The usual way in which mathematicians work with randomness is by a rigorous formulation of the idea of Brownian motion, which is the limit of a random walk as the step length goes to zero. A Brownian path is continuous but nowhere differentiable, and this nonsmoothness is associated with technical complications that can be daunting. However, there is another approach to random processes that is more elementary, involving smooth random functions defined by finite Fourier series with random coefficients or, equivalently, by trigonometric polynomial interpolation through random data values. We show here how smooth random functions can provide a very practical way to explore random effects. For example, one can solve smooth random ordinary differential equations using standard mathematical definitions and numerical algorithms, rather than having to develop new definitions and algorithms of stochastic differential equations. In the limit as the number of Fourier coefficients defining a smooth random function goes to \infty , one obtains the usual stochastic objects in what is known as their Stratonovich interpretation.
Running, walking, flying and swimming are all processes in which animals produce propulsion by executing rhythmic motions of their bodies. Dynamical stability of the locomotion is hardly automatic: millions of older people are injured by falling each year. Stability frequently requires sensory feedback. We investigate how organisms obtain the information they use in maintaining their stability. Assessing stability of a periodic orbit of a dynamical system requires information about the dynamics of the system off the orbit. For locomotion driven by a periodic orbit, perturbations that "kick" the trajectory off the orbit must occur in order to observe convergence rates toward the orbit. We propose that organisms generate excitations in order to set the gains for stabilizing feedback. We hypothesize further that these excitations are stochastic but have heavy-tailed, non-Gaussian probability distributions. Compared to Gaussian distributions, we argue that these are more effective for estimating stability characteristics of the orbit. Finally, we propose experiments to test the efficacy of these ideas.
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