The stochastic elementary formula method and approximate travelling waves for semi-linear reaction diffusion equations

In this thesis we consider approximate travelling wave solutions for stochastic and generalised KPP equations and systems by using the stochastic elementary formula method of Elworthy and Truman. We begin with the semi-classical analysis for generalised KPP equations. With a so-called "late caustic" assumption we prove that the global wave front is given by the Hamilton Jacobi function. We prove a Huygens principle on complete Riemannian manifolds without cut locus, with some bounds on their volume elements, in particular Cartan-Hadamard manifolds. Based on the semiclassical analysis we then consider the propagation of approximate travelling waves for stochastic generalised KPP equations. Three regimes of perturbation are considered: weak, mild, and strong. We show that weak perturbations have little effect on the wave like solutions of the unperturbed equations while strong perturbations essentially destroy the wave and force the solutions to decay rapidly. In the more difficult mild case we show the existence of a 'wave front', in front of which the solution is close to zero (of order exp(-c1μ-2) as μ~0 for c1 random) and behind which it has at least order exp(-c2μ-1) for some random c2 depending on the increment of the noise. For an alternative stochastic equation we classify the effect of the noise by the Lyapunov exponent of a corresponding stochastic ODE. Finally we study the asymptotic behaviour of reaction-diffusion systems with a small parameter by using the n-dimensional Feynman-Kac formula and Freidlin's large deviation theory. We obtain the travelling wave with nonlinear ergodic interactions and a special case with nonlinear reducible interactions

Random periodic processes, periodic measures and ergodicity

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincaré sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including 0 on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including 0 on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The “equivalence” of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained