带有边界条件的扩散限制凝聚模型(DLA) 研究

在本项目中，我们将研究限制在具有不同边界条件的二维无限区域内DLA模型的数学性质，并将以此为工具，推进对经典DLA模型相关理论性质的攻关。具体而言，我们将研究上半平面及二维楔形区域上带有Dirichlet以及Neumann边界条件的DLA模型，证明其增长速率、凝聚形态与局部极限稳定性。在此基础上，我们还将研究经典DLA模型在适当初值及时空尺度下的scaling limit，试图证明经典模型同样具有局部极限稳定性，并估计其无穷连通分支的数目及分布。

In this proposed research project, we are to study the Diffusion Limited Aggregation (DLA) model in non-transitive regions with different boundary conditions. And we are to utilize the results obtained to study the theoretical properties of the classic DLA model. To be specific, we are to discuss the Stationary DLA model (SDLA) in the upper half plane with Dirichlet boundary conditions, and the DLA model in the two dimensional wedges with Neumann boundary conditions, and study the growth rate, local stability, as well as the shape of the aggregation. Based on the results above, we are to prove the classic DLA stabilize, estimate the numbers of its infinite arms. We are also to prove that the classic DLA from a long line segment on the x-axis converges to the SDLA under appropriate time rescaling.