The existence phase transition for two Poisson random fractal models

The fractal cylinder process: existence and connectivity phase transition

We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension $d\geq 2,$ and a connectivity phase transition whenever $d\geq 4.$ We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point. A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this r…

Euclidean Embedding of the Poisson Weighted Infinite Tree and Application to Mobility Models

Continuous time branching models are used to create random fractals in a Euclidean space, whose Hausdorff dimension is controlled by an input parameter. Finite realizations are applied in modelling the set of sites visited in models of human and animal mobility.

布朗运动和布朗循环汤中几种随机分形的不存在性

Non-existence of several random fractals in Brownian motion and Brownian loop soup

本文通过统一方法证明了三种随机分形的不存在性，包括平面布朗运动的先驱三重点、平面和三维布朗运动的先驱双断点，以及临界强度下平面布朗循环汤簇边界上的双点，解决了相关开放问题。

在嵌套分形上带有泊松随机环境的从属布朗运动的IDS

IDS for subordinate Brownian motions in Poisson random environment on nested fractals

本文研究了在嵌套分形上带有泊松随机环境的从属布朗运动的IDS奇异性，通过将随机势能转化为合金型势能，拓展了伯恩斯坦函数的应用范围。

Selfsimilar random fractal measure using contraction method in probabilistic metric spaces

We use contraction method in probabilistic metric spaces to prove existence and uniqueness of selfsimilar random fractal measures.

Existence and space-time regularity for stochastic heat equations on p.c.f. fractals

We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued "random-field" solutions to these SPDEs exist and are jointly Hölder continuous in space and time. We calculate the respective Hölder exponents, which extend the well-known results on the Hölder exponents of the solution to SHE on the unit interval. This shows that the "curse of dimensionality" of the SHE on $\mathbb{R}^n$ depends not on the geometric dimension of the ambient space but on the anal…

Universal Behavior of Connectivity Properties in Fractal Percolation Models

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter lambda. In this paper we mainly study the transition between a phase where the random fractal sets are total…

Iteration of the lent particle method for existence of smooth densities of Poisson functionals

In previous works we have introduced a new method called the lent particle method which is an efficient tool to establish existence of densities for Poisson functionals. We now go further and iterate this method in order to prove smoothness of densities. More precisely, we construct Sobolev spaces of any order and prove a Malliavin-type criterion of existence of smooth density. We apply this approach to SDE's driven by Poisson random measures and also present some non-trivial examples to which our method applies.

Existence of a martingale solution of the stochastic Hall-MHD equations perturbed by Poisson type random forces on ${\mathbb{R}}^{3}$

Stochastic Hall-magnetohydrodynamics equations on ${\mathbb{R}}^{3}$ with random forces expressed in terms of the time homogeneous Poisson random measures are considered. We prove the existence of a global martingale solution. The construction of a solution is based on the Fourier truncation method, stochastic compactness method and a version of the Skorokhod theorem for non-metric spaces adequate for Poisson type random fields.

Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps

The aim of this paper is to establish the existence and uniqueness of the solution to a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our system is Markovian in the sense that initial and terminal values depend on solutions, and are not just fixed random variables. We establish under some monotonicity conditions, the existence and uniqueness of strong solutions of such equations by using a continuation method.

Fractal properties of Aldous-Kendall random metric

Investigating a model of scale-invariant random spatial network suggested by Aldous, Kendall constructed a random metric $T$ on $\mathbb{R}^d$, for which the distance between points is given by the optimal connection time, when travelling on the road network generated by a Poisson process of lines with a speed limit. In this paper, we look into some fractal properties of that random metric. In particular, although almost surely the metric space $\left(\mathbb{R}^d,T\right)$ is homeomorphic to the usual Euclidean $\mathbb{R}^d$, we prove that its Hausdorff dimension is given by $(γ-1)d/(γ-d)>d$…

Stochastic Lagrangian Particle Approach to Fractal Navier-Stokes Equations

In this article we study the fractal Navier-Stokes equations by using stochastic Lagrangian particle path approach in Constantin and Iyer \cite{Co-Iy}. More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential equations driven by Lévy processes. Basing on this representation, a self-contained proof for the existence of local unique solution for the fractal Navier-Stokes equation with initial data in $\mW^{1,p}$ is provided, and in the case of two dimensions or large viscosity, the existence of global solution is also obtain…

泊松跳驱动的二阶McKean-Vlasov随机发展方程：存在性、唯一性与平均原理

Second-order McKean-Vlasov stochastic evolution equation driven by Poisson jumps: existence, uniqueness and averaging principle

本文研究非Lipschitz条件下泊松跳驱动的二阶McKean-Vlasov随机发展方程，用Carathéodory逼近证明温和解的存在唯一性，并得到其与简化方程解在均方意义下的平均原理。

Transport equations with fractal noise - existence, uniqueness and regularity of the solution

The main result of the present paper is a statement on existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems on bounded smooth domains with Dirichlet boundary conditions by means of semigroup theory and fixed point arguments. Main ingredients are the definition of a product of a function and a (not too irregular) distribution as well as a corresponding norm estimate. As an application, transport stochastic partial differential equations driven by fractional Brownian…

First passage time law for some Lévy processes with compound Poisson: Existence of a conditional density with incomplete observation

We study the default risk in incomplete information. That means, we model the value of a firm by one Lévy process which is the sum of brownian motion with drift and compound Poisson process. This Lévy process can not be observed completely and we let an other process which representes the available information on the firm. We obtain an equation safisfied by the conditional density of the default time given the available information and closed form expression for the density.

Solving Poisson's Equation: Existence, Uniqueness, Martingale Structure, and CLT

The solution of Poisson's equation plays a key role in constructing the martingale through which sums of Markov correlated random variables can be analyzed. In this paper, we study two different representations for the solution in countable state space, one based on regenerative structure and the other based on an infinite sum of expectations. We also consider integrability and related uniqueness issues associated with solutions to Poisson's equation, and provide verifiable Lyapunov conditions to support our theory. Our key results include a central limit theorem and law of the iterated logari…

Term Structure Models Driven by Wiener Process and Poisson Measures: Existence and Positivity

In the spirit of Björk-DiMasi-Kabanov-Runggaldier, we investigate term structure models driven by Wiener process and Poisson measures with forward curve dependent volatilities. This includes a full existence and uniqueness proof for the corresponding Heath--Jarrow--Morton type term structure equation. Furthermore, we characterize positivity preserving models by means of the characteristic coefficients, which was open for jump-diffusions. Additionally we treat existence, uniqueness and positivity of the Brody-Hughston equation of interest rate theory with jumps, an equation which we believe to …

A fractional Poisson equation: existence, regularity and approximations

We consider a stochastic boundary value elliptic problem on a bounded domain $D\subset \mathbb{R}^k$, driven by a fractional Brownian field with Hurst parameter $H=(H_1,...,H_k)\in[{1/2},1[^k$. First we define the stochastic convolution derived from the Green kernel and prove some properties. Using monotonicity methods, we prove existence and uniqueness of solution, along with regularity of the sample paths. Finally, we propose a sequence of lattice approximations and prove its convergence to the solution of the SPDE at a given rate.

关于由Wiener过程和Poisson随机测度驱动的SPDEs存在性结果的一些改进

Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures

本文改进了由Wiener过程和Poisson随机测度驱动的SPDEs存在性结果，通过移动框架方法将SPDE问题转化为SDE问题。

Reflected Brownian motion on simple nested fractals

We prove the existence of the reflected diffusion on a complex of an arbitrary size for a large class of planar simple nested fractals. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions on the fractal under which this projection can be well defined. They are illustrated by various specific examples. We first construct a proper version of the transition probability densities for reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman-Ko…

平面Poisson道路随机度量中的测地线

Geodesics in planar Poisson roads random metric

该研究针对平面自相似Poisson道路随机度量，证明了测地线途中不使用任意低速道路、驶离道路非测地等结论，完整刻画了道路点附近的测地线局部结构，验证了Kendall的相关猜想。

Fractal Gaussian Networks: A sparse random graph model based on Gaussian Multiplicative Chaos

We propose a novel stochastic network model, called Fractal Gaussian Network (FGN), that embodies well-defined and analytically tractable fractal structures. Such fractal structures have been empirically observed in diverse applications. FGNs interpolate continuously between the popular purely random geometric graphs (a.k.a. the Poisson Boolean network), and random graphs with increasingly fractal behavior. In fact, they form a parametric family of sparse random geometric graphs that are parametrized by a fractality parameter which governs the strength of the fractal structure. FGNs are driven…

Density of states for the Anderson model on nested fractals

We prove the existence and establish the Lifschitz singularity of the integrated density of states for certain random Hamiltonians $H^ω=H_0+V^ω$ on fractal spaces of infinite diameter. The kinetic term $H_0$ is given by $ϕ(-\mathcal L),$ where $\mathcal L$ is the Laplacian on the fractal and $ϕ$ is a completely monotone function satisfying some mild regularity conditions. The random potential $V^ω$ is of alloy-type.

Isolation and connectivity in random geometric graphs with self-similar intensity measures

Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many nodes and shrinking linking range, the number of isolated nodes is Poisson distributed, and the probability of no isolated nodes is equal to the probability the whole graph is connected. Here we examine these properties for several self-similar node distributions, including smooth and fractal, uniform and nonuniform, and finitely ramified or otherwise. We show t…

Existence, uniqueness and approximation for stochastic Schrodinger equation: the Poisson case

In quantum physics, recent investigations deal with the so-called "quantum trajectory" theory. Heuristic rules are usually used to give rise to "stochastic Schrodinger equations" which are stochastic differential equations of non-usual type describing the physical models. These equations pose tedious problems in terms of mathematical justification: notion of solution, existence, uniqueness, justification... In this article, we concentrate on a particular case: the Poisson case. Random measure theory is used in order to give rigorous sense to such equations. We prove existence and uniqueness of…

分形渗流的几何函数

Geometric functionals of fractal percolation

本文研究了分形渗流的几何函数，通过引入极限几何函数来确定渗流阈值。