In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain simple Poissonian percolation clusters: Recall that the Brownian loop-soup (introduced in the paper arxiv:math.PR/0304419 with Greg Lawler) with intensity c defines a Poissonian collection of (simple if one focuses only on the outer boundary) loops in a domain. This random family of (possibly intersecting) loops is conformally invariant (and there are almost surely infinitely many small loops in any sample). We show that there exists a critical value a in (0,1] such that if one colors all the interiors of the loops, the obtained clusters are bounded when c<a, whereas when c>a, one single cluster fills the domain. We prove that for small c, the outer boundaries of the clusters are SLE-type curves where $κ\le 4$ and $c$ related by the usual relation $c=(3κ-8)(6-κ)/2κ$ (i.e. c corresponds to the central charge of the model). Conjecturally, the critical value a is equal to one and corresponds to SLE4 loops, so that this should give for any c in (0,1] a construction of a natural countable family of random disjoint SLE$_κ$ loops (i.e. $κ$ should span $(8/3,4]$), that behaves ``nicely'' under perturbation of the domain.
A precise relation between chordal SLE and the loop-soup goes as follows: Consider the sample of a certain restriction measure (i.e. a certain union of Brownian excursions) in a domain, attach to it all the above-described clusters that it intersects. The outer boundary of the obtained set is exactly an SLE$_κ$, if the restriction measure exponent is equal to the highest-weight of the corresponding representation with central charge c.
The present paper is concerned with properties of multiple Schramm--Loewner evolutions (SLEs) labelled by a parameter $κ\in (0,8]$. Specifically, we consider the solution of the multiple Loewner equation driven by a time change of Dyson's Brownian motions in the non-colliding regime. Although it is often considered that several properties of the solution can be studied by means of commutation relations of SLEs and the absolute continuity, this method is available only in the case that the curves generated by commuting SLEs are separated. Beyond this restriction, it is not even obvious that the…
We relate the formulas giving Brownian (and other) intersection exponents to the absolute continuity relations between Bessel process of different dimensions, via the two-parameter family of Schramm-Loewner Evolution processes SLE(kappa,rho) introduced in arXiv:math.PR/0209343. This allows also to compute the value of some new exponents (``hiding exponents'') related to SLEs and planar Brownian motions.
Let $D={\mathbb H}\setminus \bigcup_{j=1}^N C_j$ be a standard slit domain, where ${\mathbb H}$ is the upper half plane and $C_j,1\le j\le N,$ are mutually disjoint horizontal line segments in ${\mathbb H}$. A stochastic Komatu-Loewner evolution denoted by ${\rm SKLE}_{α,b}$ has been introduced in \cite{CF} as a family $\{F_t\}$ of random growing hulls with $F_t\subset D$ driven by a diffusion process $ξ(t)$ on $\partial {\mathbb H}$ that is determined by certain continuous homogeneous functions $α$ and $b$ defined on the space ${\cal S}$ of all labelled standard slit domains. We aim at identi…
In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone, and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane, and disk). We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane…
We investigate the large scale structure of certain sojourn sets of one dimensional Brownian motion within two-sided moving boundaries. The macroscopic Hausdorff dimension, upper mass dimension and logarithmic density of these sets are computed. We also give a uniform macroscopic dimension result for the Brownian level sets.
We consider a Markov-modulated Brownian motion $\{Y(t), ρ(t)\}$ with two boundaries at $0$ and $b > 0$, and allow for the controlling Markov chain $\{ρ(t)\}$ to instantaneously undergo a change of phase upon hitting either of the two boundaries at semi-regenerative epochs defined to be the first time the process reaches a boundary since it last hits the other boundary. We call this process a flexible Markov-modulated Brownian motion. Using the recently-established links between stochastic fluid models and Markov-modulated Brownian motions, we determine important characteristics of first exit t…
Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary $t\mapsto a+bt,\ a\geq 0,\,b\in \R,$ by a reflecting Brownian motion. The main tool hereby is Doob's formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter $δ>…
We study the long-time asymptotics of the probability P_t that the Riemann-Liouville fractional Brownian motion with Hurst index H does not escape from a fixed interval [-L,L] up to time t. We show that for any H \in ]0,1], for both subdiffusion and superdiffusion regimes, this probability obeys \ln(P_t) \sim - t^{2 H}/L^2, i.e. may decay slower than exponential (subdiffusion) or faster than exponential (superdiffusion). This implies that survival probability S_t of particles undergoing fractional Brownian motion in a one-dimensional system with randomly placed traps follows \ln(S_t) \sim - n^…
We consider various problems related to the persistence probability of fractional Brownian motion (FBM), which is the probability that the FBM $X$ stays below a certain level until time $T$. Recently, Oshanin et al. study a physical model where persistence properties of FBM are shown to be related to scaling properties of a quantity $J_N$, called steady-state current. It turns out that for this analysis it is important to determine persistence probabilities of FBM with a moving boundary. We show that one can add a boundary of logarithmic order to a FBM without changing the polynomial rate of d…
We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.
We define multiple chordal SLEs in a simply connected domain by considering a natural configurational measure on paths. We show how to construct these measures so that they are conformally covariant and satisfy certain boundary perturbation and Markov properties, as well as a cascade relation. As an example of our construction, we derive the scaling limit of Fomin's identity in the case of two paths directly; that is, we prove that the probability that an SLE(2) and a Brownian excursion do not intersect can be given in terms of the determinant of the excursion hitting matrix. Finally, we defin…
We study a classical Bayesian statistics problem of sequentially testing the sign of the drift of an arithmetic Brownian motion with the $0$-$1$ loss function and a constant cost of observation per unit of time for general prior distributions. The statistical problem is reformulated as an optimal stopping problem with the current conditional probability that the drift is non-negative as the underlying process. The volatility of this conditional probability process is shown to be non-increasing in time, which enables us to prove monotonicity and continuity of the optimal stopping boundaries as …
Semimartingale reflecting Brownian motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a perturbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A cr…
It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if $κ>4$ and a.s. cutpoints if $4<κ<8$. If $κ>4$, an appropriate version of $\SLE(κ)$ has a renewal property: it starts afresh after visiting its frontier. Thus one can give an excursion decomposition for this particular $\SLE(κ)$ ``away from its frontier''. For $4<κ<8$, there is a two-sided analogue of this situation: a particular version of $\SLE(κ)$ has a renewal property w.r.t its cutpoints; one studies excursion decompositions of this $\SLE$ ``away from its cutpoints''. For $κ=6$, this overlaps Virág's results on…
We investigate the fluctuations of cumulative density of particles in the asymmetric simple exclusion process with respect to the stationary distribution (also known as the steady state), as a stochastic process indexed by $[0,1]$. In three phases of the model and their boundaries within the fan region, we establish a complete picture of the scaling limits of the fluctuations of the density as the number of sites goes to infinity. In the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian excursion. This extends an earlier resu…
The two-dimensional Brownian loop-soup is a Poissonian random collection of loops in a planar domain with an intensity parameter c. When c is not greater than 1, we show that the outer boundaries of the loop clusters are disjoint simple loops (when c>1, there is almost surely only one cluster) that satisfy certain conformal restriction axioms. We prove various results about loop-soups, cluster sizes, and the c=1 phase transition. Combining this with the results of another paper of ours on the Markovian characterization of simple conformal loop ensembles (CLE), this proves that these outer boun…
We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein-Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear bou…
We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and $L^1$ martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy.
We consider an interacting particle system in the interval $[1,N]$ with reservoirs at the boundaries. While the dynamics in the channel is the simple symmetric exclusion process, the reservoirs are also particle systems which interact with the given system by exchanging particles. In this paper we study the case where the size of each reservoir is the same as the size of the channel . We will prove that the hydrodynamic limit equation is the heat equation with boundary conditions which relate first and second spatial derivatives at the boundaries for which we will prove the existence and uniqu…
We show that solutions to multidimensional SDEs with Lipschitz coefficients and driven by Brownian motion never reach the set where all coefficients vanish unless the initial position belongs to that set.
We derive the exact value of intersection exponents between planar Brownian motions or random walks, confirming predictions from theoretical physics by Duplantier and Kwon. Let B and B' be independent Brownian motions (or simple random walks) in the plane, started from distinct points. We prove that the probability that the paths B[0,t] and B'[0,t] do not intersect decays like t^{-5/8}. More precisely, there is a constant c>0 such that if |B(0) - B'(0)| =1, for all t \ge 1, c^{-1} t^{-5/8} \le ¶[ B[0,t] \cap B'[0,t] = \emptyset ] \le c t^{-5/8}. One consequence is that the set of cut-points of…
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…
Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying a positive distance away from each other for all time. Among other results, they showed no shy couplings could exist for reflected Brownian motions in C^2 bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al. result (a) to all bounded convex domains (whether planar and smooth or …
We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.
Non-existence of several random fractals in Brownian motion and Brownian loop soup
Westlake University（西湖大学）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Department of Mathematics, University of Hong Kong（香港大学数学系）
We consider a Brownian particle with diffusion coefficient $D$ in a $d$-dimensional ball of radius $R$ with reflecting boundaries. We study the maximum $M_x(t)$ of the trajectory of the particle along the $x$-direction at time $t$. In the long time limit, the maximum converges to the radius of the ball $M_x(t) \to R$ for $t\to \infty$. We investigate how this limit is approached and obtain an exact analytical expression for the distribution of the fluctuations $Δ(t) = [R-M_x(t)]/R$ in the limit of large $t$ in all dimensions. We find that the distribution of $Δ(t)$ exhibits a rich variety of b…
In this paper, we develop a Monte Carlo based algorithm for estimating the FPT density of a time-homogeneous SDE through a time-dependent frontier. We consider Brownian bridges as well as localized Daniels curve approximations to obtain tractable estimations of the FPT probability between successive points of a simulated path of the process. Under mild assumptions, a (unique) Daniels curve local approximation can easily be obtained by explicitly solving a non-linear system of equations.
In [4], it is proved that we can have a continuous first-passage-time density function of one dimensional standard Brownian motion when the boundary is Hölder continuous with exponent greater than 1/2. For the purpose of extending [4] into multidimensional domains, we show that there exists a continuous first-passage-time density function of standard $d$-dimensional Brownian motion in moving boundaries in $\mathbb{R}^{d}$, $d\geq 2$, under a $C^{3}$-diffeomorphism. Similarly as in [4], by using a property of local time of standard $d$-dimensional Brownian motion and the heat equation with Diri…
The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, ``large'' lattice loops from the random walk loop sou…
Consider the first exit time of one-dimensional Brownian motion $\{B_s\}_{s\geq 0}$ from a random passageway. We discuss a Brownian motion with two time-dependent random boundaries in quenched sense. Let $\{W_s\}_{s\geq 0}$ be an other one-dimensional Brownian motion independent of $\{B_s\}_{s\geq 0}$ and let $\bfP(\cdot|W)$ represent the conditional probability depending on the realization of $\{W_s\}_{s\geq 0}$. We show that $$-t^{-1}\ln\bfP^x(\forall_{s\in[0,t]}a+βW_s\leq B_s\leq b+βW_s|W)$$ converges to a finite positive constant $γ(β)(b-a)^{-2}$ almost surely and in $L^p~ (p\geq 1)$ if $a…
Fraunhofer IAIS（弗劳恩霍夫智能分析与信息系统研究所）；ML2R（机器学习研究中心）；University of Cologne（科隆大学）；IIT Bombay（印度理工学院孟买分校）
In the present work we study classifiers' decision boundaries via Brownian motion processes in ambient data space and associated probabilistic techniques. Intuitively, our ideas correspond to placing a heat source at the decision boundary and observing how effectively the sample points warm up. We are largely motivated by the search for a soft measure that sheds further light on the decision boundary's geometry. En route, we bridge aspects of potential theory and geometric analysis (Mazya, 2011, Grigoryan-Saloff-Coste, 2002) with active fields of ML research such as adversarial examples and ge…
How long does a diffusing molecule spend in a close vicinity of a confining boundary or a catalytic surface? This quantity is determined by the boundary local time, which plays thus a crucial role in the description of various surface-mediated phenomena such as heterogeneous catalysis, permeation through semi-permeable membranes, or surface relaxation in nuclear magnetic resonance. In this paper, we obtain the probability distribution of the boundary local time in terms of the spectral properties of the Dirichlet-to-Neumann operator. We investigate the short-time and long-time asymptotic behav…
This paper proves conjectures originating in the physics literature regarding the intersection exponents of Brownian motion in a half-plane. For instance, suppose that B and B' are two independent planar Brownian motions started from distinct points in a half-plane H. Then, as $t \to \infty$, $$ P [ B[0,t] \cap B'[0,t] = \emptyset, B[0,t] \cup B'[0,t] \subset H ] = t^{-5/3 + o(1)}.$$ The proofs use ideas and tools developped by the authors in previous papers. We prove that one of the stochastic Loewner evolution processes (with parameter 6, that we will call SLE_6 and which has been conjecture…
The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $λ>0$, with central charge $c=2 λ$. Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary "layering vertex operators" $\mathcal{O}_β$ with dimensions $(Δ, Δ)$, with $Δ= \fracλ{10}(1-\cosβ)$, that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions $(Δ+ k/3, Δ+ k'/3)$, for al…
We consider the random walk loop soup on the discrete half-plane corresponding to a central charge c in (0, 1]. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is the CLE(kappa) loop ensemble, with the same relation between kappa and c as in the continuum Brownian setting.
We study Conformal Loop Ensemble (CLE$_κ$) in doubly connected domains: annuli, the punctured disc, and the punctured plane. We restrict attention to CLE$_κ$ for which the loops are simple, i.e. $κ\in (8/3,4]$. In the paper "Conformal Loop Ensemble" by Sheffield and Werner, simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that…
