We consider the long-time behaviour of a branching random walk in random environment on the lattice $mathbb{Z}^d$. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of spatially dependent killing/branching rates. The main objects of our interest are the annealed moments $langle m_n^p angle $, i.e., the $p$-th moments over the medium of the $n$-th moment over the migration and killing/branching, of the local and global population sizes. For $n=1$, this is well-understood, as $m_1$ is closely connected with the parabolic Anderson model. For some special distributions, this was extended to $ngeq2$, but only as to the first term of the asymptotics, using (a recursive version of) a Feynman-Kac formula for $m_n$.In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that $langle m_n^p angle$ and $langle m_1^{np} angle$ are asymptotically equal, up to an error $e^{o(t)}$. The cornerstone of our method is a direct Feynman-Kac type formula for $m_n$, which we establish using known spine techniques.
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机译：我们考虑在随机环境中在$ mathbb {Z} ^ d $上分支随机游动的长期行为。颗粒的迁移根据连续时间中的简单随机游动而进行，而介质以空间依赖性杀灭/分支速率的随机潜力给出。我们感兴趣的主要对象是退火时刻$ langle m_n ^ p rangle $，即，在迁移和杀死/分支的$ n $时刻中间的$ p $ th时刻。当地和全球人口规模。对于$ n = 1 $，这是很好理解的，因为$ m_1 $与抛物线安德森模型紧密相关。对于某些特殊分布，它被扩展为$ n geq2 $，但仅针对渐近的第一项，使用Feynman-Kac公式（递归形式）为$ m_n $。在这项工作中，我们还得出了渐近的第二项，用于更大范围的分布。特别地，我们表明$ langle m_n ^ p rangle $和$ langle m_1 ^ {np} rangle $渐近相等，直到误差$ e ^ {o（t）} $。我们方法的基石是$ m_n $的直接Feynman-Kac类型公式，我们使用已知的脊柱技术建立了该公式。
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