Abstract

We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We discuss the relation of duality with rescalings, stochastic monotonicity, intertwining, symmetries, and quantum many-body theory, reviewing known results and establishing some new connections.

Highlights

  • Duality of Markov processes with respect to a duality function has first appeared in the literature in the late 40s and early 50s [Lev[48], KMG57, Lin52], and has been formalized and generalized over the following decades [Sie[76], HS79, CR84, CS85, EK86, Ver[88], SL95]

  • Overviews of the method of duality with respect to a function generally focus on certain aspects or applications to particular fields [DG14, Lig[05], EK86, DF90, Moh[99], S00, Asm03], and presentations of the manifold connections to fundamental structures or properties of Markov processes, such as time reversal, stochastic monotonicity, symmetries, or conserved quantities, are often restricted to specific problems

  • The interest in a general theory of duality has further increased in recent years, but even basic questions such as giving necessary and sufficient conditions for the existence of a dual process of a given Markov process have not yet been fully resolved – “finding dual processes is something of a black art” [Eth[06], p. 519]

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Summary

Introduction

Duality of Markov processes with respect to a duality function has first appeared in the literature in the late 40s and early 50s [Lev[48], KMG57, Lin52], and has been formalized and generalized over the following decades [Sie[76], HS79, CR84, CS85, EK86, Ver[88], SL95] It has since been applied in a variety of fields ranging from interacting particle systems, queueing theory, SPDEs, and superprocesses to mathematical population genetics. Overviews of the method of duality with respect to a function generally focus on certain aspects or applications to particular fields [DG14, Lig[05], EK86, DF90, Moh[99], S00, Asm03], and presentations of the manifold connections to fundamental structures or properties of Markov processes, such as time reversal, stochastic monotonicity, symmetries, or conserved quantities, are often restricted to specific problems. We hope that this article triggers new research in this multifaceted and widely applicable area of probability theory

Setting and definitions
Examples
Duality with respect to a measure
Functional analytic theory
Cone duality and lifts of non-degenerate dualities
Pathwise duality
Strong pathwise duality
Weaker notions of pathwise duality
Rescaled processes
Monotonicity
Symmetries and intertwining
Intertwining of Markov processes
Quantum many-body representations of interacting particle systems
A Non-degeneracy of some standard duality functions
B Some dual mechanisms
Part I.
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