1,721,019 research outputs found
Some Results on the Brownian Meander with Drift
No full textIn this paper we study the drifted Brownian meander that is a Brownian motion starting from u and subject to the condition that min ≤z≤t B(z) > v with u> v. The limiting process for u↓ v is analysed, and the sufficient conditions for its construction are given. We also study the distribution of the maximum of the meander with drift and the related first-passage times. The representation of the meander endowed with a drift is provided and extends the well-known result of the driftless case. The last part concerns the drifted excursion process the distribution of which coincides with the driftless case
The last zero-crossing of an iterated brownian motion with drift
No full textIn this paper, we consider the iterated Brownian motion μ1μ2I(t)=Bμ11(∣∣Bμ22(t)∣∣) where Bμjj,j=1,2 are two independent Brownian motions with drift μj. Here, we study the last zero crossing before the maximum time span travelled by the inner process of μ1μ2I(t) and for this purpose we derive the last zero-crossing distribution of the drifted Brownian motion. We derive also the joint distribution of the last zero crossing before t and of the first passage time through the zero level of a Brownian motion with drift μ after t. All these results permit us to derive explicit formulas for IμT0=sup{
Asymptotic results for the last zero crossing time of a Brownian motion with non-null drift
No full textWe consider the last zero crossing time of a Brownian motion, with drift ,
in the time interval . We prove the large deviation principle of
as tends to infinity. Moreover, motivated by the results on moderate deviations in the literature,
we also prove a class of large deviation principles for the same random variables with different
scalings, which are governed by the same rate function. Finally we compare some aspects of the
classical moderate deviation results, and the results in this paper
On the fractional wave equation
No full textIn this paper we study the time-fractional wave equation of order 1 < n < 2 and give a probabilistic interpretation of its solution. In the case 0 < n < 1, d = 1, the solution can be interpreted as a time-changed Brownian motion, while for 1 < n < 2 it coincides with the density of a symmetric stable process of order 2/n. We give here an interpretation of the fractional wave equation for d > 1 in terms of laws of stable ddimensional processes. We give a hint at the case of a fractional wave equation for n > 2 and also at space-time fractional wave equations
Elastic drifted Brownian motions and non-local boundary conditions
Full text linkWe provide a deep connection between elastic drifted Brownian motions and inverses to tempered subordinators. Based on this connection, we establish a link between multiplicative functionals and dynamical boundary conditions given in terms of non-local equations in time. Indeed, we show that the multiplicative functional associated to the elastic Brownian motion with drift is equivalent to a functional associated with non-local boundary conditions of tempered type. By exploiting such connections we write some functionals of the drifted Brownian motion in terms of a simple (positive and non-decreasing) process, the inverse of a tempered subordinator. In our view, such a representation is useful in many applications and brings new light on dynamic boundary value problems.(c) 2023 Elsevier B.V. All rights reserved
Some families of random fields related to multiparameter Lévy processes
Full text linkLet R+N=[0,∞)N. We here make new contributions concerning a class of random fields (Xt)[email protected]@232cad15Rjavax.xml.bind.JAXBElement@72133445javax.xml.bind.JAXBElement@3aa07657 which are known as multiparameter Lévy processes. Related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators. We also provide a Phillips formula concerning the composition of (Xt)[email protected]@7e3b1b80Rjavax.xml.bind.JAXBElement@15e1a91ejavax.xml.bind.JAXBElement@2c4a673b by means of subordinator fields. We finally define the composition of (Xt)[email protected]@78ac3586Rjavax.xml.bind.JAXBElement@609dcad4javax.xml.bind.JAXBElement@1f911a43 by means of the so-called inverse random fields, which gives rise to interesting long-range dependence properties. As a byproduct of our analysis, we present a model of anomalous diffusion in an anisotropic medium which extends the one treated in Beghin et al. (Stoch Proc Appl 130:6364–6387, 2020), by improving some of its shortcomings
On the sojourn time of a generalized Brownian meander
No full textIn this paper we study the sojourn time on the positive half-line up to time t of a drifted Brownian motion with starting point u and subject to the condition that min0≤z≤lB(z)>v, with u>v. This process is a drifted Brownian meander up to time l and then evolves as a free Brownian motion. We also consider the sojourn time of a bridge-type process, where we add the additional condition to return to the initial level at the end of the time interval. We analyze the weak limit of the occupation functional as u↓v. We obtain explicit distributional results when the barrier is placed at the zero level, and also in the special case when the drift is null
Telegraph random evolutions on a circle
No full textWe consider the random evolution described by the motion of a particle moving on a circle alternating the angular velocities ±c and changing rotation at Poisson random times, resulting in a telegraph process over the circle. We study the analytic properties of the semigroup it generates as well as its probability distribution. The asymptotic behavior of the wrapped process is also discussed in terms of circular Brownian motion. Besides, it is possible to derive a stochastic model for harmonic oscillators with random changes in direction and we give a diffusive approximation of this process. Furthermore, we introduce some extensions of the circular telegraph model in the asymmetric case and for non-Markovian waiting times as well. In this last case, we also provide some asymptotic results
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