Related papers: Local time for lattice paths and the associated li…

Let $S_n$ be a lattice random walk with mean zero and finite variance, and let $\Lambda^a_n$ be its occupation measure at level $a$. In this note, we prove local limit theorems for $\Pr[S_n=x,\Lambda^a_n=\ell]$ and…

Local time of a stochastic process quantifies the amount of time that sample trajectories $x(\tau)$ spend in the vicinity of an arbitrary point $x$. For a generic Hamiltonian, we employ the phase-space path-integral representation of random…

In queuing theory, it is usual to have some models with a "reset" of the queue. In terms of lattice paths, it is like having the possibility of jumping from any altitude to zero. These objects have the interesting feature that they do not…

We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the…

This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and ending at a given altitude $k$, with additional constraints such as, for example, to never…

In this paper, we study a transient spatially inhomogeneous random walk with asymptotically zero drifts on the lattice of the positive half line. We give criteria for the finiteness of the number of points having exactly the same local time…

In this paper, we obtain a local limit theorem for the Kemperman's model of oscillating random walk on $\mathbb{Z}$; it extends the existing results for classical random walks on $\mathbb Z$ or reflected random walks on $\mathbb N_0$. The…

In this paper, we study the law of the local time processes $(L_T^x(X),x\in \mathbb{R})$ associated to a spectrally negative L\'evy process $X$, in the cases $T=\tau_a^+$, the first passage time of $X$ above $a>0$ and $T=\tau(c)$, the first…

We study limit distributions for random variables defined in terms of coefficients of a power series which is determined by a certain linear functional equation. Our technique combines the method of moments with the kernel method of…

We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a c\`adl\`ag path and an occupation measure associated with this path. Using this result we define local times of fractional Brownian…

We investigate the local (or occupation) time of a discrete-time random walk on a generic graph, and present a general method for calculating sample-path averages of local time functionals in terms of the resolvent of the transition matrix.

The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We introduce a new notion…

We consider Brox's model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t;m_(log t) + x)=t; x \in R), where m_(log t) is the bottom of the deepest valley reached by the process…

These notes contains an introduction to the theory of Brownian and diffusion local time, as well as its relations to the Tanaka Formula, the extended Ito-Tanaka formula for convex functions, the running maximum process, and the theory of…

In this paper we present a computation of the mean first-passage times both for a random walk in a discrete bounded lattice, between a starting site and a target site, and for a Brownian motion in a bounded domain, where the target is a…

We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from…

Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $\Z^d$ up to time $t$. This is the $p$-norm of the vector of the walker's local times, $\ell_t$.…

I obtain the dynamics of the continuous time quantum walk on a $d$-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main…

We study the asymptotic behavior of a nonlattice random walk in a general cone of $R^d$ . Following the approach initiated by D. Denisov and V. Wachtel in [8], we use a strong approximation of random walks by the Brownian motion and prove…

We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a…