In this work, we introduce nearest neighbour q-random walks on the integers and on the two-dimensional integer lattice with transition probabilities q-varying by the number of steps, 0 < q < 1. These q-random walks are de ned as Markov chains discrete time stochastic processes. Our main results characterize under which conditions the considered q-random walks are transient or reccurent. Also, we de ne the relative continuous time q-random walks stochastic processes. Moreover, we present a q-Brownian motion as a continuous analogue of the q-random walk stochastic process on the integers. The maxima and rst hitting time of this q-Brownian motion are studied. Furthermore, we produce simulations in R of all the considered stochastic processes, indicating rst hitting times to the origin. As further study, we propose nearest neighbour q-random walks on the three-dimensional integer lattice.
Copyright ' by the paper's authors. Copying permitted for private and academic purposes. de ned as Markov chains discrete time stochastic processes. Our main results characterize under which conditions the considered q-random walks are transient or reccurent. Also, we de ne the relative continuous time q- random walks stochastic processes. Moreover, we present a q-Brownian motion as a continuous analogue of the q-random walk stochastic process on the integers. The maxima and rst hitting time of this q-Brownian motion are studied. Furthermore, we produce simulations in R of all the considered stochastic processes, indicating rst hitting times to the origin. As further study, we propose nearest neighbour q-random walks on the three-dimensional integer lattiice. q-Random walks in square or triangular lattices, where a vertex is added to one of the four or six directions respectively, according to edge (transition) probabilities varying by the number of previously visited vertices, can be applied to describe among others several real world phenomena arising in networkscience, neuroscience and statistical mechanics. 2 2.1
Preliminaries, De nitions and Notation
Markov Chains and Classi cation of States Let a discrete time stochastic process fX(t); t 2 T g; where T a countable set. If T is the set of nonnegative integers, then the process is denoted as Xn; n = 0; 1; 2; : : : : If Xn = i, then the process is said to be in state i at time n. The one-step transition probability from state i to state j; say Pi;j ; is given by
Pi;j = P (Xn+1 = j + 1=Xn = i; Xn 1 = in 1; : : : ; X0 = i0) ; for all states i0; i1; : : : ; in 1; i; j, n 0. This stochastic process is known as a Markov chain. The n-step transition probability, say Pin;j , is given by
Pin;j = P (Xn+m = j + 1=Xm = i) ; n 0; i; j 0: If Pin;j > 0 for some n 0; we say that state j is accessible from state i: States i and j communicate if state j is accessible from state i and state i is accessible from state j: The Markov chain is said to be irreducible if all states communicate with each other.
A state i is recurrent if with probability 1, the process will reenter state i. A state i is transient if with probability< 1, the process will reenter state i. A state i is recurrent if P1 n=0 Pin;i = 1. A state i is transient if P1
n=0 Pin;i < 1: If state i is recurrent and state i communicates with state j, the state j is recurrent. 2.2
q-Series Preliminaries, 0 < q < 1 The q-binomial coe cient is de ned by where n k q :=
(q; q)n (q; q)k(q; q)n k =
[n]q! [k]q![n k]q
; [n]q! = [1]q[2]q : : : [n]q = (q; q)n (1 q)n =
Qn k=1(1 (1 q)n
qk) x + y n q
n = X qk(y n+k) x k=0 k q n y k q
: (1) (2) (3) (4) is the q-factorial number of order n with [t]q = 11 qqt . the set f1; 2; : : : ; ng; weighted by qm1+m2+ +mk (k+21); The q-binomial coe cient nk q; for n and k positive integers, equals the k-combinations fm1; m2; : : : ; mkg of
X 1 m1<m2< <mk n qm1+m2+ +mk (k+21) = n k q : Let n be a positive integer and let x; y and q be real numbers, with q 6= 1. Then, a version of q-Cauchy formula is
The q-multinomial coe cient is de ned by (8) (9) By using q-Stirling formula (6), the next theorem is concluded.
Theorem 3.2. The q-random walk on the integers after 2n steps, starting from the origin 0; for for = q n; 0 < a < 1; n = 0; 1; 2; : : : is recurrent, while for qn ! 1; as n ! 1; is transient. constant or for > 0; 0 < q < 1: The distribution of the random variable (r.v.) X is called q-binomial distribution of the rst kind, with parameters n; and q (see Charalambides [Cha16]). 3 3.1
q-Random Walks on the Integers In this section, we introduce a nearest neighbour q-random walk on the integers with transition probabilities q-varying by the number of steps, 0 < q < 1. This random walk is de ned as a Markov chain discrete time stochastic process.
De nition 3.1. The Markov chain whose state space is the set of all integers with q-varying transition probabilities, 0 < q < 1, given by
Pi;i+1 = 1 + qi 1 qi 1 = 1
Pi;i 1; i = 0; 1; 2; : : : ; 0 < q < 1; 0 < < 1 is called q-random walk on the integers.
At each state of the q-random walk the chain either increases or dicreases by 1, with respective probabilities Pi;i+1 and Pi;i 1; i = 0; 1; 2; : : : ; of independent q-Bernoulli trials. Because all states communicates, they are either all transient or recurrent. Therefore, we consider state 0 and determine whether Pn1=0 P0n;0 is nite or in nite. Because it is impossible to be back in the initial state after an odd number of transitions, we have that Pn1=0 P02;n0+1 = 0; n 0: But the chain will be back in initial state after 2n transitions if n of them were increases and n of them were decreases. Because each q-Bernoulli trial results in an increased state with probability Pi;i+1; i = 1; 2; : : : ; 2n; by the p.f. (7), the desired probability is the q-binomial probability of the rst kind
P02;n0 = 2n n
n q(2) n q Qj2=n1(1 + qj 1) ; > 0; 0 < q < 1: Remark 3.3. 1 Possible realizations of the transience condition considered in the above theorem are among others the next two ones
Next, we present the relative continuous time q-random walk stochastic process. Analytically, consider the time interval (0; t]; t > 0, and partition it into parts which are geometrically decreasing with rate q, de ned by i(n; t) = (n)q 1qi 1t; i = 1; 2; : : : ; n; n 1: Also, consider the process generated by making a step of length = 1 to the right and a step of length = 1 to the left at every time period (n)q 1qi 1t, i = 1; 2; : : : ; n, with probability of success (right step) and probability of failure (left step) given respectively by De nition 3.4. The continuous time stochastic process fXn;q(t); t process with parameters q; n and , if the following properties hold (a) In each of the consecutive mutually disjoint time intervals of length 1; 2; : : : ; n; n 1, at most one event (right or left step) occurs and 0g; is called q-random walk stochastic i(n; t) = (n)q 1qi 1t; i = P P
Xn;q
Xn;q 0 < < 1:
1
= 1 + (qni)q1t ; = 1; i = 1; 2; : : : ; n;
(
Xn;q(ti 1); where ti
ti 1 = (n)q 1qi 1t; i = 1; 2; : : : ; n are independent. (c) The process starts at time t = 0 with Xn;q(0) = 0.
Remark 3.5. 2 The above q-random walk stochastic process, has been recently de ned by Vamvakari [Vam17]. This q-random walk stochastic process, has been proved that is approximated, as n ! 1; by a continuous analogue one fYq(t); t 0g, where the r.v. Yq(t) is the position after time t with probability density function f (y; t) = q 7=8 q 1 1 1=2 (2 )1=2 (log q 1)1=2 exp y > t
1 2 log q log2
q 3=2(1 tq1=2(1 q) 1=2; (1 q)1=2
(y q3=2 q)1=2 (y t t t) + q 1 t) + q 1 1=2 ; where the mean value and the variance 2 of the r.v. Yq(t) are given by t = E(Yq(t)) = ct; t2 = V (Yq(t)) =
(ct)2 + ct: 1
q q De nition 3.6. The continuous stochastic process fYq(t); t q, t and t2, if the following properties hold 0g; is called q-Brownian motion with parameters (A) The distribution of the increment Yq(t2) Yq(t1), with t2 t1 = (1 q)t, t > 0, is the linear transformed standardized Stieltjes-Wigert distribution with p.d.f (12), where t2 t1 = c(1 q)t and t22 t1 = 1 q q (c(1 q)t)2 + c(1 q)t: (B) The increments Yq(tk)
Yq(tk 1); where tk tk 1 = qk 1(1
q)t; k = 1; 2; : : : ; are independent.
0 and Yq(t) is continuous at t = 0.
Using suitably the above properties of the q-Brownian motion, the pdf (12) of the r.v.Yq(t), and its mean
value and variance (
P (Wq(T ) In this section, we introduce a nearest neighbourq- random walk on the two-dimensional integer lattice with transition probabilities q-varying by the number of steps, 0 < q < 1. This random walk is de ned as a Markov chain discrete time stochastic process.
De nition 3.8. The Markov chain in which at each transition is likely to take one step to the right, left, up or
down in the plane with q-varying transition probabilities given respectively by
(14)
(15)
(16)
(
1 P(i;j)(i 1;j) = (1 + 1qi 1)(1 + 2qj 1) ;
2qj 1 P(i;j);(i;j+1) = (1 + 1qi 1)(1 + 2qj 1) ;
1qi 1 2qj 1
P(i;j)(i;j 1) = (1 + 1qi 1)(1 + 2qj 1) ; i; j = 0; 1; 2; : : : ; where 0 < 1 < 1; 0 < 2 < 1; 0 < q < 1; is called q-random walk on the two-dimensional integer lattice.
We now study if this Markov chain is also recurrent as the q-random walk in one-dimension. Because the
qrandom walk in two-dimensions is irreducible, it follows that all states are recurrent if state 0= (0; 0) is recurrent.
Now after 2n transitions the chain will be back if for some x; x = 0; 1; : : : ; n; the 2n steps consist of x steps to
the right, x to the left, n x up, and n x down. Each step will be independenlty in any of these directions
with varying probabilities given respectively by (
Let X+ be the number of \right steps", X be the number of \left steps", Y + be the number of \up steps" and Y be the number of '\down steps" after 2n steps. We need to nd the probability function of the 4th-variate random variable (X+; X ; Y +; Y ).
Let Ai be the event of \right step" at the ith step, i = 1; 2; : : : ; n, and consider a permutation
(i1; i2; : : : ; ix; ix+1; : : : ; in) of the n positive integers. Also, let Bj be the event of \up step" at the jth step,
j = 1; 2; : : : ; n with i + j = n, and consider a permutation (j1; j2; : : : ; jn x; jn x+1; : : : ; in) of the n positive
integers. Then using the independence of each step, the q-varying transition probabilities (
Lemma 3.9. Let the 4th-variate random variable (X+; X ; Y +; Y ) , where X+ be the number of \right steps", X be the number of \left steps", Y + be the number of \up steps" and Y be the number of '\down steps" after 2n steps in the two-dimensional q-random walk starting from the origin 0. Then, it holds that P0n;0 = P
X+ = x; X = x; Y + = n = n
x = = 22nq2(n2)
2n x;x;n x q q x(n x) 1
2 Qi2=n1 (1 + 1qi 1 + 2qn i 1 + 1 2qn 2) x; Y 2x ; x = 0; 1; 2; : : : ; n:
(18)
By the above lemma 3.9, the q-Cauchy formula (4) and the q-Stirling formula (6), the next theorem is concluded.
Theorem 3.10. Let the q-random walk on the 2D dimensional integer lattice after 2n steps, starting from the origin 0; with 1= 2 = qn=2; n = 0; 1; 2; : : :. Then for 2 constant or for 2 = q n; 0 < a < 1; n = 0; 1; 2; : : : ; the 2D q-random walk is recurrent, while for 2qn ! 1; as n ! 1; is transient.
Remark 3.11. 3 Possible realizations of the transience condition considered in the above theorem are among others the next two ones
2 = q cn; c > 1 or 2 = exp(O(n)):
Next, we introduce the relative bivariate q-random walk stochastic process. Analytically, let the time interval (0; t]; t > 0, and partition it into parts which are geometrically decreasing with rate q, de ned by i(n; t) = (n)q 1qi 1t; i = 1; 2; : : : ; n; n 1: Also, let the process generated by taking one step, of length = 1; to the right, left, up or down in the plane at every time period (n)q 1qi 1t; i = 1; 2; : : : ; n n 1, with q-varying transition probabilities pi;1 = P (Xi = ; Yi = 0) = pi;2 = P (Xi =
; Yi = 0) = Pi;3 = P (Xi = 0; Yi = ) = Pi;4 = P (Xi = 0; Yi = ) = 1qi 1t=(n)q
; (1 + 1qi 1t=(n)q)(1 + 2qn i 1t=(n)q)
1 (1 + 1qi 1t=(n)q)(1 + 2qn i 1t=(n)q)
2qn i 1t=(n)q (1 + 1qi 1t=(n)q)(1 + 2qn i 1t=(n)q)
1 2qn 2t2=(n)q2 (1 + 1qi 1t=(n)q)(1 + 2qn i 1t=(n)q) ; ; ; (19) where 0 < 1 < 1; 0 < 2 < 1; 0 < q < 1 with 1= 2 = qn=2: Then, at time Pin=1 i(n; t) = t; the position of the process is the bivariate r.v. (Xn;q(t); Yn;q(t)) with Xn;q(t) = Pn i=1 Yi: i=1 Xi and Yn;q(t) = Pn De nition 3.12. The continuous time bivariate stochastic process f(Xn;q(t); Yn;q(t)) ; t > 0g, is called bivariate q-random walk stochastic process with parameters n, 1, 2 and q, 0 < q < 1, if the following properties hold (a) In each of the consecutive mutually disjoint time intervals of length i(n; t) = (n)q 1qi 1t; i = 1; 2; : : : ; n; n 1, at most one event (right or left or up or down step) occurs and
Xn;q Xn;q Xn;q Xn;q 0.4 Time 0.6 0.0 0.2 0.8 1.0 0 2 8 10 0 2 8
10 02 01 10− 20− 15 01 5− 01− 51− n itsoo 0 i P n itsoo 0 i P 51 01 5 5− 01− 51− 15 10 3.3
q-Random Walks Processes and q-Brownian Simulations in R Using de nition 3.4, the q-random walk process has been simulated in R. Figures 1-6 represent the results of these simulations for various values of the parameters q, n, and t. There is strong indication, implied by theorem 3.2, that for constant or for = q n; 0 < a < 1; even for small values of n; the 1st return to the origin occurs before or at time t. While, for = q cn; c > 1 or = exp(O(n)); there is no return to the origin. 0 2 8 10 0.0 0.2 0.8 1.0 0.0 0.2 0.8 1.0 0.4 Time 0.6 0.4 Time 0.6
Also, q-Brownian motion simulation in R has been produced by de nition 3.6. Figures 7-8 represent the results of these simulations for some values of the parameters q and t. 0.0 0.2 0.4 Time 0.6 0.8 1.0 0 2 8
10
Moerover, using de nition 3.12 the two-dimensional q-random walk process has been simulated in R-package. Figures 9-14 represent the results of these simulations for various values of the parameters q, n, 1; 2 and t. Analogously as in one-dimension, there is strong indication, implied by theorem 3.10, that for 2 constant or for 2 = q n; 0 < a < 1; with 1= 2 = qn=2; even for small values of n; the 1st return to the origin occurs before or at time t: While, for 2 = q cn; c > 1 or 2 = exp(O(n)); there is no return to the origin. Note that R-codes are available under request. y 1 0.5 0 -0.5x -1
Time 4 2 0 -2 y 0 x 0 -5 -10 -15 y 0 x 0 -5 -10 -15 q-Random walks in higher than two dimensions can also be de ned. For instance, we can introduce a nearest neighbour random walk on the three-dimensional integer lattice with transition probabilities q-varying by the number of steps, 0 < q < 1. This random walk is de ned as a Markov chain discrete time stochastic process as follows.
De nition 3.13. The Markov chain in which at each transition is likely to take one step to the six directions to the right, left, up, down, in, or out in the space, with q-varying transition probabilities given respectively by 1qi 1 2qj 1 [Cha16] C, A. Charalambides. Discrete q-Distributions. John Wiley Sons, New Jersey, 2016. [Fel64] W. Feller. An Introduction to Probability Theory and its Applications- Volume 1. Wiley, NY, 1964.