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Stochastic Master equations or quantum filtering equations for mixed states are well known objects in quantum physics. Building a mathematically rigorous theory of these equations in infinite-dimensional spaces is a long standing open problem. The first objective of this paper is to give a solution to this problem under the assumption of bounded operators providing coupling with environment (or a measurement devise). Furthermore, recently the author built the theory of the law of large number limit for continuously observed interacting quantum particle systems leading to quantum mean-field games. These limits are described by certain nontrivial extensions of quantum stochastic master equations that can be looked at as infinite-dimensional operator-valued McKean-Vlasov diffusions. The second objective of this paper is to provide a well-posedness result for these new class of McKean-Vlasov diffusions.

Ticks and tick-borne diseases present a well known threat to the health of people in many parts of the globe. The scientific literature devoted both to field observations and to modeling the propagation of ticks continues to grow. So far the majority of the mathematical studies were devoted to models based on ordinary differential equations, where spatial variability was taken into account by a discrete parameter. Only few papers use spatially nontrivial diffusion models, and they are devoted mostly to spatially homogeneous equilibria. Here we develop diffusion models for the propagation of ticks stressing spatial heterogeneity. This allows us to assess the sizes of control zones that can be created (using various available techniques) to produce a patchy territory, on which ticks will be eventually eradicated. Using averaged parameters taken from various field observations we apply our theoretical results to the concrete cases of the lone star ticks of North America and of the taiga ticks of Russia., Comment: Submitted to Journal Mathematical Biology (JMB) in October 2021

We develop the idea of non-Markovian CTRW (continuous time random walk) approximation to the evolution of interacting particle systems, which leads to a general class of fractional kinetic measure-valued evolutions with variable order. We prove the well-posedness of the resulting new equations and present a probabilistic formula for their solutions. Though our method are quite general, for simplicity we treat in detail only the fractional versions of the interacting diffusions. The paper can be considered as a development of the ideas from the works of Belavkin and Maslov devoted to Markovian (quantum and classical) systems of interacting particles., Comment: To appear in Mathematical Notes 2022

The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different however. The diffusion coefficient can be made responsible for the size of jumps or for the intensity of jumps. The "rough" diffusion limit does not feel the difference. The situation changes, if we model jump-type approximations via CTRW with non-exponential waiting times. If we make the diffusion coefficient responsible for the size of jumps and take waiting times from the domain of attraction of an $\alpha$-stable law with a constant intensity $\al$, then the standard scaling would lead in the limit of small jumps and large intensities to the most standard fractional diffusion equation. However, if we choose the CTRW approximations with fixed jump sizes and use the diffusion coefficient to distinguish intensities at different points, then we obtain in the limit the equations with variable position-dependent fractional derivatives. In this paper we build rigorously these approximations and prove their convergence to the corresponding fractional equations for the cases of multidimensional diffusions and more general Feller processes., Comment: Submitted to Proceedings of the 50th Barrette lectures Tennnessee 2021

We introduce a class of abstract nonlinear fractional pseudo-differential equations in Banach spaces that includes both the Mc-Kean-Vlasov-type equations describing nonlinear Markov processes and the Hamilton-Jacobi-Bellman(HJB)-Isaacs equation of stochastic control and games thus allowing for a unified analysis of these equations. This leads to an effective theory of coupled forward-backward systems (forward McKean-Vlasov evolution and backward HJB-Isaacs evolution) that are central to the modern theory of mean-field games., Comment: Submitted for publication to MIAN Proceedings, v. 315

Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.

There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing the limiting individual behavior of particles inside a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Scr\"odinger equations or Hartree equations, or Gross-Pitaevski equations. In this paper we extend some of these convergence results to a stochastic framework. Concretely we work with the Belavkin stochastic filtering of many particle quantum systems. The resulting limiting equation is an equation of a new type, which can be seen as a complex-valued infinite dimensional nonlinear diffusion of McKean-Vlasov type. This result is the key ingredient for the theory of quantum mean-field games developed by the author in a previous paper.

Quantum games represent the really 21st century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In the previous paper of the author the truly dynamic quantum game theory was initiated with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox) the necessary new ingredient for quantum dynamic games represented the theory of non-direct observations and the corresponding quantum filtering. Another remarkable 21st century branch of game theory represent the so-called mean-field games (MFG), with impressive and ever growing development. In this paper we are merging these two exciting new branches of game theory. Building a quantum analog of MFGs requires the full reconstruction of its foundations and methodology, because in $N$-particle quantum evolution particles are not separated in individual dynamics and the key concept of the classical MFG theory, the empirical measure defined as the sum of Dirac masses of the positions of the players, is not applicable in quantum setting. As a preliminary result we derive the new nonlinear stochastic Schr\"odinger equation, as the limit of continuously observed and controlled system of large number of interacting quantum particles, the result that may have an independent value. We then show that to a control quantum system of interacting particles there corresponds a special system of classical interacting particles with the identical limiting MFG system, defined on an appropriate Riemanian manifold. Solutions of this system are shown to specify approximate Nash equilibria for $N$-agent quantum games.

Combining intuitive probabilistic assumptions with the basic laws of classical thermodynamics, using the latter to express probabilistic parameters in terms of the thermodynamic quantities, we get a simple unified derivation of the fundamental ensembles of statistical physics avoiding any limiting procedures, quantum hypothesis and even statistical entropy maximization. This point of view leads also to some related classes of correlated particle statistics.

The general picture of game theoretic modeling dealt with here is characterized by a set of big players, also referred to as principals or major agents, acting on the background of large pools of small players, the impact of the behavior of each small player in a group on the overall evolution decreasing with the increase of the size of the group. In this Part I approach players in groups are not independent rational optimizers. They are either directly controlled by principals and serve the interests of the latter (pressure and collaboration setting) or they resist the actions of the principals (pressure and resistance setting) by evolving their strategies in an 'evolutionary manner' via interactions with other players subject to certain clear rules, deterministic or stochastic. The examples of the real world problems involved include government representatives (often referred to in the literature as benevolent dictators) chasing corrupted bureaucrats, inspectors chasing tax-paying avoidance, police acting against terrorist groups or models describing the attacks of computer or biological viruses. Other class of examples concerns appropriate (or better optimal) management of complex stochastic systems consisting of large number of interacting components (agents, mechanisms, vehicles, subsidiaries, species, police units, robot swarms, etc), which may have competitive or common interests. Such management can also deal with the processes of merging and splitting of functional units (say, firms or banks) or the coalition building of agents. The actions of the big players effectively control the distribution of small players among their possible strategies and can influence the rules of their interaction., Comment: This is the final draft of the 1st part of the authors' book 'Many agent games in socio-economic systems: corruption, inspection, coalition building, network growth, security' published by Springer in the Springer Series on Operational Research, 2019

We study the fragmentation-coagulation (or merging and splitting) evolutionary control model as introduced recently by one of the authors, where $N$ small players can form coalitions to resist to the pressure exerted by the principal. It is a Markov chain in continuous time and the players have a common reward to optimize. We study the behavior as $N$ grows and show that the problem converges to a (one player) deterministic optimization problem in continuous time, in the infinite dimensional state space.

This paper provides well-posedness results and stochastic representations for the solutions to equations involving both the right- and the left-sided generalized operators of Caputo type. As a special case, these results show the interplay between two-sided fractional differential equations and two-sided exit problems for certain L\'evy processes., Comment: 19 pages

Recently developed toy models for the mean-field games of corruption and botnet defence in cyber-security with three or four states of agents are extended to a more general mean-field-game model with $2d$ states, $d\in \N$. In order to tackle new technical difficulties arising from a larger state-space we introduce new asymptotic regimes, namely small discount and small interaction asymptotics. Moreover, the link between stationary and time-dependent solutions is established rigorously leading to a performance of the turnpike theory in a mean-field-game setting.

We investigate mean-field games from the point of view of a large number of indistinguishable players which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number $N$ of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean-Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-Nash Equilibrium for the $N$-player games., Comment: arXiv admin note: text overlap with arXiv:1403.0426

A simple model of corruption that takes into account the effect of the interaction of a large number of agents by both rational decision making and myopic behavior is developed. Its stationary version turns out to be a rare example of an exactly solvable model of mean-field-game type. The results show clearly how the presence of interaction (including social norms) influences the spread of corruption.

We investigate mean field games for players, who are weakly coupled via their empirical measure. To this end we investigate time-dependent pure jump type propagators over a finite space in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics which leads to the well-known Mckean-Vlasov dynamics in the limit as the number N of players goes to infinity. The case where one player has an individual preference different to the ones of the remaining players is also covered. The limiting system represents a 1/N-Nash Equilibrium for the approximating system of N players., Comment: 20 pages

In June 2012 on a conference in Bielefeld, after the author made the presentation of his theory of nonlinear Markov processes, Tom Kurtz asked him whether his methods would allow to get well-posedness for nonlinear McKean-Vlasov type diffusions with coefficients depending on the distribution via the median. The present paper is the answer to this question. The aim here is not to find the weakest possible conditions for the affirmative answer, but a sensible general result with a quick and transparent proof.

We review and develop a selection of models of systems with competition and cooperation, with origins in economics, where deep insights can be obtained by the mathematical methods of game theory. Some of these models were touched upon in authors' book 'Understanding Game Theory', World Scientific 2010, where also the necessary background on games can be found., Comment: 22 pages

In this paper, we investigate the mean field games with $K$ classes of agents who are weakly coupled via the empirical measure. The underlying dynamics of the representative agents is assumed to be a controlled nonlinear Markov process associated with rather general integro-differential generators of L\'evy-Khintchine type (with variable coefficients). We show that nonlinear measure-valued kinetic equations describing the dynamic law of large numbers limit for system with large number N of agents are solvable and that their solutions represent 1/N-Nash equilibria for approximating systems of N agents., Comment: 60 pages, results reported at SIAM (Contol) July 2011, Vienna ECCS'11 Sep. 2011, Warwick (Topics in Control) Dec. 2011, 2nd version just improves wording and formulations in many places

The program of studying general nonlinear Markov processes was put forward in V. N. Kolokoltsov "Nonlinear Markov Semigroups and Interacting L\'evy Type Processes" (Journ. Stat. Physics 126:3 (2007), 585-642), and was developed by the author in monograph "Nonlinear Markov processes and kinetic equations". Cambridge University Press, 2010, where, in particular, nonlinear L\'evy processes were introduced. The present paper is an invitation to the rapidly developing topic of noninear Markov processes. We provide a quick (and at the same time more abstract) introduction to the basic analytical aspects of the theory developed in Part II of the above mentioned book., Comment: 20 pages

Ito's construction of Markovian solutions to stochastic equations driven by a L\'evy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the L\'evy-Khintchine type) with variable coefficients (diffusion, drift and L\'evy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein-Kantorovich metrics. This is a nontrivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.

Functional limit theorem for continuous-time random walks (CTRW) are found in general case of dependent waiting times and jump sizes that are also position dependent. The limiting anomalous diffusion is described in terms of fractional dynamics. Probabilistic interpretation of generalized fractional evolution is given in terms of the random time change (subordination) by means of hitting times processes.