuous-time mean-field systems with input delay, which are the fundamental development of control and stabilization problems for mean-field systems. There are two main…

The Master Stability Function is a robust and useful tool for determining the conditions of synchronization stability in a network of coupled systems. While a comprehensive classification exists in the case in which the nodes are chaotic…