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Michael J. Kearney

Publications and source records attributed to Michael J. Kearney.

11 recordsLinked to original sources
arXiv2026

A first passage problem for a Poisson counting process with a linear moving boundary

The time to first crossing for the Poisson counting process with respect to a linear moving barrier with offset is a classic problem, although key results remain scattered across the literature and their equivalence is often unclear. Here we present a unified and pedagogical treatment of two approaches: the direct time-domain approach based on path-decomposition techniques and the Laplace-domain approach based on the Pollaczek-Spitzer formula. Beyond streamlining existing derivations and establishing their consistency, we leverage the complementary nature of the two methods to obtain new exact analytical results. Specifically, we derive an explicit large deviation function for the first-passage time distribution in the subcritical regime and closed-form expressions for the conditional mean first-passage time for arbitrary offset. Despite its simplicity, this first crossing process exhibits non-trivial critical behavior and provides a rare example where all the main results of interest can be derived exactly.

cond-mat.stat-mech↗
arXiv2021

Statistics of the first passage area functional for an Ornstein-Uhlenbeck process

We consider the area functional defined by the integral of an Ornstein-Uhlenbeck process which starts from a given value and ends at the time it first reaches zero (its equilibrium level). Exact results are presented for the mean, variance, skewness and kurtosis of the underlying area probability distribution, together with the covariance and correlation between the area and the first passage time. Amongst other things, the analysis demonstrates that the area distribution is asymptotically normal in the weak noise limit, which stands in contrast to the first passage time distribution. Various applications are indicated.

cond-mat.stat-mech↗
arXiv2021

A note on an absorption problem for a Brownian particle moving in a harmonic potential

An analysis is presented of a Brownian particle moving on the half-line, subject to a restoring force proportional to its displacement and an absorbing boundary at the origin. When the initial displacement is large, the central moments of the time to be absorbed tend to finite constants, as do the position moments when evaluated at the most probable absorption time. These quantities are derived explicitly.

cond-mat.stat-mech↗
arXiv2021

Record statistics for a discrete-time random walk with correlated steps

The characterization of record events is considered for a discrete-time random walk model with long-term memory arising from correlations between successive steps. An important feature is that the correlations are strong enough to give rise to super-diffusivity and transience. Various quantities related to record statistics are calculated exactly, highlighting important differences in behaviour from the simple random walk with independent steps.

cond-mat.stat-mech↗
arXiv2019

On a stochastic version of Lanchester's model of combat

Lanchester's model of combat has certain deficiencies in its standard form arising from the neglect of the influence of random fluctuations. Several approaches to rectify this have been proposed and various results are scattered throughout the literature. Here, a discrete-time stochastic version, which is amenable to exact solution, is revisited with the aim of deriving key results within one setting. The exposition simplifies and provides refinements to earlier derivations and analysis.

physics.soc-ph↗
arXiv2018

Time since maximum of Brownian motion and asymmetric Levy processes

Motivated by recent studies of record statistics in relation to strongly correlated time series, we consider explicitly the drawdown time of a Levy process, which is defined as the time since it last achieved its running maximum when observed over a fixed time period [0,T]. We show that the density function of this drawdown time, in the case of a completely asymmetric jump process, may be factored as a function of $t$ multiplied by a function of T-t. This extends a known result for the case of pure Brownian motion. We state the factors explicitly for the cases of exponential down-jumps with drift, and for the downward Inverse Gaussian Levy process with drift.

math.PR↗
arXiv2014

Statistics of the first passage time of Brownian motion conditioned by maximum value or area

We derive the moments of the first passage time for Brownian motion conditioned by either the maximum value or the area swept out by the motion. These quantities are the natural counterparts to the moments of the maximum value and area of Brownian excursions of fixed duration, which we also derive for completeness within the same mathematical framework. Various applications are indicated.

cond-mat.stat-mech↗
arXiv2008

On the time to reach maximum for a variety of constrained Brownian motions

We derive P(M,t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over M, the marginal density P(t_m) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of P(t_m) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain "agreement formulae" that are encountered more generally in probabilistic studies of Brownian motion processes.

cond-mat.stat-mech↗
arXiv2006

On the Inelastic Collapse of a Ball Bouncing on a Randomly Vibrating Platform

We study analytically the dynamics of a ball bouncing inelastically on a randomly vibrating platform, as a simple toy model of inelastic collapse. Of principal interest are the distributions of the number of flights n_f till the collapse and the total time τ_c elapsed before the collapse. In the strictly elastic case, both distributions have power law tails characterised by exponents which are universal, i.e., independent of the details of the platform noise distribution. In the inelastic case, both distributions have exponential tails: P(n_f) ~ exp[-θ_1 n_f] and P(τ_c) ~ exp[-θ_2 τ_c]. The decay exponents θ_1 and θ_2 depend continuously on the coefficient of restitution and are nonuniversal; however as one approches the elastic limit, they vanish in a universal manner that we compute exactly. An explicit expression for θ_1 is provided for a particular case of the platform noise distribution.

cond-mat.stat-mech↗
arXiv2005

On the area under a continuous time Brownian motion till its first-passage time

The area swept out under a one-dimensional Brownian motion till its first-passage time is analysed using a backward Fokker-Planck technique. We obtain an exact expression of the area distribution for the zero drift case, and provide various asymptotic results for the non-zero drift case, emphasising the critical nature of the behaviour in the limit of vanishing drift. The results offer important insights into the asymptotic behaviour of the area-perimeter generating functions in a class of discrete polygons. We also provide a succinct derivation for the distribution of the maximum displacement observed till the first-passage time.

cond-mat.stat-mech↗