Special points of the Brownian net

Exact Distribution of Verhulst process

We investigate a Verhulst process, which is the special functional of geometric Brownian motion and has many applications, among others in biology and in stochastic volatility models. We present an exact form of density of a one dimensional distribution of Verhulst process. Simple formula for the density of Verhulst process is obtained in the special case, when the drift of geometric Brownian motion is equal to -1/2. Some special properties of this process are discussed, e.g. it turns out that under Girsanov's change of measure a Verhulst process still remains a Verhulst process but with diffe…

The Brownian Net with Killing

Motivated by its relevance for the study of perturbations of one-dimensional voter models, including stochastic Potts models at low temperature, we consider diffusively rescaled coalescing random walks with branching and killing. Our main result is convergence to a new continuum process, in which the random space-time paths of the Sun-Swart Brownian net are terminated at a Poisson cloud of killing points. We also prove existence of a percolation transition as the killing rate varies. Key issues for convergence are the relations of the discrete model killing points and their Poisson intensity m…

The Brownian Net and Selection in the Spatial Lambda-Fleming-Viot Process

We obtain the Brownian net of Sun and Swart (2008) as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous …

Branching Brownian Motion, mean curvature flow and the motion of hybrid zones

We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $Λ$-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.

Stochastic flows in the Brownian web and net

Certain one-dimensional nearest-neighbor random walks in i.i.d. random space-time environments are known to have diffusive scaling limits. In the continuum limit, the random environment is represented by a `stochastic flow of kernels', which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment. The theory of stochastic flows of kernels was introduced by Le Jan and Raimond, who showed that each such flow is characterized by its n-point motions. We focus on a class of stochastic flows of kernels with Brownia…

Brownian disks and the Brownian snake

We provide a new construction of the Brownian disks, which have been defined by Bettinelli and Miermont as scaling limits of quadrangulations with a boundary when the boundary size tends to infinity. Our method is very similar to the construction of the Brownian map, but it makes use of the positive excursion measure of the Brownian snake which has been introduced recently. This excursion measure involves a random continuous tree whose vertices are assigned nonnegative labels, which correspond to distances from the boundary in our approach to the Brownian disk. We provide several applications …

一维自排斥布朗聚合物的构造

Construction of the 1d Self-repelling Brownian Polymer

该研究针对一维自排斥布朗聚合物对应的奇异SDE，通过分析相关奇异SPDE的能量解适定性，构造出方程解，证明其超扩散性及为逼近族极限分布，结果可推广至更广泛漂移项。

布朗映射中的测地线稳定性

Stability of geodesics in the Brownian map

本文研究布朗映射的测地线结构，强化测地线汇流现象，证明典型点处割迹的连续性与一致稳定性，推导测地线内部点集相关性质，分类稠密测地线网络并计算对应点对集的豪斯多夫维数。

The Brownian Castle

We introduce a $1+1$-dimensional temperature-dependent model such that the classical ballistic deposition model is recovered as its zero-temperature limit. Its $\infty$-temperature version, which we refer to as the $0$-Ballistic Deposition ($0$-BD) model, is a randomly evolving interface which, surprisingly enough, does {\it not} belong to either the Edwards--Wilkinson (EW) or the Kardar--Parisi--Zhang (KPZ) universality class. We show that $0$-BD has a scaling limit, a new stochastic process that we call {\it Brownian Castle} (BC) which, although it is "free", is distinct from EW and, like an…

Transmission and navigation on disordered lattice networks, directed spanning forests and Brownian web

Stochastic networks based on random point sets as nodes have attracted considerable interest in many applications, particularly in communication networks, including wireless sensor networks, peer-to-peer networks and so on. The study of such networks generally requires the nodes to be independently and uniformly distributed as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate the stochastic geometry of networks obtained from \textit{directed spanning forests} (DSF) based on randomly perturbed lattices, which have desirable statistical properties as…

一般线性群上乘法布朗运动的强收敛性

Strong Convergence of Multiplicative Brownian Motions on the General Linear Group

本文证明了一般线性群上由Driver-Hall-Kemp引入的乘法布朗运动族在维数趋于无穷时，其有限维边缘分布几乎必然强收敛到Hall-Ho引入的自由乘法布朗运动，推广了酉群情形的已有结果。

Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula

We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes, and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth order PDEs in which the initial function…

Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process

Using the determinantal formula of Biane, Bougerol, and O'Connell, we give multitime joint probability densities to the noncolliding Brownian motion with drift, where the number of particles is finite. We study a special case such that the initial positions of particles are equidistant with a period $a$ and the values of drift coefficients are well-ordered with a scale $σ$. We show that, at each time $t >0$, the single-time probability density of particle system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model in the Chern-Simons theory introduced by Dolivet and Tierz. …