Research for Stochastic Processes

This paper considers the effects of small random perturbations on deterministic systems of differential equations. The deterministic systems of interest have oscillatory dynamics and may undergo a bifurcation (the Hopf bifurcation). A first exit problem for experiments beginning near stable and unstable limit cycles is formulated. The unstable limit cycle is surrounded by an annulus. Of interest is the probability of first exit from the annulus through a specified boundary, conditioned on initial position. The diffusion approximation is used, so that the conditional probability satisfies a backward diffusion equation. Appropriate solutions on the backward equation are constructed by an asymptotic method. The behavior of the stochastic system in the vicinity of stable and unstable limit cycles is compared. When the deterministic system exhibits the Hopf bifurcation, the above analysis must be modified. Uniform solutions of the backward equation are constructed. The solutions are analogous to Hadamard's solution of the point source problem for the wave equation. Numerical examples are used to compare the theory with Monte Carlo experiments.

This paper considers the effects of small random perturbations in deterministic systems of differential equations. A previous version of this paper appeared as CNA Professional Paper 225.

A generalized critical point is characterized by the vanishing of certain linear relationships. In particular, the dynamics near such a point are completely non-linear. This paper analyzes fluctuations at such points of spatially homogeneous systems. Thermodynamic critical points as a special case are discussed, but the main emphasis is on stochastic kinetic equations. It is shown that fluctuations at a critical point cannot be characterized by a Gaussian density, but more sophisticated densities yield reasonable results. The theory is applied to the critical harmonic oscillator.

This research contribution consists of a series of eight memoranda originally published by the Statistical Research Group at Columbia University for the National Defense Research Committee in 1943 on methods of estimating the vulnerability of various parts of an aircraft based on damage to surviving planes. The methodology presented continues to be valuable in defense analysis and, therefore, has been reprinted by the Center for Naval Analyses in order to achieve wider dissemination.