Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization In this section we present the possible generalisation of Assumptions A-1 and A-2 as well as the exit-time result for this more general process. In this paper we did not talk in details about the problem of existence and uniqueness of the self-interacting diffusion. While in the Lipschitz case (Assumption A-1) these questions are standard, locally Lipschitzness of the potentials add some complication. In [AdMKT23], the authors provide a proof for existence and uniqueness of self-interacting diffusion under the following assumptions that we take as a baseline: Assumptions A-1′. It is worth noting that Assumption A-1′ .ii is a combination of locally Lipschitz and polynomial growth conditions. Assumption A-1′ .iii has been introduced in [AdMKT23] to control the Lyapunov functional of the diffusion process to ensure that there is no explosion within a finite time. It is important to mention that, if required, this assumption can be replaced by another one that ensures the process’s global existence and uniqueness without any impact on the exit-time result. We can also relax Assumptions A-2. Indeed, the domain G does not necessarily have to be bounded, but it is necessary to add another assumption on the level sets instead. Consider: Assumptions A-2′. Under these assumptions we can obtain Kramers’ type law and the exit location result. Acknowledgments: This work has been supported by the French ANR grant METANOLIN (ANR-19-CE40-0009). References [AdMKT23] Ashot Aleksian, Pierre del Moral, Aline Kurtzmann, and Julian Tugaut. Self-interacting diffusions: long-time behaviour and exitproblem in the convex case. preprint, March 2023. [BLR02] Michel Benaïm, Michel Ledoux, and Olivier Raimond. Selfinteracting diffusions. Probab. Theory Related Fields, 122(1):1–41, 2002. [BR03] Michel Benaïm and Olivier Raimond. Self-interacting diffusions. II. Convergence in law. Ann. Inst. H. Poincaré Probab. Statist., 39(6):1043–1055, 2003. [BR05] Michel Benaïm and Olivier Raimond. Self-interacting diffusions. III. Symmetric interactions. Ann. Probab., 33(5):1717–1759, 2005. [CGM08] P. Cattiaux, A. Guillin, and F. Malrieu. Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields, 140(1-2):19–40, 2008. [DR92] R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields, 92(3):337–349, 1992. [DZ10] Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. [FW12] Mark I. Freidlin and Alexander D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, third edition, 2012. Translated from the 1979 Russian original by Joseph Szücs. [HIP08] Samuel Herrmann, Peter Imkeller, and Dierk Peithmann. Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab., 18(4):1379–1423, 2008. [KK12] Victor Kleptsyn and Aline Kurtzmann. Ergodicity of selfattracting motion. Electron. J. Probab., 17:no. 50, 37, 2012. [Kur10] Aline Kurtzmann. The ODE method for some self-interacting diffusions on R d . Ann. Inst. Henri Poincaré Probab. Stat., 46(3):618– 643, 2010. [NRW87] J. R. Norris, L. C. G. Rogers, and David Williams. Self-avoiding random walk: a Brownian motion model with local time drift. Probab. Theory Related Fields, 74(2):271–287, 1987. [RW00] L. C. G. Rogers and David Williams. Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itô calculus, Reprint of the second (1994) edition. [TTV12] Pierre Tarrès, Bálint Tóth, and Benedek Valkó. Diffusivity bounds for 1D Brownian polymers. Ann. Probab., 40(2):695–713, 2012. [Tug12] Julian Tugaut. Exit problem of McKean-Vlasov diffusions in convex landscapes. Electron. J. Probab., 17:no. 76, 26, 2012. [Tug16] Julian Tugaut. A simple proof of a Kramers’ type law for selfstabilizing diffusions. Electron. Commun. Probab., 21:Paper No. 11, 7, 2016. [Tug18] Julian Tugaut. Exit-problem of McKean-Vlasov diffusions in double-well landscape. J. Theoret. Probab., 31(2):1013–1023, 2018. [Tug19] Julian Tugaut. A simple proof of a Kramers’ type law for selfstabilizing diffusions in double-wells landscape. ALEA Lat. Am. J. Probab. Math. Stat., 16(1):389–398, 2019. [VF70] A. D. Ventcel’ and M. I. Freıdlin. Small random perturbations of dynamical systems. Uspehi Mat. Nauk, 25(1 (151)):3–55, 1970. [Vil09] Cédric Villani. Optimal transport, volume 338 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. Old and new. This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 4 Generalization In this section we present the possible generalisation of Assumptions A-1 and A-2 as well as the exit-time result for this more general process. In this paper we did not talk in details about the problem of existence and uniqueness of the self-interacting diffusion. While in the Lipschitz case (Assumption A-1) these questions are standard, locally Lipschitzness of the potentials add some complication. In [AdMKT23], the authors provide a proof for existence and uniqueness of self-interacting diffusion under the following assumptions that we take as a baseline: Assumptions A-1′. It is worth noting that Assumption A-1′ .ii is a combination of locally Lipschitz and polynomial growth conditions. Assumption A-1′ .iii has been introduced in [AdMKT23] to control the Lyapunov functional of the diffusion process to ensure that there is no explosion within a finite time. It is important to mention that, if required, this assumption can be replaced by another one that ensures the process’s global existence and uniqueness without any impact on the exit-time result. We can also relax Assumptions A-2. Indeed, the domain G does not necessarily have to be bounded, but it is necessary to add another assumption on the level sets instead. Consider: Assumptions A-2′. Under these assumptions we can obtain Kramers’ type law and the exit location result. Acknowledgments : This work has been supported by the French ANR grant METANOLIN (ANR-19-CE40-0009). Acknowledgments This work has been supported by the French ANR grant METANOLIN (ANR-19-CE40-0009). References [AdMKT23] Ashot Aleksian, Pierre del Moral, Aline Kurtzmann, and Julian Tugaut. Self-interacting diffusions: long-time behaviour and exitproblem in the convex case. preprint, March 2023. [BLR02] Michel Benaïm, Michel Ledoux, and Olivier Raimond. Selfinteracting diffusions. Probab. Theory Related Fields, 122(1):1–41, 2002. [BR03] Michel Benaïm and Olivier Raimond. Self-interacting diffusions. II. Convergence in law. Ann. Inst. H. Poincaré Probab. Statist., 39(6):1043–1055, 2003. [BR05] Michel Benaïm and Olivier Raimond. Self-interacting diffusions. III. Symmetric interactions. Ann. Probab., 33(5):1717–1759, 2005. [CGM08] P. Cattiaux, A. Guillin, and F. Malrieu. Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields, 140(1-2):19–40, 2008. [DR92] R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields, 92(3):337–349, 1992. [DZ10] Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. [FW12] Mark I. Freidlin and Alexander D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, third edition, 2012. Translated from the 1979 Russian original by Joseph Szücs. [HIP08] Samuel Herrmann, Peter Imkeller, and Dierk Peithmann. Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab., 18(4):1379–1423, 2008. [KK12] Victor Kleptsyn and Aline Kurtzmann. Ergodicity of selfattracting motion. Electron. J. Probab., 17:no. 50, 37, 2012. [Kur10] Aline Kurtzmann. The ODE method for some self-interacting diffusions on R d . Ann. Inst. Henri Poincaré Probab. Stat., 46(3):618– 643, 2010. [NRW87] J. R. Norris, L. C. G. Rogers, and David Williams. Self-avoiding random walk: a Brownian motion model with local time drift. Probab. Theory Related Fields, 74(2):271–287, 1987. [RW00] L. C. G. Rogers and David Williams. Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itô calculus, Reprint of the second (1994) edition. [TTV12] Pierre Tarrès, Bálint Tóth, and Benedek Valkó. Diffusivity bounds for 1D Brownian polymers. Ann. Probab., 40(2):695–713, 2012. [Tug12] Julian Tugaut. Exit problem of McKean-Vlasov diffusions in convex landscapes. Electron. J. Probab., 17:no. 76, 26, 2012. [Tug16] Julian Tugaut. A simple proof of a Kramers’ type law for selfstabilizing diffusions. Electron. Commun. Probab., 21:Paper No. 11, 7, 2016. [Tug18] Julian Tugaut. Exit-problem of McKean-Vlasov diffusions in double-well landscape. J. Theoret. Probab., 31(2):1013–1023, 2018. [Tug19] Julian Tugaut. A simple proof of a Kramers’ type law for selfstabilizing diffusions in double-wells landscape. ALEA Lat. Am. J. Probab. Math. Stat., 16(1):389–398, 2019. [VF70] A. D. Ventcel’ and M. I. Freıdlin. Small random perturbations of dynamical systems. Uspehi Mat. Nauk, 25(1 (151)):3–55, 1970. [Vil09] Cédric Villani. Optimal transport, volume 338 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. Old and new. This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

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