Volume 52, Number 3, 2016 ISSN 0246-0203

Martingale defocusing and transience of a self-interacting Y. Peres, B. Schapira and P. Sousi 1009–1022 Excited random walk with periodic cookies G. Kozma, T. Orenshtein and I. Shinkar 1023–1049 Harmonic measure in the presence of a spectral gap ....I. Benjamini and A. Yadin 1050–1060 How vertex reinforced arises naturally ...... X. Zeng 1061–1075 Persistence of some additive functionals of Sinai’s walk ...... A. Devulder 1076–1105 Random directed forest and the ...... R. Roy, K. Saha and A. Sarkar 1106–1143 Slowdown in branching Brownian motion with inhomogeneous variance ...... P. Maillard and O. Zeitouni 1144–1160 Maximal displacement of critical branching symmetric stable processes ...... S. P. Lalley and Y. Shao 1161–1177 On the asymptotic behavior of the density of the supremum of Lévy processes ...... L. Chaumont and J. Małecki 1178–1195 Large deviations for non-Markovian diffusions and a path-dependent Eikonal equation ...... J.Ma,Z.Ren,N.TouziandJ.Zhang 1196–1216 Inviscid limits for a stochastically forced shell model of turbulent flow ...... S. Friedlander, N. Glatt-Holtz and V. Vicol 1217–1247

Estimate for Pt D for the Burgers equation G. Da Prato and A. Debussche 1248–1258 Skorokhod embeddings via stochastic flows on the space of Gaussian measures ...... R. Eldan 1259–1280 Liouville heat kernel: Regularity and bounds P. Maillard, R. Rhodes, V. Vargas and O. Zeitouni 1281–1320 Total length of the genealogical tree for quadratic stationary continuous-state branching processes ...... H. Bi and J.-F. Delmas 1321–1350 Weak shape theorem in first passage percolation with infinite passage times ...... R. Cerf and M. Théret 1351–1381 Critical and spanning trees partition functions ...... B. de Tilière 1382–1405

An SLE2 loop measure ...... S. Benoist and J. Dubédat 1406–1436 Independences and partial R-transforms in bi-free probability ...... P. S k o u f r a n i s 1437–1473 Precise large deviation results for products of random matrices D. Buraczewski and S. Mentemeier 1474–1513 Rédacteurs en chef / Chief Editors

Grégory MIERMONT École Normale Supérieure de Lyon CNRS UMR 5669, Unité de Mathématiques Pures et Appliquées 46, allée d’Italie 69364 Lyon Cedex 07, France [email protected]

Christophe SABOT Université Claude Bernard Lyon 1 CNRS UMR 5208, Institut Camille Jordan 43 blvd. du 11 novembre 1918 69622 Villeurbanne cedex, France [email protected] Comité de Rédaction / Editorial Board

V. BALADI (Ecole Normale Supérieure, Paris) G. BLANCHARD (Weierstrass Inst., Berlin) T. BODINEAU (École Polytechnique) P. B OURGADE (New York Univ.) P. C APUTO (Università Roma Tre) B. COLLINS (Université d’Ottawa) I. CORWIN (Columbia University) F. DELARUE (Université de Nice Sophia-Antipolis) H. DUMINIL-COPIN (Université de Genève) F. FLANDOLI (Univ. of Pisa) G. GIACOMIN (Université Paris Diderot) M. HAIRER (Warwick Univ.) M. HOFFMANN (Univ. Paris-Dauphine) Y. H U (Université Paris 13) P. M ATHIEU (Univ. de Provence) L. MYTNIK (Israel Inst. of Technology) A. NACHMIAS (Tel Aviv University) E. PERKINS (Univ. British Columbia) G. PETE (Technical Univ. of Budapest) V. WACHTEL (Universität München) L. ZAMBOTTI (Univ. Pierre et Marie Curie)

Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques (ISSN 0246-0203), Volume 52, Number 3, August 2016. Published quarterly by Association des Publications de l’Institut Henri Poincaré. POSTMASTER: Send address changes to Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, Dues and Subscriptions Office, 9650 Rockville Pike, Suite L 2310, Bethesda, Maryland 20814-3998 USA.

Copyright © 2016 Association des Publications de l’Institut Henri Poincaré Président et directeur de la publication : Cédric Villani Printed in the United States of America Périodicité : 4 nos /an Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2016, Vol. 52, No. 3, 1009–1022 DOI: 10.1214/14-AIHP667 © Association des Publications de l’Institut Henri Poincaré, 2016

Martingale defocusing and transience of a self-interacting random walk

Yuval Peresa, Bruno Schapirab and Perla Sousic

aMicrosoft Research, Redmond, Washington, USA. E-mail: [email protected] bAix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France. E-mail: [email protected] cUniversity of Cambridge, Cambridge, UK. E-mail: [email protected]

Abstract. Suppose that (X,Y,Z) is a random walk in Z3 that moves in the following way: on the first visit to a vertex only Z changes by ±1 equally likely, while on later visits to the same vertex (X, Y ) performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.

Résumé. Supposons que (X,Y,Z)soit une marche aléatoire dans Z3 qui se déplace de la façon suivante : à la première visite en un site, seule la coordonnée Z saute de ±1 avec probabilité uniforme, et aux visites suivantes en ce site (X, Y ) effectue un saut dans l’ensemble {(±1, 0), (0, ±1)} avec probabilité uniforme. Nous montrons que cette marche est transiente, répondant ainsi à une question de Benjamini, Kozma et Schapira. Un ingrédient important de la preuve est un résultat de dispersion pour les martingales.

MSC: 60K35 Keywords: Transience; Martingale; Self-interacting random walk; Excited random walk

References

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Excited random walk with periodic cookies

Gady Kozma1,3, Tal Orenshtein1 and Igor Shinkar2

Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected]; [email protected]; [email protected]

Abstract. In this paper we consider an excited random walk (ERW) on Z in identically piled periodic environment. This is a M discrete time process on Z defined by parameters (p1,...,pM ) ∈[0, 1] for some positive integer M, where the walker upon the ith visit to z ∈ Z moves to z + 1 with probability pi(mod M), and moves to z − 1 with probability 1 − pi(mod M).Wegivean explicit formula in terms of the parameters (p1,...,pM ) which determines whether the walk is recurrent, transient to the left, or 1 M = 1 transient to the right. In particular, in the case that M i=1 pi 2 all behaviors are possible, and may depend on the order of the pi. Our framework allows us to reprove some known results on ERW and branching processes with migration with no additional effort.

Résumé. Dans ce papier, nous considérons une marche aléatoire excitée (MAE) sur Z en environnement empilé de manière iden- M tique et périodique. Il s’agit d’un processus à temps discret sur Z défini par des paramètres (p1,...,pM ) ∈[0, 1] pour un certain entier strictement positif M, où le marcheur, après la iième visite au site z ∈ Z se déplace soit en z + 1 avec probabi- lité pi(mod M),soitenz − 1 avec probabilité 1 − pi(mod M). Nous donnons une formule explicite en fonction des paramètres (p1,...,pM) qui détermine si la marche est récurrente, transiente vers la gauche, ou transiente vers la droite. En particulier, dans 1 M = 1 le cas où M i=1 pi 2 , tous les comportements sont possibles et peuvent dépendre de l’ordre des pi. Notre approche permet de retrouver directement certains résultats connus sur les MAE et les processus de branchement sans immigration.

MSC: 60K35; 60J85 Keywords: Excited random walk; Cookie walk; Recurrence; Transience; ; Lyapunov function; with migration

References

[1] G. Amir, N. Berger and T. Orenshtein. Zero–one law for directional transience of one dimensional excited random walks. Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 47–57. MR3449293 [2] A.-L. Basdevant and A. Singh. On the speed of a cookie random walk. Probab. Theory Related Fields 141 (3) (2008) 625–645. MR2391167 [3] I. Benjamini and D. B. Wilson. Excited random walk. Electron. Commun. Probab. 8 (9) (2003) 86–92. MR1987097 [4] A. C. Berry. The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49 (1) (1941) 122–136. MR0003498 [5] C. G. Esseen. On the Liapunoff limit of error in the theory of probability. Ark. Mat. Astr. Fys. A28 (1942) 1–19. MR0011909 [6] T. E. Harris. First passage and recurrence distributions. Trans. Amer. Math. Soc. 73 (3) (1952) 471–486. MR0052057 [7] H. Kesten, M. V. Kozlov and F. Spitzer. A limit law for random walk in a random environment. Compos. Math. 30 (1975) 145–168. MR0380998 [8] E. Kosygina and T. Mountford. Limit laws of transient excited random walks on integers. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2) (2011) 575–600. MR2814424 [9] E. Kosygina and M. P. W. Zerner. Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13 (2008) 1952–1979. MR2453552 [10] E. Kosygina and M. P. W. Zerner. Excited random walks: Results, methods, open problems. Bull. Inst. Math. Acad. Sin. (N.S.) 8 (1) (2013) 105–107. MR3097419 [11] E. Kosygina and M. P. W. Zerner. Excursions of excited random walks on integers. Electron. J. Probab. 19 (2014) 1–25. MR3174837 [12] J. Lamperti. Criteria for the recurrence or transience of . I. J. Math. Anal. Appl. 1 (1960) 314–330. MR0126872 [13] D. A. Levin, Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI, 2009. MR2466937 [14] M. V. Menshikov, I. M. Asymonth and R. Iasnogorodski. Markov processes with asymptotically zero drift. Probl. Inf. Transm. 31 (3) (1995) 248–261. [15] M. P. W. Zerner. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (1) (2005) 98–122. MR2197139 Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2016, Vol. 52, No. 3, 1050–1060 DOI: 10.1214/15-AIHP670 © Association des Publications de l’Institut Henri Poincaré, 2016

Harmonic measure in the presence of a spectral gap

Itai Benjaminia and Ariel Yadinb

aFaculty of Mathematics and Computer Science, The Weizmann Institute of Science, 234 Herzl Street, Rehovot 76100, Israel. E-mail: [email protected] bDepartment of Mathematics, Ben Gurion-University of the Negev, PO Box 653, Be’er Sheva 8410501, Israel. E-mail: [email protected]

Abstract. We study harmonic measure in finite graphs with an emphasis on expanders, that is, positive spectral gap. It is shown that if the spectral gap is positive then for all sets that are not too large the harmonic measure from a uniform starting point is not more than a constant factor of the uniform measure on the set. For large sets there is a tight logarithmic correction factor. We also show that positive spectral gap does not allow for a fixed proportion of the harmonic measure of sets to be supported on small subsets, in contrast to the situation in Euclidean space. The results are quantitative as a function of the spectral gap, and apply also when the spectral gap decays to 0 as the size of the graph grows to infinity. As an application we consider a model of diffusion limited aggregation,orDLA, on finite graphs, obtaining upper bounds on the growth rate of the aggregate.

Résumé. On étudie la mesure harmonique sur les graphes finis en s’intéressant de près au cas des expanseurs, c’est à dire des graphes dont le trou spectral est positif. On montrera que dans ce cas, pour tout sous-ensemble pas trop gros, la mesure harmonique vue d’un point uniforme est bornée par un facteur multiplicatif fois la mesure uniforme sur l’ensemble. Pour les gros ensembles il y a une correction logarithmique tendue. On montrera aussi que dans le cas d’un trou spectral positif, une proportion constante de la mesure harmonique ne peut pas être supportée par de petits sous-ensembles, contrairement à ce qui se passe dans le cas euclidien. Des résultats quantitatifs sont présentés en fonction de la taille du trou spectral, et s’appliquent aussi lorsque cette taille tend vers 0 lorsque la taille du graphe tend vers l’infini. En application, on considèrera un modèle d’agrégation limitée par diffusion (DLA) sur des graphes finis, pour obtenir des bornes supérieures sur la croissance de l’aggrégat.

MSC: 05C81; 31A15 Keywords: Harmonic measure; Spectral gap; Buerling estimate; DLA

References

[1] D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. Unpublished book, 1999. Available at http://www.stat. berkeley.edu/~aldous/RWG/book.html. [2] N. Alon and J. H. Spencer. The Probabilistic Method. Wiley, Hoboken, NJ, 2004. [3] M. T. Barlow, R. Pemantle and E. A. Perkins. Diffusion limited aggregation on a tree. Probab. Theory Related Fields 107 (1997) 1–60. MR1427716 [4] I. Benjamini. On the support of harmonic measure for the random walk. Israel J. Math. 100 (1997) 1–6. MR1469102 [5] I. Benjamini, H. Finucane and R. Tessera. On the scaling limit of finite vertex transitive graphs with large diameter. Combinatorica. To appear. Available at arXiv:1203.5624. [6] J. Bourgain. On the Hausdorff dimension of harmonic measure in higher dimension. Invent. Math. 87 (3) (1987) 477–483. MR0874032 [7] R. Eldan. Diffusion limited aggregation on the hyperbolic plane. Preprint, 2013. Available at arXiv:1306.3129. [8] J. B. Garnett and D. E. Marshall. Harmonic Measure, 2. Cambridge Univ. Press, Cambridge, 2005. MR2150803 [9] P. W. Jones and T. H. Wolff. Hausdorff dimension of harmonic measures in the plane. Acta Math. 161 (1) (1988) 131–144. MR0962097 [10] H. Kesten. How long are the arms in DLA? J. Phys. A: Math. Gen. 20 (1) (1987) L29–L33. MR0873177 [11] H. Kesten. Upper bounds for the growth rate of DLA. Phys. A 168 (1) (1990) 529–535. MR1077203 [12] G. F. Lawler. A discrete analogue of a theorem of Makarov. Combin. Probab. Comput. 2 (2) (1993) 181–199. MR1249129 [13] G. F. Lawler. Intersections of Random Walks. Springer, New York, 2013. MR2985195 [14] D. A. Levin, Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. AMS, Providence, RI, 2009. MR2466937 [15] N. G. Makarov. On the distortion of boundary sets under conformal mappings. Proc. London Math. Soc. (3) 51 (2) (1985) 369–384. MR0794117 [16] T. A. Witten and L. M. Sander. Diffusion-limited aggregation. Phys.Rev.B27 (9) (1983) 5686–5697. MR0704464 [17] W. Woess. Lamplighters, Diestel–Leader graphs, random walks, and harmonic functions. Combin. Probab. Comput. 14 (3) (2005) 415–433. MR2138121 Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2016, Vol. 52, No. 3, 1061–1075 DOI: 10.1214/15-AIHP671 © Association des Publications de l’Institut Henri Poincaré, 2016

How vertex reinforced jump process arises naturally

Xiaolin Zeng

Institut Camille Jordan, Université Lyon 1, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. E-mail: [email protected]

Abstract. We prove that the only nearest neighbor jump process with local dependence on the occupation times satisfying the partially exchangeable property is the vertex reinforced jump process, under some technical conditions (Theorem 4). This result gives a counterpart to the characterization of edge reinforced random walk given by Rolles (Probab. Theory Related Fields 126 (2003) 243–260).

Résumé. Nous montrons que le seul processus de saut sur les plus proches voisins avec une dépendance locale par rapport aux temps d’occupation et satisfaisant la propriété d’échangeabilité partielle est, sous quelques conditions techniques, le processus de saut avec renforcement par sommet (Théorème 4). Ce résultat donne une contrepartie à la caractérisation de la marche aléatoire avec renforcement par arête obtenue par Rolles (Probab. Theory Related Fields 126 (2003) 243–260).

MSC: 60G Keywords: Partial exchangeability; Vertex reinforced jump processes

References

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Persistence of some additive functionals of Sinai’s walk1

Alexis Devulder

Université de Versailles Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Bât. Fermat, 45 avenue des Etats-Unis, 78035 Versailles Cedex, France. E-mail: [email protected]

 n Abstract. We are interested in Sinai’s walk√(Sn)n∈N. We prove that the annealed probability that k=0 f(Sk) is strictly positive (3− 5)/2+o(1) for all n ∈[1,N] is equal√ to 1/(log N) , for a large class of functions f , and in particular for f(x)= x. The per- 3− 5 sistence exponent 2 first appears in a nonrigorous paper of Le Doussal, Monthus and Fischer, with motivations coming from physics. The proof relies on techniques of localization for Sinai’s walk and uses results of Cheliotis about the sign changes of the bottom of valleys of a two-sided Brownian motion.  n Résumé. Nous nous intéressons à la marche de Sinai (Sn)n∈N. Nous√ prouvons que la probabilité annealed que k=0 f(Sk) soit (3− 5)/2+o(1) strictement positive pour tout n ∈[1,N] est égale√ à 1/(log N) , pour une large classe de fonctions f , et en particulier = 3− 5 pour f(x) x. L’exposant de persistance 2 est d’abord apparu dans un article non rigoureux de Le Doussal, Monthus et Fischer, avec des motivations venant de la physique. La preuve est basée sur des techniques de localisation pour la marche de Sinai et utilise des résultats de Cheliotis sur les changements de signe des fonds de vallées d’un mouvement Brownien indexé par R.

MSC: 60K37; 60J55 Keywords: Random walk in random environment; Sinai’s walk; Integrated random walk; One-sided exit problem; Persistence; Survival exponent

References

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Random directed forest and the Brownian web

Rahul Roy, Kumarjit Saha and Anish Sarkar

Indian Statistical Institute, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India. E-mail: [email protected]; [email protected]; [email protected]

Abstract. Consider the d dimensional lattice Zd where each vertex is open or closed with probability p or 1 − p respectively. An open vertex u := (u(1), u(2),...,u(d)) is connected by an edge to another open vertex which has the minimum L1 distance among all the open vertices x with x(d) > u(d). It is shown that this is a tree almost surely for d = 2and3anditis an infinite collection of disjoint trees for d ≥ 4. In addition, for d = 2, we show that when properly scaled, the family of its paths converges in distribution to the Brownian web.

Résumé. Nous considérons le réseau Zd dont les sommets sont ouverts ou fermés, respectivement avec probabilité p et 1 − p. Chaque sommet ouvert u = (u(1), u(2),...,u(d)) est connecté par une arête au sommet ouvert x le plus proche de lui, pour la distance L1, et satisfaisant x(d) > u(d). Nous montrons que le graphe aléatoire résultant est presque sûrement un arbre pour d = 2 et 3, et qu’il est une collection infinie d’arbres disjoints pour d ≥ 4. De plus, pour d = 2, nous montrons que la famille de ses trajectoires correctement renormalisées converge en loi vers la toile Brownienne.

MSC: 60D05; 60K35

Keywords: ; Random walk; Directed spanning forest; Brownian web

References

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Slowdown in branching Brownian motion with inhomogeneous variance

Pascal Maillarda,1 and Ofer Zeitounib,2

aDépartement de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France. bDepartment of Mathematics, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel and Courant Institute, New-York University. E-mail: [email protected]

Abstract. We consider the distribution of the maximum MT of branching Brownian motion with time-inhomogeneous variance of the form σ 2(t/T ),whereσ(·) is a strictly decreasing function. This corresponds to the study of the time-inhomogeneous 2 Fisher–Kolmogorov–Petrovskii–Piskunov (F-KPP) equation Ft (x, t) = σ (1−t/T)Fxx(x, t)/2+g(F(x,t)), for appropriate non- · − 1/3 linearities g( ). Fang and Zeitouni (J. Stat. Phys. 149 (2012) 1–9) showed that MT vσ T is negative of order T ,where   v = 1 σ(s)ds. In this paper, we show the existence of a function m , such that M − m converges in law, as T →∞.Fur- σ 0 T  T T  = − 1/3 − + = −1/3 1 1/3|  |2/3 − =− thermore, mT vσ T wσ T σ(1) log T O(1) with wσ 2 α1 0 σ(s) σ (s) ds. Here, α1 2.33811 ...is the largest zero of the Airy function Ai. The proof uses a mixture of probabilistic and analytic arguments.

Résumé. Nous étudions la loi du maximum MT d’un mouvement brownien branchant avec une variance inhomogène en temps de la form σ 2(t/T ),oùσ(·) est une fonction strictement décroissante. Ceci correspond à étudier l’équation Fisher–Kolmogorov– 2 Petrovskii–Piskunov (F-KPP) inhomogène en temps, Ft (x, t) = σ (1 − t/T)Fxx(x, t)/2 + g(F(x,t)), pour des nonlinéarités · − 1/3 g( ) appropriées. Fang et Zeitouni (J. Stat. Phys. 149 (2012) 1–9) ont montré que MT vσ T est negatif de l’ordre T ,où   v = 1 σ(s)ds. Dans cet article, nous montrons l’existence d’une fonction m telle que M −m converge en loi quand T →∞. σ 0  T T T  = − 1/3 − + = −1/3 1 1/3|  |2/3 − =− De plus, mT vσ T wσ T σ(1) log T O(1) avec wσ 2 α1 0 σ(s) σ (s) ds.Ici, α1 2.33811 ...est la plus grande racine de la fonction d’Airy Ai. La démonstration repose sur un mélange d’arguments probabilistes et analytiques.

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Maximal displacement of critical branching symmetric stable processes

Steven P. Lalleya,1 andYuanShaob

aDepartment of Statistics, University of Chicago, Chicago, IL 60637, USA. E-mail: [email protected]; url: www.statistics.uchicago.edu/~lalley bDepartment of Mathematics, University of Chicago, Chicago, IL 60637, USA. E-mail: [email protected]

Abstract. We consider a critical continuous-time branching process (a Yule process) in which the individuals independently exe- cute symmetric α-stable random motions on the real line starting at their birth points. Because the branching process is critical, it will eventually die out, and so there is a well-defined maximal location M ever visited by an individual particle of the process. We − prove that the distribution of M satisfies the asymptotic relation P {M ≥ x}∼(2/α)1/2x α/2 as x →∞.

Résumé. Nous considérons un processus de branchement en temps continu critique (processus de Yule) dont les individus suivent indépendamment un processus α-stable symétrique réel issu de leur point de naissance. Le processus de branchement étant critique, il s’éteint presque surement et nous pouvons définir la valeur M qui représente la position maximale jamais atteinte par un individu. − Nous montrons que la distribution de M satisfait l’estimée asymptotique suivante : P {M ≥ x}∼(2/α)1/2x α/2.

MSC: Primary 60J80; secondary 60J15 Keywords: Branching ; Critical branching process; Nonlinear convolution equation; Feynman–Kac formula; Fractional Laplacian

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On the asymptotic behavior of the density of the supremum of Lévy processes

Loïc Chaumonta,1 and Jacek Małeckib,2

aLAREMA UMR CNRS 6093, Université d’Angers, 2, Bd Lavoisier, Angers Cedex 01, 49045, France. E-mail: [email protected] bFaculty of Fundamental Problems of Technology, Department of Mathematics, Wrocław University of Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław, Poland. E-mail: [email protected]

Abstract. Let us consider a real valued Lévy process X whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of X before any deterministic time t is absolutely continuous on (0, ∞). We show that  its density ft (x) is continuous on (0, ∞) if and only if the potential density h of the upward ladder height process is continuous  on (0, ∞). Then we prove that ft behaves at 0 as h . We also describe the asymptotic behaviour of ft ,whent tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the Lévy process conditioned to stay positive.

Résumé. Soit X un processus de Lévy réel dont les probabilités de transition sont absolument continues par rapport à la mesure de Lebesgue. Dans ce cas, il est connu que la loi du supremum passé avant un temps déterministe t est elle-même absolument continue sur (0, ∞). En supposant de plus que les densités sont bornées, nous montrons que la densité ft (x) du supremum passé  est continue en x sur (0, ∞), si et seulement si la densité potentielle h (x) du subordinateur des hauteurs d’échelle ascendant est  continue sur (0, ∞). Nous montrons alors que ft se comporte en 0 de la même manière que h . Nous donnons également une description du comportement asymptotique de ft , lorsque t tend vers l’infini. Enfin nous appliquons ces résultats pour étudier le comportement asymptotique de la densité du méandre des processus de Lévy et de la densité de la loi d’entrée des processus de Lévy conditionnés à rester positifs.

MSC: 60G51; 60J75 Keywords: Density; Past supremum; Asymptotic behaviour; Renewal function; Conditioning to stay positive; Meander

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Large deviations for non-Markovian diffusions and a path-dependent Eikonal equation

Jin Maa,1, Zhenjie Renb,2, Nizar Touzib,3 and Jianfeng Zhanga,4

aDepartment of Mathematics, University of Southern California, 3620 S. Vermont Ave, KAP 108, Los Angeles, CA 90089-1113, USA. E-mail: [email protected]; [email protected] bCMAP, Ecole Polytechnique Paris, UMR CNRS 7641, 91128 Palaiseau Cedex, France. E-mail: [email protected]; [email protected]

Abstract. This paper provides a large deviation principle for non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao and Liu (Stoch. Dyn. 6 (2006) 487–520), this extends the corresponding results collected in Freidlin and Wentzell (Random Perturbations of Dynamical Systems (1984) Springer). However, we use a different line of argument, adapting the PDE method of Fleming (Appl. Math. Optim. 4 (1978) 329–346) and Evans and Ishii (Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 1–20) to the path-dependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path- dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied surface in financial mathematics.

Résumé. Nous montrons un principe de grandes déviations pour les équations différentielles stochastiques non-markoviennes, dirigées par un mouvement brownien, et à coefficients aléatoires dépendant de l’ensemble du passé. Comme dans Gao et Liu (Stoch. Dyn. 6 (2006) 487–520), ceci étend les résultats correspondants dans Freidlin et Wentzell (Random Perturbations of Dyna- mical Systems (1984) Springer). Cependant, nous utilisons un argument différent, adaptant la méthode d’EDP de Fleming (Appl. Math. Optim. 4 (1978) 329–346) et Evans et Ishii (Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 1–20) au cas des équations dépendant des trajectoires, en utilisant des techniques d’équations différentielles stochastiques rétrogrades. Comme dans le cas markovien, nous obtenons une caractérisation de la fonction d’action comme l’unique solution bornée d’une version non marko- vienne de l’équation eikonale. Enfin, nous proposons une application à l’analyse asymptotique, en maturité courte, de la surface de volatilité implicite en mathématiques financières.

MSC: 35D40; 35K10; 60H10; 60H30 Keywords: Large deviations; Backward stochastic differential equations; Viscosity solutions of path-dependent PDEs

References

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Inviscid limits for a stochastically forced shell model of turbulent flow

Susan Friedlandera, Nathan Glatt-Holtzb andVladVicolc

aUniversity of Southern California, Los Angeles, CA 90089, USA. E-mail: [email protected] bVirginia Tech, Blacksburg, VA 24061, USA. E-mail: [email protected] cPrinceton University, Princeton, NJ 08544, USA. E-mail: [email protected]

Abstract. We establish the anomalous mean dissipation rate of energy in the inviscid limit for a stochastic shell model of turbulent fluid flow. The proof relies on viscosity independent bounds for stationary solutions and on establishing ergodic and mixing properties for the viscous model. The shell model is subject to a degenerate stochastic forcing in the sense that noise acts directly only through one wavenumber. We show that it is hypo-elliptic (in the sense of Hörmander) and use this property to prove a gradient bound on the Markov semigroup.

Résumé. Nous étudions le taux anormal de la dissipation moyenne de l’énergie dans la limite non visqueuse d’un modèle en couche de fluide turbulent. La preuve se base sur des estimations indépendantes de la viscosité pour des solutions stationnaires, ainsi que sur des propriétés ergodiques et de mélange pour le modèle visqueux. Le modèle en couche subit un forçage aléatoire dégénéré, c’est à dire que le bruit n’agit seulement que sur un mode. Nous montrons que le système est hypoelliptique au sens d’Hörmander et utilisons cette propriété pour prouver une borne sur le gradient du semigroupe de Markov.

MSC: 76F05; 35Q35; 37L40; 37L55 Keywords: Inviscid limits; Invariant measures; Dissipation anomaly; Shell models;

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MR0562801 Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2016, Vol. 52, No. 3, 1248–1258 DOI: 10.1214/15-AIHP685 © Association des Publications de l’Institut Henri Poincaré, 2016

Estimate for PtD for the stochastic Burgers equation

Giuseppe Da Pratoa,1 and Arnaud Debusscheb,2

aScuola Normale Superiore, 56126, Pisa, Italy. E-mail: [email protected] bIRMAR and École Normale Supérieure de Rennes, Campus de Ker Lann, 37170 Bruz, France. E-mail: [email protected]

Abstract. We consider the Burgers equation on H = L2(0, 1) perturbed by and the corresponding transition semi- group Pt .WeproveanewformulaforPt Dxϕ which depends on ϕ but not on its derivative. This formula allows us to provide 2 a bound on Dxϕ in L (H, ν) where ν is the invariant measure of Pt . Some new consequences for the invariant measure ν of Pt are discussed as its Fomin differentiability and an integration by parts formula which generalises the classical one for Gaussian measures.

Résumé. Nous considèrons l’équation de Burgers stochastique sur H = L2(0, 1) dirigée par un bruit blanc, de semi-groupe de transition Pt , et démontrons une nouvelle formule qui permet d’exprimer Pt Dxϕ en terme de ϕ mais pas de sa différentielle. Celle- 2 ci nous permet d’obtenir des estimations sur Dxϕ dans L (H, ν),oùν est la mesure invariante de Pt , dont découlent quelques conséquences telles que l’existence de dérivées de Fomin pour ν ou encore une formule d’intégration par partie qui généralise celle bien connue pour les mesures gaussiennes.

MSC: 60H15; 35R15 Keywords: Stochastic Burgers equation; invariant measure; Fomin differentiability

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Skorokhod embeddings via stochastic flows on the space of Gaussian measures

Ronen Eldan

Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel. E-mail: [email protected]

Abstract. We present a new construction of a Skorokhod embedding, namely, given a probability measure μ with zero expectation and finite variance, we construct an integrable T adapted to a filtration Ft , such that WT has the law μ,whereWt is a standard adapted to the same filtration. We find several sufficient conditions for the stopping time T to be bounded or to have a sub-exponential tail. In particular, our embedding seems rather natural for the case that μ is a log-concave measure and T satisfies several tight bounds in that case. Our embedding admits the property that the stochastic measure-valued process {μt }t≥0,whereμt is as the law of WT conditioned on Ft , is a Markov process. In view of this property, we will consider a more general family of Skorokhod embeddings which can be constructed via a kernel generating a stochastic flow on the space of measures. This family includes existing constructions such as the ones by Azéma–Yor (In Séminaire de Probabilités XIII (1979) 90–115 Springer) and by Bass (In Séminaire de Probabilités XVII (1983) 221–224 Springer), and thus suggests a new point of view on these constructions.

Résumé. Nous proposons une nouvelle construction d’un plongement de Skorokhod: étant donnée une mesure de probabilité μ avec espérance nulle et variance finie, nous construisons un temps d’arrêt intégrable T adapté à la filtration Ft , tel que WT possède la loi μ et W est un processus de Wiener standard adapté à la même filtration. Nous trouvons plusieurs conditions suffisantes pour que le temps d’arrêt T soit borné ou ait des queues sous-exponentielles. En particulier, notre plongement semble assez naturel dans le cas où μ est log-concave et T satisfait plusieurs estimations fortes. Notre plongement a la propriété suivante : le processus stochastique à valeur dans les mesures {μt }t≥0,oùμt est la loi de WT conditionnée par Ft , est un processus de Markov. Compte tenu de cette propriété, nous allons considérer une famille plus générale de plongements de Skorokhod qui peuvent être construits à l’aide d’un noyau générant un flot stochastique sur l’espace des mesures. Cette famille inclut des constructions déjà existantes comme celle d’Azéma–Yor (In Séminaire de Probabilités XIII (1979) 90–115 Springer) et celle de Bass (In Séminaire de Probabilités XVII (1983) 221–224 Springer), suggérant ainsi un point de vue nouveau sur ces constructions.

MSC: Primary 60G40; 60G44; secondary 60H20

Keywords: Skorokhod embedding; Brownian motion; Log concave measure; Markov chain

References

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Liouville heat kernel: Regularity and bounds

P. Maillarda,1, R. Rhodesb,2,V.Vargasc,2 and O. Zeitounia,1

aWeizmann Institute of Science, Rehovot, Israel bUniversité Paris-Dauphine, Ceremade, F-75016 Paris, France cEcole Normale Supérieure, DMA, 45 rue d’Ulm, 75005 Paris, France

Abstract. We initiate in this paper the study of analytic properties of the Liouville heat kernel. In particular, we establish regularity estimates on the heat kernel and derive non-trivial lower and upper bounds.

Résumé. Dans ce papier, nous initions l’étude des propriétés analytiques du noyau de la chaleur de Liouville. En particulier, nous établissons des estimées de régularité pour le noyau et nous l’encadrons par des bornes inférieures et supérieures non triviales.

MSC: 35K08; 60J60; 60K37; 60J55; 60J70 Keywords: Liouville quantum gravity; Heat kernel; Liouville Brownian motion; Gaussian multiplicative chaos

References

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Probab. 38 (2) (2010) 605–631. MR2642887 [40] S. Sheffield. Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 (2007) 521–541. MR2322706 [41] A.-S. Sznitman. Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998. MR1717054 [42] Y. Watabiki. Analytic study of fractal structure of quantized surface in two-dimensional quantum gravity. Progr. Theoret. Phys. 114 (Suppl.) (1993) 1–17. MR1268911 Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2016, Vol. 52, No. 3, 1321–1350 DOI: 10.1214/15-AIHP683 © Association des Publications de l’Institut Henri Poincaré, 2016

Total length of the genealogical tree for quadratic stationary continuous-state branching processes

Hongwei Bia,1 and Jean-François Delmasb aSchool of Insurance and Economics, University of International Business and Economics, Beijing 100029, China. E-mail: [email protected] bUniversité Paris-Est, CERMICS (ENPC), F-77455 Marne La Vallee, France. E-mail: [email protected]

Abstract. We prove the existence of the total length process for the genealogical tree of a population model with random size given by quadratic stationary continuous-state branching processes. We also give, for the one-dimensional marginal, its Laplace transform as well as the fluctuation of the corresponding convergence. This result is to be compared with the one obtained by Pfaffelhuber and Wakolbinger for a constant size population associated to the Kingman coalescent. We also give a time reversal property of the number of ancestors process at all times, and a description of the so-called lineage tree in this model.

Résumé. Nous démonstrons l’existence du processus de longueur totale renormalisée pour l’arbre généalogique dans un modèle de population dont la taille évolue suivant un processus de branchement continu quadratique (diffusion de Feller). Nous donnons également la loi unidimensionnelle de la longueur totale de l’arbre généalogique ainsi que les fluctuations associées à la renor- malisation. Ce résultat est à rapprocher de ceux obtenus par Pfaffelhuber et Wakolbinger dans le cadre d’une population de taille constante associée au processus de coalescence de Kingman. Nous établissons également une propriété d’invariance par retourne- ment du temps pour le processus du nombre des ancêtres qui permet d’obtenir en particulier une description du processus ancestral dans ce modèle. MSC: Primary 60J80; 92D25; secondary 60G10; 60G55 Keywords: Branching process; Population model; Genealogical tree; Lineage tree; Time-reversal

References

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Weak shape theorem in first passage percolation with infinite passage times

Raphaël Cerfa and Marie Théretb

aDMA, Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France. E-mail: [email protected]; url: http://www.math.u-psud.fr/~cerf bLPMA, Université Paris Diderot, Bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris Cedex 13, France. E-mail: [email protected];url:http://www.proba.jussieu.fr/~theret

Abstract. We consider the model of i.i.d. first passage percolation on Zd : we associate with each edge e of the graph a passage time t(e) taking values in [0, +∞], such that P[t(e)<+∞] >pc(d). Equivalently, we consider a standard (finite) i.i.d. first passage percolation model on a super-critical Bernoulli percolation performed independently. We prove a weak shape theorem without any moment assumption. We also prove that the corresponding time constant is positive if and only if P[t(e)= 0]

Résumé. Nous considérons le modèle standard de percolation de premier passage sur Zd : nous associons à chaque arête e du graphe un temps de passage t(e) à valeurs dans [0, +∞], tel que P[t(e)<+∞] >pc(d). De façon équivalente, nous considérons un modèle de percolation de premier passage standard (fini) sur le graphe obtenu par une percolation de Bernoulli sur-critique réalisée indépendamment. Nous prouvons un théorème de forme faible sans aucune hypothèse de moment. Nous prouvons aussi que la constante de temps correspondante est strictement positive si et seulement si P[t(e)= 0]

MSC: Primary 60K35; 60K35; secondary 82B20 Keywords: First passage percolation; Time constant; Shape theorem

References

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Critical Ising model and spanning trees partition functions

Béatrice de Tilière1

UPMC Univ. Paris 06, UMR 7599, Laboratoire de Probabilités et Modèles Aléatoires, 4 place Jussieu, F-75005 Paris, France. E-mail: [email protected]

Abstract. We prove that the squared partition function of the two-dimensional critical Ising model defined on a finite, isoradial | | graph G = (V, E), is equal to 2 V times the partition function of spanning trees of the graph G¯ ,whereG¯ is the graph G extended along the boundary; edges of G are assigned Kenyon’s (Invent. Math. 150 (2) (2002) 409–439) critical weights, and boundary edges of G¯ have specific weights. The proof is an explicit construction, providing a new relation on the level of configurations between two classical, critical models of statistical mechanics.

Résumé. Nous montrons que le carré de la fonction de partition du modèle d’Ising critique en dimension deux, défini sur un graphe | | isoradial G = (V, E) fini, est égale à 2 V fois la fonction de partition des arbres couvrants du graphe G¯ , où le graphe G¯ est le graphe G prolongé le long du bord; les arêtes de G sont munies des poids critiques de Kenyon (Invent. Math. 150 (2) (2002) 409–439), et les arêtes du bord de G¯ ont des poids spécifiques. La preuve consiste en une construction explicite, qui donne une nouvelle relation, au niveau des configurations, entre deux modèles classiques de mécanique statistique au point critique.

MSC: 82B20; 82B27; 05A19

Keywords: Critical two-dimensional Ising model; Critical spanning trees; Isoradial graphs; Partition functions

References

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An SLE2 loop measure Stéphane Benoist and Julien Dubédat

Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027. E-mail: [email protected]; [email protected]

Abstract. There is an essentially unique way to associate to any Riemann surface a measure on its simple loops, such that the collection of measures satisfy a strong conformal invariance property (see (J. Amer. Math. Soc. 21 (2008) 137–169)). These random loops are constructed as the boundary of Brownian loops, and so correspond in the zoo of statistical mechanics models to central charge 0, or Schramm–Loewner Evolution (SLE) parameter κ = 8/3. The goal of this paper is to construct a family of measures on simple loops on Riemann surfaces that satisfies a conformal covariance property, and that would correspond to SLE parameter κ = 2 (central charge −2). On planar annuli, this loop measure was already built by Adrien Kassel and Rick Kenyon in (Random curves on surfaces induced from the Laplacian determinant (2012) ArXiv e-prints). We will give an alternative construction of this loop measure on planar annuli, investigate its conformal covariance, and finally extend this measure to general Riemann surfaces. This gives an example of a Malliavin–Kontsevich–Suhov loop measure (Tr. Mat. Inst. Steklova 258 (2007) 107–153) in non-zero central charge.

Résumé. Il y a une manière essentiellement unique d’associer à toute surface de Riemann une mesure sur ses boucles simples de telle sorte que la collection de mesures soit invariante conforme en un sens fort (voir (J. Amer. Math. Soc. 21 (2008) 137– 169)). Ces boucles aléatoires peuvent être construites comme frontières de boucles browniennes, et correpondent donc, dans la classification des modèles de mécanique statistique à la valeur de charge centrale 0, ou alternativement au paramètre d’évolution de Schramm–Loewner (SLE) κ = 8/3. Dans cet article, nous construisons une famille de mesures sur les boucles simples dans les surfaces de Riemann qui est covariante conforme, et qui correpond à la valeur 2 du paramètre SLE (ou de maniére équivalente, à la charge centrale −2). Cette mesure de boucles avait été précédemment construite dans les anneaux planaires par Adrien Kassel et Rick Kenyon (Random curves on surfaces induced from the Laplacian determinant (2012) ArXiv e-prints). Nous donnerons une construction alternative de cette mesure de boucles dans les anneaux planaires et étudierons ses propriétés de covariance conforme, avant d’étendre cette mesure à toute surface de Riemann. En particulier, cela donne un exemple de mesure de Malliavin– Kontsevich–Suhov (Tr. Mat. Inst. Steklova 258 (2007) 107–153) en charge centrale non nulle.

MSC: 60J67; 82B20 Keywords: SLE; UST; Loop; Restriction

References

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Independences and partial R-transforms in bi-free probability

Paul Skoufranis

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA. E-mail: [email protected]

Abstract. In this paper, we examine how various notions of independence in non-commutative probability theory arise in bi-free probability. We exhibit how Boolean and monotone independence occur from bi-free pairs of faces and establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove that bi-freeness is preserved under tensoring with matrices. Finally, via combinatorial arguments, we construct partial R-transforms in two settings relating the moments and cumulants of a left–right pair of operators.

Résumé. Dans cet article, nous examinons comment diverses notions d’indépendance en théorie des probabilités non commuta- tives se traduisent en probabilités bi-libres. Nous montrons comment l’indépendance booléenne et monotone se produisent à partir de paires de faces bi-libres, et établissons un théorème de Kac/Loève pour l’indépendance bi-libre. En outre, nous prouvons que l’indépendance bi-libre est préservée par tensorisation avec des matrices. Enfin, par des arguments combinatoires, nous construi- sons deux types de R-transformations partielles, reliant les moments et les cumulants d’une paire gauche-droite des opérateurs.

MSC: 46L54; 46L53 Keywords: Bi-free probability; Free independence; Boolean independence; Monotone independence; Bi-free independence over matrices; Partial R-transforms

References

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Precise large deviation results for products of random matrices

Dariusz Buraczewskia,1 and Sebastian Mentemeierb,2

aUniwersytet Wrocławski, Instytut Matematyczny, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. E-mail: [email protected] bTU Dortmund, Fakultät für Mathematik, Lehrstuhl IV, Vogelpothsweg 87, 44227 Dortmund, Germany. E-mail: [email protected]

Abstract. The theorem of Furstenberg and Kesten provides a strong for the norm of a product of random matrices. This can be extended under various assumptions, covering nonnegative as well as invertible matrices, to a law of large numbers for the norm of a vector on which the matrices act. We prove corresponding precise large deviation results, generalizing the Bahadur–Rao theorem to this situation. Therefore, we obtain a third-order Edgeworth expansion for the cumulative distribution function of the vector norm. This result in turn relies on an application of the Nagaev–Guivarch method. Our result is then used to study matrix recursions, arising e.g. in financial , and to provide precise large deviation estimates there.

Résumé. Le théorème de Furstenberg et Kesten établit une loi forte des grands nombres pour la norme d’un produit de matrices aléatoires. Cela peut être étendu sous des hypothèses variées, dans le cas des matrices positives ou inversibles, à une loi des grand nombres pour la norme d’un vecteur sur lequel les matrices agissent. Dans ce cadre, nous prouvons des résultats de grandes déviations précis, en généralisant le théorème de Bahadur–Rao à cette situation. Ainsi, nous obtenons une expansion de Edgeworth au troisième ordre pour la fonction de répartition de la norme du vecteur. Ce résultat se base sur une application de la méthode de Nagaev–Guivarch. Notre résultat est utilisé ensuite pour étudier des récurrences matricielles, qui apparaissent par exemple dans les séries temporelles en finance, et pour donner des estimations précises de grandes déviations.

MSC: Primary 60F10; secondary 60H25 Keywords: Products of random matrices; Limit theorems; Large deviations; Random difference equations; Edgeworth expansion; Fourier techniques; Markov chains with general state space; Markov random walks; Heavy tails

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