h -transform of Doob and nonlocal branching processes
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h-transform of Doob and nonlocal branching processes Lucian Beznea1,2 · Ana-Maria Boeangiu1 · Oana Lupa¸scu-Stamate3 Received: 7 October 2019 / Revised: 15 July 2020 / Accepted: 4 September 2020 © Springer Nature Switzerland AG 2020
Abstract We study the h-transform of Doob for nonlocal branching processes, we show that the branching property is preserved provided that h is a coherent state, and we emphasize the probabilistic representation of the solution to the associated nonlinear evolution equation. The tools are from the analytic and probabilistic potential theory. We also investigate the h-transform of a subordinate C0 -semigroup of sub-Markovian operators on an L p space. Keywords h-transform of Doob · Branching process · L p -semigroup · Transition function · Excessive function · Right process · Nonlinear evolution equation · Weak generator Mathematics Subject Classification 60J80 · 60J45 · 60J35 · 60J57 · 47D07 · 60J25 · 60J40 · 31C05
Introduction In the study of the boundary behaviour of the harmonic functions in an arbitrary Euclidean domain, Martin [29] considered kernel functions given by a quotient of two
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Lucian Beznea [email protected] Ana-Maria Boeangiu [email protected] Oana Lupa¸scu-Stamate [email protected]
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Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
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Faculty of Mathematics and Computer Science, University of Bucharest and Centre Francophone en Mathématique de Bucarest, Bucharest, Romania
3
Gheorghe Mihoc–Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie 13, 050711 Bucharest, Romania 0123456789().: V,-vol
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superharmonic functions. This is the origin of the h-transform of Doob. An abundant literature was then initiated from both analytic and probabilistic perspectives, we only mention the monographs of Meyer [30] and Doob [16]. The aim of this work is to study the h-transform from the point of view of the current state of Markov processes, with application to the nonlocal discrete branching processes on the set of all finite configuration of a given set. It turns out that by this transformation we obtain a new branching process with a smaller state space of finite configurations. Since this application requires to work on an infinite dimensional state space, it is convenient for our approach to use suitable, recent methods from the potential theory of a right (Markov) process with general state space; cf. [3,5,7,9– 11,34]. The paper is organised as follows. We give in Sect. 1 (Proposition 1.1) a transparent proof for the construction of the h-transformed process and we deduce then results on the excessive functions, the fine topology, the reduced function, and the polar sets. We point out that the tightness property of the capacities induced by the process is preserved by a h-transform. Complementary bibliographical comments are exposed in Remark 1.2. As a byproduct of the h-transform we can t
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