# Current Trends in Random Walks on Random Lattices

^{*}

## Abstract

**:**

## 1. Introduction

**S${}_{n}=\phantom{\rule{3.33333pt}{0ex}}$X${}_{1}+\dots +$X${}_{n}$**made of a sequence of iid ${\mathbb{Z}}^{d}$-valued r.v.’s

**$\mathfrak{X}=\{$X${}_{1},$X${}_{2},\dots .\}$**In its simple form,

**X${}_{i}\in \left[\mathrm{X}\right]$**and

**X**is uniformly distributed on ${\left\{-1,1\right\}}^{d}$, such that if

**S${}_{n}=\phantom{\rule{0.277778em}{0ex}}$x**$\in {\mathbb{Z}}^{d},$ the particle or walker moves from state

**x**to state

**y**within the integer hypercube

**x**$+{\left\{-1,1\right\}}^{d}$ equally likely in any direction with probability $\frac{1}{2d}$. So here the walker moves randomly within the d-dimensional integer lattice. One of the key objectives is to find the probability distribution of r.v.’s $\nu =\phantom{\rule{0.277778em}{0ex}}$inf$\left\{n:{\mathbf{S}}_{n}\in {A}^{c}\right\}$ and

**S${}_{\nu}$**, where A is a bounded subset of ${\mathbb{Z}}^{d}$. That is, the position of the walker when it escapes from set A.

**X**to be ${\mathbb{R}}^{d}$-valued and arbitrarily distributed with the entries $\left({X}_{1},\dots ,{X}_{d}\right)$ of

**X**not independent as assumed above, then such random walk is largely embellished. Here we can think of a randomly generated grid replacing ${\mathbb{Z}}^{d}$, so that if

**S${}_{n}=\phantom{\rule{0.277778em}{0ex}}$x$\in {\mathbb{R}}^{d}$**, the walker moves from state

**x**to state

**y**according to random increment

**X${}_{n}\in \left[\mathrm{X}\right]$**(the equivalence class of all r.v.’s a.s. equal

**X**) and it turns out that the grid along which the particle moves is randomly generated each time the walker lands at some

**x**. Furthermore, we take into account the time that the walker takes to move from

**x**to

**y**in one step thereby forming a point process $T=\left\{{t}_{1},{t}_{2},\dots \right\}$ and the associated marked point process $\mathcal{S}={\sum}_{n=0}^{\infty}$

**X**${}_{n}{\epsilon}_{{t}_{n}}$ (${\epsilon}_{\alpha}$ is the unit measure at point a). The escape parameters of such random walk now require more specifications. If A is a bounded subset of ${\mathbb{R}}^{d},$ then $\nu =\phantom{\rule{0.277778em}{0ex}}$inf$\left\{n:{\mathrm{S}}_{n}\in {A}^{c}\right\}$ (exit index), ${t}_{\nu}$ is the first passage time or escape time, and

**S${}_{\nu}$**is the position of the walker on its escape (escape position).

**X**to be ${\mathbb{R}}_{+}^{d}$-valued is helpful and still enough practical. The random walk terminology is appropriate to describe the physical motion of a walker regardless of where the walker moves, although other descriptive terms like a marked point process or marked random measure or recurrent or renewal process are also common in the literature. Furthermore, the additive components or jumps

**X**${}_{i}$’s of

**S**need not be iid and can form Markov or semi-Markov processes, although these classes of $\mathcal{S}$ are out of scope and interest of this paper that targets only cases that lead to analytically closed forms.

**X**can also be integer-valued, whereas we retain all other assumptions on $\mathcal{S}$. Then if the walker at step n at time ${t}_{n}$ lands at some

**S${}_{n}=\phantom{\rule{0.277778em}{0ex}}$X${}_{0}+\dots +$X${}_{n}$**, then at time ${t}_{n+1},$ the walker moves to state

**S${}_{n+1}=\phantom{\rule{0.277778em}{0ex}}$S${}_{n}+$X${}_{n+1}$**, where

**X${}_{n+1}$**is a ${\mathbb{Z}}_{+}^{d}$-valued r.v. that generates a path on ${\mathbb{Z}}_{+}^{d}$ running in all non-negative directions from

**S${}_{n}$**.

**X**${}_{n}{\epsilon}_{{t}_{n}}$ of an underlying random walk is an atomic random measure that is often a convenient alternative interpretation.

#### Related Literature

**$\left\{{\mathrm{X}}_{0},{\mathrm{X}}_{1},\dots \right\}$**of iid r.v.’s. Note that if

**X**${}_{i}$’s are non-negative, then $\left\{{\mathbf{S}}_{n}\right\}$ is referred to as a renewal process. If

**X**${}_{i}$’s are real-valued, $\left\{{\mathbf{S}}_{n}\right\}$ is recurrent (cf. Tak$\stackrel{\xb4}{\mathrm{a}}$cs [2]).

**X**${}_{n}$’s can be position dependent, that is when

**X**${}_{n}$ depends on ${t}_{n}-{t}_{n-1}$ only, $n=1,2,\dots $, i.e., the time since the previous jump. Now because very often, one is concerned about escape of a random walk from a bounded set, the underlying analysis of the escape is referred to fluctuations of sums of random variables. However, the “sums” are not always a traditional random walk with independent jumps (cf. Andersen [4,5]). Besides, the fluctuations are also mentioned in reference to processes with continuous paths like Brownian motion and here the escape from a set A means crossing its boundary with a location next to A. The latter differs from leaving A and landing at a point distant from A as it takes place under non simple random walk. Hence, to include certain work in the literature we will use the common sense and space constraints.

**x**$\in {\mathbb{Z}}^{d}$ at step n and the large-n asymptotics of the functional

**X**${}_{n}{\epsilon}_{{t}_{n}}$ (${\epsilon}_{\alpha}$ is the unit measure at point a) be a marked signed random measure. Suppose

**X${}_{0},$X${}_{1},$X${}_{2},\dots $**are independent and for $n=1,2,\dots ,\phantom{\rule{0.166667em}{0ex}}$identically distributed r.v.’s valued in ${\mathbb{R}}^{d}$ while $\mathcal{T}={\sum}_{n=0}^{\infty}{\epsilon}_{{t}_{n}}$ is the associated support counting measure. Thus, ${N}_{t}=\mathcal{T}\left[0,t\right]$ is the associated counting process. Then, the CTRW is $S\left(t\right)=\mathcal{S}\left[0,t\right]$. The inter-renewal times ${t}_{1}-{t}_{0},{t}_{2}-{t}_{1},\dots $ are referred to as waiting times. If $\mathcal{S}$ is with position independent marking, then $S\left(t\right)$ is called decoupled. A coupled CTRW is $S\left(t\right)$ such that $\mathcal{S}$ is with position dependent marking, that is

**X**${}_{n}$ depends on ${t}_{n}-{t}_{n-1}$. The marks

**X**${}_{n}$’s are called jumps and in physics, they represent instantaneous jumps of a diffusing walker. (A so-called CTRW characteristic function pertains to fractional diffusion equations.) CTRW’s find applications in physics, insurance, and finance. The literature on CTRW’s is contained within its own terminology distinct from that in on random walks. It needs more scrutiny to see one and the same notions. See interesting surveys in Kutner and Masoliver [18] and Scalas [19]. See Balakrishnan and Khantha paper about the first passage time in CTRW [20].

## 2. The Operational Calculus of One-Dimensional Random Walks

**Theorem**

**1.**

**Proof.**

- (i)
- ${\mathcal{D}}^{k}$ is a linear functional on the space of all functions analytic at zero.
- (ii)
- ${\mathcal{D}}_{x}^{k}\left(\mathbf{1}\left(x\right)\right)=1,\phantom{\rule{0.166667em}{0ex}}$where 1$\left(x\right)=1$ for all $x\in \mathbb{R}$.
- (iii)
- Let g be an analytic function at zero. Then,$$\begin{array}{c}\hfill {\mathcal{D}}_{x}^{k}\left({x}^{j}g\left(x\right)\right)={\mathcal{D}}_{x}^{k-j}g\left(x\right)\end{array}$$

**Proof.**

- (iv)
- In particular, if $j=k$, we have ${\mathcal{D}}_{x}^{k}\left({x}^{k}g\left(x\right)\right)=g\left(0\right)$.
- (v)
- If ${x}^{-s}f\left(x\right)$ is analytic at zero, then$$\begin{array}{c}\hfill {\mathcal{D}}_{x}^{k}\left({x}^{-s}f\left(x\right)\right)={\mathcal{D}}_{x}^{k+s}\left(f\left(x\right)\right)\end{array}$$
- (vi)
- If $a\left(x\right)={\sum}_{i=0}^{\infty}{a}_{i}{x}^{i}$, then$$\begin{array}{c}\hfill {\mathcal{D}}_{x}^{k}a\left(x\right)=\sum _{i=0}^{k}{a}_{i}\end{array}$$$$\begin{array}{c}\hfill {\mathcal{D}}_{x}^{k}a\left(xy\right)=\sum _{i=0}^{k}{a}_{i}{y}^{i}.\end{array}$$

- (vii)
- For any real number a and for a positive integer n, except for $a=n=1$, it holds true that$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\mathcal{D}}_{x}^{k}\left\{\frac{1}{{\left(1-ax\right)}^{n}}\right\}=\left\{\begin{array}{cc}k+1,\hfill & a=n=1\hfill \\ \sum _{j=0}^{k}\left(\genfrac{}{}{0pt}{}{n+j-1}{j}\right){a}^{j},\hfill & \mathrm{else}.\hfill \end{array}\right.\hfill \end{array}$$
- (viii)
- For any two real numbers $a,b$ and two positive integers n and r (except for $a=n=1$ and $b=r=1$) it holds that$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\mathcal{D}}_{x}^{k}\left\{\frac{1}{{\left(1-ax\right)}^{n}}\frac{1}{{(1-bx)}^{r}}\right\}=\sum _{j=0}^{k}\left(\genfrac{}{}{0pt}{}{n+j-1}{j}\right){a}^{j}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\sum _{i=0}^{k-j}\left(\genfrac{}{}{0pt}{}{r+i-1}{i}\right){b}^{i}.\hfill \end{array}$$

**Proof.**

- (ix)
- If X be an integer-valued non-negative r.v. with $h\left(z\right)=E{z}^{X}$ and k is a positive integer, then$$\begin{array}{c}\hfill E{z}^{X\wedge k}={\mathcal{D}}_{x}^{k}\left\{h\left(xz\right)+{z}^{k}[1-h\left(x\right)]\right\}.\end{array}$$
- (x)
- If X be an integer-valued non-negative r.v. with $h\left(z\right)=E{z}^{X}$ and k is a positive integer, then$$\begin{array}{c}\hfill E{z}^{{(X-k)}^{+}}={\mathcal{D}}_{x}^{k}\left\{h\left(x\right)+{z}^{-k}[h\left(z\right)-h\left(xz\right)]\right\}.\end{array}$$

**Example**

**1.**

## 3. Random Walks on Infinite Graphs and Cybersecurity: A Bivariate Model

**Theorem**

**2.**

**Theorem**

**3.**

**Example**

**2.**

- As previously, the attack times $\left\{{t}_{1},{t}_{2},\dots \right\}$ form a Poisson point process of rate λ.
- Inter-observation times are constant, i.e.,${\Delta}_{k}={\tau}_{k}-{\tau}_{k-1}=c$ a.s., $L\left(\theta \right)={e}^{-\theta c}$.
- Nodes lost per strike have an arbitrary finite discrete distribution, i.e., $P\{{n}_{k}=j\}={p}_{j}$, $\phantom{\rule{0.166667em}{0ex}}j=1,\dots ,R$ and PGF $g\left(z\right)={\sum}_{s=1}^{R}{p}_{s}{z}^{s}$, with
**p**$=\left({p}_{1},\dots ,{p}_{R}\right)$ - Weight per node ${w}_{jk}\in \left[Gamma\left(\alpha ,\beta \right)\right]$, so we have LST $l\left(z\right)={\left(\frac{\beta}{z+\beta}\right)}^{\alpha}$.
- The initial functional ${\gamma}_{0}=1$ (i.e., zero initial damage).

**Remark**

**2.**

**Theorem**

**4.**

## 4. Time Insensitive Random Walk and Applications

**Example**

**3.**

**u**$=\left({u}_{1},\cdots ,{u}_{d}\right)$,

**v**$({v}_{1},\cdots ,{v}_{d})\in {\mathbb{C}}^{d}$ with ${u}_{i},{v}_{i}\in \overline{B}\left(0,1\right)$ for $i=1,\cdots ,d$, $\mathbf{\eta},\mathbf{\xi}\in {\mathbb{R}}^{l}$, and $\vartheta ,\theta \in {\mathbb{C}}_{+}$. (i.e., Re$\left(\vartheta \right)\ge 0$ and Re$\left(\theta \right)\ge 0$.)

**Theorem**

**5.**

**u∘v**is the Hadamard product of vectors

**u**and

**v**.

**Example**

**4.**

**P**and ${t}_{1}$. (With position independent marking, that is, assuming

**P**and ${t}_{1}$ independent, the computation can be straightforward.) Now applying Theorem 5, we have

## 5. Higher Dimensional Random Walks

**Theorem**

**6.**

**Example**

**5.**

**Example**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

## 6. Time Sensitive Analysis of Random Walks

**Theorem**

**9.**

**Theorem**

**10.**

**Example**

**7.**

- The jump times $\left\{{t}_{1},{t}_{2},\dots \right\}$ form a Poisson point process of rate λ.
- Inter-observation times ${\Delta}_{n}$ are exponentially distributed with parameter μ, so their LST is $L\left(z\right)=\frac{\mu}{\mu +z}$.
- The marks of the real time process are geometrically distributed with parameter a, so their PGF is $g\left(z\right)=\frac{az}{1-bz}$ where $b=1-a$.
- The initial functional ${\gamma}_{0}=1$ (i.e., zero initial state and time).

**Proposition**

**1.**

**Theorem**

**11.**

**Corollary**

**1.**

**Theorem**

**12.**

**Theorem**

**13.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Pearson, K. The Problem of the Random Walk. Nature
**1905**, 72, 294. [Google Scholar] [CrossRef] - Takács, L. On fluctuations of sums of random variables. In Sudies in Probability and Ergodic Theory. Advances in Mathematics. Supplementary Studies; Rota, G.C., Ed.; Academic Press: New York, NY, USA, 1978; Volume 2, pp. 45–93. [Google Scholar]
- Unver, I.; Tundzh, Y.S.; Ibaev, E. Laplace-Stieltjes transform of the distribution of the first moment of crossing the level a (a > 0) by a semi-Markovian random walk with positive drift and negative jumps. Autom. Control. Comput. Sci.
**2014**, 48, 144–149. [Google Scholar] [CrossRef] - Andersen, E.S. On the fluctuations of sums of random variables. Math. Scand.
**1953**, 1, 263. [Google Scholar] [CrossRef][Green Version] - Andersen, E.S. On the fluctuations of sums of random variables II. Math. Scand.
**1954**, 2, 194. [Google Scholar] [CrossRef][Green Version] - Rayleigh, J.W.S. The Problem of the Random Walk. Nature
**1905**, 72, 318. [Google Scholar] [CrossRef][Green Version] - Takács, L. Random walk on a finite group. Acta Sci. Math.
**1983**, 45, 395–408. [Google Scholar] - Takacs, C. Biased random walks on directed trees. Probab. Theory Relat. Fields
**1983**, 111, 123–139. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Syski, R. Lajos Takács and his work. J. Appl. Math. Stoch. Anal.
**1994**, 7, 215–237. [Google Scholar] [CrossRef][Green Version] - Van den Berg, M. Exit and Return of a Simple Random Walk. Potential Anal.
**2005**, 23, 45–53. [Google Scholar] [CrossRef] - Mogul’skiĭ, A.A. Local limit theorem for the first crossing time of a fixed level by a random walk. Sib. Adv. Math.
**2010**, 20, 191–200. [Google Scholar] [CrossRef] - Becker, M.; König, W. Moments and Distribution of the Local Times of a Transient Random Walk on ℤ
^{d}. J. Theor. Probab.**2008**. [Google Scholar] [CrossRef] - Csáki, E.; Földes, A.; Révész, P. Maximal Local Time of a d-dimensional Simple Random Walk on Subsets. J. Theor. Probab.
**2005**, 18, 687–717. [Google Scholar] [CrossRef][Green Version] - Gluck, D. First hitting times for some random walks on finite groups. J. Theor. Probab.
**1999**, 12, 739–755. [Google Scholar] [CrossRef] - Fayolle, G.; Iasnogorodski, R.; Malyshev, V. Random Walks in the Quarter Plane; Springer: Berlin/Heisenberg, Germany, 2017. [Google Scholar] [CrossRef]
- Hildebrand, M. A survey of results on random random walks on finite groups. Probab. Surv.
**2005**, 2. [Google Scholar] [CrossRef] - Montroll, E.W.; Weiss, G.H. Random Walks on Lattices. II. J. Math. Phys.
**1965**, 6, 167–181. [Google Scholar] [CrossRef] - Kutner, R.; Masoliver, J. The continuous time random walk, still trendy: Fifty-year history, state of art and outlook. Eur. Phys. J. B
**2017**, 90. [Google Scholar] [CrossRef][Green Version] - Scalas, E. The application of continuous-time random walks in finance and economics. Phys. A Stat. Mech. Its Appl.
**2006**, 362, 225–239. [Google Scholar] [CrossRef] - Balakrishnan, V.; Khantha, M. First passage time and escape time distributions for continuous time random walks. Pramana
**1983**, 21, 187–200. [Google Scholar] [CrossRef] - Blanchard, P.; Volchenkov, D. Random Walks and Diffusions on Graphs and Databases; Springer: Berlin/Heisenberg, Germany, 2011. [Google Scholar] [CrossRef][Green Version]
- Brémaud, P. Discrete Probability Models and Methods; Springer: Berlin/Heisenberg, Germany, 2017. [Google Scholar] [CrossRef]
- Fujie, F.; Zhang, P. Covering Walks in Graphs; Springer: New York, NY, USA, 2014. [Google Scholar] [CrossRef]
- Sarkar, P.; Moore, A.W. Random Walks in Social Networks and their Applications: A Survey. In Social Network Data Analytics; Springer: New York, NY, USA, 2011; Chapter 3. [Google Scholar] [CrossRef]
- Shi, Z. Branching Random Walks; Springer: Berlin/Heisenberg, Germany, 2015. [Google Scholar] [CrossRef]
- Telcs, A. Random Walks on graphs, electric networks and fractals. Probab. Theory Relat. Fields
**1989**, 82, 435–449. [Google Scholar] [CrossRef] - Abolnikov, L.; Dshalalow, J.H. Ergodicity conditions and invariant probability measure for an embedded Markov chain in a controlled bulk queueing system with a bilevel service delay discipline part I. Appl. Math. Lett.
**1992**, 5, 25–27. [Google Scholar] [CrossRef][Green Version] - Abolnikov, L.; Dshalalow, J.H. A first passage problem and its applications to the analysis of a class of stochastic models. J. Appl. Math. Stoch. Anal.
**1992**, 5, 83–97. [Google Scholar] [CrossRef][Green Version] - Abolnikov, L.; Dshalalow, J.H. On a multilevel controlled bulk queueing system MX/Gr,R/1. J. Appl. Math. Stoch. Anal.
**1992**, 5, 237–260. [Google Scholar] [CrossRef][Green Version] - Abolnikov, L.M.; Dshalalow, J.H. Semi-regenerative analysis of controlled bulk queueing systems with a bilevel service delay discipline and some ergodic theorems. Comput. Math. Appl.
**1993**, 25, 107–116. [Google Scholar] [CrossRef][Green Version] - Abolnikov, L.; Agarwal, R.P.; Dshalalow, J.H. Random walk analysis of parallel queueing stations. Math. Comput. Model.
**2008**, 47, 452–468. [Google Scholar] [CrossRef] - Abolnikov, L.M.; Dshalalow, J.H.; Dukhovny, A.M. On stochastic processes in a multilevel control bulk queueing system. Stoch. Anal. Appl.
**1992**, 10, 155–179. [Google Scholar] [CrossRef] - Abolnikov, L.M.; Dshalalow, J.H.; Dukhovny, A.M. Stochastic analysis of a controlled bulk queueing system with continuously operating server: Continuous time parameter queueing process. Stat. Probab. Lett.
**1993**, 16, 121–128. [Google Scholar] [CrossRef] - Abolnikov, L.M.; Dshalalow, J.H.; Dukhovny, A.M. A multilevel control bulk queueing system with vacationing server. Oper. Res. Lett.
**1993**, 13, 183–188. [Google Scholar] [CrossRef] - Dshalalow, J.H. On a first passage problem in general queueing systems with multiple vacations. J. Appl. Math. Stoch. Anal.
**1992**, 5, 177–192. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Tadj, L. On applications of first excess level random processes to queueing systems with random server capacity and capacity dependent service time. Stochastics Stoch. Rep.
**1993**, 45, 45–60. [Google Scholar] [CrossRef] - Dshalalow, J.H. First excess levels of vector processes. J. Appl. Math. Stoch. Anal.
**1994**, 7, 457–464. [Google Scholar] [CrossRef][Green Version] - Dshalalow, J. First excess level analysis of random processes in a class of stochastic servicing systems with global control. Stoch. Anal. Appl.
**1994**, 12, 75–101. [Google Scholar] [CrossRef] - Dshalalow, J. Excess level processes in queueing. In Advances in Queueing; Dshalalow, J., Ed.; CRC Press: Boca Raton, FL, USA, 1995; pp. 243–262. [Google Scholar]
- Dshalalow, J. On the level crossing of multi-dimensional delayed renewal processes. J. Appl. Math. Stoch. Anal.
**1997**, 10, 355–361. [Google Scholar] [CrossRef][Green Version] - Dshalalow, J.H.; Motir, R. Random Walk Processes in a Bilevel (M-N)-Policy Queue with Multiple Vacations. Qual. Technol. Quant. Manag.
**2011**, 8, 303–332. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Russell, G. On a single-server queue with fixed accumulation level, state dependent service, and semi-Markov modulated input flow. Int. J. Math. Math. Sci.
**1992**, 15, 593–600. [Google Scholar] [CrossRef][Green Version] - Dshalalow, J.H.; Yellen, J. Bulk input queues with quorum and multiple vacations. Math. Probl. Eng.
**1996**, 2, 95–106. [Google Scholar] [CrossRef] - Agarwal, R.; Dshalalow, J. New fluctuation analysis of D-policy bulk queues with multiple vacations. Math. Comput. Model.
**2005**, 41, 253–269. [Google Scholar] [CrossRef] - Abolnikov, L.; Dshalalow, J.H.; Treerattrakoon, A. On a dual hybrid queueing system. Nonlinear Anal. Hybrid Syst.
**2008**, 2, 96–109. [Google Scholar] [CrossRef] - Dshalalow, J.H. Queues with hysteretic control by vacation and post-vacation periods. Queueing Syst.
**1998**, 29, 231–268. [Google Scholar] [CrossRef] - Dshalalow, J.; Dikong, E. On generalized hysteretic control queues with modulated input and state dependent service. Stoch. Anal. Appl.
**1999**, 17, 937–961. [Google Scholar] [CrossRef] - Dikong, E.E.; Dshalalow, J.H. Bulk input queues with hysteretic control. Queueing Syst.
**1999**, 32, 287–304. [Google Scholar] [CrossRef] - Dshalalow, J.; Kim, S.; Tadj, L. Hybrid queueing systems with hysteretic bilevel control policies. Nonlinear Anal. Theory Methods Appl.
**2006**, 65, 2153–2168. [Google Scholar] [CrossRef] - Bacot, J.B.; Dshalalow, J. A bulk input queueing system with batch gated service and multiple vacation policy. Math. Comput. Model.
**2001**, 34, 873–886. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Merie, A. Fluctuation analysis in queues with several operational modes and priority customers. Top
**2018**, 26, 309–333. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Merie, A.; White, R.T. Fluctuation Analysis in Parallel Queues with Hysteretic Control. Methodol. Comput. Appl. Probab.
**2019**, 22, 295–327. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Huang, W. A stochastic games with a two-phase conflict. In Jubilee Volume: Legacy of the Legend, Professor V. Lakshmikantham; Cambridge Scientific Publishers: Cottenham, UK, 2009; pp. 201–209. [Google Scholar]
- Dshalalow, J.H.; Huang, W. Tandem antagonistic games. Nonlinear Anal. Ser. Theory Methods
**2009**, 71, 259–270. [Google Scholar] - Dshalalow, J.H.; Huang, W. Sequential antagonistic games with an auxiliary initial phase. In Functional Equations, Difference Inequalities, and Ulam Stability Notions (F.U.N.); Nova Science Publishers: New York, NY, USA, 2010; Chapter 2; pp. 15–36. [Google Scholar]
- Dshalalow, J.H. Fluctuations of Recurrent Processes and Their Applications to the Stock Market. Stoch. Anal. Appl.
**2004**, 22, 67–79. [Google Scholar] [CrossRef] - Dshalalow, J.H. On exit times of a multivariate random walk with some applications to finance. Nonlinear Anal.
**2005**, 63, 569–577. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Liew, A. Level crossings of an oscillating marked random walk. Comput. Math. Appl.
**2006**, 52, 917–932. [Google Scholar] [CrossRef][Green Version] - Dshalalow, J.; Liew, A. On fluctuations of a multivariate random walk with some applications to stock options trading and hedging. Math. Comput. Model.
**2006**, 44, 931–944. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Liew, A. On exit times of a multivariate random walk and its embedding in a quasi Poisson process. Stoch. Anal. Appl.
**2006**, 24, 451–474. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Iwezulu, K. Discrete versus continuous operational calculus in antagonistic stochastic games. São Paulo J. Math. Sci.
**2017**, 11, 471–489. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Ke, H.J. Layers of noncooperative games. Nonlinear Anal. Ser. Theory Methods
**2009**, 71, 283–291. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Ke, H.J. Multilayers in a modulated stochastic game. J. Math. Anal. Appl.
**2009**, 353, 553–565. [Google Scholar] [CrossRef][Green Version] - Dshalalow, J.H.; Treerattrakoon, A. Set-theoretic inequalities in stochastic noncooperative games with coalition. J. Inequalities Appl.
**2008**, 2008, 1–14. [Google Scholar] [CrossRef] - Dshalalow, J.H.; White, R. On Reliability of Stochastic Networks. Neural Parallel Sci. Comput.
**2013**, 21, 141–160. [Google Scholar] - Dshalalow, J.H.; White, R. On Strategic Defense in Stochastic Networks. Stoch. Anal. Appl.
**2014**, 32, 365–396. [Google Scholar] [CrossRef] - Dshalalow, J.H. Time dependent analysis of multivariate marked renewal processes. J. Appl. Probab.
**2001**, 38, 707–721. [Google Scholar] [CrossRef] - Agarwal, R.P.; Dshalalow, J.H.; O’Regan, D. Time sensitive functionals of marked Cox processes. J. Math. Anal. Appl.
**2004**, 293, 14–27. [Google Scholar] [CrossRef][Green Version] - Al-Matar, N.; Dshalalow, J.H. Time sensitive functionals in a queue with sequential maintenance. Stoch. Model.
**2011**, 27, 687–704. [Google Scholar] [CrossRef] - Dshalalow, J.H. Random Walk Analysis in Antagonistic Stochastic Games. Stoch. Anal. Appl.
**2008**, 26, 738–783. [Google Scholar] [CrossRef] - Dshalalow, J.H. On multivariate antagonistic marked point processes. Math. Comput. Model.
**2009**, 49, 432–452. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Bacot, J.B. On functionals of a marked Poisson process observed by a renewal process. Int. J. Math. Math. Sci.
**2001**, 26, 427–436. [Google Scholar] [CrossRef][Green Version] - Dshalalow, J.H.; Nandyose, K.M. Continous time interpolation of monotone marked random measures with applications. Neural Parallel Sci. Comput.
**2018**, 26, 119–141. [Google Scholar] [CrossRef] - Dshalalow, J.H.; Nandyose, K.M. Real time analysis of signed marked random measures with applications to finance and insurance. Nonlinear Dyn. Syst. Theory
**2018**, 19, 36–54. [Google Scholar] - Dshalalow, J.H.; Nandyose, K.M.; White, R.T. Time dependent analysis of stochastic games of three players with applications. Math. Stat.
**2021**. pending minor revision. [Google Scholar] - White, R.T.; Dshalalow, J.H. Characterizations of random walks on random lattices and their ramifications. Stoch. Anal. Appl.
**2019**, 38, 307–342. [Google Scholar] [CrossRef] - Antal, T.; Redner, S. Escape of a Uniform Random Walk from an Interval. J. Stat. Phys.
**2006**, 123, 1129–1144. [Google Scholar] [CrossRef][Green Version] - Hughes, B.D. Random Walks and Random Environments; Clarendon Press Oxford University Press: Oxford, NY, USA, 1995. [Google Scholar]
- Redner, S. A Guide to First-Passage Processes; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar] [CrossRef]
- Kyprianou, A.E.; Pistorius, M.R. Perpetual options and Canadization through fluctuation theory. Ann. Appl. Probab.
**2003**, 13, 1077–1098. [Google Scholar] [CrossRef][Green Version] - Muzy, J.; Delour, J.; Bacry, E. Modelling fluctuations of financial time series: From cascade process to stochastic volatility model. Eur. Phys. J. B
**2000**, 17, 537–548. [Google Scholar] [CrossRef][Green Version] - Uchaikin, V.V.; Gusarov, G.G. Analysis of the structure function for the spatial distribution of galaxies in the random-walk model. Russ. Phys. J.
**1997**, 40, 707–710. [Google Scholar] [CrossRef] - Zhou, J.L.; Sun, Y.S.; Zhou, L.Y. Evidence for Lévy Random Walks in the Evolution of Comets from the Oort Cloud. Celest. Mech. Dyn. Astron.
**2002**, 84, 409–427. [Google Scholar] [CrossRef] - Odagaki, T.; Kasuya, K. Alzheimer random walk. Eur. Phys. J. B
**2017**, 90. [Google Scholar] [CrossRef] - Jabbari, B.; Zhou, Y.; Hillier, F.S. A decomposable random walk model for mobility in wireless communications. Telecommun. Syst.
**2001**, 16, 523–537. [Google Scholar] [CrossRef] - Asmussen, S. Phase-Type Representations in Random Walk and Queueing Problems. Ann. Probab.
**1992**, 20, 772–789. [Google Scholar] [CrossRef] - Bayer, N.; Boxma, O.J. Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks. Queueing Syst.
**1996**, 23, 301–316. [Google Scholar] [CrossRef][Green Version] - Cohen, J. Random Walk with a Heavy-Tailed Jump Distribution. Queueing Syst.
**2002**, 40, 35–73. [Google Scholar] [CrossRef] - Gannon, M.; Pechersky, E.; Suhov, Y.; Yambartsev, A. Random walks in a queueing network environment. J. Appl. Probab.
**2016**, 53, 448–462. [Google Scholar] [CrossRef][Green Version] - Guillemin, F.; van Leeuwaarden, J.S.H. Rare event asymptotics for a random walk in the quarter plane. Queueing Syst.
**2010**, 67, 1–32. [Google Scholar] [CrossRef][Green Version] - Janssen, A.; van Leeuwaarden, J. Spitzer’s identity for discrete random walks. Oper. Res. Lett.
**2018**, 46, 168–172. [Google Scholar] [CrossRef][Green Version] - Lemoine, A.J. On Random Walks and StableGI/G/1 Queues. Math. Oper. Res.
**1976**, 1, 159–164. [Google Scholar] [CrossRef] - Stadje, W. The embedded random walk in the stationary M/M/1 queue. Methodol. Comput. Appl. Probab.
**2002**, 4, 143–151. [Google Scholar] [CrossRef] - Zorine, A.V. Study of a Service Process by a Loop Algorithm by Means of a Stopped Random Walk. In Information Technologies and Mathematical Modelling. Queueing Theory and Applications; Springer: Berlin/Heisenberg, Germany, 2019; pp. 121–135. [Google Scholar] [CrossRef]
- Bingham, N.H. Fluctuation theory in continuous time. Adv. Appl. Probab.
**1975**, 7, 705–766. [Google Scholar] [CrossRef] - Bingham, N.H. Random walk and fluctuation theory. In Handbook of Statistics; Shanbhag, D.N., Rao, C.R., Eds.; Elsevier: Amsterdam, The Netherlands, 2001; Volume 19, pp. 171–213. [Google Scholar]
- Bladt, M.; Nielsen, B.F. Matrix-Exponential Distributions in Applied Probability; Springer: New York, NY, USA, 2017. [Google Scholar] [CrossRef]
- Foss, S.; Korshunov, D.; Zachary, S. An Introduction to Heavy-Tailed and Subexponential Distributions; Springer: New York, NY, USA, 2013. [Google Scholar] [CrossRef]
- Gut, A. Stopped Random Walks; Springer: New York, NY, USA, 2009. [Google Scholar] [CrossRef]
- Iksanov, A. Renewal Theory for Perturbed Random Walks and Similar Processes; Springer: Berlin/Heisenberg, Germany, 2016. [Google Scholar] [CrossRef]
- Lawler, G.F. Intersections of Random Walks; Springer: New York, NY, USA, 2013. [Google Scholar] [CrossRef]
- Slade, G. The Lace Expansion and Its Applications; Springer: Berlin/Heisenberg, Germany, 2006. [Google Scholar] [CrossRef][Green Version]
- Telcs, A. The Art of Random Walks; Springer: Berlin/Heisenberg, Germany, 2006. [Google Scholar] [CrossRef]
- Wijesundera, I.; Halgamuge, M.N.; Nanayakkara, T.; Nirmalathas, T. Natural Disasters, When Will They Reach Me? Springer: Singapore, 2016. [Google Scholar] [CrossRef]
- Dshalalow, J.H. Single-server queues with controlled bulk service, random accumulation level, and modulated input. Stoch. Anal. Appl.
**1993**, 11, 29–41. [Google Scholar] [CrossRef] - White, R.T. Reliability of networks under stochastic attacks. manuscript in progress.
- Agarwal, R.P.; Dshalalow, J.H.; O’Regan, D. Random observations of marked Cox processes. Time insensitive functionals. J. Math. Anal. Appl.
**2004**, 293, 1–13. [Google Scholar] [CrossRef][Green Version] - Agarwal, R.P.; Dshalalow, J.H. On multivariate delayed recurrent processes. Pan Am. Math. J.
**2005**, 15, 35–49. [Google Scholar] - White, R.T. On the exiting patterns of sums of independent random vectors with an application to stochastic networks.
**2021**. submitted. [Google Scholar] - Talbot, A. The Accurate Numerical Inversion of Laplace Transforms. IMA J. Appl. Math.
**1979**, 23, 97–120. [Google Scholar] [CrossRef] - Abate, J.; Whitt, W. A Unified Framework for Numerically Inverting Laplace Transforms. Informs J. Comput.
**2006**, 18, 408–421. [Google Scholar] [CrossRef] - White, R.T. On exits and overshoots of dependent jump processes. manuscript in progress.
- Dshalalow, J.H.; White, R.T. Time sensitive analysis of independent and stationary increment processes. J. Math. Anal. Appl.
**2016**, 443, 817–833. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Predicted and empirical (simulated) probabilities for parameters $(1,[0.25,0.5,0.25],[1.5,1.5],1,{M}_{1},1000,1000)$.

**Figure 3.**Predicted and empirical (simulated) probabilities for parameters $(1,[0.25,0.5,0.25],[1.5,1.5],$$1,{M}_{1},100,50)$.

**Figure 4.**Predicted and empirical (simulated) probabilities $P({\nu}_{1}<{\nu}_{2})$ with parameters $\mu =1$, $p=0.5$, and $0\le {M}_{2}\le 20$ for various ${M}_{1}$ values.

**Figure 5.**Predicted and empirical (simulated) probabilities $P({\nu}_{1}={\nu}_{2})$ with parameters $\mu =1$, $p=0.5$, and $0\le {M}_{2}\le 20$ for various ${M}_{1}$ values.

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Dshalalow, J.H.; White, R.T. Current Trends in Random Walks on Random Lattices. *Mathematics* **2021**, *9*, 1148.
https://doi.org/10.3390/math9101148

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Dshalalow JH, White RT. Current Trends in Random Walks on Random Lattices. *Mathematics*. 2021; 9(10):1148.
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Dshalalow, Jewgeni H., and Ryan T. White. 2021. "Current Trends in Random Walks on Random Lattices" *Mathematics* 9, no. 10: 1148.
https://doi.org/10.3390/math9101148