A solar eclipse will grace the skies over North America on Monday. Learn more about what times it&#x27;s visible, how to watch and where the path of totality goes.

Solar eclipse path, map and time guide 2024: Where and when to watch the total eclipse

Astronomy fans figuring out where and what time to see the April 8, 2024, total solar eclipse could check out maps showing the path of totality.

Solar eclipse maps show 2024 totality path, peak times and how much of the eclipse people could see across the U.S. - CBS News

A rare total solar eclipse is Monday, April 8. Here&#x27;s everything you need to know about its path, duration and how to view it safely.

The total solar eclipse is Monday: Here's everything to know, including time, path, safety

To witness the April 8, 2024, total solar eclipse, viewers must be within the 115-mile-wide path that will sweep across the U.S.

Total solar eclipse April 8, 2024 facts: Path, time and the best places to view - ABC News

And even if you don’t live in the path of totality or you aren’t one of the millions of people traveling to see the major event, there are multiple ways for you to join in and watch the total solar eclipse online. You can also view Monday&#x27;s best solar eclipse photos right here.

Total Solar Eclipse of April 8, 2024: Watch Online, What Time, Path of Totality | WIRED

Sky-gazers across North America are in for a treat on April 8 when a total solar eclipse will pass over Mexico, the United States and Canada.

What to know for the total solar eclipse: Time, path of totality, eclipse glasses | CNN

The 2024 solar eclipse will bring an incredible spectacle to skies on Monday, marking a rare and historic moment in the U.S., but also in Illinois.

2024 solar eclipse guide: Timing, path, glasses, forecast and more – NBC Chicago

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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.

A Generalization of Our Exit-problem for a Class of Non-Markov Processes With Path Dependency Paper

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 3.3 Proofs of auxiliary lemmas 3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction 3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards a Then, by Grönwall’s inequality: That guarantees exponentially fast convergence towards 0: .3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards a And this concludes the proof of the lemma. 3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between Bρ(a) and ∂G 3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure Proof. The proof of the following lemma follow the same logic as the one of Lemma 3.3. We have to show now that This completes the proof. Now we are ready to prove Lemma 3.5 itself. That gives us: We recall that Lemma 3.9 establishes the following asymptotic behaviour for small σ: Let us first deal with the term B. By (3.22), we have Consider now A. Separate each probability inside the sum the following way 3.3.6 Proof of Lemma 3.6: Exit before nearing a First, we show that it is possible to establish a lower bound for By Lemma 2.5, which proves the lemma by taking ρ → 0. 3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals Use the Jensen’s inequality, integrate the second part of the last integral over s, and get: We apply Grönwall’s inequality and get: Thus, it follows that, for the absolute value of the Brownian motion itself, Taking into account these remarks, we can now use the Schilder theorem [DZ10, Lemma 5.2.1] that provides the LDP for the path of the Brownian motion. Hence, for some fixed ϵ and 0 < T < 1: where d is the dimension of the space. Note that for any ϵ > 0 the following limit holds: 3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit from G 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 3.3 Proofs of auxiliary lemmas 3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction 3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards a Then, by Grönwall’s inequality: That guarantees exponentially fast convergence towards 0: .3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards a And this concludes the proof of the lemma. 3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between Bρ(a) and ∂G 3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure Proof . The proof of the following lemma follow the same logic as the one of Lemma 3.3. Proof We have to show now that This completes the proof. Now we are ready to prove Lemma 3.5 itself. That gives us: We recall that Lemma 3.9 establishes the following asymptotic behaviour for small σ: Let us first deal with the term B. By (3.22), we have Consider now A. Separate each probability inside the sum the following way 3.3.6 Proof of Lemma 3.6: Exit before nearing a First, we show that it is possible to establish a lower bound for By Lemma 2.5, which proves the lemma by taking ρ → 0. 3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals Use the Jensen’s inequality, integrate the second part of the last integral over s, and get: We apply Grönwall’s inequality and get: Thus, it follows that, for the absolute value of the Brownian motion itself, Taking into account these remarks, we can now use the Schilder theorem [DZ10, Lemma 5.2.1] that provides the LDP for the path of the Brownian motion. Hence, for some fixed ϵ and 0 < T < 1: where d is the dimension of the space. Note that for any ϵ > 0 the following limit holds: 3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit from G 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.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proofs of Auxiliary Lemmas

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 3.2 Proof of the main theorem 3.2.1 Kramers’ law Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times That gives us: Therefore, by induction in k, we get: By Markov’s inequality, we get: We remark that by Lemmas 3.1, 3.5 and derivations above, we have This concludes the proof. 3.2.2 Exit-location 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 3.2 Proof of the main theorem 3.2.1 Kramers’ law Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times That gives us: Therefore, by induction in k, we get: By Markov’s inequality, we get: We remark that by Lemmas 3.1, 3.5 and derivations above, we have This concludes the proof. 3.2.2 Exit-location 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.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proof of the Main Theorem

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 3.1 Auxiliary results 3.1.1 Initial descent to the point of attraction 3.1.2 Convergence of deterministic process towards a 3.1.3 Attraction of stochastic process towards a 3.1.4 Behaviour in the annulus between Bρ(a) and ∂G 3.1.5 Stabilization of the occupation measure 3.1.6 Exit before nearing a 3.1.7 Control of dynamics for small time intervals 3.1.8 Uniform lower bound for probability of exit from G 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 3.1 Auxiliary results 3.1.1 Initial descent to the point of attraction 3.1.2 Convergence of deterministic process towards a 3.1.3 Attraction of stochastic process towards a 3.1.4 Behaviour in the annulus between Bρ(a) and ∂G 3.1.5 Stabilization of the occupation measure 3.1.6 Exit before nearing a 3.1.7 Control of dynamics for small time intervals 3.1.8 Uniform lower bound for probability of exit from G 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.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: The Auxiliary Results

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 3 Exit-time First, we present the following assumptions on domain G and geometry of the potentials that drive the system inside the domain. Assumptions A-2. Now, let us remind the main theorem before proving it. Under the sets of Assumptions A-1 and A-2 the following theorem holds: 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 3 Exit-time First, we present the following assumptions on domain G and geometry of the potentials that drive the system inside the domain. Assumptions A-2. Assumptions A-2. Now, let us remind the main theorem before proving it. Under the sets of Assumptions A-1 and A-2 the following theorem holds: 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.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: A Peek at the Exit-Time

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 2.3 Compactness results This completes the proof. 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 2.3 Compactness results This completes the proof. 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.

The Compactness Results of Our Non-Markov Processes With Path Dependency Paper

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 1.4 List of notations The paper contains number of technical results. To make them more reader-friendly, we present here a list of notions that are used later in the paper. All the equations in the paper are meant to hold a.s. unless stated otherwise. 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 1.4 List of notations The paper contains number of technical results. To make them more reader-friendly, we present here a list of notions that are used later in the paper. All the equations in the paper are meant to hold a.s. unless stated otherwise. 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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