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Integration by Parts Formulae for the Laws of Bessel Bridges, and Bessel Stochastic Pdes Henri Elad Altman

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Integration by Parts Formulae for the Laws of Bessel Bridges, and Bessel Stochastic Pdes Henri Elad Altman Integration by parts formulae for the laws of Bessel bridges, and Bessel stochastic PDEs Henri Elad Altman To cite this version: Henri Elad Altman. Integration by parts formulae for the laws of Bessel bridges, and Bessel stochastic PDEs. Probability [math.PR]. Sorbonne Université, 2019. English. NNT : 2019SORUS441. tel- 02284974v2 HAL Id: tel-02284974 https://tel.archives-ouvertes.fr/tel-02284974v2 Submitted on 24 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sorbonne Université Ecole doctorale 386 : Sciences Mathématiques de Paris Centre Laboratoire de Probabilités, Statistique et Modélisation Integration by parts formulae for the laws of Bessel bridges, and Bessel stochastic PDEs Par Henri ELAD ALTMAN Thèse de doctorat de Mathématiques Dirigée par Lorenzo ZAMBOTTI Rapporteurs : M. Arnaud DEBUSSCHE Ecole Normale Supérieure de Rennes M. Tadahisa FUNAKI Université de Tokyo et Université Waseda Présentée et soutenue publiquement le 18 avril 2019, devant un jury composé de : Thomas DUQUESNE, président du jury Lorenzo ZAMBOTTI, directeur de thèse Arnaud DEBUSSCHE, rapporteur Paul GASSIAT, examinateur Massimiliano GUBINELLI, examinateur Martin HAIRER, examinateur Integration by parts formulae for the laws of Bessel bridges, and Bessel stochastic PDEs Henri Elad Altman, supervized by Lorenzo Zambotti February 20, 2019 2 R´esum´e Dans cette th`ese,nous obtenons des formules d'int´egrationpar parties pour les lois de ponts de Bessel de dimension δ > 0, ´etendant ainsi les formules pr´ec´edemment obtenues par Zambotti dans le cas δ ≥ 3. Ceci nous permet d'identifier la structure de certaines EDP stochastiques (EDPS) ayant la loi d'un pont de Bessel de di- mension δ 2 (0; 3) pour mesure invariant, et qui ´etendent de mani`erenaturelle les EDPS consid´er´eespr´ec´edemment par Zambotti dans le cas δ ≥ 3. Nous nommons ces ´equationsEDPS de Bessel, et les ´ecrivons `al'aide de temps locaux renormalis´es. Dans les cas particuliers δ = 1; 2, en utilisant la th´eoriedes formes de Dirichlet, nous construisons une solution d'une version faible de ces EDPS. Nous prouvons ´egalement plusieurs r´esultatspartiels qui sugg`erent que les EDPS de Bessel de param`etre δ < 3 poss`edent certaines propri´et´esimportantes: propri´et´ede Feller forte, existence de temps locaux. Enfin, nous prouvons des r´esultatsde tensions pour diff´erents mod`elesde pinning critiques dynamiques, dont nous conjecturons que la limite d'´echelle est d´ecritepar l'EDPS de Bessel associ´ee`a δ = 1 Abstract In this thesis, we derive integration by parts formulae (IbPF) for the laws of Bessel bridges of dimension δ > 0, thus extending previous formulae obtained by Zambotti in the case δ ≥ 3. This allows us to identify the structure of some stochastic PDEs (SPDEs) having the law of a Bessel bridge of dimension δ < 3 as invariant measure, and which extend in a natural way the family of SPDEs previously considered by Zambotti for δ ≥ 3. We call these equations Bessel SPDEs, and write them using renormalized local times. In the particular cases δ = 1; 2, using Dirichlet forms, we construct a solution to a weak version of these SPDEs. We also provide several partial results suggesting that the SPDEs associated with δ < 3 should have several important properties: strong Feller property, existence of local times. Finally, we prove tightness results for several dynamical critical wetting models, and conjecture that the scaling limit should be described by the Bessel SPDE associated with δ = 1. Contents Contents 2 1 Introduction 7 Introduction 7 1.1 From Bessel SDEs to Bessel SPDEs . 12 1.2 The main problem: what SPDEs for δ < 3? . 16 1.3 Integration by parts formulae for the laws of Bessel bridges . 17 1.4 The IbPF for the laws of Bessel bridges of dimension δ < 3 . 18 1.5 The structure of the SPDEs for δ < 3................. 21 1.6 Properties of the SPDEs for δ < 3................... 23 1.7 Application to scaling limits of dynamical critical wetting models . 25 2 Bessel processes and Bessel bridges 27 2.1 An important family of distributions . 27 2.2 Bessel processes and associated bridges . 31 2.2.1 Squared Bessel processes and Bessel processes . 31 2.2.2 Squared Bessel bridges and Bessel bridges . 35 2.2.3 Pinned bridges . 36 2.3 Proof of Prop 2.1.9 . 39 3 Integration by parts formulae for the laws of Bessel bridges 41 3.1 The statement . 41 3.2 Proof of the IbPF : the case of homogeneous Dirichlet boundary values . 44 3.3 The case of general boundary values . 48 3.3.1 Case of unconstrained Bessel processes . 48 3.3.2 The case of bridges . 54 3.4 The IbPF via hypergeometric functions . 55 3.5 A slightly more general class of functionals . 62 3 4 CONTENTS 4 Bessel SPDEs: conjectures, and existence of solutions for δ = 1; 2 63 4.1 From the IbPF to the SPDEs . 63 4.2 The case δ =1 ............................. 66 4.2.1 The one-dimensional random string . 67 4.2.2 Gradient Dirichlet form associated with the 1-dimensional Bessel bridge . 69 4.2.3 Convergence of one-potentials . 70 4.2.4 The dynamics for δ =1..................... 74 4.2.5 A distinction result . 77 4.2.6 Proofs of two technical results . 80 4.3 The case δ =2 ............................. 83 4.3.1 The 2-dimensional random string . 83 4.3.2 Gradient Dirichlet form associated with the 2-dimensional Bessel bridge . 84 4.3.3 Convergence of one-potentials . 86 4.3.4 The dynamics for δ =2..................... 88 4.4 Proof of two technical result . 89 4.5 Open problems . 93 5 Taylor estimates on the laws of pinned Bessel bridges, and appli- cation to integration by parts 95 5.1 Statement of the results . 95 5.2 Notations and basic facts . 99 5.2.1 Notations . 99 5.2.2 Squared Bessel bridges and Bessel bridges . 100 5.2.3 Laws of pinned squared Bessel bridges as a convolution semi- group on C+([0; 1]) . 101 5.3 Density of S in a large space of functionals on L2(0; 1) . 102 5.4 Taylor estimates for the laws of pinned Bessel bridges . 113 5.4.1 Taylor estimates at order 0 . 113 5.4.2 Differentiability properties of conditional expectations . 118 5.4.3 A second-order Taylor estimate . 124 5.5 Extension of the integration by parts formulae to general functionals 127 5.5.1 Extension of the IbPF for δ 2 (1; 3) . 129 5.5.2 Extension of the IbPF for δ =1................ 130 5.5.3 Extension of the IbPF for δ 2 (0; 1) . 132 5.6 Proofs of the approximation results . 133 6 The case of integer dimensions 141 6.1 Link with an already known formula . 141 6.2 Proof of Proposition 6.1.1 . 143 CONTENTS 5 6.3 IbPF for the laws of integer-dimensional Bessel bridges . 149 6.4 Conjectures for the dynamics . 154 7 Strong Feller property in the case of non-dissipative drifts: the example of Bessel processes 155 7.1 Classical Bismut-Elworthy-Li formula for one-dimensional diffusions 156 7.2 Bessel processes: notations and basic facts . 159 7.3 Derivative in space of the Bessel semi-group . 160 7.4 Differentiability of the flow . 164 7.5 Properties of η ............................. 166 7.5.1 Regularity of the sample paths of η .............. 166 7.5.2 Study of a martingale related to η ............... 169 7.5.3 Continuity of (Dt)t≥0 ...................... 170 7.5.4 Martingale property of (Dt)t≥0 ................. 172 7.5.5 Moment estimates for the martingale (Dt)t≥0 ........ 174 7.6 A Bismut-Elworthy-Li formula for the Bessel processes . 177 7.7 Proof of a technical result . 182 8 Towards local times for solutions to SPDEs 189 8.1 Local times of the solution to the stochastic heat equation . 189 8.2 A perturbation method . 190 8.3 A more probabilistic approach . 192 8.4 Comparison of the two methods . 193 9 Application to the scaling limit of dynamical critical pinning mod- els 195 9.1 The δ-pinning case . 195 9.2 Case of the wetting model with a shrinking strip . 197 9.2.1 A tightness result . 197 9.2.2 An integration by parts formula . 200 9.2.3 Conjecture for the scaling limit . 202 9.3 A wetting model in the continuum . 202 9.3.1 Motivation: wetting model and local times . 203 9.3.2 A continuous wetting model . 204 9.3.3 The corresponding dynamics . 211 9.3.4 Convergence of the whole dynamics . 212 6 CONTENTS Chapter 1 Introduction In 1827, the Scottish botanist Robert Brown observed that some organic particles on a droplet he was looking at through his microscope were moving in a chaotic way along unusual and irregular paths. This movement, which was later baptized Brownian motion, was understood only much later, due to the work, at the turn of the 20th century, of Einstein and Smoluchowski, who argued that the chaotic tra- jectories one could see on large scales were due to the presence of highly-energetic atoms hitting the particles in all directions.
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