Markov Processes, Feller Semigroups and Evolution Equations

Casteren, Jan A. van

doi:10.1142/7871

Cited by 48 publications

(21 citation statements)

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“…For linear propagators, respectively evolutions see [9] Chapter 2, it means the application of linear operators, see [2] Chapter 3 and [9] Chapter 2. In the particular case when the propagator is given by a non-autonomous jump Feller process X on R d , i.e.…”

Section: Preliminary Resultsmentioning

confidence: 99%

An approximate Nash equilibrium for pure jump Markov games of mean-field-type on continuous state space

2017

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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 Nplayer games.Mathematics Subject Classification (2010): 91A22, 91A13, 91A15, 60J75, 60J25.

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Section: Preliminary Resultsmentioning

confidence: 99%

An approximate Nash equilibrium for pure jump Markov games of mean-field-type on continuous state space

2017

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“…More to the point, the Markov processes of [36] actually emerge as particular cases of reversible diffusions that belong to the larger class of the so-called reciprocal or Bernstein processes, whose theory was launched many years ago in [2] following Schrödinger's seminal contribution in [27]. The theory of Bernstein processes was subsequently further developed and systematically investigated in [19], and since then has played an important rôle in relating various fields such as the Malliavin calculus and Euclidean quantum mechanics, or Markov bridges with jumps and Lévy processes, to name only a few (see for instance [7], [8], [16], [25], [32] and the references therein for a more complete account).…”

Section: Introduction and Outlinementioning

confidence: 99%

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

Vuillermot¹,

Zambrini²

2012

J Theor Probab

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In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of non-autonomous linear parabolic initial-and final-boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two Itô equations for some suitably constructed Wiener processes, and from that analysis derive Feynman-Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a twodimensional disk.

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“…Hence by [33] this Feller evolution is associated to a Hunt process with state space E. For a countable state space E such result was obtained by martingale techniques in [35]. We show that U (s, t) provides existence and uniqueness of solutions to the Kolmogorov equations and establish the relation to the jump process by the associated Martingale problem.…”

Section: Introductionmentioning

confidence: 62%

Non-autonomous interacting particle systems in continuum

Friesen¹

2016

Preprint

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A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.

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Markov Processes, Feller Semigroups and Evolution Equations

Cited by 48 publications

References 0 publications

An approximate Nash equilibrium for pure jump Markov games of mean-field-type on continuous state space

An approximate Nash equilibrium for pure jump Markov games of mean-field-type on continuous state space

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

Non-autonomous interacting particle systems in continuum

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