Vitalii Konarovskyi, Victor Marx
We introduce the notion of a conditional distribution to a zero-probability event in a given direction of approximation, and prove that the conditional distribution of a family of independent Brownian particles to the event that their paths coalesce after the meeting coincides with the law of a modified massive Arratia flow, defined in [arXiv:1408.0628].
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We introduce several martingale changes of measure of the law of the exit measure of super Brownian motion. These changes of measure include and generalize one arising by conditioning the exit measures to charge a point on the boun dary of a 2-dimensional domain. In the case we discuss this is a non-degenerate conditioning. We give characterizations of the new processes in terms of "immortal particle" branching processes with immigration of mass, and give application s to the study of solutions to Lu = cu^2 in D. The representations are related to those in an earlier paper, which treated the c…
We generalize the notion of Gaussian bridges by conditioning Gaussian processes given that certain linear functionals of the sample paths vanish. We show the equivalence of the laws of the unconditioned and the conditioned process and by an application of Girsanov's theorem, we show that the conditioned process follows a stochastic differential equation (SDE) whenever the unconditioned process does. In the Markovian case, we are able to determine the coefficients in the SDE of the conditioned process explicitly. Our main example is Brownian motion on $[0,1]$ pinned down in 0 at time 1 and cond…
We consider a continuous population whose dynamics is described by the standard stationary Fleming-Viot process, so that the genealogy of $n$ uniformly sampled individuals is distributed as the Kingman $n$-coalescent. In this note, we study some genealogical properties of this population when the sample is conditioned to fall entirely into a subpopulation with most recent common ancestor (MRCA) shorter than $\varepsilon$. First, using the comb representation of the total genealogy (Lambert & Uribe Bravo 2016), we show that the genealogy of the descendance of the MRCA of the sample on the times…
We condition super-Brownian motion on "boundary statistics" of the exit measure $X_D$ from a bounded domain $D$. These are random variables defined on an auxiliary probability space generated by sampling from the exit measure $X_D$. Two particular examples are: conditioning on a Poisson random measure with intensity $βX_D$ and conditioning on $X_D$ itself. We find the conditional laws as $h$-transforms of the original SBM law using Dynkin's formulation of $X$-harmonic functions. We give explicit expression for the (extended) $X$-harmonic functions considered. We also obtain explicit constructi…
A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the tim…
We introduce a system of one-dimensional coalescing nonsimple random walks with long range jumps allowing crossing paths and exibiting dependence before coalescence. We show that under diffusive scaling this system converges in distribution to the Brownian Web.
We extend earlier results on conditioning of super-Brownian motion to general branching rules. We obtain representations of the conditioned process, both as an $h$-transform, and as an unconditioned superprocess with immigration along a branching tree. Unlike the finite-variance branching setting, these trees are no longer binary, and strictly positive mass can be created at branch points. This construction is singular in the case of stable branching. We analyze this singularity first by approaching the stable branching function via analytic approximations. In this context the singularity of t…
We calculate the regular conditional future law of the fractional Brownian motion with index $H\in(0,1)$ conditioned on its past. We show that the conditional law is continuous with respect to the conditioning path. We investigate the path properties of the conditional process and the asymptotic behavior of the conditional covariance.
Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations.
Conditioning the tanh-drift process on first-passage times: Exact drifts, bridges, and process equivalences
本文研究了具有漂移μ(x)=αtanh(αx+β)的Benes过程，探讨了在无限和有限时间范围内对首次通过时间分布的条件化，揭示了Benes过程与布朗运动在桥梁结构上的等价性，并连接了最近关于漂移布朗运动和Benes过程的条件化结果。
We prove tightness of a family of path measures $ν_{\varepsilon}$ on tubes $L(\varepsilon)$ of small diameters around a closed and connected submanifold $L$ of another Riemannian manifold $M$. Together with a convergence result for Dirichlet semigroups on tubular neighbourhoods, that implies weak convergence of the measures as the tube radius $\varepsilon$ tends to zero to a measure supported by the path space of the submanifold. As a consequence, we obtain weak convergence of the measures obtained by conditioning Brownian motion to stay within the tubes $L(\varepsilon)$ up to a finite time $T…
We consider the tributary structure of Howard's drainage model studied by Gangopadhyay et. al. Conditional on the event that the tributary survives up to time $n$, we show that, as a sequence of random metric spaces, scaled tributary converges in distribution to a continuum random tree with respect to Gromov Hausdorff topology. This verifies a prediction made by Aldous for a simpler model (where paths are independent till they coalesce) but for a different conditional set up. The limiting continuum tree is slightly different from what was surmised earlier. Our proof uses the fact that there ex…
We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $l$ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $l$ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $l$ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among o…
In this paper we present a martingale related to the exit measures of super-Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain. The results are similar in flavor to the "immortal particle" picture of conditioned super-Brownian motion …
Consider a locally finite Dawson-Watanabe superprocess $ξ=(ξ_t)$ in $\mathsf{R}^d$ with $d\geq2$. Our main results include some recursive formulas for the moment measures of $ξ$, with connections to the uniform Brownian tree, a Brownian snake representation of Palm measures, continuity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of $ξ_t$ by a stationary cluster $\tildeη$ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditiona…
We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order $(\log N)^3$, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the …
We study the zero temperature limit for interacting Brownian particles in one dimension with a pairwise potential which is of finite range and attains a unique minimum when the distance of two particles becomes a>0. We say a chain is formed when the particles are arranged in an ``almost equal'' distance a. If a chain is formed at time 0, so is for positive time as the temperature of the system decreases to 0 and, under a suitable macroscopic space-time scaling, the center of mass of the chain performs the Brownian motion with the speed inversely proportional to the total mass. If there are two…
Brownian motion with drift conditioned to have restricted $L^2$-norm
该研究推导了带漂移布朗运动的二次加性泛函的小偏差渐近式，证明其在受限L²范数条件下弱收敛到Ornstein-Uhlenbeck过程，且极限动力学与原始漂移无关。
Coalescing random walks via the coalescence determinant
本文通过凝聚行列式研究了随机游走的凝聚问题，推导了幸存者分布和间隙密度的精确公式，并适用于任意最近邻随机游走及其布朗运动极限。
Power-law scaling of the effective population size in a branching particle system for moderate mutation-selection
本文研究了具有正漂移的分支布朗运动在中等突变-选择下的种群规模变化，证明在大N极限下，种群波动遵循多项式时间尺度的Yaglom定律，且基因谱仅包含二元合并。
Spectral analysis and $k$-spine decomposition of inhomogeneous branching Brownian motions. Genealogies in fully pushed fronts
本文研究完全推动型波动前沿中非均匀分支布朗运动的谱系结构，通过建立Yaglom律、脊分解和谱分析方法，证明了粒子谱系收敛于Brownian Coalescent Point Process，并确定了脊的不变测度与混合时间。
Area fluctuations of a Brownian excursion are described by the Airy distribution, which found applications in different areas of physics, mathematics and computer science. Here we generalize this distribution to describe the area fluctuations on a \emph{subinterval} of a Brownian excursion. In the first version of the problem (Model 1) no additional conditions are imposed. In the second version (Model 2) we study the distribution of the area fluctuations on a subinterval given the excursion area on the whole interval. Both versions admit convenient path-integral formulations. In Model 1 we obt…
We consider two independent identical diffusion processes that annihilate upon meeting in order to study their conditioning with respect to their first-encounter properties. For the case of finite horizon $T<+\infty$, the maximum conditioning consists in imposing the probability $P^*(x,y,T ) $ that the two particles are surviving at positions $x$ and $y$ at time $T$, as well as the probability $γ^*(z,t) $ of annihilation at position $z$ at the intermediate times $t \in [0,T]$. The adaptation to various conditioning constraints that are less-detailed than these full distributions is analyzed vi…
Scaling limits of multitype Bienaymé trees
研究临界多类型Bienaymé树在两种条件框架下的标度极限，证明在有限指数矩条件下，极限为布朗连续随机树，并建立高度的强非渐近尾界。
We study asymptotic properties of the system of interacting diffusion particles on the real line which transfer a mass [arXiv:1408.0628]. The system is a natural generalization of the coalescing Brownian motions. The main difference is that diffusion particles coalesce summing their mass and changing their diffusion rate inversely proportional to the mass. First we construct the system in the case where the initial mass distribution has the moment of the order greater then two as an $L_2$-valued martingale with a suitable quadratic variation. Then we find the relationship between the asymptoti…
When the unconditioned process is a diffusion living on the half-line $x \in ]-\infty,a[$ in the presence of an absorbing boundary condition at position $x=a$, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite $T<+\infty$, the conditioning consists in imposing the probability $P^*(y,T ) $ to be surviving at time $T$ and at the position $y \in ]-\infty,a[$, as well as the probability $γ^*(T_a ) $ to have been absorbed at the previous time $T_a \in [0,T]$. When the time horizon is infinite $T=+\infty$, the conditioning consist…
Bañuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) analyse the asymptotic tail distribution of the first time a stable (Lévy) process in dimension $d\geq 2$ exists a cone. We use these results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone). As self-similar Markov processes we examine some of their fundamental properties through the lens of its Lamperti-Kiu decomposition. In particular we are interested to understand the under…
We study the convergence of $N-$particle systems described by SDEs driven by Brownian motion and Poisson random measure, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position and on the empirical measure of the system. Jumps are simultaneous, that is, at each jump time, all particles of the system are affected by this jump and receive a random jump height that is centred and scaled in $N^{-1/2}.$ This particular scaling implies that the limit of the empirical measures of the system is random, describing the conditi…
Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes
该研究分析了扩散过程在两个位置的局部时的联合分布及大时间渐近特性，据此构造了满足双局部时约束的各类条件过程，并推广了单局部时条件化的已有结果，还将其应用于均匀漂移布朗运动场景。
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