Master report:

Ph.D. Thesis:

Defended on March 19, 1992 at the University of Lyon. Composition of the jury: J. Brossard, A. Goldman, J.-F. Le Gall, E. Pardoux.
Abstract    Thesis

       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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Thesis of “Habilitation à diriger des recherches”:

Defended on July 3, 1995 at the University of Lyon. Composition of the jury: J. Brossard, A. Goldman, J.-P. Imhof, J.-F. Le Gall, P. McGill, E. Pardoux, B. Roynette, M. Yor.
Abstract    Thesis

       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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On the first passage time for integrated Brownian motion (in French)
Annales de l'I. H. P. Sect. B 27(3) (1991), 385-405.
MathSciNet reference: MR1131839 (92m:60073)
Abstract   Article

       This paper concerns a famous problem of Kac, Rice and Potter on the windings of Brownian motion in the phase plane. One looks to compute explicitly the joint law of the hitting time and hitting distribution for the process \((x+ty+ \int_0^t B_s\,ds, B_t+y\)) on vertical lines \(\{a\}\times \mathbb{R}.\) The problem can be reduced to solving a PDE with boundary conditions, which is solved here by taking a Kantorovich-Lebedev transform; McKean looked at this in the case \(x=a\). The analogue for (one or two) horizontal lines is easier; joint laws can be computed by using the Feynman-Kac formula.

Pierre-Auguste Renoir (1841-1919) – Luncheon of the Boating Party, 1880-1881

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Integrated Brownian motion (in French)
Journal of Applied Probability 30 (1993), 17-27.
MathSciNet reference: MR1206349 (94a:60120)   DOI:10.2307/3214618
Abstract   Article

       Let \((B_t)_{t\ge 0}\) denote the Brownian motion process starting at the origin, let \(X_t=\int^t_0B_sds\) be its primitive and let \(U_t=(X_t+x+ty,B_t+y)\), \(t\ge0\), be the associated bidimensional process starting from a point \((x,y)\in \mathbb{R}^2\). In this paper we present an elementary procedure for rederiving the formula of M. Lefebvre [SIAM J. Appl. Math. 49 (1989), no. 5, 1514-1523] giving the Laplace-Fourier transform of the distribution of the pair \((\sigma_a,U_{\sigma_a})\), as well as Lachal's formulae (1991) giving the explicit Laplace-Fourier transform of the law of the couple \((\sigma_{ab},U_{\sigma_{ab}})\), where \(\sigma_a\) and \(\sigma_{ab}\) denote, respectively, the first hitting time of \(\mathbb{R}\times \{a\}\) from the right and the first hitting time of the double-sided barrier \(\mathbb{R}\times \{a,b\}\) by the process \((U_t)_{t\ge 0}\). This method, which unifies and considerably simplifies the proofs of these results, is in fact a `vectorial' extension of the classical technique of D. A. Darling and A. J. F. Siegert [Ann. Math. Statist. 24 (1953), 624-639]. It rests on an essential observation [A. Lachal, “Étude des trajectoires de la primitive du mouvement brownien”, Thèse de Doctorat, 1992; per bibl.] of the Markovian character of the bidimensional process \((U_t)_{t\ge 0}\). Using the same procedure, we subsequently determine the Laplace-Fourier transform of the conjoint law of the quadruplet \((\sigma_a,U_{\sigma_a},\sigma_b,U_{\sigma_b})\).

Edouard Manet (1832-1883) – Monet in his Studio Boat, 1874

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Last passage time for integrated Brownian motion (in French)
Stochastic Processes and their Applications 49 (1994), 57-64.
MathSciNet reference: MR1258281 (95a:60109)   DOI:10.1016/0304-4149(94)90111-2
Abstract   Article

       Let \((B_t)_{t\ge 0}\) be the standard Brownian motion. Fix \(T > 0\) and look at the excursion of the process \(X_t=x+\int_0^t B_s\,ds\), \(t\ge0\), from \(a\) which straddles \(T\). The paper shows how to compute the joint law of the excursion interval and the terminal value of \(B\). In fact, since the semigroup of \(((X_t, B_t))_{t\ge 0}\) is known from the work of McKean, the problem reduces to calculating the law of \((B_t)_{t\ge 0}\) when \((X_t)_{t\ge0}\) hits \(a\). But this the author already knows. The answers are explicit and yield a simple form for the law of the excursion interval spanning \(T\) in terms of the Bessel function \(K_0\).

Alfred Sisley (1839-1899) – Flood at Port-Marley, 1876

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On the distribution of some functionals of integrated Brownian motion with parabolic and cubic drifts (in French)
Communications on Pure and Applied Mathematics XliX (1996), 1299-1338.
MathSciNet reference: MR1414588 (97j:60150)   DOI:10.1002/(SICI)1097-0312(199612)49:12<1299::AID-CPA4>3.0.CO;2-5
Abstract

       Let \((B_t)_{t\ge0}\) be the Brownian motion starting at 0 and \(X_t=\int^t_0 B_udu\). Define the process \(U^{z,w}_t=(X_t+{1\over2}t^2z+{1\over6}t^3w, B_t+tz+t^2w),\ t\ge s\ (s>0\) fixed), consisting of a Brownian motion with parabolic drift and the integral of the Brownian motion with cubic drift.
A series of explicit formulas for distributions of functionals of \(U^{z,w}_t\) are derived, such as the distribution (depending on the starting point \((x,y)\)) of the following:

\((\tau_a,B_{\tau_a})\), where \(\tau_a=\inf\{t\ge s\colon X_t=a\}\) (\(a\in{\mathbb{R}}\) is fixed);

\((\tau_M,M)\), where \(M=\sup_{t\ge s}X_t\) (here \(x\le a\) and \(w\le0\));

\((\sigma_b,B_{\sigma_b})\), where \(\sigma_b=\inf\{t\ge s\colon B_t=b\}\) (\(b\in\mathbb{R}\) is fixed);

\((\sigma_N,X_{\sigma_N},N)\), where \(N=\sup_{t\ge s}B_t\) (here \(y\le b\) and \(w\le 0)\);

\((\tau^-_a,B_{\tau^-_a},\tau^+_a,B_{\tau^+_a})\), where \(\tau^-_a=\sup\{t\in[s,T]\colon X_t=a\}\) and \(\tau^+_a=\inf\{t\ge T\colon X_t=a\}\) (\(T>s\) fixed);

\((\sigma^-_b,X_{\sigma^-_b},\sigma^+_b,X_{\sigma^+_b})\), where \(\sigma^-_b=\sup\{t\in[s,T]\colon B_t=a\}\) and \(\sigma^+_b=\inf\{t\ge T\colon B_t=a\}\).

Also, special results for \(w=0\) are presented.
Further, the process \(\Pi_t=(X_t| U^{0,0}_1=(0,0))\), called the “bridge of the integral of the Brownian motion”, is considered and some of its properties as well as the distribution of \(\max_{\alpha\leq t\leq1}\Pi_t, 0\leq\alpha<1\), are given.

Vincent Van Gogh (1853-1890) – The Church at Auvers, 1890

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Some martingales associated with integrated Ornstein-Uhlenbeck process. Application to the study of the first passage times (in French)
Stochastics and Stochastics Reports 58 (1996), 285-302.
MathSciNet reference: MR1424696 (98e:60059)   DOI:10.1080/17442509608834078
Abstract

       Let \(O_t\) be the Ornstein-Uhlenbeck process, starting at \(y\), and satisfying \(dO_t={-\beta}O_tdt+dB_t,\ B\) being a one-dimensional Brownian motion. Let \(\sigma_a\) denote its first hitting time of \(a\) and set \(X_t=x+\int_0^tO_sds\). Making use of martingales, we compute the Laplace-Fourier transform of \((\sigma_a,X_{\sigma_a})\): \[E[\exp(-\lambda\sigma_a+i\mu X_{\sigma_a})]=e^{i\mu x}H(y)/H(a),\] where \(H(z)=e^{\beta z^2/2}D_\gamma(\epsilon\sqrt{2\beta} (z-iz/\beta^2)),\ \epsilon={\rm sign}(y-a),\ \gamma={-\mu}^2/(2\beta^3)-\lambda/\beta,\ \lambda>0,\ \mu\in{\bf R},\ D_\gamma\) is the parabolic cylinder function of Weber's equation, with index \(\gamma\). Using the same approach, we evaluate the Laplace-Fourier transform of \((\sigma_{a,b},X_{\sigma_{a,b}})\) and \((\sigma_a,X_{\sigma_a,}, \sigma_b,X_{\sigma_b})\), where \(\sigma_{a,b}=\sigma_a\wedge\sigma_b\). Some previous results are extended to an Ornstein-Uhlenbeck process driven by the hyperbolic drift \((z/\beta)\sinh(\beta t)+(w/\beta^2)(\cosh(\beta t)-1)\).

Joseph Mallord William Turner (1775-1851) – Rain, steam and speed - The great western railway painted, 1844

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Successive passage times of integrated Brownian motion (in French)
Annales de l'I. H. P. Sect. B 33(1) (1997), 1-36.
MathSciNet reference: MR1440254 (98g:60148)   DOI:10.1016/S0246-0203(97)80114-8
Abstract   Article

       Let \((B_t)_{t\ge 0}\) be linear Brownian motion, and set \(X_t=\int^t_0 B_sds\) and \(U_t=(X_t,B_t)\). Then \((U_t)_{t\ge 0}\) is a temporally homogeneous Markov process, already studied in 1963 by H.P. McKean. Let \(\tau_0=0\), and for \(n\geq 0\), let \(\tau_{n+1} = \min\{t>\tau_n\colon X_t=0\}\), \(\beta_n=|B_{\tau_n}|\), relative to \(\mathbb{P}_{(0,b)}\), \(b>0\), but, for \(b=0\), let \(\tau^+_0=\tau^-_0=1\) and define \(\tau^+_{n+1}=\min\{t>\tau^+_n\colon X_t = 0\}\), \(\tau^-_{n+1}=\max\{t<\tau^-_n\colon X_t=0\}\). The law of \((\tau_1,\beta_1)\) was given by McKean, along with Markov properties of the sequences \((\tau_n,\beta_n)\), \((\tau^+_n,\beta^+_n)\), \((\tau^-_n,\beta^-_n)\), and asymptotics. The present work fills in the study of these sequences at the following points: (a) a greatly simplified expression is obtained for the law of \((\tau_n,\beta_n)\), \(n\ge 1\); (b) the laws of \((\tau^+_n,\beta^+_n)\) and \((\tau^-_n,\beta^-_n)\), \(n\geq 1\), are derived. The law of \(\beta_n\) is given explicitly as a series. The paper involves a heavy use of integral (Lebedev) transforms. Some new probabilistic phenomena are relegated to later work by Lachal [''Quelques applications de la théorie des excursions à l'intégrale du mouvement brownien'', to appear].

Jean-Baptiste Olive (1848-1936) – Port of Marseille

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Regular points for the successive primitives of Brownian motion
Journal of Mathematics of Kyoto University 37(1) (1997), 99-119.
MathSciNet reference: MR1447364 (98g:60149)   DOI:10.1215/kjm/1250518399
Abstract   Article

       Let \((B(t))_{t\geq 0}\) be the linear Brownian motion starting at \(0\), and set \(X_n(t) = {1\over n!}\int_0^ t (t-s)^ n\, dB(s)\). In this paper we write out a Wiener's test about regular points for the \((n+1)\)-dimensional process \((B,X_1,\ldots ,X_n)\), and we next apply this test to two examples of thornshaped sets.

Pierre-Auguste Renoir (1841-1919) – Dance at Le Moulin de la Galette, 1876

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Local asymptotic classes for the successive primitives of Brownian motion
Annals of Probability 25(4) (1997), 1712-1734.
MathSciNet reference: MR1487433 (98j:60112)   DOI:10.1214/aop/1023481108
Abstract   Article

       Let \((B(t))_{t\geq 0}\) be the linear Brownian motion starting at \(0\), and set \(X_n(t) ={1\over n!}\int_0^ t (t-s)^ n\, dB(s)\). H. Watanabe [Trans. Amer. Math. Soc., 148 (1970), pp. 233-248] stated a law of the iterated logarithm for the process \((X_1(t))_{t\geq 0}\) among other things. In this paper is proposed an elementary proof of this fact, which can be extended to the general case \(n\geq 1\). Next, we study the local asymptotic classes (upper and lower) of the \((n+1)\)-dimensional process \(U_n=(B,X_1,\ldots ,X_n)\) near zero and infinity, and the obtained results are extended to the case where \(B\) is the \(d\)-dimensional Brownian motion.

Berthe Morisot (1841-1895) – Port of Lorient, 1869

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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.
MathSciNet reference: MR1607384 (99f:60146)   DOI:10.1023/A:1021646925303
Abstract

       Let \((B_t)_{t\geq 0}\) be the standard linear Brownian motion started at \(y\) and set \(X_t=x+\int_0^ t B_s\,ds\), \(U_t=(X_t,B_t)\). In this paper we introduce some martingales related to the Markov process \((U_t)_{t\geq 0}\), which allow us to calculate explicitly the probability laws of several passage times associated to \(U\) in a probabilistic way. With the aid of an appropriate supermartingale, we also establish the transience of the process \((U_t)_{t\geq 0}\).

Balthasar van der Ast (1593-1657) – Still life with fruit, 1620

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First exit time from a bounded interval for a certain class of additive functionals of Brownian motion
Journal of Theoretical Probability 13(3) (2000), 733-775.
MathSciNet reference: MR1785528 (2001i:60126)   DOI:10.1023/A:1007810528683
Abstract

       Let \((B_t)_{t\ge 0}\) be standard Brownian motion starting at \(y\), \(X_t = x+\int_0^t V(B_s) \, ds\), \(x\in (a,b)\), with \(V(y)= y^{\gamma}\) if \(y\ge 0\), \(V(y)= -K(-y)^{\gamma}\) if \(y\le 0\), where \(\gamma >0\) and \(K\) is a given positive constant. Set \(\tau_{ab} = \inf \{ t>0 : X_t \notin (a,b) \}\) and \(\sigma_0 = \inf\{ t>0 : B_t = 0\}\).

       In this paper we give several informations about the random variable \(\tau_{ab}\). We namely evaluate the moments of the random variables \(B_{\tau_{ab}}\) and \(B_{\tau_{ab}\wedge \sigma_0}\), and also show how to calculate the expectations \(\mathbb{E} (\tau_{ab}^m B_{\tau_{ab}}^n)\) and \(\mathbb{E} ((\tau_{ab}\wedge\sigma_0)^m B_{\tau_{ab}\wedge\sigma_0}^n)\). Then, we explicitly determine the probability laws of the random variables \(B_{\tau_{ab}}\) and \(B_{\tau_{ab}\wedge \sigma_0}\) as well as the probability \(\mathbb{P}\{ X_{\tau_{ab}}=a \mbox{ (or \(b\))}\}\) by means of special functions.

Johan Barthold Jongkind (1819-1891) – The Harbor: the Brussels Warehouse District, 1874

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Study of some new integrated statistics: computation of Bahadur efficiency, relation with non-standard boundary value problems
Mathematical Methods of Statistics (2001) 10(1) 73-104.
MathSciNet reference: MR1841809 (2002j:62066)
Abstract

       Let \(p\) be a positive integer. This paper deals with \(p\)-fold integrated empirical process the limiting process of which being the \(p\)-fold integrated Brownian bridge, together with some related Kolmogorov-Smirnov, Cramér-von Mises and Watson-type statistics. The Bahadur theory is applied to those statistics in case of shift alternatives and comparisons with classical non-integrated case (\(p=0\)) are made. Several boundary value problems arise in this study and are stated on the one hand in the context of relationships between covariance and Green functions, and on the other hand in the context of optimality for the Bahadur-Raghavachari inequality. The results that are displayed here extend certain of Henze and Nikitin [1,2] who considered the case \(p=1\) and others of the author.

[1] Henze, N. and Nikitin, Ya. Yu. A new approach to goodness-of-fit tests based on the integrated empirical process, J. Nonpar. Statist. 12 (2000), 391-416.

[2] Henze, N. and Nikitin, Ya. Yu. Watson-type goodness-of-fit tests based on the integrated empirical process, preprint of University of Karlsruhe, 1999.

Paul Cézanne (1839-1906) – Still life with fruit dish, 1879-1882

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Bridges of certain Wiener integrals. Prediction properties, relation with polynomial interpolation and differential equations. Application to goodness-of-fit testing
in Bolyai Society Mathematical Studies X. Limit Theorems, Balatonlelle (Hungary), 1999. Budapest, 2002, 1-51.
MathSciNet reference: MR1979998 (2004e:60140)
Abstract

       Let \((W(t))_{t\ge 0}\) and \((\beta(t))_{t\ge 0}\) be respectively standard Wiener process started at 0 and standard Brownian bridge, and denote by \(W_n(t)=\frac{1}{n!} \int_0^t (t-s)^n\, dW(s)\) and \(\beta_n(t)=\frac{1}{n!} \int_0^t (t-s)^n\, d\beta(s)\) their \(n\)-fold primitives. This paper deals with the conditioned process \(B_n = (W_n(t)| W_0(1)=\ldots = W_n(1)=0)_{0\le t\le 1}\) which shall be called `bridge' associated with \(W_n\). Many properties are drawn for this process. We give several representations by means of time-inversion, random drift and conditioning \(\beta_n\). The prediction problem for \(B_n\) is solved, involving some interpolation poynomials; this may provide a probabilistic interpretation for those poynomials. We also found a connexion between the covariance functions for \(B_n\) and \(\beta_n\) and the Green functions for various boundary value problems on the interval [0,1] related to the differential operator \(d^{2n+2}/dx^{2n+2}\). On the other hand, some formulas are derived for the probability distributions of the maximum for the random variables \(B_n\), \(|B_n|\), \(\beta_n\), \(|\beta_n|\). Next we propose some statistics of the type of empirical distributions whose limiting processes are \(B_n\) and \(\beta_n\). Finally, we elaborate several Kolmogorov-Smirnov-like tests of goodness-of-fit.

Paul Cézanne (1839-1906) – Jas de Bouffan, the pool, 1876

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Some probability distributions in modeling DNA replication
Annals of Applied Probability 13(3) (2003), 1207-1230.
MathSciNet reference: MR1994049 (2004k:60239)   DOI:10.1214/aoap/1060202839
Abstract   Article

       By using some quasi-renewal equations and functional differential equations, we explicitly compute the Laplace transforms of some random variables introduced by Cowan and Chiu in modeling the mechanism of replication of a DNA molecule, [1] and [2]. These Laplace transforms are expressed by means of infinite product arising in the theory of partitions.

[1] R. Cowan. Stochastic models for DNA replication, in The Handbook of Statistics, Vol. 20, eds. C.R. Rao and D.N. Shanbhag, Elsevier, 2001.

[2] R. Cowan and S.N. Chiu. Stochastic model of fragment formation when DNA replicates, J. Appl. Prob. 31 (1994), 301-308.

Vincent Van Gogh (1853-1890) – Green wheat field with cypresses, 1889

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Applications of excursions theory to integrated Brownian motion (in French)
Seminar of Probability XXXVIII, Lecture Notes in Mathematics 1801 (2003), 109-195.
MathSciNet reference: MR2053045 (2005g:60128)   DOI:10.1007/b94376
Abstract

       Let \((B_t)_{t\ge 0}\) be linear Brownian motion started at \(y\) and set \(X_t=x+\int_0^t B_s ds\) and \(U_t =(X_t,B_t)\). We introduce the following hitting times straddling a deterministic time \(T>0\):

• \(\tau_{a,T}^- = \sup\{t\le T: X_t=a\}\),
• \(\tau_{a,T}^+ = \inf\{t\ge T: X_t=a\}\),
• \(\sigma_{b,T}^- = \sup\{t\le T: B_t=b\}\),
• \(\sigma_{b,T}^+ = \inf\{t\ge T: B_t=b\}\),
• \(\sigma_{ab,T}^- = \sup\{t\le T: B_t\in\{a,b\}\}\),
• \(\sigma_{ab,T}^+ = \inf\{t\ge T: B_t\in\{a,b\}\}\).

By making use of the excursion theory of a Markov process, we derive the probability distributions of the excursion process straddling time \(T\) associated with the process \((U_t)_{t\ge 0}\), in relation with the foregoing hitting times. We deduce the distributions of the random vectors

• \((\tau_{a,T}^-,B_{\tau_{a,T}^-},\tau_{a,T}^+,B_{\tau_{a,T}^+})\),
• \((\sigma_{b,T}^-,X_{\sigma_{b,T}^-},\sigma_{b,T}^+,X_{\sigma_{b,T}^+})\),
• \((\sigma^-_{ab,T},X_{\sigma^-_{ab,T}},\sigma^+_{ab,T},X_{\sigma^+_{ab,T}})\),

as well as the distributions of some related functionals.

Johan Barthold Jongkind (1819-1891) – The Church of Overschie, 1866

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Distributions of sojourn time, maximum and minimum for pseudo-processes governed by higher-order heat-type equations
Electronic Journal of Probability 8 (2003), paper no. 20, 1-53.
MathSciNet reference: MR2041821 (2005j:60082)   DOI:10.1214/EJP.v8-178
Abstract   Article

       The higher-order heat-type equation \(\partial u/\partial t=\pm\partial^{n} u/ \partial x^{n}\) has been investigated by many authors. With this equation is associated a pseudo-process \((X_t)_{t\ge 0}\) which is governed by a signed measure. In the even-order case, Krylov, [2], proved that the classical arc-sine law of Paul Lévy for standard Brownian motion holds for the pseudo-process \((X_t)_{t\ge 0}\), that is, if \(T_t\) is the sojourn time of \((X_t)_{t\ge 0}\) in the half line \((0,+\infty)\), then \(\mathbb{P}(T_t\in\,ds)=\frac{ds}{\pi\sqrt{s(t-s)}}\). Orsingher, [4], and next Hochberg and Orsingher, [1], obtained a counterpart to that law in the odd cases \(n=3,5,7.\) Actually Hochberg and Orsingher, [1], proposed a more or less explicit expression for that new law in the odd-order general case and conjectured a quite simple formula for it. The distribution of \(T_t\) subject to some conditioning has also been studied by Nikitin and Orsingher, [3], in the cases \(n=3,4.\) In this paper, we prove that the conjecture of Hochberg and Orsingher, [1], is true and we extend the results of Nikitin and Orsingher for any integer \(n\). We also investigate the distributions of maximal and minimal functionals of \((X_t)_{t\ge 0}\).

[1] Hochberg, K.J. and Orsingher, E. The arc-sine law and its analogs for processes governed by signed and complex measures. Stochastic Process. Appl. 52 (1994), no. 2, 273-292.

[2] Krylov, V. Yu. Some properties of the distribution corresponding to the equation \(\frac{\partial u}{\partial t}=(-1)^{q+1} \frac{\partial^{2q} u}{\partial^{2q} x}\). Soviet Math. Dokl. 1 (1960), 760-763.

[3] Nikitin, Ya. Yu. and Orsingher, E. On sojourn distributions of processes related to some higher-order heat-type equations. J. Theoret. Probab. 13 (2000), no.4, 997-1012.

[4] Orsingher, E. Processes governed by signed measures connected with third-order `heat-type' equations. Lithuanian Math. J. 31 (1991), no. 2, 220-231.

Alfred Sisley (1839-1899) – Autumn: Banks of the Seine near Bougival, 1873

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Probabilistic approach to Page's formula for the entropy of a quantum subsystem
Stochastics 78(3) (2006), 157-178.
MathSciNet reference: MR2241914 (2007j:81020)   DOI:10.1080/17442500600737133
Abstract

       Page conjectured [Phys. Rev. Lett. 71, 1291-1294 (1993)] that if a quantum system of Hilbert space dimension \(mn\) is in a random pure state, the average entropy of a subsystem of dimension \(m\le n\) should be given by the simple and elegant formula \(S_{m,n}=\sum_{k=n+1}^{mn}\frac 1k-\frac{m-1}{2n}\). This formula appeared to be true and was first proved by Foong and Kanno [Phys. Rev. Lett. 72, 1148-1151 (1994)] by using Fourier transform, and next by Sánchez-Ruiz [Phys. Rev. E 52, 5653-5655 (1995)] and by Sen [Phys. Rev. Lett. 77, 1-3 (1996)] by using random matrix theory connected with generalized Laguerre polynomials.

       Adopting this last viewpoint, we detail the probabilistic approach to this problem. Especially, viewing any Gaussian vector as a product of a uniformly distributed unitary vector by an Erlang distribution, we give a new insight for the different entropies introduced by Page.

Claude Monet (1840-1926) – The highway bridge at Argenteuil, 1874

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Some explicit ditributions related to the first exit time from a bounded interval for certain functionals of Brownian motion
Journal of Theoretical Probability 19(4) (2006), 757-771.
MathSciNet reference: MR2279602   DOI:10.1007/s10959-006-0039-9
Abstract

       Let \((B_t)_{t\ge 0}\) be standard Brownian motion starting at \(y\) and set \(X_t=x+\int_0^t V(B_s) \, ds\) for \(x\in (a,b)\), with \(V(y)= y^{\gamma}\) if \(y\ge 0\), \(V(y)= -K(-y)^{\gamma}\) if \(y\le 0\), where \(\gamma\) and \(K\) are some given positive constants. Set \(\tau_{ab} = \inf \{t>0:X_t \notin (a,b)\}\). In this paper we provide some formulas for the probability distribution of the random variable \(B_{\tau_{ab}}\) as well as for the probability \(\mathbb{P}\{X_{\tau_{ab}}=a\) (or \(b)\}\). The formulas corresponding to the particular cases \(x=a\) or \(b\) are explicitly expressed by means of hypergeometric functions.

Claude Monet (1840-1926) – Impression, sunrise, 1872

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Minimal cyclic random motion in \(\mathbb{R}^n\) and hyper-Bessel functions (with E. Orsingher and S. Leorato, Università di Roma “La Sapienza”)
Annales de l'I. H. P. Sect. B 42(6) (2006), 753-772.
MathSciNet reference: MR2269237   DOI:10.1016/j.anihpb.2005.11.002
Abstract   Article

       We obtain the explicit distribution of the position of a particle performing a cyclic, minimal, random motion with constant velocity \(c\) in \(\mathbb{R}^n\). The \(n+1\) possible directions of motion as well as the support of the distribution form a regular hyperpolyhedron (the first one having constant sides and the other expanding with time \(t\)), the geometrical features of which are here investigated.

       The distribution is obtained by using order statistics and is expressed in terms of hyper-Bessel functions of order \(n+1\). These distributions are proved to be connected with \((n+1)\)th order p.d.e. which can be reduced to Bessel equations of higher order.

       Some properties of the distributions obtained are examined. This research has been inspired by a conjecture formulated in Orsingher and Sommella [1] which is here proved to be false.

[1] E. Orsingher and A.M. Sommella (2004). A cyclic random motion in \(R^3\) with four directions and finite velocity, Stoch. Stoch. Rep, 76(2), 113-133.

John Singer Sargent (1856-1925) – Piazza Navona, Rome, 1907

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Cyclic random motions in space \(\mathbb{R}^d\) with \(n\) directions
ESAIM: Probability and Statistics 10 (2006), 277-316.
MathSciNet reference: MR2247923   DOI:10.1051/ps:2006012
Abstract   Article

       We study the probability distribution of the location of a particle performing a cyclic random motion in \(\mathbb{R}^d\). The particle can take \(n\) possible directions with different velocities and the changes of direction occur at random times. The speed-vectors as well as the support of the distribution form a polyhedron (the first one having constant sides and the other expanding with time \(t\)). The distribution of the location of the particle is made up of two components: a singular component (corresponding to the beginning of the travel of the particle) and an absolutely continuous component.

       We completely describe the singular component and exhibit an integral representation for the absolutely continuous one. The distribution is obtained by using a suitable expression of the location of the particle as well as some probability calculus together with some linear algebra. The particular case of the minimal cyclic motion (\(n=d+1\)) with Erlangian switching times is also investigated and the related distribution can be expressed in terms of hyper-Bessel functions with several arguments.

Vincent Van Gogh (1853-1890) – Irises, 1889

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First hitting time and place, monopoles and multipoles for the pseudo-process driven by the equation \(\partial/\partial t=\pm\partial^N/\partial x^N\)
Electronic Journal of Probability 12 (2007), paper no. 11, 300-353.
MathSciNet reference: MR2299920   DOI:10.1214/EJP.v12-399
Abstract   Article

       Consider the high-order heat-type equation \(\partial u/\partial t=\pm\partial^N u/\partial x^N\) for an integer \(N>2\) and introduce the related Markov pseudo-process \((X(t))_{t\ge 0}\). In this paper, we study several functionals related to \((X(t))_{t\ge 0}\): the maximum \(M(t)\) and minimum \(m(t)\) up to time \(t\); the hitting times \(\tau_a^+\) and \(\tau_a^-\) of the half lines \((a,+\infty)\) and \((-\infty,a)\) respectively. We provide explicit expressions for the distributions of the vectors \((X(t),M(t))\) and \((X(t),m(t))\), as well as those of the vectors \((\tau_a^+,X(\tau_a^+))\) and \((\tau_a^-,X(\tau_a^-))\).

John Singer Sargent (1856-1925) – Palazzo Labia and San Geremia, Venise, 1913

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A stochastic continuation approach to piecewise constant reconstruction (with M. Robini and I. Magnin, INSA Lyon)
IEEE Transactions on Image Processing 16(10) (2007), 2576-2589.
MathSciNet reference: MR2467787   DOI:10.1109/TIP.2007.904975
Abstract

       We address the problem of reconstructing a piecewise constant 3-D object from a few noisy 2-D line-integral projections. More generally, the theory developed here readily applies to the recovery of an ideal n-D signal from indirect measurements corrupted by noise. Stabilization of this ill-conditioned inverse problem is achieved with the Potts prior model, which leads to the minimization of a discontinuous, highly multimodal cost function. To carry out this challenging optimization task, we introduce a new class of annealing-type algorithms we call stochastic continuation (SC). We first prove that, under mild assumptions, SC inherits the desirable finite-time convergence properties of generalized simulated annealing. Then, we show that SC can be successfully applied to our 3-D reconstruction problem. In addition, we look into the concave distortion acceleration method introduced for standard simulated annealing and we derive an explicit formula for choosing the free parameter of the cost function. Numerical experiments using both synthetic data and real radiographic testing data show that SC outperforms standard simulated annealing.

John Singer Sargent (1856-1925) – Venice in Gray Weather, 1880

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First hitting time and place for the pseudo-process driven by the equation \(\partial/\partial t=\pm\partial^N/\partial x^N\) subject to a linear drift
Stochastic Processes and their Applications 118 (2008), 1-27.
MathSciNet reference: MR2376250   DOI:10.1016/j.spa.2007.03.009
Abstract   Article

       Consider the high-order heat-type equation \(\partial u/\partial t=(-1)^{1+N/2}\partial^N u/\partial x^N\) for an even integer \(N>2\) and introduce the related Markov pseudo-process \((X(t))_{t\ge 0}\). Let us define the drifted pseudo-process \((X^b(t))_{t\ge 0}\) by \(X^b(t)=X(t)+bt\). In this paper, we study the following functionals related to \((X^b(t))_{t\ge 0}\): the maximum \(M^b(t)\) up to time \(t\); the first hitting time \(\tau_a^b\) of the half line \((a,+\infty)\); the hitting place \(X^b(\tau_a^b)\) at this time. We provide explicit expressions for the Laplace-Fourier transforms of the distributions of the vectors \((X^b(t),M^b(t))\) and \((\tau_a^b,X^b(\tau_a^b))\), from which we deduce remarkable expressions for the distribution of \(X^b(\tau_a^b)\) as well as for the escape pseudo-probability: \(P\{\tau_a^b=+\infty\}\).

John Singer Sargent (1856-1925) – Rio dei Mendicanti, Venise, 1909

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A note on Spitzer identity for random walk
Statistics and Probability Letters 78 (2008), 2576-2589.
MathSciNet reference: MR2382062   DOI:10.1016/j.spl.2007.05.025
Abstract   Article

       Let \((S_n)_{n\ge 0}\) be a random walk evolving on the real line and introduce the first hitting time of the half-line \((a,+\infty)\) for any real \(a\): \(\tau_a=\min\{n\ge 1:S_n>a\}\). The classical Spitzer identity (1960) supplies an expression for the generating function of the couple \((\tau_0,S_{\tau_0})\). In (1998), Nakajima [Kodai Math. J. 21, 192-200] derived a relationship between the generating functions of the random couples \((\tau_0,S_{\tau_0})\) and \((\tau_a,S_{\tau_a})\) for any positive number \(a\). In this note, we propose a new and shorter proof for this relationship and complement this analysis by considering the case of an increasing random walk. We especially investigate the Erlangian case and provide an explicit expression for the joint distribution of \((\tau_a,S_{\tau_a})\) in this situation.

John Singer Sargent (1856-1925) – The entrance to the Grand Canal, Venice, 1907

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Some Darling-Siegert relationships connected with random flights (with V. Cammarota and E. Orsingher, Università di Roma “La Sapienza”)
Statistics and Probability Letters 79(2) (2009), 243-254.
MathSciNet reference: MR2483547   DOI:10.1016/j.spl.2008.08.002
Abstract   Article

       We derive in detail four important results on integrals of Bessel functions from which three combinatorial identities are extracted. We present the probabilistic interpretation of these identities in terms of different types of random walks, including asymmetric ones. This work extends the results of a previous paper concerning the Darling-Siegert interpretation of similar formulas emerging in the analysis of random flights.

Ippolito Caffi (1809-1866) – Rome, view of the holes with the column of Traiano

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Chung's law for homogeneous Brownian functionals (with T. Simon, University of Évry)
Rocky Mountain Journal of Mathematics 40(2) (2010), 561-579.
MathSciNet reference: MR2646458   DOI:10.1216/RMJ-2010-40-2-561
Abstract   Article

       Consider the first exit time \(T_{a,b}\) from a finite interval \([-a,b]\) for an homogeneous fluctuating functional \(X\) of a linear Brownian motion. We show the existence of a finite positive constant \(\kappa\) such that \[\lim_{t\to\infty}t^{-1}\log \mathbb{P}[ T_{ab}> t]\; =\; -\kappa.\] Following Chung's original approach [1], we deduce a “liminf” law of the iterated logarithm for the two-sided supremum of \(X\). This extends and gives a new point of view on a result of Khoshnevisan and Shi [2].

[1] K. L. Chung. On the maximum partial sums of sequences of independent random variables. Trans. Amer. Math. Soc. 64 (1948), 205-233.

[2] D. Khoshnevisan and Z. Shi. Chung's law for integrated Brownian motion. Trans. Amer. Math. Soc. 350(10) (1998), 4253-4264.

Francesco Guardi (1712-1793) – Santa Maria della Salute, 1783

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Joint distribution of the process and its sojourn time in the positive half-line for pseudo-processes governed by higher-order heat-type equations (with V. Cammarota, Università di Roma “La Sapienza”)
Electronic Journal of Probability 15 (2010), Paper no. 28, 895-931.
MathSciNet reference: MR2653948   DOI:10.1214/EJP.v15-782
Abstract   Article

       Consider the high-order heat-type equation \(\partial u/\partial t=\pm\,\partial^N u/\partial x^N\) for an integer \(N>2\) and introduce the related Markov pseudo-process \((X(t))_{t\ge 0}\). In this paper, we study the sojourn time \(T(t)\) in the positive half-line \([0,+\infty)\) up to a fixed time \(t\) for this pseudo-process. We provide explicit expressions for the joint distribution of the couple \((T(t),X(t))\).

Ippolito Caffi (1809-1866) – Rome with a view of the Tiber River, Castel Sant'Angelo and St. Peter

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A class of bridges of iterated integrals of Brownian motion related to various boundary value problems involving the one-dimensional polyharmonic operator
International Journal of Stochastic Analysis 2011 (2011), Art. ID 762486, 32 p.
MathSciNet reference: MR2861121   DOI:10.1155/2011/762486
Abstract   Article

       Let \((B(t))_{t\in [0,1]}\) be the linear Brownian motion and \((X_n(t))_{t\in [0,1]}\) be the \((n-1)\)-fold integral of Brownian motion, \(n\) being a positive integer: \[X_n(t)=\int_0^t \frac{(t-s)^{n-1}}{(n-1)!} \,\mathrm{d} B(s) \text{ for any } t\in[0,1].\] In this paper we construct several bridges between times \(0\) and \(1\) of the process \((X_n(t))_{t\in [0,1]}\) involving conditions on the successive derivatives of \(X_n\) at times \(0\) and \(1\). For this family of bridges, we make a correspondance with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical problem of prediction. Our results involve various Hermite interpolation polynomials.

Emmanuel Costa (1833-1921) – The monastery of Saint-Pons

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A random walk model related to the clustering of membrane receptors
In: Skogseid, A. and Fasano, V. (eds) “Statistical Mechanics and Random Walks: Principles, Processes and Applications”, Chapter 18, 545-580, Nova Science publishers, 2012.
Abstract   Article

       In a cellular medium, the plasmic membrane is a place of interactions between the cell and its direct external environment. A classic model describes it as a fluid mosaic. The fluid phase of the membrane allows a lateral degree of freedom to its constituents: they seem to be driven by random motions along the membrane. On the other hand, experimentations bring to light inhomogeneities on the membrane; these micro-domains (the so-called rafts) are very rich in proteins and phospholipids. Nevertheless, few functional properties of these micro-domains have been shown and it appears necessary to build appropriate models of the membrane for recreating the biological mechanism.

       In this article, we propose a random walk model simulating the evolution of certain constituents–the so-called ligands–along a heterogeneous membrane. Inhomogeneities–the rafts–are described as being still clustered receptors. An important variable of interest to biologists is the time that ligands and receptors bind during a fixed amount of time. This stochastic time can be interpreted as a measurement of affinity/sentivity of ligands for receptors. It corresponds to the sojourn time in a suitable set for a certain random walk.

       We provide a method of calculation for the probability distribution of this random variable and we next determine explicitly this distribution in the simple case when we are dealing with only one ligand and one receptor. We finally address some further more realistic models.

Structure of the glutamate receptor’s ligand binding domain, highlighting residues (red) that make connection across the cleft. The central residue forms a bond within the binding domain, stabilizing and locking the domain in the closed state.

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Sojourn time in \(\mathbb{Z}^+\) for the Bernoulli random walk on \(\mathbb{Z}\)
ESAIM: Probability and Statistics 16 (2012), 324-351.
MathSciNet reference: MR2966167   DOI:10.1051/ps/2010013
Abstract   Article

       Let \((S_k)_{k\ge 1}\) be the classical Bernoulli random walk on the integer line with jump parameters \(p\in(0,1)\) and \(q=1-p\). The probability distribution of the sojourn time of the walk in the set of non-negative integers up to a fixed time is well-known, but its expression is not simple. By modifying slightly this sojourn time–through a particular counting process of the zeros of the walk as done by Chung and Feller ["On fluctuations in coin-tossings", Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 605-608]–, simpler representations may be obtained for its probability distribution. In the aforementioned article, only the symmetric case (\(p=q=1/2\)) is considered. This is the discrete counterpart to the famous Paul Lévy's arcsine law for Brownian motion.

Francesco Guardi (1712-1793) – The Grand Canal with San Simeone Piccolo and Santa Lucia, 1780

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Joint distribution of the process and its sojourn time in a half-line \([a,+\infty)\) for pseudo-processes governed by higher-order heat-type equations (with V. Cammarota, Università di Roma “La Sapienza”)
Stochastic Processes and their Applications 122 (2012), 217-249.
MathSciNet reference: MR2860448   DOI:10.1016/j.spa.2011.08.004
Abstract   Article

       Let \((X(t))_{t \ge 0}\) be the pseudo-process driven by the high-order heat-type equation \(\partial u/\partial t=\pm\,\partial^N u/\partial x^N\), where \(N\) is an integer greater than 2. Let us introduce the sojourn time spent by \((X(t))_{t \ge 0}\) in \([a,+\infty)\) (\(a\in \mathbb{R}\)), up to a fixed time \(t>0\): \(T_a(t)=\int_0^t 1\!\!\mathrm{l}_{[a,+\infty)}(X(s))\,\mathrm{d}s\). The purpose of this paper is to explicit the joint pseudo-distribution of the vector \((T_a(t),X(t))\) when the pseudo-process starts at a point \(x\in \mathbb{R}\) at time \(0\). The method consists of solving a boundary value problem satisfied by the Laplace transform of the aforementioned distribution.

Ippolito Caffi (1809-1866) – View of Santa Maria Maggiore and Santa Prassede

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A survey on the pseudo-process driven by the high-order heat-type equation \(\partial/\partial t=\pm\partial^N/\partial x^N\) concerning the first hitting times and sojourn times
Methodology and Computing in Applied Probability 14(3) (2012), 549-566.
MathSciNet reference: MR2966309   DOI:10.1007/s11009-011-9245-8
Abstract   Article

       Fix an integer \(n>2\) and let \((X(t))_{t\ge 0}\) be the pseudo-process driven by the high-order heat-type equation \(\partial/\partial t=\pm\partial^n/\partial x^n\). The denomination “pseudo-process” means that \((X(t))_{t\ge 0}\) is related to a signed measure (which is not a probability measure) with total mass equal to 1.

In this note, we present some results and discuss some problems concerning the pseudo-distributions of the first overshooting times of a single barrier \(\{a\}\) or a double barrier \(\{a,b\}\) by \((X(t))_{t\ge 0}\), as well as those of the sojourn times of \((X(t))_{t\ge 0}\) in the intervals \([a,+\infty)\) and \([a,b]\) up to a fixed time.

Emmanuel Costa (1833-1921) – The Cathedral of St. Nicolas in Nice

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Sojourn time in an union of intervals for diffusions
Methodology and Computing in Applied Probability 15(4) (2013), 743-771.
MathSciNet reference: MR3117625   DOI:10.1007/s11009-012-9280-0
Abstract   Article

       We give a method for computing the iterated Laplace-transform of the sojourn time in an union of intervals for linear diffusion processes. This random variable comes from a model occurring in biology concerning the clustering of membrane receptors. The way used hinges on solving differential equations. Mask the abstract

From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval
International Journal of Stochastic Analysis 2014 (2014), Art. ID 520136, 49 p.
MathSciNet reference: MR3191097   DOI:10.1155/2014/520136
Abstract   Article

       Let \(N\) be a positive integer, \(c\) be a positive constant and \((U_n)_{n\ge 1}\) be a sequence of independent identically distributed pseudo-random variables. We assume that the \(U_n\)'s take their values in the discrete set \(\{-N,-N+1,\dots,N-1,N\}\) and that their common pseudo-distribution is characterized by the (positive or negative) real numbers \[\mathbb{P}\{U_n=k\}=\delta_{k0}+(-1)^{k-1} c{2N\choose k+N}\] for any \(k\in\{-N,-N+1,\dots,N-1,N\}\). Let us finally introduce \((S_n)_{n\ge 0}\) the associated pseudo-random walk defined on \(\mathbb{Z}\) by \(S_0=0\) and \(S_n=\sum_{j=1}^n U_j\) for \(n\ge 1\).

In this paper, we exhibit some properties of \((S_n)_{n\ge 0}\). In particular, we explicitly determine the pseudo-distribution of the first overshooting time of a given threshold for \((S_n)_{n\ge 0}\) as well as that of the first exit time from a bounded interval.

Next, with an appropriate normalization, we pass from the pseudo-random walk to the pseudo-Brownian motion driven by the high-order heat-type equation \(\partial/\partial t=(-1)^{N-1} c\;\partial^{2N}\!/\partial x^{2N}\). We retrieve the corresponding pseudo-distribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, A.: First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation \(\partial/\partial t=\pm \partial^N/\partial x^N\). Electron. J. Probab. 12 (2007), 300-353 [MR2299920]). In the same way, we get the pseudo-distribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudo-process.

Vincent Van Gogh (1853-1890) – Almond blossoms, 1890

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First exit time from a bounded interval for pseudo-processes driven by the equation \(\partial/\partial t=(-1)^{N-1} \partial^{2N}\!/\partial x^{2N}\)
Stochastic Processes and their Applications 124(2) (2014), 1084-1111.
MathSciNet reference: MR3138608   DOI:10.1016/j.spa.2013.09.016
Abstract   Article

       Let \(N\) be a positive integer. We consider pseudo-Brownian motion \(X=(X(t))_{t \ge 0}\) driven by the high-order heat-type equation \(\partial/\partial t=(-1)^{N-1} \partial^{2N}\!/\partial x^{2N}\). Let us introduce the first exit time \(\tau_{ab}\) from a bounded interval \((a,b)\) by \(X\) (\(a,b\in \mathbb{R}\)) together with the related location, namely \(X_{\tau_{ab}}\).

In this paper, we provide a representation of the joint pseudo-distribution of the vector \((\tau_{ab},X(\tau_{ab}))\) by means of some determinants. The method we use is based on the Feynman-Kac functional related to pseudo-Brownian motion which leads to a boundary value problem. In particular, the pseudo-distribution of \(X(\tau_{ab})\) admits a fine expression involving famous Hermite interpolating polynomials.

Federico Moja (1802–1885) – The courtyard of the Doge's Palace, Venice

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Entrance and sojourn times for Markov chains. Application to \((L,R)\)-random walks (with V. Cammarota, Università di Roma “Tor Vergata”)
Markov Processes and Related Fields 21(4) (2015), 887-938.
MathSciNet reference: MR3496230
Abstract   Article

       In this paper, we provide a methodology for computing the probability distribution of certain sojourn times for Markov chains. Our methodology hinges on matrix equations for various generating functions. As an example, we apply this methodology to a class of random walks with bounded integer-valued jumps.

Giovanni Paolo Panini (1691–1765) – Roman Capriccio showing the Colosseum, 1735

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Some asymptotic results for the integrated empirical processes with applications to statistical tests (with S. Alvarez-Andrade and S. Bouzebda, Technologic University of Compiègne)
Communications in Statistics – Theory and Methods 46(7) (2017), 3365-3392
MathSciNet reference: MR3589103   DOI:10.1080/03610926.2015.1060346
Abstract   Article

       The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al.'s results, [1]. Applications include the two-sample testing procedures together with the change-point problems. We consider also the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial sums process representation of the integrated empirical process.

[1] Komlós, J., Major, P. and Tusnády, G. An approximation of partial sums of independent RV's and the sample DF (I). Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 111-131.

Carle Vernet – Napoleon on a hunt in the Compiègne forest, 1811

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Strong approximations for the \(p\)-fold integrated empirical processes with applications to statistical tests (with S. Alvarez-Andrade and S. Bouzebda, University of Technology of Compiègne)
TEST 27(4) (2018), 826-849
MathSciNet reference: MR3878363   DOI:10.1007/s11749-017-0572-0
Abstract   Article

       The main purpose of this paper is to investigate the strong approximation of a class of integrated empirical processes. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al.'s results, [1]. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated.

[1] Komlós, J., Major, P. and Tusnády, G. An approximation of partial sums of independent RV's and the sample DF (I). Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 111-131.

Raymond Fournier-Sarlovèze (1836-1916) – 23 may 1430, The capture of Jeanne d'Arc ahead of Compiègne

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On integrated Brownian motion (in French)
t. 311 (1990), 461-464.   MathSciNet reference: MR1075671 (91i:60207)
Abstract

       Let \((B_t)_{t\ge0}\), be the standard Brownian motion in \(\mathbb{R}\). Define \(X_t=\int^t_0B_s\,ds\), \(U_t=(X_t+x+ty,B_t+y)\), \((x,y) \in\mathbb{R}^2\), and \(\tau_a=\inf\{t>0\); \(U_t\in\{a\}\times\mathbb{R}\}\). In this note we compute explicitly the joint distribution of \(\tau_a\) and \(U_{\tau_a}\). We also indicate a simple proof of a recent result of M. Lefebvre with some improvements.

Edouard Manet (1832-1883) – The bench (garden of Versailles), 1880-81

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On the excursions of integrated Brownian motion (in French)
t. 314 (1992), 1053-1056.   MathSciNet reference: MR1168534 (93c:60123)
Abstract

       Let \((B(t))_{t\ge 0}\) be the standard Brownian motion in \(\mathbb{R}\) started at \(0\), and let \(X(t)=\int^t_0 B(s)\,ds+x+ty\), where \((x,y)\in\mathbb{R}^2\) is a fixed point. In this note we compute explicitly the law of the excursion process straddling a fixed instant \(T>0\) related to \((X(t))_{t\ge 0}\).

Pierre-Auguste Renoir (1841-1919) – Rocky crags at l'Estaque, 1882

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On the successive passage times of integrated Brownian motion (in French)
t. 321 (1995), 903-908.   MathSciNet reference: MR1355850 (96k:60207)
Abstract

       In this short note, we explicitly present several probability laws related to the successive passage times to the level \(0\) for the integrated Brownian motion.

Jean-Baptiste Camille Corot (1796-1875) – View of Genoa from the Promenade of Acqua Sola, 1834

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Exit time from a bounded interval for integrated Brownian motion (in French)
t. 324 (1997), 559-564.   MathSciNet reference: MR1443994 (98a:60116)   DOI:10.1016/S0764-4442(99)80390-5
Abstract

       Let \((B_t)_{t\ge0}\) be a standard Brownian motion starting at \(y, X_ t=x+\int^t_0B_sds, x\in(a,b)\). Set \(\tau_{ab}=\inf\{t>0\colon X_ t\notin(a,b)\}\). In this paper, we compute the moments of the random variable \(B_{\tau_{ab}}\), and deduce the probability law of \(B_{\tau_{ab}}\). We show how to obtain the expectation \({\bf E}_{(x,y)}(\tau^m_ {ab}B^n_{\tau_{ab}}\)). We also determine explicitly the probabilities \({\bf P}_{(x,y)}\{X_{\tau_{ab}}=a\}\) and \({\bf P}_{(x,y)}\{X_{\tau_{ab}}=b\}\).

Henri Fantin-Latour (1836-1904) – Still life with flowers, 1881

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Some probability distributions in modeling DNA replication (in French)
t. 336 (2003), 175-180.   MathSciNet reference: MR1969574 (2004c:62030)   DOI:10.1016/S1631-073X(03)00006-2
Abstract   Article

       By using some quasi-renewal-like equations and functional differential equations, we explicitly compute the Laplace transforms of some random variables introduced by Cowan and Chiu in modelling the process of replication of a DNA molecule [J. Appl. Prob. 31 (1994) 301-308]. These Laplace transforms are expressed by means of infinite products arising in the theory of partitions.

[1] R. Cowan. Stochastic models for DNA replication, in The Handbook of Statistics, Vol. 20, eds. C.R. Rao and D.N. Shanbhag, Elsevier, 2001.

[2] R. Cowan and S.N. Chiu. Stochastic model of fragment formation when DNA replicates, J. Appl. Prob. 31 (1994), 301-308.

Paul Cézanne (1839-1906) – Mount Sainte-Victoire, 1885

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Joint laws of the process and its maximum, the first hitting time and location of a half-line for the pseudo-process governed by the equation \(\partial/\partial t=\pm\partial^n/\partial x^n\)
t. 343 (2006), 525-530.   MathSciNet reference: MR2267588   DOI:10.1016/j.crma.2006.09.027
Abstract   Article

       In this Note, we obtain explicit formulas for the joint distribution of the pseudo-process driven by the equation \(\frac{\partial}{\partial t}=\pm\frac{\partial^N}{\partial x^N}\) coupled together with its maximum, as well as that of the first time when this pseudo-process overshoots a fixed level coupled together with the corresponding overshooting place.

John Singer Sargent (1856-1925) – Rio dell'Angelo

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The Kheops pyramid and some equations of degree 4 (in French)
Quadrature 69 (2008), 36-47.   Zentralblatt MATH reference: Zbl 1242.51013   DOI:10.1051/quadrature:2008004
Abstract   Article

       Dans un article publié dans l'édition française de la revue australienne Nexus, James Colmer présentait une hypothèse originale sur la fonction de la pyramide de Kheops et sur l'existence présumée d'un ensemble de galeries et chambres cachées, symétriques à celles connues. La démarche de J. Colmer, dans sa recherche, passait par un tracé géométrique impliquant un certaine valeur de l'angle formé par la face de la pyramide avec sa base horizontale et la triple intersection d'une circonférence avec deux segments de droites spécifiques.

       André Dufour, traducteur pour Nexus France et architecte de métier, ayant eu la charge de traduire cet article, s'aperçut après avoir refait sur ordinateur le tracé géométrique de J. Colmer que ce dernier était faux. Cela n'enlevait rien à l'intérêt de l'hypothèse de J. Colmer, objet principal de l'article, mais posait un problème intéressant de géométrie. Après en avoir fait une note de traducteur publiée simultanément avec l'article traduit de J. Colmer, plusieurs lecteurs, F. De Ligt, J.-F. Pioche et l'auteur du présent article réagirent au problème en fournissant des valeurs exactes pour l'angle mis en cause par le biais de diverses méthodes calculatoires.

       Par une démarche empirique, A. Dufour tenta une construction graphique à l'aide d'un logiciel de dessin vectoriel afin d'obtenir la position du sommet de la pyramide requise par la théorie de J. Colmer. Cette approche, qu'il a soumise à l'auteur, repose sur la construction d'une courbe en polyligne lissée dont l'intersection avec une droite adéquate fournit avec une excellente précision la position du sommet recherché (en fait deux sommets conviendront comme cela apparaîtra ultérieurement). Par une approche analytique, l'auteur prouvera alors que cette construction est parfaitement exacte. Il est intéressant de noter que, bien au-delà d'un exercice de géométrie élémentaire en apparence, ce problème suscitera en fait une analyse mathématique particulièrement riche touchant à des domaines diversifiés.

       Tous les éléments des diverses correspondances entre A. Dufour, F. De Ligt, J.-F. Pioche et l'auteur sont rassemblées dans le présent article.

David Roberts (1796-1864) – The pyramids of Cheops and Chephren

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Perfect cards shuffles (I). In-shuffles and out-shuffles (in French)
Quadrature 76 (2010), 13-25.   Zentralblatt MATH reference: Zbl 1214.91014   DOI:10.1051/quadrature/2010001
Abstract   Article

       In this paper followed by a companion paper, we study some cards shuffles which are used by magicians. We focus ourselves on the possibility to hit eventually the initial state after several shuffles. This is a classical problem arising in discrete dynamical systems. The computations are performed through an elementary approach, so the paper is easily accessible.

Paul Cézanne (1839-1906) – The card players, 1890-1892

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Perfect cards shuffles (II). Monge shuffles (in French)
Quadrature 77 (2010), 23-29.   Zentralblatt MATH reference: Zbl 1262.91038   DOI:10.1051/quadrature/2010008
Abstract   Article

       This paper is the second part of a work dealing with some cards shuffles which are used by magicians. In this part, we consider the so-called Monge shuffles and we focus ourselves on the possibility to hit eventually the initial state after several shuffles. This is a classical problem arising in discrete dynamical systems. The computations are performed through an elementary approach, so the paper is easily accessible.

Paul Cézanne (1839-1906) – The card players, 1890-1892

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Cardmagic: principles of Gilbreath (I). Enumeration of American shuffles (in French; with P. Schott, ESIEA of Paris)
Quadrature 85 (2012), 24-35.   Zentralblatt MATH reference: Zbl 06131689
Abstract   Article

       The magic principles of Gilbreath allow, from a beforehand classified deck of cards to keep, after an American mixture, its properties of classification by block of cards but in a possibly disordered way (the cards of the same block can be sorted in a different order). Such properties allow to see coming true predictions in spite of a real mixture! We exhibit in three parts these principles of cardmagic. In this first article, we present American mixtures and detail their enumeration. In two forthcoming companion papers, we shall present magic applications of the principles of Gilbreath, as well as their mathematical proofs.

Adriaen Brouwer (1605-1638) – The card players

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Cardmagic: principles of Gilbreath (II). Some applications (in French; with P. Schott, ESIEA of Paris)
Quadrature 86 (2012), 31-37.   Zentralblatt MATH reference: Zbl 1254.97015   MathSciNet reference: MR3059190
Abstract   Article

       The magic principles of Gilbreath allow, from a beforehand classified deck of cards to keep, after an American mixture, its properties of classification by block of cards but in a possibly muddled way (the cards of the same block could be sorted in a different order). Such properties allow to see coming true predictions in spite of a real mixture! We proposed in a companion paper published in the previous number of Quadrature a calculation of the enumeration of all possible American mixtures as well as an algorithm realizing them from a given deck of cards. In this second part, we give applications of American mixtures based on the magic principles of Gilbreath. For each of them, we propose a magic trick. The proofs of the associated principles will appear in a forthcoming issue.

Michelangelo Merisi da Caravaggio (1571-1610) – The cardsharps, 1594

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Cardmagic: principles of Gilbreath (III). Various proofs (in French; with P. Schott, ESIEA of Paris)
Quadrature 87 (2013), 30-37.   Zentralblatt MATH reference: Zbl 1284.00003   MathSciNet reference: MR3059800
Abstract   Article

       The magic principles of Gilbreath allow, from a beforehand classified deck of cards to keep, after an American mixture, its properties of classification by block of cards but in a possibly muddled way (the cards of the same block could be sorted in a different order). Such properties allow to see coming true predictions in spite of a real mixture! We proposed in two companion papers published in the two previous issues of Quadrature a calculation of the enumeration of all possible American mixtures as well as several tricks appealing to the Gilbreath principles. In this last part, we provide the proofs of these principles.

Georges de La Tour (1593-1652) – The cardsharp with the ace of diamonds, 1632

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A trick around Fibonacci, Lucas and Chebyshev (in French)
Quadrature 93 (2014), 27-35.     MathSciNet Reference: MR3237161
Abstract   Article

       In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of this trick to more general sequences. This study leads us to interesting connections concerning connections between Fibonacci, Lucas sequences and Chebyshev polynomials.

Leonardo Fibonacci (1175-1250)

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Some stochastic reproduction models (in French)
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Stochastic phylogenetic models (in French)
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(in French) Sequence alignments: stochastic models
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Bridges of certain Wiener integrals: some properties and application to goodness-of-fit testing
Fourth Hungarian Colloquium on limit theorems of Probability and Statistics, Balatonlelle (Hungary) and seminary of Aarhus (Danemark), 1999
Slides

Study of some new integrated statistics
Journées de probabilités, Luminy (France), 2000
Slides

Some probabilistic models in molecular biology (in French)
Seminary of Dijon (France), 2002
Slides

Distribution of the sojourn time and the maximum for a pseudo-process governed by the higher-order heat-type equation \(\frac{\partial u}{\partial t} = \pm \frac{\partial^n u}{\partial x^n}\) (in French)
Seminary of Nancy (France), 2002
Slides

Modeling in DNA replication (in French)
Seminary of Saint-Étienne (France), 2003
Slides

Some passage time problems for certain non-Markov processes (in French)
Journées SMAI : Modélisation aléatoire et statistique, Nancy (France), 2004
Slides

Maximum, first hitting time and first hitting place of a half-line for a pseudo-process driven by a high-order heat-type equation
Seminary of Rome (Italy), 2006
Slides

Overview on remarkable distributions related to the pseudo-process driven by the heat-type equation \(\frac{\partial u}{\partial t}=\pm\frac{\partial^nu}{\partial x^n}\) of order \(n>2\) (in French)
Seminary of Évry (France), 2008
Slides

A survey on the pseudo-process driven by the high-order heat-type equation \(\partial/\partial t=\pm\partial^N/\partial x^N\) concerning the first hitting times and sojourn times
Fifth International Workshop on Applied Probability, Madrid (Spain), 2010
Slides

Some stochastic processes related to the polyharmonic differential operator
Self-Similarity and Stochastic Analysis, le Touquet (France), 2011
Slides

Brownian motion, pseudo-Brownian motion and PDE. Applications (in French)
Journées Modélisation Mathématique et Calcul Scientifique, Sainte Foy-lès-Lyon (France), 2011
Slides

Math&magic : A didactic approach to mathematics and science by Magic (in French; with P. Schott, ESIEA of Paris)
Cycle de conférences mathématiques, Insa de Lyon (France), 2012
Slides: Presentation - A video

Pseudo-random walk and pseudo-Brownian motion : first overshooting time of a single or double threshold (in French)
Seminary of Lille (France), 2014
Slides

Overview on integrated Brownian motion and other integrated processes (in French)
Seminary of Lille (France), 2014
Slides

Entrance and sojourn times for Markov chains. Application to \((L,R)\)-random walks
A probability afternoon, Lille (France), 2015
Stochastic processes under constraints, Augsburg (Germany), 2016
Slides

Some distributions on pseudo-Brownian motion and pseudo-random walk
Recent developments in probability theory and stochastic processes – A conference in honour of Enzo Orsingher on the occasion of his 70th birthday, Rome (Italy), 2016
Slides

Math&Magic (II) : Mathematics for Magic or Magic for Mathematics? (in French; with P. Schott, ESIEA of Paris)
Cycle de conférences mathématiques, Insa de Lyon (France), 2016
Slides: Part 1 - Part 2

Magic squares (in French; with P. Schott, ESIEA of Paris)
Exhibition «Magimatique», Maison des Mathématiques et de l'Informatique, Lyon (France), 2016/2017
Equipments: slideshow - Poster 1 - Poster 2 - Poster 3 - Poster 4

Math&Magic (III) : Are Mathematics Magical ? or is Magic Mathematical ? (in French; with P. Schott, ESIEA of Paris)
Exhibition «Magimatique», Maison des Mathématiques et de l'Informatique, Lyon, 2017
Slides

Some asymptotic results for integrated empirical processes with applications to statistical tests
10th International Conference of the ERCIM WG on Computational and Methodological Statistics, London (UK), 2017
Slides

Math&Magic (IV) : Are Mathematics Magical ? or is Magic Mathematical ? (in French; with P. Schott, ESIEA of Paris)
Cycle de conférences mathématiques, Insa de Lyon, 2018
Slides

Faro shuffles and cryptography (in French)
Rallye mathématique de l’académie de Lyon, Insa Lyon, 2023
Slides

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Vincent Van Gogh (1853-1890) – The starry night, 1889