Erhan Çınlar

Probability and Stochastics

Preface.- Measure and Integration.- Probability Spaces.- Convergence.- Conditioning.- Martingales and Stochastics.- Poisson Random Measures.- Levy Processes.- Index.- Bibliography

Fatigue Crack Growth

Metal fatigue is a major cause for failure of mechanical and structural components. We review the fracture mechanics of fatigue and Paris-Erdogan law for the mean behavior. After a consideration of experimental data reported by Virkler et al. (1979), we propose a continuous semimarkov process to model crack growth. The model accounts for the material randomness and sees crack as a motion in a random field.

On lifetimes influenced by a common environment

Consider the lifelengths T1,..., Tk of k components subjected to a randomly varying environment. They are dependent on each other because of their common dependence of the environment. The parameters of the model are the distribution of the random process which describes the environment and a set of rate functions which determine the probability law of Ti,..., Tk as a function of the distribution of the environment. We find conditions on the parameters of the model which imply that T1,..., Tk are associated. Other conditions which imply that T1,..., Tk have the multivariate aging properties IHR (increasing hazard rate) and NBU (new better than used) are also described. Also two such models are compared. In particular, we characterize the parameters of these models so that stochastic ordering between the two vectors of resulting lifetimes can be obtained.

Mass Transport by Brownian Flows

We consider the motion of a mass distribution in a random velocity field which is “δ-correlated” in time but which has arbitrary spatial correlations. We discuss the rigorous formulation of this problem in terms of Brownian flows and stochastic calculus. We use numerical simulations to illustrate the effect of the flow on the mass. We present results concerning the evolution of the mass distribution and, in particular, the long-time asymptotics of the center of mass and relative dispersion.

Sunset over Brownistan

Consider a Brownian motion with a downward drift of rate a. Its maximum over all time has the exponential distribution with parameter 2a. Our aim is to study this maximum as a stochastic process indexed by a. That process is related to the concave majorant of the standard Brownian motion and, through the latter, to a Poisson random measure. This connection is exploited to obtain distributional results. The results are of interest in queueing theory.

Lyapunov exponents of Poisson shot-noise velocity fields

We consider the Lyapunov exponents of flows generated by a class of Markovian velocity fields. The existence of the exponents is obtained for flows on a compact set, but with the most general form of the velocity field. As a particular class, we study the homogeneous and incompressible flows. In this case, the exponents are nonrandom, free of the initial position of the particle path, and their sum is zero. We numerically compute the top Lyapunov exponent on for a range of parameters to conjecture that it is strictly positive.

Birth and Death on a Flow

In recent years there has been much interest in the equilibrium behavior of stochastic flows; see for instance Baxendale [1], Carverhill [2], Le Jan [6], [7] and [8]. Most of the work seems to be concentrated on the limiting distribution, as t → ∞, of the random measure

Measure and Integration

{\mu _t}(\omega ,A) = {\mu _0}\{ x:F_{0,t}^\omega x \in A\}