In this paper, we establish a relationship between the asymptotic form of conditional boundary crossing probabilities and first passage time densities for diffusion processes. Namely, we show that, under broad assumptions, the first crossing time density of a general curvilinear boundary by a general time-homogeneous diffusion process has a product-form, the factors being the transition density of the process and the coefficient of the leading term in the asymptotic representation of the non-crossing probability of the boundary by the respective diffusion bridge (as the end-point of the bridge approaches the boundary). Using a similar technique, we also demonstrate that the boundary crossing probability is a Gateaux differentiable function of the boundary and give an explicit representation of its derivative.
In this paper we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product we solve a minimization problem related to the norm of the trend function.
We consider metric graph Gaussian free field (GFF) defined on polygons of $δ\mathbb{Z}^2$ with alternating boundary data. The crossing probabilities for level-set percolation of metric graph GFF have scaling limits. When the boundary data is well-chosen, the scaling limits of crossing probabilities can be explicitly constructed as "fusion" of multiple SLE$_4$ pure partition functions.
We prove that the standard Russo-Seymour-Welsh theory is valid for Voronoi percolation. This implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio. This result has many consequences, such as the polynomial decay of the one-arm event at criticality.
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.
This paper studies the queue length process in series Jackson networks with external input to the first station. We show that its Markov transition probabilities can be written as a finite sum of non-crossing probabilities, so that questions on time-dependent queueing behavior are translated to questions on non-crossing probabilities. This makes previous work on non-crossing probabilities relevant to queueing systems and allows new queueing results to be established. To illustrate the latter, we prove that the relaxation time (i.e., the reciprocal of the `spectral gap') of a positive recurrent…
We propose an approach to approximate the boundary crossing probabilities for general one-dimensional diffusion processes, and derive the convergence rate for this approximation scheme. There results are based on the explicit expression of the Laplace transforms of the first passage densities for diffusions with piecewise linear drifts. The proposed method is applied to a reliability problem where the standard degradation model based on Wiener process is extended to diffusion processes with piecewise linear drifts.
Motivated by an approximation problem from mathematical finance, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad assumptions on the nature of the boundary, including the Lipschitz condition (in a Hausdorff-type metric) on its time cross-sections, we obtain an analogue of the Borovkov and Novikov (2005) upper bound for the difference between boundary hitting probabilities for "close boundaries" in the univariate case. We also obtained upper bounds for the first boundary crossin…
Prompted by an example arising in critical percolation, we study some reflected Brownian motions in symmetric planar domains and show that they are intertwined with one-dimensional diffusions. In the case of a wedge, the reflected Brownian motion is intertwined with the 3-dimensional Bessel process. This implies some simple hitting distributions and sheds some light on the formula proposed by Watts for double-crossing probabilities in critical percolation.
We propose a discrete time discrete space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the p…
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…
In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph $G$. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of Kesten's incipient infinite cluster. We show that our assumptions are satisfied if $G$ is a slab $\mathbb Z^2\times\{0,\ldots,k\}^{d-2}$ ($d\geq 2$, $k\geq 0$). We also argue that the quasi-multiplicativity assumption is fulfilled for $G=\mathbb Z^d$ if and only if $d<6$.
We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to whole real line this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogene…
We consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit lower bounds for the first-crossing-time density and for the first-crossing-time distribution function. In the case of the distribution function, the bound is improved by use of processes comparison based on the usual stochastic order. The special case of constant jumps driven by a Poisson process is thoroughly discussed.
We analyze here different forms of fractional relaxation equations of order ν\in(0,1) and we derive their solutions both in analytical and in probabilistic forms. In particular we show that these solutions can be expressed as crossing probabilities of random boundaries by various types of stochastic processes, which are all related to the Brownian motion B. In the special case ν=1/2, the fractional relaxation is proved to coincide with Pr{sup_{0\leqs\leqt} B(s)<U}, for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the Ga…
Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain ``clumps'' around high-level crossings. We also discuss how these results are related t…
We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing symmetry the (sharp) inequality P(|X+Y| <= 1) < 2 P(|X-Y| <= 1) for independent identically distributed X,Y is used. In part II we shall discuss the connection of this result to 'polygonal recurrence' of higher-dimensional walks and some conjectures on directionally random walks in the sense of Mauldin, Monticino and v.Weizsaecker [5].
We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm P\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous centered Gaussian process $X$ indexed by some arbitrary compact separable metric space $\mathbb T$. We obtain both upper and lower bounds for $P_{f,u}$. The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $P_{y f,u}$, $y \to+\infty$, which are two-term approximations (up to $o(y)$). The asymptotics are formulated in terms of the solution $\tilde f$ to the constrained optimization problem $$ \|h\|_{…
The purpose of this paper is to prove ergodicity and provide asymptotic formulae for probabilities of threshold crossing related to smooth approximations of three fundamental nonlinear mechanical models: (a) an elasto-plastic oscillator, (b) an oscillator with dry friction, (c) an oscillator constrained by an obstacle (one sided or two sided) and subject to impacts, all three in presence of white or colored noise. Relying on a groundbreaking result on density estimates for degenerate diffusions by Delarue and Menozzi (2010), we identify Lyapunov functions that satisfy appropriate conditions le…
We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman-Kac formula, where the underlying stochastic processes satisfy variational inequalities modelling elasto-plastic and obstacle…
In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z^3 in the…
The study of crossing probabilities - i.e. probabilities of existence of paths crossing rectangles - has been at the heart of the theory of two-dimensional percolation since its beginning. They may be used to prove a number of results on the model, including speed of mixing, tails of decay of the connectivity probabilities, scaling relations, etc. In this article, we develop a renormalization scheme for crossing probabilities in the two-dimensional random-cluster model. The outcome of the process is a precise description of an alternative between four behaviors: - Subcritical: Crossing probabi…
In this contribution we derive an explicit formula for the boundary non-crossing probabilities for Slepian processes associated with the piecewise linear boundary function. This formula is used to develop an approximation formula to the boundary non-crossing probabilities for general continuous boundaries. The formulas we developed are easy to implement in calculation the boundary non-crossing probabilities.
We prove the super-exponential decay of probabilities that there exist $n$ crossings of a given quadrilateral in a simple $\text{CLE}_κ(Ω)$, $\frac{8}{3}<κ\le 4$, as $n$ goes to infinity. Besides being of independent interest, this also provides the missing ingredient in arXiv:1809.00690 for proving the convergence of probabilities of cylindrical events for the double-dimer loop ensembles to those for the nested $\text{CLE}_4(Ω)$.
For the site percolation model on the triangular lattice and certain generalizations for which Cardy's Formula has been established we acquire a power law estimate for the \emph{rate} of convergence of the crossing probabilities to Cardy's Formula.
The boundary crossing probability of a Poisson process with $n$ jumps is a fundamental quantity with numerous applications. We present a fast $O(n^2 \log n)$ algorithm to calculate this probability for arbitrary upper and lower boundaries.
We consider the FK-Ising model in two dimensions at criticality. We obtain bounds on crossing probabilities of arbitrary topological rectangles, uniform with respect to the boundary conditions, generalizing results of [DCHN11] and [CS12]. Our result relies on new discrete complex analysis techniques, introduced in [Che12]. We detail some applications, in particular the computation of so-called universal exponents, the proof of quasi-multiplicativity properties of arm probabilities, and bounds on crossing probabilities for the classical Ising model.
We prove that in the critical Bernoulli percolation on two dimensional lattice slabs the probabilities of open left-right crossings of rectangles with any given aspect ratio are uniformly positive.
We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval $[0,T]$ via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class of barriers on $[0,T]$ by proving a Schwartz distribution version of the method of images. Analytic expressions for crossing probabilities and related densities are given for new explicit and semi-explicit barriers.
We prove a general Russo-Seymour-Welsh result valid for any invariant planar percolation process satisfying positive association. This means that the probability of crossing a rectangle in the long direction is related by a homeomorphism to the probability of crossing it in the short direction. This homeomorphism is universal in the sense that it depends only on the aspect ratio of the rectangle, and is uniform in the scale and the considered model.
We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin. We do so by establishing the convergence of certain exploration processes towards SLE$(3,\frac{-3}2,\frac{-3}2)$. We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield.
Consider a diffusion process corresponding to the operator $L=\frac12a\frac{d^2}{dx^2}+b\frac d{dx}$ and which is transient to $+\infty$. For $c>0$, we give an explicit criterion in terms of the coefficients $a$ and $b$ which determines whether or not the diffusion almost surely eventually stops making down-crossings of length $c$. As a particular case, we show that if $a=1$, then the diffusion almost surely stops making down-crossings of length $c$ if $b(x)\ge\frac1{2c}\log x+\fracγc\log\log x$, for some $γ>1$ and for large $x$, but makes down-crossings of length $c$ at arbitrarily large time…
We prove convergence results for variants of Smirnov's fermionic observable in the critical Ising model in presence of free boundary conditions. One application of our analysis is a simple proof of a theorem by Hongler and Kytölä on convergence of critical Ising interfaces with plus-minus-free boundary conditions to dipolar SLE(3), and generalization of this result to arbitrary number of arcs carrying plus, minus or free boundary conditions. Another application is a computation of scaling limits of crossing probabilities in FK-Ising model with arbitrary number of alternating wired/free boundar…
This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if Fokker--Planck--Kolmogorov equation has non-trivial Lie symmetry, then boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found necessary and sufficient conditions of symmetries' existence. This paper shows that if drift function satisfy one of a family of Ricatti equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. B…
For a multivariate random walk with i.i.d. jumps satisfying the Cramer moment condition and having a mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results from a paper by F. Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a recent series of papers by A. Borovkov and A. Mogulskii with new auxiliary constructions, whic…
We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic Löwner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension 1/3), follows from a …
For a given barrier $S$ and a one-dimensional jump-diffusion process $X(t),$ starting from $x<S,$ we study the probability distribution of the integral $A_S(x)= \int_0 ^ {τ_S(x)}X(t) \ dt$ determined by $X(t)$ till its first-crossing time $τ_S(x)$ over $S.$ In particular, we show that the Laplace transform and the moments of $A_S(x)$ are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of $X(t)$ in $[0, τ_S(x)]$ is also studied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by $X…
We review and study a one-parameter family of functional transformations, denoted by $(S^{(β)})_{β\in \R}$, which, in the case $β<0$, provides a path realization of bridges associated to the family of diffusion processes enjoying the time inversion property. This family includes the Brownian motion, Bessel processes with a positive dimension and their conservative $h$-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary crossing times for these diffusions over a given function $f$ to those over the image of $f$ by the …
We study the inverse boundary crossing problem for diffusions. Given a diffusion process $X_t$, and a survival distribution $p$ on $[0,\infty)$, we demonstrate that there exists a boundary $b(t)$ such that $p(t)=\mathbb{P}[τ>t]$, where $τ$ is the first hitting time of $X_t$ to the boundary $b(t)$. The approach taken is analytic, based on solving a parabolic variational inequality to find $b$. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary $b$ does indeed have $p$ as its boundary cros…
We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.
We consider diffusion processes in Hilbert spaces with constant non-degenerate diffusion operators and show that, under broad assumptions on the drift, the transition probabilities of the process are positive on ellipsoids associated with the diffusion operator. This is an infinite-dimensional analogue of positivity of densities of transition probabilities. Our results apply to diffusions corresponding to stochastic partial differential equations.
We develop the mathematics of a filtration shrinkage model that has recently been considered in the credit risk modeling literature. Given a finite collection of points $x_1<...<x_N$ in $\mathbb{R}$, the region indicator function $R(x)$ assumes the value $i$ if $x\in(x_{i-1},x_i]$. We take $\mathbb{F}$ to be the filtration generated by $(R(X_t))_{t\geq0}$, where $X$ is a diffusion with infinitesimal generator $\mathcal{A}$. We prove a martingale representation theorem for $\mathbb{F}$ in terms of stochastic integrals with respect to $N$ random measures whose compensators have a simple form giv…
We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and …
We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model, which allows us to get precise estimates on boundary connection probabilities. It remains purely discrete, in particular we do not make use of any continuum limit, and it can be used to derive directly several noteworthy results - some new and some not - among which the fact that there is no spontaneous magnetization at criticality, tightness properties for the …
We construct a planar diffusion process whose infinitesimal generator depends only on the order of the components of the process. Speaking informally and a bit imprecisely for the moment, imagine you run two Brownian-like particles on the real line. At any given time, you assign positive drift g and diffusion σ to the laggard; and you assign negative drift -h and diffusion ρ to the leader. We compute the transition probabilities of this process, discuss its realization in terms of appropriate systems of stochastic differential equations, study its dynamics under a time reversal, and note that …
Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show that these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times.…
We determine the decay rate of the bottom crossing probability for symmetric jump processes under the condition on heat kernel estimates. Our results are applicable to symmetric stable-like processes and stable-subordinated diffusion processes on a class of (unbounded) fractals and fractal-like spaces.
Assume that you observe trajectories of a non-diffusive non-stationary process and that you are interested in the average number of times where the process crosses some threshold (in dimension $d=1$) or hypersurface (in dimension $d\geq2$). Of course, you can actually estimate this quantity by its empirical version counting the number of observed crossings. But is there a better way? In this paper, for a wide class of piecewise smooth processes, we propose estimators of the average number of continuous crossings of an hypersurface based on Kac-Rice formulae. We revisit these formulae in the un…
