In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if Sn is a random walk with negative mean and finite variance then there is a constant c~ so that (St,.j/c~nl/2[N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES~ = a < 0 , ES~ < oo, and there is a slowly var...

We establish the variational principle of Kolmogorov-PetrovskyPiskunov (KPP) front speeds in a one dimensional random drift which is a mean zero stationary ergodic process with mixing property and local Lipschitz continuity. To prove the variational principle, we use the path integral representation of solutions, hitting time and large deviation estimates of the associated stochastic flows. The...

Let fSn; nX0g be a random walk starting from 0 and drifting to 1, and let tðxÞ be the first time when the random walk crosses a given level xX0. Some asymptotics for the tail probability of the overshoot StðxÞ x, associated with the event ðtðxÞo1Þ, are derived for the cases of heavy-tailed and light-tailed increments. In particular, the formulae obtained fulfill certain uniform requirements. r ...

Brownian motion with drift is constructed on the real line as the almost sure limit of a sequence of random walks. Central to the construction is an embedded varying environment branching process, which encodes the sample path behaviour of the limiting diiusion. We show how a single small time bound on the normed limit of the branching process leads to diierent small and large time bounds on th...

We noted in Chapter 2 that when operating as the sole evolutionary force, random genetic drift leads inevitably to the loss of alleles within populations as well as to the fixation of alternative alleles in different populations. These conclusions extend logically to quantitative characters. Following a reduction in population size, for example, we expect the genetic variance within populations...

We study the homogenization limit of solutions to the G-equation with random drift. This Hamilton-Jacobi equation is a model for flame propagation in a turbulent fluid in the regime of thin flames. For a fluid velocity field that is statistically stationary and ergodic, we prove sufficient conditions for homogenization to hold with probability one. These conditions are expressed in terms of tra...

The equilibrium state of a diffusion model for random genetic drift in a cline is analyzed numerically. The monoecious organism occupies an unbounded linear habitat with constant, uniform population density. Migration is homogeneous, symmetric and independent of genotype. A single diallelic locus with a step environment is investigated in the absence of dominance and mutation. The flattening of...

We derive transport equations for the propagation of water wave action in the presence of subsurface random flows. Using the Wigner distribution W(x, k, t) to represent the envelope of the wave amplitude at position x, time t contained in high frequency waves with wave vector k/ε (where ε is a small parameter compared to a characteristic distance of propagation), we describe surface wave transp...

To understand the relative importance of natural selection and random genetic drift in finite but growing populations, the asymptotic behavior of a class of generalized Polya urns is studied using the method of ordinary differential equation (ODE). Of particular interest is the replicator process: two balls (individuals) are chosen from an urn (the population) at random with replacement and bal...

In this paper, the performance of a Ring Laser Gyro based inertial navigation is investigated. Dynamic and stochastic modeling are applied to gyro simulation and performance evaluation. In the dynamic model, some parameters such as scale factor and environmental sensitivity have been determined, whereas in the stochastic model, the other parameters such as random drift and measurement noise...