Martin kernels for Markov processes with jumps

Juszczyszyn, Tomasz; Kwaśnicki, Mateusz

doi:10.48550/arxiv.1509.05677

Cited by 2 publications

(4 citation statements)

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“…Since the conditions of the present paper are implied by the conditions of [26], we refer the readers to that paper for details. Here we will focus on certain symmetric and isotropic Lévy processes where we can say more regarding accessible boundary points, and a class of subordinate Brownian motions not covered by [15].…”

Section: Examplesmentioning

confidence: 99%

“…The preliminary version of the results of this paper (and the forthcoming paper [27]) was presented at the 11th Workshop on Markov Processes and Related Topics held in Shanghai Jiaotong University from June 27 to June 30 2015, and at the International Conference on Stochastic Analysis and Related Topics held in Wuhan University from August 3 to August 8 2015. In the recent preprint [15], Juszczyszyn and Kwaśnicki independently considered similar problems as those in Corollary 1.2 for bounded D. Our main motivation for the current paper was to investigate the Martin boundary at infinity. The investigation starts with the result stating that there is only one Martin boundary point associated with ∞ which should be understood as a local result about the Martin boundary in the sense that no other information about the remaining part of the boundary is required.…”

Section: Introduction and Setupmentioning

confidence: 99%

“…The case of inaccessible boundary points will be discussed in the forthcoming paper [27], the main reason being that the treatment of inaccessible points requires additional assumptions on j(x, y) -see E1 and E2 in [27], and Theorem 3.1(ii) in [15].…”

Section: Introduction and Setupmentioning

confidence: 99%

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Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces

2018

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In this paper we study the Martin boundary at infinity for a large class of purely discontinuous Feller processes in metric measure spaces. We show that if ∞ is accessible from an open set D, then there is only one Martin boundary point of D associated with it, and this point is minimal. We also prove the analogous result for finite boundary points. As a consequence, we show that minimal thinness of a set is a local property. ∞ 0 1 (Xt∈A) dt = A G(x, y)m(dy). Moreover, G(x, y) = G(y, x) for all x, y ∈ X, cf. VI.1 in [3] for details. Further, let D be an open subset of X and τ D = inf{t > 0 : X t / ∈ D} the exit time from D. The killed process X D is defined bywhere ∂ is an extra point added to X. The killed process X D is defined analogously. By Hunt's switching identity (Theorem 1.16 for all x, y ∈ X which implies that X D and X D are in duality, see p.43 in [13]. Again by VI.1 in [3], X D admits a unique (possibly infinite) Green function (potential kernel) G D (x, y) such that for every non-negative Borel function f ,the Green function of X D . It is assumed throughout the paper that G D (x, y) = 0 for (x, y) ∈ (D × D) c . We also note that the killed process X D is strongly Feller, see e.g. the first part of the proof of Theorem on pp. 68-69 in [11]. From now on, we will always assume that D is Greenian, that is, the Green function G D (x, y) is finite for all x, y ∈ D, x = y. Under this assumption, the killed process X D is transient in the sense that there exists a non-negative Borel function f on D such that 0 < G D f < ∞ (and the same is true for X).Recall that z ∈ ∂D is said to be regular with respect to X if P z (τ D = 0) = 1 and irregular otherwise. We will denote the set of regular (respectively irregular) points of ∂D with respect to X by D reg (respectively D irr ). D reg (respectively D irr ) stands for the sets of regular (respectively irregular) points of ∂D with respect to X respectively. It is well known that D irr and D irr are semipolar, hence polar under A.The process X, being a Hunt process, admits a Lévy system (J, H) where J(x, dy) is a kernel on the state space X (called the Lévy kernel of X), and H = (H t ) t≥0 is a positive continuous additive functional of X. We assume that H t = t so that for every function f : X × X → [0, ∞) vanishing on the diagonal and every stopping time T ,By using τ D in the displayed formula above and taking f (where ζ is the life time of X. Similar formulae hold for the dual process X and J(x, dy)m(dx) = J(y, dx)m(dy).Assumption C: The Lévy kernels of X and X are of the form J(x, dy) = j(x, y)m(dy), J(x, dy) = j(x, y)m(dy), where j(x, y) = j(y, x) > 0 for all x, y ∈ X, x = y. The next two related assumptions control the decay of the density j. Assumption C1(z 0 , R): Let z 0 ∈ X and R ≤ R 0 . For all 0 < r 1 < r 2 < R, there exists a constant c = c(z 0 , r 2 /r 1 ) > 0 such that for all x ∈ B(z 0 , r 1 ) and all y ∈ X \ B(z 0 , r 2 ),In the next assumption we require that the localization radius R 0 = ∞. Assumption C2(z 0 , R): Let z 0 ∈ X and R > 0. For all...

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Section: Examplesmentioning

confidence: 99%

Section: Introduction and Setupmentioning

confidence: 99%

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Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces

2018

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“…The background and recent progress on the Martin boundary is explained in the companion paper [14]. Martin kernels of bounded open sets D associated with both accessible and inaccessible boundary points of D have been studied in the recent preprint [5]. In this paper, we are mainly concerned with the Martin kernels of unbounded open sets associated with ∞ when ∞ is inaccessible from D. For completeness, we also spell out some of the details of the argument for dealing with the Martin kernels of unbounded open sets associated with inaccessible boundary points of D. To accomplish our task of studying the Martin kernels of general open sets, we follow the ideas of [1,7] and first study the oscillation reduction of ratios of positive harmonic functions.…”

Section: Introduction and Setupmentioning

confidence: 99%