Two Properties of Stochastic KPP Equations: Ergodicity and Pathwise Property

In this paper we study the random approximate travelling wave solutions of the stochastic KPP equations. Two new properties of the stochastic KPP equations are obtained. We prove the ergodicity that for almost all sample paths, behind the wave front x = γt, the lower limit of 1t ∫ t 0 u(s, x)ds as t → ∞ is positive, and ahead of the wave front, the limit is zero. In some cases, behind the wave front, the limit of 1t ∫ t 0 u(s, x)ds as t → ∞ exists and is positive almost surely. We also prove that behind the wave front, for almost each ω, the solution of some special stochastic KPP equations converges to a stationary trajectory of the corresponding stochastic differential equation. In front of wave front, the solution converges to 0 which is another stationary trajectory of the corresponding SDE. We also study the space derivative of the solution for large time. We show that away from the wave front, for almost all large t the solution is flat in the x-direction for almost all sample paths.