A stochastic distance measure is defined for a general diffusion process on a parameter space X. This distance is defined by where (gij) is the inverse of the covariance matrix of the diffusion equation. This permits the study of the geometry associated with a diffusion equation, since the matrix (gij) is the fundamental tensor of the Riemannian space (X, gij), and of a diffusion process in terms of Brownian motion. For the diffusion equation approximation to random drift with n alleles the covariance matrix is that of a multinomial distribution. The resulting stochastic distance is equal to twice the genetic distance as defined by Cavalli-Sforza and Edwards and is a generalization of the angular transformation of Fisher to n alleles. The geometry associated with the diffusion equation for random drift with n alleles is that of a part of an (n − 1)-sphere of radius two. We also show that the diffusion equation for random drift is not spherical Brownian motion, although it is approximated by it near the centroid of frequency space.

It has been conjectured that a certain transformation of gene frequency space due to Fisher and Bhattacharyya will map the random genetic drift process, or its diffusion approximation, into one with radial symmetry. This paper proves rigorously that the Fisher–Bhattacharyya mapping does not do this. This implies that the initial state of an evolving ensemble can only be unbiasedly estimated from the means of a sample if we weight by the proper divergence times. If the ensemble is known not to have begun at the centroid of frequency space, the estimate of the initial state vector is not simply the arithmetic average, as symmetry analysis of the Christoffel velocity field shows.

A new diffusion model for random genetic drift of a two-locus di-allelic system is proposed. The Christoffel velocity field and the intrinsic geometry of the diffusion is computed for the equilibrium surface. It is seen to be radically non-spherical and to depend explicitly on the recombination fraction. The model has not been shown to be a limit of discrete Markov chains. For large values of the recombination, the present model is radically different from that of Ohta and Kimura, which is an approximation to the discrete process of random mating in the limit as the value of the recombination fraction goes to zero.

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C0 is a constant depending on the migration law, K0(y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log mi in two. For is known in one dimension and C0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

Several problems are considered in the theory of first-passage percolation on the two-dimensional integer lattice. The results include: (i) necessary and sufficient conditions for the existence of moments of first-passage times; (ii) determination of an upper bound for the time constant; (iii) an initial result concerning the maximum height of routes for first-passage times; (iv) ergodic theorems for a class of reach processes.

This paper provides a new approach for solving a wide class of Markov decision problems including problems in which the space is general and the system can be continuously controlled. The optimality criterion is the long-run average cost per unit time. We decompose the decision processes into a common underlying stochastic process and a sequence of interventions so that the decision processes can be embedded upon a reduced set of states. Consequently, in the policy-iteration algorithm resulting from this approach the number of equations to be solved in any iteration step can be substantially reduced. Further, by its flexibility, this algorithm allows us to exploit any structure of the particular problem to be solved.

In a preceding paper [2] we have introduced a new approach for solving a wide class of Markov decision problems in which the state-space may be general and the system may be continuously controlled. The criterion is the average cost. This paper discusses two applications of this approach. The first application concerns a house-selling problem in which a constructor builds houses which may be sold at any stage of the construction and potential customers make offers depending on the stage of the construction. The second application considers an M/M/c queueing problem in which the number of operating servers can be controlled by turning servers on or off.

A stability theorem determines the extent to which the conclusions of a given theorem are affected if the assumptions of the theorem are not exactly but only approximately satisfied. The meaning of the word ‘approximately’ has to be defined exactly. The stability of decomposition theorems, of characterizations by independence and by regression properties are the primary object of the paper.

This paper contains a review of the results of the martingale approach to point processes and its applications to the theory of dynamical systems (in the engineering sense), mainly estimation problems.

Some of the topics reviewed are: stochastic intensity, absolutely continuous changes of measure, martingale representations, changes of times, filtering, queues, etc.

Let be a process with stationary, independent increments and no negative jumps. Let be this same process modified by a reflecting barrier at zero (a storage process). Assuming that – and denote by ψ(s) the exponent function of X. A simple formula is derived for the Laplace transform of as a function of W(0). Using the fact that the distribution of M is the unique stationary distribution of the Markov process W, this yields an elementary proof that the Laplace transform of M is µs/ψ(s). If it follows that These surprisingly simple formulas were originally obtained by Zolotarev using analytical methods.