Effect of noise on front propagation in reaction-diffusion equations of KPP type
The authors study randomly perturbed Kolmogorov-Petrovskij-Piscounov (KPP) equations, \[ \partial_t u = \partial_x^2 u + f(u) + \varepsilon\sigma(u) \dot W(t,x), \quad t\geq 0,\; x\in\mathbb{R},\; u\geq 0, \] where \(\dot W\) is space-time white noise, \(f\) is Lipschitz satisfying the usual KPP conditions as well as the growth condition \(f(u)\leq 2-u\) for all \(u\geq 1\); \(\sigma^2\) is also Lipschitz, grows at most linearly and \(\sigma^2(u)-\sigma^2(v)\geq a^*(u-v)\) for some \(a^*\) and all \(0\leq u\leq v\leq u^*\) with \(u^*\in (0,1)\) suitable. The initial condition \(u(0,x)=u_0(x)\) is assumed to satisfy for some \(x_0\) the conditions \(u_0|[0,x_0) \geq \theta > 0\) and \(\int_{x_0}^\infty u_0(x)\,dx<\infty\), and that \(u_0\) is contained in some subset \(\hat C\) of the set of non-negative continuous exponentially bounded functions. In particular it is assumed that if \(u_0\in\hat C\) then \(u(t,\cdot)\in\hat C\) and that one has weak uniqueness on \(\hat C\). Typical examples are \(f(u)=u(1-u)\) and \(\sigma^2(u)=u\), which leads to the Fisher-KPP equations \[ \partial_tu = \partial_x^2 u + u(1-u)+\varepsilon\sqrt{u(1-u)}\,\dot W \] and \[ \partial_tu = \partial_x^2 u + u(1-u)+\varepsilon\sqrt{u}\,\dot W. \] For such equations the Brunet-Derrida conjecture is proved, i.e., it is shown that the speed of travelling fronts for small \(\varepsilon\) is \[ 2-\pi^2 |\log\varepsilon^2|^{-1} + \mathsf{O}\big((\log|\log\varepsilon|)|\log\varepsilon|^{-3}\big). \]
stochastic partial differential equation
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