论文标题
langevin的约束langevin近似是自催化反应的togashi-kaneko模型
Constrained Langevin approximation for the Togashi-Kaneko model of autocatalytic reactions
论文作者
论文摘要
Togashi和Kaneko在2001年推出的Togashi Kaneko模型（TK模型）是一个简单的随机反应网络，它显示了离散性诱导的元稳定模式之间的过渡。在这里，我们研究了该模型的限制性langevin近似（CLA）。 CLA，由Anderson等人获得。在2019年，是一个倾斜的扩散过程，在正骨上，因此尊重化学浓度永远不会为阴性的约束。我们表明，CLA是一个砍伐的过程，是正式的harris复发，并以指数级的快速收敛到唯一的固定分布。我们还表征了固定分布，并表明它具有有限的时刻。此外，我们在各个维度上都模拟了TK模型及其CLA。例如，我们描述了TK模型如何在维度6中的元稳定模式之间切换。我们的仿真表明，在经典缩放下，CLA在固定分布和模式之间的过渡时间方面是与TK模型的良好近似值。
The Togashi Kaneko model (TK model), introduced by Togashi and Kaneko in 2001, is a simple stochastic reaction network that displays discreteness-induced transitions between meta-stable patterns. Here we study a constrained Langevin approximation (CLA) of this model. The CLA, obtained by Anderson et al. in 2019, is an obliquely reflected diffusion process on the positive orthant and hence it respects the constrain that chemical concentrations are never negative. We show that the CLA is a Feller process, is positive Harris recurrent, and converges exponentially fast to the unique stationary distribution. We also characterize the stationary distribution and show that it has finite moments. In addition, we simulate both the TK model and its CLA in various dimensions. For example, we describe how the TK model switches between meta-stable patterns in dimension 6. Our simulations suggest that, under the classical scaling, the CLA is a good approximation to the TK model in terms of both the stationary distribution and the transition times between patterns.
