Several researchers, including M. Gell-Mann, argue that the notion of Kolmogorov complexity, developed in the algorithmic information theory, is useful in physics (i.e., in the description of the physical world). Their arguments are rather convincing, but there seems to be a gap between traditional physical equations and Kolmogorov complexity: namely, it is not clear how the standard equations ...

This work, Part II, together with its companion-Part I develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear differential equations depending on the current as well past states. Because of complexity problems, it is natural to divide our contributions into two parts answer long-standing question in biology and ecology. What minimal conditions long-term per...

The reliability of dynamic systems modeled by white noise excited, stochastic, ordinary di erential equations can be computed from deterministic backward Kolmogorov equations. This paper discusses and compares some numerical solution methods for boundary value problems involving backward Kolmogorov equations. The numerical examples concern single degree of freedom oscillators subjected to white...

This paper finds fundamental solutions to the backward Kolmogorov equations, usually interpretable as transition density functions for variables x that follow certain stochastic processes of the form dx = A(x, t)dt + cxγdX and dx = A(x, t)dt +α1 +α2x+α3xdX . This is achieved by first reducing the relevant PDEs that the density functions satisfy to their canonical form. These stochastic processe...

We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on Rd which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as t → +∞. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients.

In this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic diierential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an \ordinary" sense over an arbitrarily prescribed time duration, via a direct \Four Step Scheme". Using this scheme, we further p...