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 wavefront x = gammat, the lower limit of 1/t integral (t)(0) u(s, x) ds as t --> infinity is positive, and ahead of the wavefront, the limit is zero. In some cases, behind the wavefront, the limit of 1/t integral (t)(0) u(s, x) ds as t --> infinity exists and is positive almost surely. We also prove that behind the wavefront, for almost every omega, the solution of some special stochastic KPP equations converges to a stationary trajectory of the corresponding stochastic differential equation. In front of the wavefront, 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 times. We show that away from the wavefront, for almost all large t the solution is flat in the x-direction for almost all sample paths.