# Positive random walks and an identity for half-space SPDEs

@article{Parekh2022PositiveRW,
title={Positive random walks and an identity for half-space SPDEs},
author={Shalin Parekh},
journal={Electronic Journal of Probability},
year={2022}
}
• Shalin Parekh
• Published 27 January 2019
• Mathematics
• Electronic Journal of Probability
The purpose of this article is threefold. First, we introduce a new type of boundary condition for the multiplicative-noise stochastic heat equation on the half space. This is essentially a Dirichlet boundary condition but with a nontrivial normalization near the boundary which leads to inhomogeneous transition densities (roughly, those of a Brownian \textit{meander}) within the associated chaos series. Secondly, we prove a new convergence result of the directed-polymer partition function in an…

## Figures from this paper

### Stationary measures for the log-gamma polymer and KPZ equation in half-space

• Mathematics
• 2022
. We construct explicit one-parameter families of stationary measures for the Kardar-Parisi-Zhang equation in half-space with Neumann boundary conditions at the origin, as well as for the log-gamma

### Intermediate Disorder Regime for Half-Space Directed Polymers

• Xuan Wu
• Mathematics
Journal of Statistical Physics
• 2020
We consider the convergence of partition functions and endpoint density for the half-space directed polymer model in dimension $1+1$ in the intermediate disorder regime as considered for the full

### Delta-Bose gas on a half-line and the Kardar–Parisi–Zhang equation: boundary bound states and unbinding transitions

• Mathematics
Journal of Statistical Mechanics: Theory and Experiment
• 2020
We revisit the Lieb–Liniger model for n bosons in one dimension with attractive delta interaction in a half-space with diagonal boundary conditions. This model is integrable for the arbitrary value

### An identity in distribution between full-space and half-space log-gamma polymers

• Materials Science
• 2021
We prove an identity in distribution between two kinds of partition functions for the log-gamma directed polymer model: (1) the point-to-point partition function in a quadrant, (2) the point-to-line

### Lyapunov Exponents of the Half-Line SHE

We consider the half-line stochastic heat equation (SHE) with Robin boundary parameter $A = -\frac{1}{2}$. Under narrow wedge initial condition, we compute every positive (including non-integer)

### Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions

• Mathematics
• 2020
We revisit the Lieb-Liniger model for $n$ bosons in one dimension with attractive delta interaction in a half-space $\mathbb{R}^+$ with diagonal boundary conditions. This model is integrable for

### Time-time covariance for last passage percolation in half-space

• Mathematics
• 2022
This article studies several properties of the half-space last passage percolation, in particular the two-time covariance. We show that, when the two end-points are at small macroscopic distance,

### The lower tail of the half-space KPZ equation

• Yujin H. Kim
• Mathematics
Stochastic Processes and their Applications
• 2021

### Half-Space Stationary Kardar–Parisi–Zhang Equation

• Materials Science
Journal of statistical physics
• 2020
We study the solution of the Kardar–Parisi–Zhang (KPZ) equation for the stochastic growth of an interface of height h(x, t) on the positive half line, equivalently the free energy of the continuum

### Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

• Mathematics
SciPost Physics
• 2020
<jats:p>We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height <jats:inline-formula><jats:alternatives><jats:tex-math>h(x,t)</jats:tex-math><mml:math

## References

SHOWING 1-10 OF 51 REFERENCES

### Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

• Mathematics
• 2017
We consider the weakly asymmetric simple exclusion process on the discrete space $\{1,...,n-1\}$, in contact with stochastic reservoirs, both with density $\rho\in{(0,1)}$ at the extremity points,

### HALF-SPACE MACDONALD PROCESSES

• Mathematics
Forum of Mathematics, Pi
• 2020
Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the

### Pfaffian Schur processes and last passage percolation in a half-quadrant

• Mathematics
The Annals of Probability
• 2018
We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the

### Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

• Mathematics
Duke Mathematical Journal
• 2018
We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary

### Intermediate Disorder Regime for Half-Space Directed Polymers

• Xuan Wu
• Mathematics
Journal of Statistical Physics
• 2020
We consider the convergence of partition functions and endpoint density for the half-space directed polymer model in dimension $1+1$ in the intermediate disorder regime as considered for the full

### Singular SPDEs in domains with boundaries

• Mathematics
• 2017
We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories

### Multiplicative stochastic heat equations on the whole space

• Mathematics
• 2015
We carry out the construction of some ill-posed multiplicative stochastic heat equations on unbounded domains. The two main equations our result covers are, on the one hand the parabolic Anderson

### A Functional Limit Theorem for Random Walk Conditioned to Stay Non‐Negative

• Mathematics
• 2006
In this paper we consider an aperiodic integer‐valued random walk S and a process S* that is a harmonic transform of S killed when it first enters the negative half; informally, S* is ‘S conditioned