We consider the system of one-sided reflected Brownian motions which is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling invariant Markov process defined in [MQR17] and believed to govern the long time large scale fluctuations for all models in the KPZ universality class. Brownian last passage percolation was shown recently in [DOV18] to converge to the Airy sheet (or directed landscape), defined there as a strong limit of a functional of the Airy line ensemble. This establishes the variational formula for the KPZ fixed point in terms of the Airy sheet.
Let H(t,x) be the Hopf-Cole solution at time t of the Kardar-Parisi-Zhang (KPZ) equation starting with narrow wedge initial condition, i.e. the logarithm of the solution of the multiplicative stochastic heat equation starting from a Dirac delta. Also let H^{eq}(t,x) be the solution at time t of the KPZ equation with the same noise, but with initial condition given by a standard two-sided Brownian motion, so that H^{eq}(t,x)-H^{eq}(0,x) is itself distributed as a standard two-sided Brownian motion. We provide a simple proof of the following fact: for fixed t, H(t,x)-(H(t,x)-H^{eq}(t,0)) is loca…
We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data H(0,X)=B(X), for B(X) a two-sided standard Brownian motion) and show that as time T goes to infinity, the fluctuations of the height function H(T,X) grow like T^{1/3} and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula fo…
The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process, recently introduced by Matetski, Quastel, Remenik (arXiv:1701.00018), that describes the limit fluctuations of the height function associated to the totally asymmetric simple exclusion process (TASEP), and it is conjectured to be at the centre of the KPZ universality class. Our main result is that the KPZ incremental process converges weakly to its invariant measure, given by a two-sided Brownian motion with zero drift and diffusion coefficient 2. The heart of the proof is the coupling method that allows us to compare the TASEP hei…
In this paper, we prove the existence and uniqueness of the solution to reflected backward doubly stochastic differential equations driven by Teugels martingales associated with a Lévy process where the barrier process is not necessarily right continuous by approximating such equations by a new version of penalization method.
We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one …
In this paper we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain h…
In this article we consider the KPZ fixed point starting from a two-sided Brownian motion with an arbitrary diffusion coefficient. We apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point (covariance) function of the spatial derivative process and the location of the maximum of an Airy process plus Brownian motion minus a parabola. Integration by parts also allows us to deduce the density of this location in terms of the second derivative of the variance of the KPZ fixed point. In the stationary regime, we find the same density related…
An obvious way to simulate a Lévy process $X$ is to sample its increments over time $1/n$, thus constructing an approximating random walk $X^{(n)}$. This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resultant process $Y$ and regulators $L,U$ at the lower and upper barriers at some fixed time. Under the weak assumption that $X_\varepsilon/a_\varepsilon$ has a non-trivial weak limit for some scaling function $a_\varepsilon$ as $\varepsilon\downarrow 0$, it is proved in particular that $(Y_1-Y^{(n)}_n)/a_{1/n}$ conve…
We present a new approach to fluctuation identities for reflected Lévy processes with one-sided jumps. This approach is based on a number of easy to understand observations and does not involve excursion theory or Itô calculus. It also leads to more general results.
The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because the nonlinearity is marginally defined with respect to the roughness of the forcing noise. However, its Cole-Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer [13] has recently proved that the solution of SHE can actually be derived through the Cole-Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic bou…
We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütz-type formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, that is, particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the pa…
In this note, we study the asymptotical frontier behavior of a branching reflected Brownian motion. There is essentially no difference in maximal displacement between a branching Brownian motion and its reflected counterpart. We provide two proofs of this fact, one via a soft argument on the dependance of two-sided extremal particles in a branching Brownian motion and the other based on direct computations as in Roberts. The asymptotics of minimal displacement is also given.
We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $ρ_-<ρ_+$ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf--Cole solution of the Kardar-Parisi-Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466-537] and [Nuclear Phys. B 83…
We consider exit problems for general Lévy processes, where the first passage over a threshold is detected either immediately or at an epoch of an independent homogeneous Poisson process. It is shown that the two corresponding one-sided problems are related through a surprisingly simple identity. Moreover, we identify a simple link between two-sided exit problems with one continuous and one Poisson exit. Finally, Poisson exit of a reflected process is connected to the continuous exit of a process reflected at Poisson epochs, and a link between some Parisian type exit problems is established. W…
We give sharp two-sided estimates for the functions $g_M(t,x,y)$ and $g_M(t,x,y)-g(t,x,y)$, where $g_M(t,x,y)$ are the transition probability densities of the reflected Brownian motion on a $M$-complex of size $M \in \mathbb{Z}$ of an unbounded planar simple nested fractal and $g(t,x,y)$ are the transition probability densities of the `free' Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.
We establish several existence, uniqueness and comparison results for $L^1$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under the assumption that the generator $g$ satisfies a one-sided Osgood condition together with a very general growth condition in $y$, a uniform continuity condition and/or a sub-linear growth condition in $z$, and a generalized Mokobodzki condition which relates the growth of $g$ and that of the barriers. This generalized Mokobodzki condition is proved to be necessary for existence of $L^1$ solutions of the reflected BSDEs. We …
We extend to Markov-modulated Brownian motion (MMBM) the renewal approach which has been successfully applied to the analysis of Markov-modulated fluid models. It has recently been shown that MMBM may be expressed as the limit of a parameterized family of Markov-modulated fluid models. We prove that the weak convergence also holds for systems with two reflecting boundaries, one at zero and one at $b >0$, and that the stationary distributions of the approximating fluid models converge to the stationary distribution of the two-sided reflected MMBM. Thus, we obtain a new representation for the st…
For a spectrally one-sided Lévy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas.
Consider the $n$th iterated Brownian motion $I^{(n)}=B_n \circ\cdots \circ B_1$. Curien and Konstantopoulos proved that for any distinct numbers $t_i\neq 0$, $(I^{(n)}(t_1),\dots,I^{(n)}(t_k))$ converges in distribution to a limit $I[k]$ independent of the $t_i$'s, exchangeable, and gave some elements on the limit occupation measure of $I^{(n)}$. Here, we prove under some conditions, finite dimensional distributions of $n$th iterated two-sided stable processes converge, and the same holds the reflected Brownian motions. We give a description of the law of $I[k]$, of the finite dimensional dist…
First passage problems for spectrally negative Lévy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator. In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interv…
