One-dimensional long-range Diffusion Limited Aggregation II: the transient case

One-dimensional long-range diffusion-limited aggregation I

We examine diffusion-limited aggregation generated by a random walk on Z with long jumps. We derive upper and lower bounds on the growth rate of the aggregate as a function of the number moments a single step of the walk has. Under various regularity conditions on the tail of the step distribution, we prove that the diameter grows as n^{beta+o(1)}, with an explicitly given beta. The growth rate of the aggregate is shown to have three phase transitions, when the walk steps have finite third moment, finite variance, and, conjecturally, finite half moment.

Internal Diffusion Limited Aggregation on discrete groups having exponential growth

The Internal Diffusion Limited Aggregation has been introduced by Diaconis and Fulton in 1991. It is a growth model defined on an infinite set and associated to a Markov chain on this set. We focus here on sets which are finitely generated groups with exponential growth. We prove a shape theorem for the Internal DLA on such groups associated to symmetric random walks. For that purpose, we introduce a new distance associated to the Green function, which happens to have some interesting properties. In the case of homogeneous trees, we also get the right order for the fluctuations of that model a…

Logarithmic fluctuations for the Internal Diffusion Limited Aggregation

This paper has been withdrawn by the author due to an error in the computation of E(n,x) on page 6 which appears to be essential for the result. The author is currently trying to correct this proof

A slow transient diffusion in a drifted stable potential

We consider a diffusion process $X$ in a random potential $\V$ of the form $\V_x = §_x -δx$ where $δ$ is a positive drift and $§$ is a strictly stable process of index $α\in (1,2)$ with positive jumps. Then the diffusion is transient and $X_t / \log^αt$ converges in law towards an exponential distribution. This behaviour contrasts with the case where $\V$ is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as "slow" as in the recurrent setting.

Containing Internal Diffusion Limited Aggregation

Internal Diffusion Limited Aggregation (IDLA) is a model that describes the growth of a random aggregate of particles from the inside out. Shellef proved that IDLA processes on supercritical percolation clusters of integer-lattices fill Euclidean balls, with high probability. In this article, we complete the picture and prove a limit-shape theorem for IDLA on such percolation clusters, by providing the corresponding upper bound. The technique to prove upper bounds is new and robust: it only requires the existence of a "good" lower bound. Specifically, this way of proving upper bounds on IDLA c…

Diffusion limited aggregation on a tree

We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to alpha^{-n}, where alpha<1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements known results that show the height increases only logarithmically when alpha>=1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation comes from two other ways in which the model may be viewed: as a problem …

源自生物学的一类随机双侧退化聚集-扩散方程的全局弱鞅解

Global weak martingale solutions to a stochastic two-sidedly degenerate aggregation-diffusion equation issued from biology

本文研究生物与公共卫生领域的随机双侧退化聚集-扩散方程，结合辅助非退化系统与Faedo--Galerkin方法证明鞅解的存在唯一性，并通过数值模拟揭示环境噪声对系统行为的影响，为公共卫生相关研究提供参考。

Learning interacting particle systems: diffusion parameter estimation for aggregation equations

In this article, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation and Brownian diffusion. Specifically, we construct an estimator $\widehatν$ with partial observed data to approximate the diffusion parameter $ν$, and the estimation error is achieved. Furthermore, we extend this result to general aggregation equations with a bounded Lipschitz interaction field.

内部扩散限制凝聚与临界分支随机游走

Internal Diffusion Limited Aggregation with Critical Branching Random Walks

本文研究由临界分支随机游走驱动的内部扩散限制凝聚，证明其在维度上存在相变：高维（d≥3）有球形形状定理，低维（d≤2）无，且偏差界为多项式性质。

Optimal Prediction of the Last-Passage Time of a Transient Diffusion

We identify the integrable stopping time $τ_*$ with minimal $L^1$-distance to the last-passage time $γ_z$ to a given level $z>0$, for an arbitrary non-negative time-homogeneous transient diffusion $X$. We demonstrate that $τ_*$ is in fact the first time that $X$ assumes a value outside a half-open interval $[0,r_*)$. The upper boundary $r_*>z$ of this interval is characterised either as the solution for a one-dimensional optimisation problem, or as part of the solution for a free-boundary problem. A number of concrete examples illustrate the result.

A note on fluctuations for internal diffusion limited aggregation

We consider a cluster growth model on Z^d, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. Also, when dimension is 2 or more, and when the cluster has volume $n^d$, it is known that fluctuations of the radius are at most of order $n^{1/3}$. We improve this estimate to $n^{1/(d+1)}$, in dimension 3 or more. In so doing, we introduce a closely related cluster growth model, that we call…

Global solutions of aggregation equations and other flows with random diffusion

Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence vs. finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. arXiv:1806.03734 showing that, with high probability, the inviscid SQG equation with random diffusion has global …

具有有界核的扩散–聚集方程的适定性及其平均场近似

Well-posedness of diffusion-aggregation equations with bounded kernels and their mean-field approximations

本文针对有界相互作用力核，在全空间上建立扩散–聚集方程的适定性与正则性，并给出粒子系统与 McKean–Vlasov SDE 间任意代数速率的定量概率估计，从而证明混沌传播。

Diffusion-limited aggregation on the hyperbolic plane

We consider an analogous version of the diffusion-limited aggregation model defined on the hyperbolic plane. We prove that almost surely the aggregate viewed at time infinity will have a positive density.

The Limiting Shape for Drifted Internal Diffusion Limited Aggregation is a True Heat Ball

We build the iDLA cluster using drifted random walks, and study the limiting shapes they exhibit, with the help of sandpile models. For constant drift, the normalised cluster converges to a canonical shape S, which can be termed a true heat ball, in that it gives rise to a mean value property for caloric functions. The existence and boundedness of such a shape answers a natural yet open question in PDE theory.

Diffusion limited aggregation in the Boolean lattice

In the Diffusion Limited Aggregation (DLA) process on on $\mathbb{Z}^2$, or more generally $\mathbb{Z}^d$, particles aggregate to an initially occupied origin by arrivals on a random walk. The scaling limit of the result, empirically, is a fractal with dimension strictly less than $d$. Very little has been shown rigorously about the process, however. We study an analogous process on the Boolean lattice $\{0,1\}^n$, in which particles take random decreasing walks from $(1,\dots,1)$, and stick at the last vertex before they encounter an occupied site for the first time; the vertex $(0,\dots,0)$ …

无扩散极限下的聚合方程研究

Zero-diffusion Limit for Aggregation Equations over Bounded Domains

本文研究了在有界域上连续和离散聚合模型的无扩散极限，采用耦合方法连接PDE与对应的SDE，并放宽了相互作用和外部势能的正则性假设，提高了收敛速率。

One-dimensional long-range diffusion-limited aggregation III -- The limit aggregate

In this paper we study the structure of the limit aggregate $A_\infty = \bigcup_{n\geq 0} A_n$ of the one-dimensional long range diffusion limited aggregation process defined in [AABK09]. We show (under some regularity conditions) that for walks with finite third moment $A_\infty$ has renewal structure and positive density, while for walks with finite variance the renewal structure no longer exists and $A_\infty$ has 0 density. We define a tree structure on the aggregates and show some results on the degrees and number of ends of these random trees. We introduce a new "harmonic competition" mo…

Upper bounds on the growth rate of Diffusion Limited Aggregation

We revisit Kesten's argument for the upper bound on the growth rate of DLA. We are able to make the argument robust enough so that it applies to many graphs, where only control of the heat kernel is required. We apply this to many examples including transitive graphs of polynomial growth, graphs of exponential growth, non-amenable graphs, super-critical percolation on Z^d, high dimensional pre-Sierpinski carpets. We also observe that a careful analysis shows that Kesten's original bound on Z^3 can be improved from t^{2/3} to (t log t)^{1/2} .

随机聚合扩散方程：基于 Dirichlet 形式的分析

Stochastic Aggregation Diffusion-Equation : Analysis via Dirichlet Forms

本文研究了带奇异漂移的随机聚合扩散方程，利用 Dirichlet 形式理论和 H_2 条件证明了扩散过程弱解的存在性并深入分析了解的唯一性问题，同时给出了广义 Schrödinger 算子的显式表达式。

Transient anomalous sub-diffusion on bounded domains

This paper develops strong solutions and stochastic solutions for the tempered fractional diffusion equation on bounded domains. First the eigenvalue problem for tempered fractional derivatives is solved. Then a separation of variables, and eigenfunction expansions in time and space, are used to write strong solutions. Finally, stochastic solutions are written in terms of an inverse subordinator.