We study the asymptotic speed of a random front for solutions \(u_t(x)\) to stochastic reaction–diffusion equations of the form

arising in population genetics. Here, f is a continuous function with \(f(0)=f(1)=0\), and such that \(|f(u)|\le K|u(1-u)|^\gamma \) with \(\gamma \ge 1/2\), and \({\dot{W}}(t,x)\) is a space-time Gaussian white noise. We assume that the initial condition \(u_0(x)\) satisfies \(0\le u_0(x)\le 1\) for all \(x\in {\mathbb {R}}\), \(u_0(x)=1\) for \(x<L_0\) and \( u_0(x)=0\) for \(x>R_0\). We show that when \(\sigma >0\), for each \(t>0\) there exist \(R(u_t)<+\infty \) and \(L(u_t)<-\infty \) such that \(u_t(x)=0\) for \(x>R(u_t)\) and \(u_t(x)=1\) for \(x<L(u_t)\) even if f is not Lipschitz. We also show that for all \(\sigma >0\) there exists a finite deterministic speed \(V(\sigma )\in {\mathbb {R}}\) so that \(R(u_t)/t\rightarrow V(\sigma )\) as \(t\rightarrow +\infty \), almost surely. This is in dramatic contrast with the deterministic case \(\sigma =0\) for nonlinearities of the type \(f(u)=u^m(1-u)\) with \(0<m<1\) when solutions converge to 1 uniformly on \({\mathbb {R}}\) as \(t\rightarrow +\infty \). Finally, we prove that when \(\gamma >1/2\) there exists \(c_f\in {\mathbb {R}}\), so that \(\sigma ^2V(\sigma )\rightarrow c_f\) as \(\sigma \rightarrow +\infty \) and give a characterization of \(c_f\). The last result complements a lower bound obtained by Conlon and Doering (J Stat Phys 120(3–4):421–477, 2005) for the special case of \(f(u)=u(1-u)\) where a duality argument is available.

我们研究形式为随机反应-扩散方程的解\（u_t（x）\）的随机前沿的渐近速度

在人口遗传学中产生。在这里，f是具有\（f（0）= f（1）= 0 \）的连续函数，并且使得 \（| f（u）| \ le K | u（1-u）| ^ \ gamma \ ），带有 \（\ gamma \ ge 1/2 \）和\（{\ dot {W}}（t，x）\）是时空高斯白噪声。我们假设初始条件\ {u_0（x）\）满足所有\ {x \ in {\ mathbb {R}} \）中的\（0 \ le u_0（x）\ le 1 \），\（u_0（ X）= 1 \）为 \（X <L_0 \）和\（U_0（X）= 0 \）为 \（X> R_0 \） 。我们证明当\（\ sigma> 0 \）时，对于每个\（t> 0 \）都存在 \（R（u_t）<+ \ infty \）和 \（L（u_t）< - \ infty \） ，使得\（u_t（X）= 0 \）为\（X> R（u_t）\）和\ （f <L（u_t）\）的（u_t（x）= 1 \）， 即使f不是Lipschitz。我们还表明，对于所有\（\西格玛> 0 \）存在有限的确定性速度 \（V（\西格马）\在{\ mathbb {R}} \） ，使得 \（R（u_t）/ T \ RIGHTARROW V（\ sigma）\）为\（t \ rightarrow + \ infty \），几乎可以肯定。这与类型为\（f（u）= u ^ m（1-u）\）且类型为\（0 <m <1 \）的非线性的确定性情况\（\ sigma = 0 \）形成了鲜明的对比。当解收敛到1上均匀地\（{\ mathbb {R}} \）作为\（T \ RIGHTARROW + \ infty \） 。最后，我们证明当\（\ gamma> 1/2 \）时存在\（c_f \在{\ mathbb {R}} \）中，因此 \（\ sigma ^ 2V（\ sigma）\ rightarrow c_f \）为 \（\ sigma \ rightarrow + \ infty \）并给出\（c_f \）的特征。最后的结果的补充的下限由Conlon的和多林（j统计物理学120（3-4）：421-477，2005）获得的特殊情况\（F（U）= U（1-U）\） ，其中对偶参数可用。