PROBABILISTIC ANALYSIS OF THE TELEGRAPHER ’ S PROCESS WITH DRIFT BY MEANS OF RELATIVISTIC TRANSFORMATIONS

The telegrapher's process with drift is here examined and its distribution is 
obtained by applying the Lorentz transformation. The related characteristic function as well as the distribution are also derived by solving an initial 
value problem for the generalized telegraph equation.


Introduction
In this paper we consider the two-valued integrated telegraph signal with rightward velocity c 1 and leftward velocity -c 2 (cl,c2 > 0) and rates 11,12 of the occurrence of velocity switches, (1 when the current velocity is (-1) i+ lci, 1,2).
The classical case (c 1 c 2 c;I 12 1) has been studied in many papers and important probabilistic distributions and representations have been obtained independently by various authors and by different methods (for example, Orsingher [8], Foong [2], Foong and Kanno [3], Kabanov [4]).
When c 1 :/: c and 11 12, the motion differs from that in the classical case in that it displays a drift whose components have also been studied (see [1,6,7]).One component of the drift depends on the different velocities and the other on the different rates.These components differ substantially in the mathematical treatment they necessitate.
In particular, when 11 12, the elimination of the drift requires the Lorentz transformation of Special Relativity Theory.This was first noted by Cane [1] and further examined in [6, 7] but nowhere has an accurate analysis of the transformation and its probabilistic implications been carried out.
Here we discuss the random motion in the original frame of reference (x, t) and in the related relativistic one, (x', t') where the drift has been eliminated.12 L. BEGHIN, L. NIEDDU and E. ORSINGHER The space coordinate x' must move with velocity Vr 2 ('k2 1)(Cl + c2) "k2Cl Ale2 2(A2 + A1 A1 + A2 with respect to the original frame of reference and the time t' must either be speeded up or slowed down with respect to t, in order to elimi- nate the drift. 2(Cl q_ c2),klA2 In the frame (x',t'), the particle moves with velocities c'-+ Therefore, the probabilist, in the reference {a', attributes to the random position of the particle, a symmetric distribution p-p{a', t'}.Returning to the original coor- dinates and writing down the asymmetric distribution p-p(a,t requires careful attention due to the fact that here, differently from the Special Relativity theory, the adjustment of time depends on the random changes of the rates (and thus of the velo- city of the particle}.
In the last section of the paper we obtain the distribution p-p(a, t} by means of the usual approach, based on Fourier transforms.This also enables us to present the characteristic function in the case where a general form of drift is assumed.
The reader can easily judge how significant the simpification using the relativistic transformation is and how deep an insight into the intimate structure of the random motion is afforded.
The relativistic approach also immediately yields the form of the flow function and therefore the joint distributions of the position and of the velocity of the particle.

Features of Motion and the Governing Equation
We assume that at time t-0, a particle starts from the origin and that its initial velocity is the two-valued r.v.The current velocity V V(t), t > 0 switches from C 1 to -c 2 after an exponential- ly distributed time (with parameter 1) and from -c2 to c I after a random time with exponential distribution with parameter 2" The time intervals separated by velocity changes are independent r.v.s (also inde- pendent from V(0)).
Thus the particle moves forward with velocity c 1 and backward with velocity -c 2 and the changes are governed by a non-homogeneous Poisson process.
For the probabilistic description of the random position X-X(t)-f toV(s)ds we need the following distributions fl(x,t)dx Pr{X(t) dx, V(t)-Cl} f2(x, t)dx Pr{X(t) dx, V(t) c2}. (2.1) It is well known that the functions (2.1) are solutions of the following differential system (see [5]) + 1fl A2f2" The system (2.2) by means of the transformation P fl + f2'w fl-f2 can equivalently be written down as Op_ Ot--OW The distribution p(x, t)dx Pr{X(t)E dx} consists of a singular component con- centrated in xclt (with probability 1/2e-Alt) and in x--ct (with probability 1/2e-,t) and an absolutely continuous part spread over the interval (-c2t clt).
The absolutely continuous part of the distribution is a solution of the second-order hyperbolic equation (extracted from the differential system (2.3) by means of sub- sequent differentiations and substitutions): ) is clearly related to the drift of motion.

Efimination of the Drift by Means of a Relativistic Transformation
The elimination of the drift necessitates the use of the Lorentz transformation x'-cx + t' 7x + 5t (3.1)where the constants c, 3, 7, 5 are to be determined in such a way that the coefficients of Ox'Ot' 02p and 0" vanish.In order to evaluate the four parameters in (3.1), clearly, L. BEGHIN, L. NIEDDU and E. ORSINGHER two further conditions must be introduced.
We now have our first theorem.
Theorem 3.1: A linear transformation from the frame of reference (x,t) into (x', t'), capable of eliminating the drift in (2.4) is Proof: We first remark that any transformation of the form (3.2) Op and 02p eliminates the coefficients of -x ox'ot'" Assuming that a 1 and that the Jacobian of (3.3) is equal to 1, we get that Or,2 (h I + ,)4 Oz,2 h + h Ot --w" (3.4) Proof: We first observe that in the frame (x', t'), after the elimination of the drift, equation (2.4) is transformed into the following one 02p and therefore (3.7) Substituting (3 7) into (3 5) and then dividing by (1 + 2) 2 4A1A2 we ,readily obtain equa- tion (3.4).Remark 3.1: We are now able to infer from equation (3.4) some important features of the random motion in the frame (x', t').
(3.9) lemark 3.2: The connection between the velocities in (x, t) and in (x', t')is given by the formula which can be straightforwardly obtained from (3.2).From (3.10) it is easy to see that dx' dx which tells us how the time t' changes with (as a function of the velocity -3T)" We have the following picture: Formula (3.12) shows that the times t' and t grow at the same rate if )1 2" This explains the fact that the part of the drift due to different rates must be canceled by suitably speeding up (or slowing down) the clock in (x', t').For example, if 1 > 2 (and the current velocity is c in (x, t) and c' in (x', t')) the time t' must be speeded up in order to compensate for the fact that switches from c I to c2 occur more frequently then those from c2 to c 1.
mk 3.4: The connection between the interval of possible positions in (x, t) and in (x', t') can be discussed observing that: + ' c' ' 1 i+i mark .:In the frame (','), the velocity I'1 is lwys inferior to the mean of the velocities c and c.In fact we have, from (a.8), that c'l Cl + c 41 The Lorentz transformation discussed in Section 3 permits us to derive the distribu- tion p(x, t)dx Pr{X(t) e dx} from that of p--p(x',t').By exploiting well known results in literature (see [3, 8]) we can express p(x', t') as follows: is the zero-order Bessel function with imaginary argument.In view of a result in [8]  we can also write the expression for the flow function w-w(x',t') We now present what we consider the most important result in this paper.
For the transformation of the singular component we must bear in mind that and A1 -- Since the Dirac delta function is concentrated in x-c it and x--c2t we have 2Al'k2 {((X'c't') --(X'--c't')} e ((X C lt) --e ((X --c2t 2 2 This concludes the proof of Theorem 4.1.ttemark 4.1: On the basis of the same reasoning it is possible to obtain the expres- sion of the flow function in (x, t) from (4.2).
After some calculations we have that dx' dt' 0 0 0 dx dt' dt cox'-Otcox dt" (4.8)In view of Remark 3.3 and formula (3.8) we get Subtracting the identities in (4.9) yields 0 Ox'-Ox and thus the flow function in the frame (x, t) can be written as for --c2t < x < Clt.
Remark 4.2: We note that formulas (4.3), (4.10)and (4.11)coincide with the well-known distributions when no drift is assumed (c I c 2 c, A A2 A).Even more important is the fact that clt p(x, )dx 1, t c2t whose verification involves intriguing, yet cumbersome, calculations.This will be done in the next section since the necessary formulas will be extracted from general ones.
It may appear strange that the distribution (4.3), in the special case where the asymmetry is due only to different rates A1 :/= A2, Cl c2 c, is much simpler than in the case where )1 /2 A, but C 1 C 2.
In effect, in this case, formula (4.3) reduces to +ff---iIo(YffclA2v/c2t 2 (4.12) 20 L. BEGHIN, L. NIEDDU and E. ORSINGHER The reason for the simple structure of (4.12) is that the support of the distribution is symmetric and the asymmetry of the distribution is due only to the factor A 2 A 1 exp 2 x.
5. Derivation of the Distribution P(x, t) by Solving an Initial-Value

Problem
The classical approach based on Fourier transforms permits us to obtain the charac- teristic function + F(, t) / eifXdP(x, t) (5.1) of the distribution P(x,t)-P(X(t)<_x}.

c 1 c 2
c 2 are positive, real numbers.

3 )
This completes the proof of (3.2).Theorem 3.2:  In force of the Lorentz transformation (3.2), equation (3.2) is converted to the following telegraph equation with respect to the space-time coordinates (x', t'):02p 4(c 1 -I-c2)2h12A 02p 4hlA 20p mark .:If I I I, c c, the transformation (a.2) Distribution of the Position in the Frame (x, t)

(4. 4 )-
Of fundamental importance is to obtain the connection between t' and 0__ o Ox Ot"We first note that Or'-Ot dt' + 0-d