TY - INPR A1 - Conforti, Giovanni A1 - Léonard, Christian A1 - Murr, Rüdiger A1 - Roelly, Sylvie T1 - Bridges of Markov counting processes : reciprocal classes and duality formulas N2 - Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 9 KW - counting process KW - bridge KW - reciprocal class KW - duality formula Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71855 SN - 2193-6943 VL - 3 IS - 9 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Conforti, Giovanni A1 - Dai Pra, Paolo A1 - Roelly, Sylvie T1 - Reciprocal class of jump processes N2 - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set A in R^d. We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of A plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 6 KW - reciprocal processes KW - stochastic bridges KW - jump processes KW - compound Poisson processes Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70776 SN - 2193-6943 VL - 3 IS - 6 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Conforti, Giovanni A1 - Roelly, Sylvie T1 - Reciprocal class of random walks on an Abelian group N2 - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 1 KW - reciprocal class KW - stochastic bridge KW - random walk on Abelian group Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72604 SN - 2193-6943 VL - 4 IS - 1 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Conforti, Giovanni A1 - Pra, Paolo Dai A1 - Roelly, Sylvie T1 - Reciprocal Class of Jump Processes JF - Journal of theoretical probability N2 - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set . We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state. KW - Reciprocal processes KW - Stochastic bridges KW - Jump processes KW - Compound Poisson processes Y1 - 2015 U6 - https://doi.org/10.1007/s10959-015-0655-3 SN - 0894-9840 SN - 1572-9230 VL - 30 SP - 551 EP - 580 PB - Springer CY - New York ER - TY - THES A1 - Conforti, Giovanni T1 - Reciprocal classes of continuous time Markov Chains T1 - Reziproke Klassen zeitkontinuierlicher Markov-Ketten N2 - In this thesis we study reciprocal classes of Markov chains. Given a continuous time Markov chain on a countable state space, acting as reference dynamics, the associated reciprocal class is the set of all probability measures on path space that can be written as a mixture of its bridges. These processes possess a conditional independence property that generalizes the Markov property, and evolved from an idea of Schrödinger, who wanted to obtain a probabilistic interpretation of quantum mechanics. Associated to a reciprocal class is a set of reciprocal characteristics, which are space-time functions that determine the reciprocal class. We compute explicitly these characteristics, and divide them into two main families: arc characteristics and cycle characteristics. As a byproduct, we obtain an explicit criterion to check when two different Markov chains share their bridges. Starting from the characteristics we offer two different descriptions of the reciprocal class, including its non-Markov probabilities. The first one is based on a pathwise approach and the second one on short time asymptotic. With the first approach one produces a family of functional equations whose only solutions are precisely the elements of the reciprocal class. These equations are integration by parts on path space associated with derivative operators which perturb the paths by mean of the addition of random loops. Several geometrical tools are employed to construct such formulas. The problem of obtaining sharp characterizations is also considered, showing some interesting connections with discrete geometry. Examples of such formulas are given in the framework of counting processes and random walks on Abelian groups, where the set of loops has a group structure. In addition to this global description, we propose a second approach by looking at the short time behavior of a reciprocal process. In the same way as the Markov property and short time expansions of transition probabilities characterize Markov chains, we show that a reciprocal class is characterized by imposing the reciprocal property and two families of short time expansions for the bridges. Such local approach is suitable to study reciprocal processes on general countable graphs. As application of our characterization, we considered several interesting graphs, such as lattices, planar graphs, the complete graph, and the hypercube. Finally, we obtain some first results about concentration of measure implied by lower bounds on the reciprocal characteristics. N2 - Diese Dissertation behandelt die reziproke zufällige Prozesse mit Sprüngen. Gegeben eine zeitkontinuierliche Markovkette als Referenzdynamik, ist die assoziierte reziproke Klasse die Menge aller Wahrscheinlichkeiten auf dem Pfadraum, die als eine Mischung ihrer Brücken geschrieben werden kann. Reziproke Prozesse zeichnen sich durch eine Form der bedingten Unabhängigkeit aus, die die Markoveigenschaft verallgemeinert. Ursprünglich ist diese Idee auf Schrödinger zurückzuführen, der nach einer probabilistischen Interpretation für die Quantenmechanik suchte. Einer reziproken Klasse wird eine Familie reziproker Charakteristiken assoziiert. Dies sind Raum-Zeit Abbildungen, die die reziproke Klasse eindeutig definieren. Wir berechnen diese Charakteristiken explizit und unterteilen sie in zwei Typen: Bogen-Charakteristiken und Kreis-Charakteristiken. Zusätzlich erhalten wir ein klares Kriterium zur Prüfung wann die Brücken von zwei verschiedenen Markovketten übereinstimmen. Wir beschreiben auf zwei verschiedene Arten reziproken Klasse und berücksichtigen auch ihre nicht-Markov Elemente. Die erste Charakterisierung basiert auf einem pfadweisen Ansatz, während die zweite kurzzeit Asymptotik benutzt. Der erste Ansatz liefert eine Familie funktionaler Gleichungen deren einzige Lösungen die Elemente der reziproken Klasse sind. Die Gleichungen können als partielle Integration auf dem Pfadraum mit einem Ableitungsoperator, der eine St¨orung der Pfade durch zusätzliche zufällige Kreise hervorruft, interpretiert werden. Die Konstruktion dieser Gleichungen benötigt eine geometrische Analyse des Problems. Wir behandeln außerdem die Fragestellung einer scharfen Charakterisierung und zeigen interessante Verbindungen zur diskreten Geometrie. Beispiele, für die wir eine solche Formel finden konnten, sind für Zählprozesse und für Irrfahrte auf abelschen Gruppen, in denen die Menge der Kreise eine Gruppenstruktur erweist. Zusätzlich zu diesem globalen Zugang, erforschen wir eine lokale Beschreibung durch die Analyse des kurzfristigen Verhaltens eines reziproken Prozesses. Analog zur Markoveigenschaft und kurzzeit Entwicklung ihrer Übergangswahrscheinlichkeit Markovketten charakterisieren, zeigen wir, dass eine reziproke Klasse charakterisiert werden kann indem wir ihre reziproke Eigenschaft und zwei Familien von Kurzzeit Entwicklungen der Brücken voraussetzen. Solche lokalen Ansatz ist geeignet, um Sprungprozesse auf allgemeine zählbaren Graphen zu studieren. Als Beispiele unserer Charakterisierung, betrachten wir Gitter, planare Graphen, komplette Graphen und die Hyperwürfel. Zusätzlich präsentieren wir erste Ergebnisse über Maßenkonzentration eines reziproken Prozesses, als Konsequenz unterer Schranken seiner Charakteristiken. N2 - In questa tesi si studiano le classi reciproche delle catene di Markov. Data una catena di Markov a tempo continuo su uno spazio numerabile, che svolge il ruolo di dinamica di riferimento, la sua classe reciproca é costituita da tutte le leggi sullo spazio dei cammini che si possono scrivere come un miscuglio dei ponti della legge di riferimento. Questi processi stocastici godono di una propriet`a di independenza condizionale che generalizza la proprietá di Markov ed é ispirata ad un’idea avuta da Schrödinger nel tentativo di derivare un’interpretazione stocastica della meccanica quantistica. A ciascuna classe reciproca é associato un insieme di caratteristiche reciproche. Una caratteristica reciproca é una proprietá della dinamica di riferimento che viene trasmessa a tutti gli elementi della classe, e viene espressa matematicamente da un opportuna combinazione di funzionali del generatore della catena di riferimento. Nella tesi, le caratteristiche vengono calcolate esplicitamente e suddivise in due famiglie principali: le caratteristiche di arco e le caratteristice di ciclo. Come sottoprodotto, otteniamo un criterio esplicito per decidere quando due catene di Markov hanno gli stessi ponti. A partire dalle caratteristiche reciproche, vengono proposte due caratterizzazioni della classe reciproca, compresi i suoi elementi non Markoviani. La prima é basata su un approccio traiettoriale, mentre la seconda si basa sul comportamento asintotico locale dei processi reciproci. Utilizzando il primo approccio, si ottiene una famiglia di equazioni funzionali che ammette come soluzioni tutti e soli gli elementi della classe reciproca. Queste equazioni sono integrazioni per parti sullo spazio dei cammini associate ad operatori differenziali che perturbano le traiettorie del processo canonico con l’aggiunta di loops casuali. Nella costruzione di queste equazioni si impiegano tecniche di geometria discreta, stabilendo un interessante collegamento con risultati recenti in questo campo. Le caratterizzazioni ottenute sono ottimali, in quanto impiegano un numero minimo di equazioni per descrivere la classe. Con questo metodo vengono studiate le classi reciproche di processi di conteggio, di camminate aleatorie su gruppi Abeliani, dove l’insieme dei cicli gode anch’esso di una struttura di gruppo. Il secondo approccio, di natura locale, si basa su stime asintotiche in tempo corto. É ben noto come una catena di Markov sia caratterizzata dal fatto di possedere la propriet`a di Markov e dal comportamento in tempo corto delle probabilitá di transizione. In questa tesi mostriamo che una classe reciproca é caratterizzata dalla propriet`a reciproca, e da due famiglie di stime asintotiche per i ponti del processo. Questo approccio locale permette di analizzare le classi reciproche di passeggiate aleatorie su grafi generali. Come applicazione dei risultati teorici, consideriamo i lattici, i grafi planari, il grafo completo, e l’ipercubo discreto. Infine, otteniamo delle stime di concentrazione della misura e sul comportamento globale dei ponti, sotto l’ipotesi di un limite inferiore per le caratteristiche reciproche. KW - reciprocal characteristics KW - random walks on graphs KW - reziproke Invarianten KW - reziproke Klassen KW - Schrödinger Problem KW - partielle Integration auf dem Pfadraum KW - Irrfahrten auf Graphen Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-82255 ER - TY - INPR A1 - Conforti, Giovanni T1 - Reciprocal classes of continuous time Markov Chains N2 - In this work we study reciprocal classes of Markov walks on graphs. Given a continuous time reference Markov chain on a graph, its reciprocal class is the set of all probability measures which can be represented as a mixture of the bridges of the reference walks. We characterize reciprocal classes with two different approaches. With the first approach we found it as the set of solutions to duality formulae on path space, where the differential operators have the interpretation of the addition of infinitesimal random loops to the paths of the canonical process. With the second approach we look at short time asymptotics of bridges. Both approaches allow an explicit computation of reciprocal characteristics, which are divided into two families, the loop characteristics and the arc characteristics. They are those specific functionals of the generator of the reference chain which determine its reciprocal class. We look at the specific examples such as Cayley graphs, the hypercube and planar graphs. Finally we establish the first concentration of measure results for the bridges of a continuous time Markov chain based on the reciprocal characteristics. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 8 KW - random walks on graphs KW - bridges of random walks KW - reciprocal characteristics KW - Schrödinger problem KW - integration by parts on path space Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-78234 SN - 2193-6943 VL - 4 IS - 8 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Conforti, Giovanni A1 - Leonard, Christian A1 - Murr, Rüdiger A1 - Roelly, Sylvie T1 - Bridges of Markov counting processes. Reciprocal classes and duality formulas JF - Electronic communications in probability N2 - Processes sharing the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula. KW - Counting process KW - bridge KW - reciprocal class KW - duality formula Y1 - 2015 U6 - https://doi.org/10.1214/ECP.v20-3697 SN - 1083-589X VL - 20 PB - Univ. of Washington, Mathematics Dep. CY - Seattle ER - TY - JOUR A1 - Conforti, Giovanni A1 - Roelly, Sylvie T1 - Bridge mixtures of random walks on an Abelian group JF - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability KW - random walk on Abelian group KW - reciprocal class KW - stochastic bridge Y1 - 2017 U6 - https://doi.org/10.3150/15-BEJ783 SN - 1350-7265 SN - 1573-9759 VL - 23 SP - 1518 EP - 1537 PB - International Statistical Institute CY - Voorburg ER - TY - JOUR A1 - Conforti, Giovanni A1 - Kosenkova, Tetiana A1 - Roelly, Sylvie T1 - Conditioned Point Processes with Application to Levy Bridges JF - Journal of theoretical probability N2 - Our first result concerns a characterization by means of a functional equation of Poisson point processes conditioned by the value of their first moment. It leads to a generalized version of Mecke’s formula. En passant, it also allows us to gain quantitative results about stochastic domination for Poisson point processes under linear constraints. Since bridges of a pure jump Lévy process in Rd with a height a can be interpreted as a Poisson point process on space–time conditioned by pinning its first moment to a, our approach allows us to characterize bridges of Lévy processes by means of a functional equation. The latter result has two direct applications: First, we obtain a constructive and simple way to sample Lévy bridge dynamics; second, it allows us to estimate the number of jumps for such bridges. We finally show that our method remains valid for linearly perturbed Lévy processes like periodic Ornstein–Uhlenbeck processes driven by Lévy noise. KW - Ornstein-Uhlenbeck Y1 - 2019 U6 - https://doi.org/10.1007/s10959-018-0863-8 SN - 0894-9840 SN - 1572-9230 VL - 32 IS - 4 SP - 2111 EP - 2134 PB - Springer CY - New York ER -