Potential theory of random walks on Abelian groups

Port, Sidney C.; Stone, Charles J.

doi:10.1007/bf02392007

Cited by 53 publications

(48 citation statements)

References 9 publications

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“…In the general case it follows by the same argument used in proving the corresponding result in discrete time, Theorem 5.8 of [7]. Proof.…”

Section: Gx /^) = ^ (^(-^ -A\v -^)F(y) Dy + ^ V\ {Dy}f{x + Y)mentioning

confidence: 74%

“…Then the random walk obtained by looking at the process at integer times is a recurrent random walk on @. By Theorem 5 1 of [7]^( r^,))" …”

Section: It Follows Thatmentioning

confidence: 99%

“…Since 9(6) is a logarithm of p- (6), the proof of the proposition is complete. The proof of this result is an obvious modification of the proof of Theorem 3.1 of [7] and will be omitted.…”

Section: Some Fourier Analysismentioning

confidence: 99%

“…Properties of this family of functions were developed in [7]. Let 9 denote the collection of differences of elements of S^".…”

Section: The Recurrent Potential Operatormentioning

confidence: 99%

See 3 more Smart Citations

Infinitely divisible processes and their potential theory. II

Port¹,

Stone²

1971

Annales De L’institut Fourier

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“…In the general case it follows by the same argument used in proving the corresponding result in discrete time, Theorem 5.8 of [7]. Proof.…”

Section: Gx /^) = ^ (^(-^ -A\v -^)F(y) Dy + ^ V\ {Dy}f{x + Y)mentioning

confidence: 74%

“…Then the random walk obtained by looking at the process at integer times is a recurrent random walk on @. By Theorem 5 1 of [7]^( r^,))" …”

Section: It Follows Thatmentioning

confidence: 99%

“…Since 9(6) is a logarithm of p- (6), the proof of the proposition is complete. The proof of this result is an obvious modification of the proof of Theorem 3.1 of [7] and will be omitted.…”

Section: Some Fourier Analysismentioning

confidence: 99%

“…Properties of this family of functions were developed in [7]. Let 9 denote the collection of differences of elements of S^".…”

Section: The Recurrent Potential Operatormentioning

confidence: 99%

See 2 more Smart Citations

Infinitely divisible processes and their potential theory. II

Port¹,

Stone²

1971

Annales De L’institut Fourier

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“…Except when discussing the renewal theorem we will always assume that 2 = ®. Very briefly, our purpose is to show that results in [3] and [4] for random walks on ® go over to i.d. processes on ®, and this we do rather completely.…”

mentioning

confidence: 99%

Potential theory for infinitely divisible processes on Abelian groups

Port¹,

Stone²

1969

Bull. Amer. Math. Soc.

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Let © be a 2nd countable locally compact, noncompact Abelian group, and let X t be an infinitely divisible process (henceforth called an i.d. process) with stationary increments taking values in ®. The transition functions P l (x, dy) of such a process have the Feller property, so via a fundamental theorem in the theory of Markov processes (see [l, p. 44 ff.]) there is a realization of the process as a standard Markov process. Henceforth we shall assume that X t is this version of the process. A point #£® is called possible if for each neighborhood N of 0 there is a *>0 such that P<(0, N+x) >0. The set 2 of all possible points is a closed subsemigroup of ©. Except when discussing the renewal theorem we will always assume that 2 = ®. Very briefly, our purpose is to show that results in [3] and [4] for random walks on ® go over to i.d. processes on ®, and this we do rather completely. These results are new even for i.d. processes on Euclidean spaces. Space does not permit a detailed description of all these results (much less their proofs) so we will only sketch what has been done. We point out here, however, that although most of the results on random walks have their counterpart for i.d. processes, the proofs of these results in many cases require significant new ideas. Some of these ideas involve indirect reductions to the discrete time case.Let B be a relatively compact set, and let Ts^ini {t>0: X t ÇzB} denote the first hitting time of B. We show that a theory of X-capacities can be developed for any i.d. process. If the X-capacity &(B) = 0 for one X>0, then this is true for all X>0, and then P X (TB< oo) =0 a.e. On the other hand, if C x (B)>0 for some X>0, then this is true

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Verallgemeinerung eines Satzes von DOBRUSCHIN V

Liemant

1975

Mathematische Nachrichten

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Eingega.ngen a m 3.2. 1975) Die vorliegende Kote ist eine unmittelbare Fortsetzung der Arbeit [I]. Dort wurde im 4. Abschnitt eine notwendige und hinreichende Bedingung ( e ) dafiir formuliert, daB fur ein Schauerverteilungsgesetz D aus D eine der Bedingung 0 < i, < + 00 genugende stationare Losung der Funktionalgleichung Flu, = P esistiert. Wir verschkfen nun diese Stabilitiitseigenschaft dahingehend, da13 wil s o g x &e Existenz einer stationgren Losung P verlangen, die neben i, > 0 fur alle kompakten Teilmengen X von G der Bedingung ( @ ( X ) ) 2 P ( d @ ) < + 03 geniigt. I n diesem Fall sagen wir, D sei stccbil von zweiter Ordnung. Es zeigt sich nun, da13 unter der zusiitzlichen Annahme (x(G))z D(dx) < + 03 die Stabilitlit zweiter Ordnung auf einfache Weise durch das IntensitiitsmaB eD ausgedriickt w-erden kann: Sie liegt genau dann vor, wenn die durch das syinmetrisierte Verteilungsgesetz Oen bestimmte zufllllige Irrfahrt in der Gruppe G transien,t ist. Unser Ergebnis macht es moglich, die bekannten analytischen Kriterien fur die Transienz einer zufillligen Irrfahrt auf lokalkompakten abelschen topologischen Gruppen (vgl. 131) zu nutzen, um zu einem Kriteriuin fur die Stabilit.at zweiter Ordnung zu kommen, in das nur die Fouriertransformierte des spmmetrisierten IntensitiitsmaBes O e D eingeht. Das in [2] von .K. FLEISCHMAKK abgeleitete Resultat, wonach im Falle G = R*, s 2 3, jedes der Bediiigung(x(G))2 D(dx) < + 00 genugende Verteilungsgesetz D aus D stabil zweiter Ordnung ist, liiBt sich auf diesem Wege als Konsequenz unserer Resultate erhalten. Jedem Verteilungsgesetz e auf @ konnen wir ein symnietrisiertes Verteilnngsgesetz O e auf @ gem513 O@e((.)) = e ((.) + x) e(dz) zuordnen. Die Spnimetrisierung ist niit der Faltungsoperation vertauschbar. Es gilt O ( e * p') = Op * O(p') und deshalb auch (Op)" = O(e"). Wir betrachten nun folgende zufallige Irrfahrt, {qn, n = 0, 1, . . .} auf den1 Phasenrauni G . Es sei D g = o , 17," = ti + t 2 + * . . + t" ( n 2 1) 9 wobei ti, la, . . . eine Folge unabhiingiger identisch gemail3 Opu verteilter zufalliger Gruppeneleniente sei. Diese zufallige Irrfahrt wird durch das Intensit5tsmaB e 25'

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Potential theory of random walks on Abelian groups

Cited by 53 publications

References 9 publications

Infinitely divisible processes and their potential theory. II

Infinitely divisible processes and their potential theory. II

Potential theory for infinitely divisible processes on Abelian groups

Verallgemeinerung eines Satzes von DOBRUSCHIN V

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