Abstract:
This paper develops some general calculus for GGC and Dirichlet process means functionals. It then proceeds via an investigation of positive Linnik random variables, and more generally random variables derived from compositions of a stable subordinator with GGC subordinators, to establish various distributional equivalences between these models and phenomena connected to local times and occupation times of what are defined as randomly skewed Bessel processes and bridges. This yields a host of interesting identities and explicit density formula for these models. Randomly skewed Bessel processes and bridges may be seen as a randomization of their p-skewed counterparts developed in Barlow, Pitman and Yor (1989) and Pitman and Yor (1997), and are shown to naturally arise via exponential tilting. As a special result it is shown that the occupation time of a p-skewed random Bessel process or (generalized) bridge is equivalent in distribution to the occupation time of a non-trivial randomly skewed process.

Abstract:
In this paper excursions of a stationary diffusion in stationary state are studied. In particular, we compute the joint distribution of the occupation times $I^{(+)}_t$ and $I^{(-)}_t$ above and below, respectively, the observed level at time $t$ during an excursion. We consider also the starting time $g_t$ and the ending time $d_t$ of the excursion (straddling $t$) and discuss their relations to the Levy measure of the inverse local time. It is seen that the pairs $(I^{(+)}_t, I^{(-)}_t)$ and $(t-g_t, d_t-t)$ are identically distributed. Moreover, conditionally on $I^{(+)}_t + I^{(-)}_t =v$, the variables $I^{(+)}_t$ and $I^{(-)}_t$ are uniformly distributed on $(0,v)$. Using the theory of the Palm measures, we derive an analoguous result for excursion bridges.

Abstract:
We study properties of occupation times by Brownian excursions and Brownian loops in two-dimensional domains. This allows for instance to interpret some Gaussian fields, such as the Gaussian Free Fields as (properly normalized) fluctuations of the total occupation time of a Poisson cloud of Brownian excursions when the intensity of the cloud goes to infinity.

Abstract:
We exhibit, in the form of some identities in law, some connections between tilted stable subordinators, time-changed by independent Gamma processes and the occupation times of Bessel spiders, or their bridges. These identities in law are then explained thanks to excursion theory.

Abstract:
The paper considers excited random walks (ERWs) on integers in i.i.d. environments with a bounded number of excitations per site. The emphasis is primarily on the critical case for the transition between recurrence and transience which occurs when the total expected drift $\delta$ at each site of the environment is equal to 1 in absolute value. Several crucial estimates for ERWs fail in the critical case and require a separate treatment. The main results discuss the depth and duration of excursions from the origin for $|\delta|=1$ as well as occupation times of negative and positive semi-axes and scaling limits of ERW indexed by these occupation times. It is also pointed out that the limiting proportions of the time spent by a non-critical recurrent ERW (i.e. when $|\delta|<1$) above or below zero converge to beta random variables with explicit parameters given in terms of $\delta$.

Abstract:
This study of occupation time densities for continuous-time Markov processes was inspired by the work of E.Nir et al (2006) in the field of Single Molecule FRET spectroscopy. There, a single molecule fluctuates between two or more states, and the experimental observable depends on the state's occupation time distribution. To mathematically describe the observable there was a need to calculate a single state occupation time distribution. In this paper, we consider a Markov process with countably many states. In order to find a one-stete occupation time density, we use a combination of Fourier and Laplace transforms in the way that allows for inversion of the Fourier transform. We derive an explicit expression for an occupation time density in the case of a simple continuous time random walk on Z. Also we examine the spectral measures in Karlin-McGregor diagonalization in an attempt to represent occupation time densities via modified Bessel functions.

Abstract:
In this paper we establish relationships between four important concepts: (a) hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c) Schr\"odinger's equation with linear potential, and (d) heat equation problems with moving boundary. We relate (a) and (b) by means of Girsanov's theorem, which suggests a strategy to extend our ideas to problems in $\mathbb{R}^n$ and general diffusions. This approach also leads to (c) because we may relate, through a Feynman-Kac representation, functionals of a Bessel bridge with two Schr\"odinger-type problems. In particular, we also find a fundamental solution to a class of parabolic partial differential equations with linear potential. Finally, the relationship between (c) and (d) suggests a possible link between Generalized Airy processes and their hitting times.

Abstract:
We address the question about the velocity of signals carried by Bessel beams wave packets propagating in vacuum and having well defined wavefronts in time. We find that this problem is analogous to that of propagation of usual plane wave packets within dispersive media and conclude that the signal velocity can not be superluminal.

Abstract:
We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then in the non-drifted case we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated to this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. On a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. This permits in particular to get directly access to the joint distribution of the value at t of the process and its integral. We finally give some financial applications to illustrate the panel of applications of our results.

Abstract:
It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in long-time asymptotics of moments of height of 2-watermelons are completely determined. For the height of 2-watermelons with a wall, the average value was recently studied by Fulmek by a method of enumerative combinatorics.