If $L^x$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal{L}$, called the boundary local time measure, as a rescaling of $L^x e^{-λL^x} dx$ as $λ\to \infty$, thus confirming a conjecture of \cite{MP17} and further show that the support of $\mathcal{L}$ equals the topological boundary of the range of $X$, $\partial\mathcal{R}$. This latter result uses a second construction of a boundary local time $\widetilde{\mathcal{L}}$ given in terms of exit measures and we prove that $\widetilde{\mathcal{L}}=c\mathcal{L}$ a.s. for some constant $c>0$. We derive reasonably explicit first and second moment measures for $\mathcal{L}$ in terms of negative dimensional Bessel processes and use it with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of $\partial\mathcal{R}$ in \cite{HMP18}. The construction requires a refinement of the $L^2$ upper bounds in \cite{MP17} and \cite{HMP18} to exact $L^2$ asymptotics. The methods also refine the left tail bounds for $L^x$ in \cite{MP17} to exact asymptotics. We conjecture that the Minkowski content of $\partial\mathcal{R}$ is equal to the total mass of the boundary local time $\mathcal{L}$ up to some constant.
In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of $κ$-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.
We show that the local time of one-dimensional super-Brownian motion is locally $γ$-Hölder continuous near the boundary if $0<γ<3$ and fails to be locally $γ$-Hölder continuous if $γ>3$.
We show that the Hausdorff dimension of the boundary of $d$-dimensional super-Brownian motion is $0$, if $d=1$, $4-2\sqrt2$, if $d=2$, and $(9-\sqrt{17})/2$, if $d=3$.
(i) Uncountably many synchronized reflected Brownian motions can hit the boundary of a $C^2$ domain at the same time. (ii) Measures associated to local times of two synchronized reflected Brownian motions are mutually singular until the time when the normal vectors at the reflection locations become identical. (iii) Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.
We establish a boundary Harnack principle for a large class of subordinate Brownian motion, including mixtures of symmetric stable processes, in bounded $κ$-fat open set (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded $κ$-fat open sets with respect to these processes with their Euclidean boundary.
In this paper, we study a two-point boundary value problem consisting of the heat equation on the open interval $(0,1)$ with boundary conditions which relate first and second spatial derivatives at the boundary points. Moreover, the unique solution to this problem can be represented probabilistically in terms of a sticky Brownian motion. This probabilistic representation is attained from the stochastic differential equation for a sticky Brownian motion on the bounded interval $[0,1]$.
A classification for Brownian motions on metric graphs, that is, right continuous strong Markov processes which behave like a one-dimensional Brownian motion on the edges and feature effects like Walsh skewness, stickiness and jumps at the vertices, is obtained. The Feller property of these processes is proved, and the boundary conditions of their generators are identified as non-local Feller-Wentzell boundary conditions. By using a technique of successive revivals, a complete description of the generator is achieved for Brownian motions on star graphs.
In this paper we consider a (reflected) Brownian motion with broken drift hitting a random boundary. Some dedicated calculations allow us to obtain the formula on the joint Laplace transform of the hitting time and hitting position. These develop the research on first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes by Perry et al. (2004).
In this paper, we introduce the idea of stochastic integrals with respect to an increasing process in the $G$-framework and extend $G$-Itô's formula. Moreover, we study the solvability of the scalar valued stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions (RGSDEs).
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 first consider the additive Brownian motion process $(X(s_1,s_2),\ (s_1,s_2) \in \mathbb{R}^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability one, the Hausdorff dimension of the boundary of any connected component of the random set $\{(s_1,s_2)\in \mathbb{R}^2: X(s_1,s_2) >0\}$ is equal to $$ \frac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421\, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.
A pathwise construction of discontinuous Brownian motions on metric graphs is given for every possible set of non-local Feller-Wentzell boundary conditions. This construction is achieved by locally decomposing the metric graphs into star graphs, establishing local solutions on these partial graphs, pasting the solutions together, introducing non-local jumps, and verifying the generator of the resulting process.
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…
We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This construction shows in particular that distances from the distinguished point evolve along the boundary as a five-dimensional Bessel bridge. As an important ingredient of our proofs, we show that the uniform measure on the boundary, as defined in the earlier work of Bettinelli and Miermont, is the limit of the suitably normalized volume measure on a small tubul…
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 …
We use reflecting Brownian motion (RBM) to prove the well known Gauss-Bonnet-Chern theorem for a compact Riemannian manifold with boundary. The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.
In this paper, we consider first order Sobolev spaces with Robin boundary condition on unbounded Lipschitz domains. Hunt processes are associated with these spaces. We prove that the semigroup of these processes are doubly Feller. As a corollary, we provide a condition for semigroups generated by these processes being compact.
Brownian motions on star graphs in the sense of Itô-McKean, that is, Walsh processes admitting a generalized boundary behavior including stickiness and jumps and having an angular distribution with finite support, are examined. Their generators are identified as Laplace operators on the graph subject to non-local Feller-Wentzell boundary conditions. A pathwise description is achieved for every admissible boundary condition: For finite jump measures, a construction of Kostrykin, Potthoff and Schrader in the continuous setting is expanded via a technique of successive killings and revivals; for …
The trace of a Markov process is the time changed process of the original process on the support of the Revuz measure used in the time change. In this paper, we will concentrate on the reflecting Brownian motions on certain closed strips. On one hand, we will formulate the concrete expression of the Dirichlet forms associated with the traces of such reflecting Brownian motions on the boundary. On the other hand, the limits of these traces as the distance between the upper and lower boundaries tends to $0$ or $\infty$ will be further obtained.
Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models--related to the SABR model in mathematical finance--which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change …
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.
In this paper we consider the Brownian motion with jump boundary and present a new proof of a recent result of Li, Leung and Rakesh concerning the exact convergence rate in the one-dimensional case. Our methods are different and mainly probabilistic relying on coupling methods adapted to the special situation under investigation. Moreover, we answer a question raised by Ben-Ari and Pinsky concerning the dependence of the spectral gap on the jump distribution in a multi-dimensional setting.
We analyze the stationary distribution of regulated Markov modulated Brownian motions (MMBM) modified so that their evolution is slowed down when the process reaches level zero --- level zero is said to be {\em sticky}. To determine the stationary distribution, we extend to MMBMs a construction of Brownian motion with sticky boundary, and we follow a Markov-regenerative approach similar to the one developed in past years in the context of quasi-birth-and-death processes and fluid queues. We also rely on recent work showing that Markov-modulated Brownian motions may be analyzed as limits of a p…
Under some weak conditions, the first-passage time of the Brownian motion to a continuous curved boundary is an almost surely finite stopping time. Its probability density function (pdf) is explicitly known only in few particular cases. Several mathematical studies proposed to approximate the pdf in a quite general framework or even to simulate this hitting time using a discrete time approximation of the Brownian motion. The authors study a new algorithm which permits to simulate the first-passage time using an iterating procedure. The convergence rate presented in this paper suggests that the…
We revisit a result of Uchiyama (1980): given that a certain integral test is satisfied, the rate of the probability that Brownian motion remains below the moving boundary $f$ is asymptotically the same as for the constant boundary. The integral test for $f$ is also necessary in some sense. After Uchiyama's result, a number of different proofs appeared simplifying the original arguments, which strongly rely on some known identities for Brownian motion. In particular, Novikov (1996) gives an elementary proof in the case of an increasing boundary. Here, we provide an elementary, half-page proof …
We study the density X(t,x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0<X(t,x)<a) as a approaches 0, as well as the Hausdorff dimension of the boundary of the support of X(t). The answers are in terms of the lead eigenvalue of the Ornstein-Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.
We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting $0$, when $0$ is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolve…
In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in $C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bo…
We show that if $\partial\mathcal{R}$ is the boundary of the range of super-Brownian motion and dim denotes Hausdorff dimension, then with probability one, for any open set $U$, $\partial\mathcal{R}\cap U\neq\emptyset$ implies $$\text{dim}(U\cap\partial\mathcal{R})=\begin{cases} 4-2\sqrt2\approx1.17&\text{ if }d=2\\ \frac{9-\sqrt{17}}{2}\approx 2.44&\text{ if }d=3. \end{cases}$$ This improves recent results of the last two authors (arxiv:1711.03486) by working with the actual topological boundary, rather than the boundary of the zero set of the local time, and establishing a local result for t…
Stochastic variational inequalities provide a unified treatment for stochastic differential equations living in a closed domain with normal reflection and (or) singular repellent drift. When the domain is a polyhedron, we prove that the reflected-repelled Brownian motion does not hit the non-smooth part of the boundary. A sufficient condition for non-hitting a face of the polyhedron is derived from the one-dimensional case. A complete answer to the question of attainability of the walls of the Weyl chamber may be given for a radial Dunkl process.
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…
We develop an excursion theory for Brownian motion indexed by the Brownian tree, which in many respects is analogous to the classical Itô theory for linear Brownian motion. Each excursion is associated with a connected component of the complement of the zero set of the tree-indexed Brownian motion. Each such connectedcomponent is itself a continuous tree, and we introduce a quantity measuring the length of its boundary. The collection of boundary lengths coincides with the collection of jumps of a continuous-state branching process with branching mechanism $ψ(u)=\sqrt{8/3}\,u^{3/2}$. Furthermo…
We consider the model of Brownian motion indexed by the Brownian tree. For every $r\geq 0$ and every connected component of the set of points where Brownian motion is greater than $r$, we define the boundary size of this component, and we then show that the collection of these boundary sizes evolves when $r$ varies like a well-identified growth-fragmentation process. We then prove that the same growth-fragmentation process appears when slicing a Brownian disk at height $r$ and considering the perimeters of the resulting connected components.
We study an inverse first-passage-time problem for Wiener process $X(t)$ subject to hold and jump from a boundary $c.$ Let be given a threshold $S>X(0) \ge c,$ and a distribution function $F$ on $[0, + \infty ).$ The problem consists in finding the distribution of the holding time at $c$ and the distribution of jumps from $c,$ so that the first-passage time of $X(t)$ through $S$ has distribution $F.$
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…
We compute a closed-form expression for the moment generating function $\hat{f}(x;λ,α)=\frac{1}λ\mathbb{E}_x(e^{αL_τ})$, where $L_t$ is the local time at zero for standard Brownian motion with reflecting barriers at $0$ and $b$, and $τ\sim \mathrm{Exp}(λ)$ is independent of $W$. By analyzing how and where $\hat{f}(x;\cdot,α)$ blows up in $λ$, a large-time large deviation principle (LDP) for $L_t/t$ is established using a Tauberian result and the Gärtner-Ellis Theorem.
Brownian motions on a metric graph are defined. Their generators are characterized as Laplace operators subject to Wentzell boundary at every vertex. Conversely, given a set of Wentzell boundary conditions at the vertices of a metric graph, a Brownian motion is constructed pathwise on this graph so that its generator satisfies the given boundary conditions.
We use a renormalization of the total mass of the exit measure from the complement of a small ball centered at $x\in \mathbb{R}^d$ for $d\leq 3$ to give a new construction of the total local time $L^x$ of super-Brownian motion at $x$. In \cite{Hong20} a more singular renormalization of the total mass of the exit measure concentrating on $x$, where the exit measure is positive but unusually small, is used to build a boundary local time supported on the topological boundary of the range of super-Brownian motion. Our exit measure construction of $L^x$ motivates this renormalization. We give an im…
If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that $\text{dim}(\partial\{x:X(t,x)>0\})=2-2λ_0\in(0,1)$ a.s. on $\{X_t\neq 0\}$, where $-λ_0\in(-1,-1/2)$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck process. This confirms a conjecture of Mueller, Mytnik and Perkins who proved the above with positive probability. To establish this result we derive some new basic properties of a recently introduced boundary local time and analyze the behaviour of $X(t,\cdot)$ near the upper edge of its support. Numerical estimates of $λ_0$ suggest that the above Hausdorff …
Let $(B(t))_{t\in [0,1]}$ be the linear Brownian motion and $(X_n(t))_{t\in [0,1]}$ be the $(n-1)$-fold integral of Brownian motion, $n$ being a positive integer: $$ X_n(t)=\int_0^t \frac{(t-s)^{n-1}}{(n-1)!} \,\dd B(s) for any $t\in[0,1]$. $$ In this paper we construct several bridges between times 0 and 1 of the process $(X_n(t))_{t\in [0,1]}$ involving conditions on the successive derivatives of $X_n$ at times 0 and 1. For this family of bridges, we make a correspondance with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical pr…
Let $π$ be the occupancy density of an obliquely reflected Brownian motion in the half plane and let ($ρ$, $α$) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of $π$($ρ$, $α$) as $ρ$ $\rightarrow$ $\infty$ with $α$ $\in$ (0, $π$). We find explicit functions a, b, c such that $π$($ρ$, $α$) $\sim$ $ρ$$\rightarrow$$\infty$ a($α$)$ρ$ b($α$) e --c($α$)$ρ$. This closes an open problem first stated by Professor J. Michael Harrison in August 2013. We also compute the exact asymptotics for the tail distribution of the boundary occupancy m…
The binary branching Brownian motion in the boundary case is a particle system on the real line behaving as follows. It starts with a unique particle positioned at the origin at time $0$. The particle moves according to a Brownian motion with drift $μ= 2$ and diffusion coefficient $σ^2 = 2$, until an independent exponential time of parameter $1$. At that time, the particle dies giving birth to two children who then start independent copies of the same process from their birth place. It is well-known that in this system, the cloud of particles eventually drifts to $\infty$. The aim of this note…
In this paper we study the effect of stochastic perturbations on a common type of moving boundary value PDE's which endorse Stefan boundary conditions, or Stefan problems, and show the existence and uniqueness of the solutions to a number of stochastic equations of this kind. Moreover we also derive the space and time regularities of the solutions and the associated boundaries via Kolmogorov's Continuity Theorem in an appropriately defined normed space. The paper first conveys our previous results where randomness is smoothly correlated in space and Brownian in time, then introduces a new meth…
In this paper we consider $β[0; s]$, Brownian motion of time length $s > 0$, in $m$-dimensional Euclidean space $\mathbb R^m$ and on the $m$-dimensional torus $\mathbb T^m$. We compute the expectation of (i) the heat content at time $t$ of $\mathbb R^m\setminus β[0; s]$ for fixed $s$ and $m = 2,3$ in the limit $t \downarrow 0$, when $β[0; s]$ is kept at temperature 1 for all $t > 0$ and $\mathbb R^m\setminus β[0; s]$ has initial temperature 0, and (ii) the inradius of $\mathbb R^m\setminus β[0; s]$ for $m = 2,3,\cdots$ in the limit $s \rightarrow \infty$.
We define a class a metric spaces we call Brownian surfaces, arising as the scaling limits of random maps on general orientable surfaces with a boundary and we study the geodesics from a uniformly chosen random point. These metric spaces generalize the well-known Brownian map and our results generalize the properties shown by Le Gall on geodesics in the latter space. We use a different approach based on two ingredients: we first study typical geodesics and then all geodesics by an "entrapment" strategy. In particular, we give geometrical characterizations of some subsets of interest, in terms …
We study a free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of an interacting particle system involving branching Brownian motion with selection, the so-called Brownian bees model which is studied in a companion paper. In this paper we prove existence and uniqueness of the solution to the free boundary problem, and we characterise the behaviour of the solution in the large time limit.
Consider a metric graph G with set of vertices V. Assume that for every vertex in V one is given a Wentzell boundary condition. It is shown how one can construct the paths of a Brownian motion on G such that its generator - viewed as an operator on the space of continuous functions vanishing at infinity - has a domain consisting of twice continuously differentiable functions satisfying these boundary conditions.
We provide short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. The boundary behaviour is precisely described. Presented results may be considered as a complement or a generalization of the famous "principle of not feeling the boundary" in case of a ball.
For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polyno-mials. To that purpose we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.
