In this work we gather several results we obtained on the behavior of the integral of linear Brownian motion, and more particularly on the various distributions related to the first passage times of the trajectories by fixed thresholds.

       For instance, we were able to explicitly determine the joint law of the couple made up of the first passage time of the integrated process by a fixed point and of the related location of Brownian motion. We retrieved in particular the marginal laws of this couple discovered by M. Goldman (1971) and Ju. P. Gor'kov (1975), as well as the law of the first return time to the origin obtained by H.P. McKean (1963). This result enabled us to resolve several open problems. In particular, we obtained the distributions of several functionals associated with the integral of Brownian motion: successive passage times, last passage time, sojourn time, excursions...

       We next studied the location of the primitive of Brownian motion when this latter reaches a single or double barrier. Such functionals naturally arise in some optimization problems studied by M. Lefèbvre (1989). A new approach enabled us to find and improve its results.

       We finally derived the distribution of certain functionals related to the integral of Brownian motion, this latter being subjected to a parabolic or cubic drift. We retrieved in particular a result of P. Groeneboom (1989) concerning Brownian motion with a parabolic drift.

       An exhibition of some still open problems completes this work.

Table of contents Introduction Chapter I: Langevin equation Introduction Description of the physical experience A solution to Langevin equation: a stationary Gaussian process \((x_t)_{t\geqslant 0}\) The number of zeros of \((x_t)_{t\geqslant 0}\) Chapter II: Study of a particulier case: integrated Brownian motion Introduction Joint distribution of the couple \((\tau_a,B_{\tau_a})\) related to the probability \(\mathbb{P}_{(a,y)}\) Distributions of the random variables \(\tau_a\) and \(B_{\tau_a}\) related to the probability \(\mathbb{P}_{(a,y)}\) Joint distribution of the couple \((\tau_a,B_{\tau_a})\) related to the probability \(\mathbb{P}_{(x,y)}\) On integrated Brownian motion On the first passage time for integrated Brownian motion Successive passage times of integrated Brownian motion Last passage time for integrated Brownian motion About excursions of integrated Brownian motion Excursions of integrated Brownian motion The moments of the sojourn time for integrated Brownian motion Joint distributions of the couples \((\sigma_b,X_{\sigma_b})\) and \((\sigma_{ab},X_{\sigma_{ab}})\) related to the probability \(\mathbb{P}_{(x,y)}\) An optimization problem Various extensions About integrated Brownian motion On the distribution of some functionals of integrated Brownian motion with parabolic and cubic drifts Chapter III: Open problems First hitting time of a two-sided barrier \(\{a,b\}\) for the process \((X_t)_{t\geqslant 0}\) First exit time of a quadrant for the process \((X_t,B_t)_{t\geqslant 0}\) Sojourn time for the process \((X_t)_{t\geqslant 0}\) Iterated primitives of Brownian motion The differential operator \(d^4/dx^4\)

Publications from the thesis (in order of appearance):

• Comptes Rendus de l'Académie des Sciences, t. 311 (1990), 461-464.
• Annales de l'Institut Henri Poincaré Section B 27(3) (1991), 385-405.
• Comptes Rendus de l'Académie des Sciences, t. 321 (1995), 903-908.
• Stochastic Processes and their Applications 49 (1994), 57-64.
• Comptes Rendus de l'Académie des Sciences, t. 314 (1992), 1053-1056.
• Journal of Applied Probability 30 (1993), 17-27.
• Communications on Pure and Applied Mathematics XLIX (1996), 1299-1338.

Eugène Boudin (1824-1898) – The Coast of Portrieux, 1874

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       The whole of our work is primarily devoted to the study of the primitive of Brownian motion; more particularly it deals with the explicit determination of the probability distributions of various functionals associated with this process. The motivations and the historical context of this study, which begins mainly with a work by P. Langevin, H.P. McKean and M. Kac, are described in detail in the introduction of our PhD. thesis.

       The starting point of our study was the explicit determination of the law of the first passage time \(\tau_a\) for the integral of Brownian motion \((X_t)_{t\ge 0}\) through a fixed threshold \(a\), coupled with the related location of Brownian motion \((B_t)_{t\ge 0}\), when the two-dimensional Markovian process \((X_t, B_t)_{t\ge 0}\) starts at a point \((x,y)\in \mathbb{R}^2\). This result, which solved an open problem posed in the paper by H.P. McKean “A winding problem for a resonator driven by a white noise”, going back to 1963, allowed to resolve many questions and played a determining role in our forthcoming researches.

       To quote a significant example, the knowledge of the joint distribution of the couple \((\tau_a, B_{\tau_a})\) led us to that of the last hitting time \(\tau_{a,T}^-\) of the point \(a\) by the process \((X_t)_{t\ge 0}\), before a fixed instant \(T\). Consequently, the explicit expression of this last law made it possible to describe its asymptotic behavior as \(T\) tends to zero. This estimate, in particular, was exploited by S. Aspandiiarov and J.-F. Gall in a work related to the study of Burgers' equation.

       In addition, we carried out a thorough study of various excursions of the process \((X_t, B_t)_{t\ge 0}\), having always for objective the exact and explicit determination of the law of certain functionals. By invoking the general theory of the excursions of a Markov process, we have been able to derive for example the law of the quadruple \((\tau_{a,T}^-, B_{\tau_{a,T}^-}, \tau_{a,T}^+, B_{\tau_{a,T}^+})\) made up of the last and first passage times of \(a\) respectively posterior and former to deterministic instant \(T\), and of the related locations of Brownian motion. By a Markovian technique, we had before obtained from them only some marginal laws. A research related to the area of a loop of excursion associated with the process \((X_t, B_t)_{t\ge 0}\) then led us to the primitive of Ornstein-Uhlenbeck process, for which we also derived some related distributions.

       More generally, the excursions of the integral of Brownian motion away of a point \(a\) (without temporal restriction now) involve the sequence of successive passage times \((\mathbf{t}_n)_{n \ge 1}\) through a point \(a\) by the process \((X_t)_{t\ge 0}\). This sequence of excursions is very different from the Brownian excursions, this latter case for which such a sequence cannot be defined because of the irregularity of Brownian paths. By using the Kontorovich-Lebedev transform, we have been able to obtain for the joint law of the couple \((\mathbf{t}_n, B_{\mathbf{t}_n}), n \ge 1\) a simple formula requiring no multiple integral.

       Various problems remain currently unsolved. In particular: the law of the first exit time from a bounded interval \([a, b]\) by the primitive of Brownian motion remains unknown; the distribution of the sojourn time in \([a, b]\) by \((X_t)_{t\ge 0}\) is still not clarified.

       Other questions, of geometrical nature, also arise: do the trajectories of the two-dimensional process \((X_t, B_t)_{t\ge 0}\) have multiple points? Is it possible to characterize the polar sets for this process? Which is the exact Hausdorff measure of the curve \(t\mapsto (X_t, B_t)\)?

Table of contents Introduction On integrated Brownian motion Comptes Rendus de l'Académie des Sciences, t. 311 (1990), 461-464. On the first passage time for integrated Brownian motion Annales de l'Institut Henri Poincaré Section B 27(3) (1991), 385-405. Integrated Brownian motion Journal of Applied Probability 30 (1993), 17-27. Last passage time for integrated Brownian motion Stochastic Processes and their Applications 49 (1994), 57-64. About excursions of integrated Brownian motion Comptes Rendus de l'Académie des Sciences, t. 314 (1992), 1053-1056. Some applications of excursion theory to integrated Brownian motion Séminaire de Probabilités XXXVIII, Lecture Notes in Mathematics 1801 (2003), 109-195. On successive passage times of integrated Brownian motion Comptes Rendus de l'Académie des Sciences, t. 321 (1995), 903-908. Successive passage times of integrated Brownian motion Annales de l'Institut Henri Poincaré Section B 33(1) (1997), 1-36. Some martingales related to the integral of Brownian motion. Application to passage times and transience Stochastics and Stochastics Reports 58 (1996), 285-302. On the distribution of some functionals of integrated Brownian motion with parabolic and cubic drifts Communications on Pure and Applied Mathematics XLIX (1996), 1299-1338. Some martingales related to the integral of Brownian motion. Application to the passage times and transience Journal of Theoretical Probability 11(1) (1998), 127-156. Appendix: Summary References

Claude Monet (1840-1926) – Floating ice on the Seine, 1880

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https://math.univ-lyon1.fr/~alachal