Two properties of stochastic KPP equations: Ergodicity and pathwise property (Q2722674)

The authors consider the stochastic KPP equation NEWLINE\[NEWLINEdu(t,x)= (\textstyle{{1\over 2}} D\Delta u(t, x)+ u(t,x) c(u(t, x))) dt+ k(t) u(t,x) dW_t,\qquad u_{t= t_0}= 1_{(-\infty,0]},\tag{\(*\)}NEWLINE\]NEWLINE where \(W_t\) is the Brownian motion on the probability space \((\Omega,{\mathcal F},P)\). They assume that \(k^2_m=: \lim_{t\to\infty} {1\over t} \int^t_0 k^2(s) ds\in [0,\infty)\) and some verifiable conditions on \(c(\cdot)\), \(a\), \(b\) (see p. 647). Theorem 3.1 states that NEWLINE\[NEWLINE\limsup_{t\to\infty} {1\over t} \int^t_0 \sup_{x\in\mathbb{R}} u(s,x) ds\leq {1\over b} \Biggl(c(0)-{1\over 2} k^2_m\Biggr)\quad\text{a.e.}.NEWLINE\]NEWLINE The difficult Theorem 3.4 states that given \(h> 0\), \(\varepsilon> 0\) there exist \(t_0> 0\), \(\delta> 0\) such that NEWLINE\[NEWLINE\begin{multlined} P\Biggl\{{1\over t} \int^t_0 \inf_{x< s(\gamma b/a- h)} u(s,x) dx> {1\over a} \Biggl(c(0)- {1\over 2} k^2_m\Biggr)- \varepsilon\;(\forall t\geq T)\Biggr\}\\ > 1-\exp\{- \delta T\}\qquad (\forall T\geq t_0),\end{multlined}NEWLINE\]NEWLINE where \(\gamma=: \sqrt{D(2c(0)- k^2_m)}\). The authors also prove that behind the wavefront, for a.e. \(\omega\), the solution of \((*)\) with \(c(u)= \text{const}\cdot(1-u)\) converges to a stationary trajectory of the corresponding stochastic equation. They prove also that away from the wavefront, for a.a. large \(t\), the solution is flat in the \(x\)-direction for a.a. sample paths.