Markov Processes, Feller Semigroups and Evolution Equations

Casteren, Jan A. van

doi:10.1142/9789814322195

Cited by 15 publications

(18 citation statements)

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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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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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“…It is known (see [4,19,28]) that there exists the (unique) family of bounded operators U σ s,t on C ∞ which satisfies…”

Section: Evolution Kernelmentioning

confidence: 99%

“…[5,[12][13][14][15]31] and the monographs by Stroock and Varadhan [27], and van Casteren [4]). The aim of this paper is to get a Gaussian-type upper bound for the transition kernel of a particular kind of diffusion process (evolution) on a nilpotent meta-abelian group N .…”

Section: Introductionmentioning

confidence: 99%

Gaussian-type upper bound for the evolution kernels on nilpotent meta-abelian groups

Penney

Urban

2015

Positivity

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We give a Gaussian-type upper bound for the transition kernels of the timeinhomogeneous diffusion processes on a nilpotent meta-abelian Lie group N generated by the family of time dependent second order left-invariant differential operators. These evolution kernels are related to the heat kernel for the left-invariant second order differential operators on higher rank N A groups.Keywords Left invariant differential operators · Time-dependent parabolic opearators · Brownian motion · Evolution kernel · Diffusion process · Meta-abelian nilpotent Lie groups · Borell-TIS inequality

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“…[7], [16]. Since all agents are assumed to be subject to the same equation, the generator A being of special form (1.2), one investigates the dynamics for one representative of N agents given by the time inhomogeneous Markov process…”

Section: Hypothesis Amentioning

confidence: 99%

“…For linear propagators or evolutions it means the application of linear operators, see [2] and [7]. The family {Λ(t, s, ·) | 0 ≤ t ≤ s ≤ T } generated by the operator Au defines a positive, strongly continuous linear propagator or evolution on the set of Euclidean k-vectors which trivially coincides with the set of (continuous) real valued functions on the discrete set X.…”

Section: Pure Jump Markov Processesmentioning

confidence: 99%

An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces

Basna¹,

Hilbert²,

Kolokoltsov³

2014

COSA

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Abstract. 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 individual players is governed by pure jump type propagators over a finite space. 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 a 1 N -Nash Equilibrium for the approximating system of N players.

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Markov Processes, Feller Semigroups and Evolution Equations

Cited by 15 publications

References 0 publications

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

Gaussian-type upper bound for the evolution kernels on nilpotent meta-abelian groups

An epsilon-Nash equilibrium for non-linear Markov games of mean-field-type on finite spaces

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