Brownian motion at the speed of light: a Lorentz invariant family of processes
aa r X i v : . [ phy s i c s . c l a ss - ph ] A p r Brownian motion at the speed of light
Maurizio ServaApril 28, 2020
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit`a dell’Aquila, L’Aquila, Italy.
Abstract
In this paper we construct a class of processes which transform one in the other by Lorentz boost. Theparticle only moves at light speed and the velocity performs a non uniform Wiener process on the surface ofa sphere of radius c . Therefore, the trajectories of the velocity are almost everywhere continuous but theyare not differentiable, on the contrary the trajectories of the positions are continuous and differentiable. Forlarge times the behavior of the position is diffusive while for short times is ballistic.This class of processes generalizes to 3+1 dimensions the 1956 idea of the Polish physicist and mathe-matician Marek Kac which considered a (1+1)-dimensional class of processes where the particle travels atlight speed (left or right) and randomly inverts its velocity. Although the generalization to 3+1 dimensionscould be obtained in a different way, for example considering a velocity which performs jumps, it seems tous that our proposal is the most natural having as a constitutive ingredient the Wiener process. Keywords:
Brownian motion, Wiener process, relativity, Lorentz boost, Ito calculus.
Brownian motion is a physical phenomenon which can be mathematically modeled by the Wiener process, bothdirectly identifying the particle trajectories with the realizations of the process and indirectly via the Langevinequation (Ornstein-Uhlenbeck process). Due to its historical connection with the physical phenomenon, theWiener process is often simply called Brownian motion. Although in the last century it has been used todescribe a veriety of phenomenon in finance, biology, engineering, electronics and so on, its name remainsstrictly tied to the description of the motion of random particles.If one tries to extend its use to the description of the random motion of relativistic particles he clashesagainst one of its more characteristic properties: trajectories are not differentiable, which means infinite speedwhile relativity only allows luminal or subluminal velocities. This fact doesn’t imply that it is useless, on thecontrary the relativistic Brownian motion can be still modeled via a variety of modified Langevin equationswhich produce trajectories with a speed which is never superluminal. We just quote a few of the studies whichfollowed this strategy in the last fifty years: [1-18], more bibliography can be found therein.One more reason for searching a Wiener description of relativistic random particles is the possibility to extendthe analogy between the Feynman integral and the Wiener integral (Feynmam-Kac formula) to the relativisticquantum domain. The Schr¨odinger equation is solved by the Feynman integral while the heat equation, whichis connected to the first by analytic continuation, is solved by the Wiener integral. The relativistic versions ofthe Schr¨odinger equation are the Klein-Gordon (zero spin particles) and Dirac (spin one half) equations. Bothare hyperbolic equations (Dirac equation in its second order formulation). Analytic continuation gives rise toelliptic equations, the point is: which process is associated to the elliptic equations?Indeed, the answer is again the Wiener process, but a four dimensional one where both position and timefollow trajectories which are the realizations of a Wiener process with the proper time as index [19, 20, 21].The proper time is then eliminated by a procedure based on hitting times. Nevertheless, the resulting processis unphysical since the speed is not bounded and the whole construction only results in a tool for obtaining1 probabilistic solution of some elliptic equations. If one forces the approach to the realm of physics has toabandon Markov property and the single particle picture [22].There is a third way to approach the relativistic problem with a process which is physical and allows toconstruct the solution of the quantum hyperbolic equations. In 1956 the Polish physicist and mathematicianMarek Kac considered a (1+1)-dimensional process where the particle travels at light speed (left or right) andrandomly inverts its velocity and he proved that the associated probability density satisfies the telegrapherequation [23].Later Gaveau et al. noticed that the telegrapher equation can be easily associated both to the Dirac equationin 1+1 dimensions (first order formulation) and to the Klein-Gordon equation also in 1+1 dimensions (secondorder formulation). Using this equivalence they were able to give a probabilistic solution (by the backwardKolmogorov equation) to these fundamental quantum equations [24]. The weak point was that both Kac andGaveau constructions only worked for particles in 1+1 dimensions (one space dimension + time).Indeed, the process considered by Gaveau et al. is part of a larger class, in fact by Lorentz boosts newprocesses can be obtained with particles moving at the speed of light (a simple consequence of the fact that alight speed particle in an inertial frame is also light speed in any other inertial frame). The processes of thisgeneralized class have in general an unbalanced probability rate of velocity inversion i.e, the inversions fromright to left occour with a different probability rate of those from left to right, as a consequence, the particlemay have a non vanishing average velocity.The class of these one-dimensional light speed process was further extended by considering inversion rateswhich not only depend on the sign of the velocity but also on the position. This extension gave the possibilityto reformulate the quantum mechanics of a relativistic particle in terms of stochastic processes [25] in the spiritof Nelson’s Stochastic Mechanics [26]. Again, this construction was limited to 1+1 dimensions.In this paper we propose a process which generalize the Kac approach to the 3+1 dimensional case. The goalis to construct a Brownian motion which is the most similar to the Wiener process among all those processeswhich do not conflict with relativity. Our particle moves constantly at the speed of light c , but velocity performsa uniform Brownian motion on the surface of a sphere of radius c . In this way the speed is always c which is thelargest among those compatible with relativity, but velocity direction changes. As a consequence, the positiontrajectories are always differentiable.While the particle is ballistic at short times (position changes proportionally to time), the behavior of theposition, at large times, is ordinary diffusive ( E [ x ] ∼ t ) and the average velocity vanishes. This is our process’at rest’ i.e, a process with a velocity which performs a uniform Brownian motion on the surface of the sphereso that the average velocity vanishes at large times. Then we consider the class all those process which aregenerated by Lorentz boosts. The instantaneous velocity of these process must be also luminal, because aluminal particle is luminal in any inertial frame. Therefore, the velocity still remains on the surface of thesphere, nevertheless, the Brownian motion is not homogeneous and average velocity is different from zero atlarge times. In order to characterize this class of processes which transform one in the other by Lorentz boostwe have to use Ito calculus.The paper is simply organized: in section 2 we introduce the process ’at rest’. The velocity process on thesphere is formulated in a new and more economic way which allow a simpler use of Ito calculus. In section 3 wecharacterize the entire class of processes generated by Lorentz boosts. Nevertheless, the very long applicationof Ito calculus to reach this goal is postponed in an Appendix that eventually the reader can skip. Averages arecomputed in section 4 where we also highlight some relevant variables which can be isolated from the others.Summary and outlook can be finally found in section 5. We assume that the particle velocity performs a uniform Brownian motion on the surface of a sphere of radius c . In this way, while the velocity direction changes, the speed always equals c which is the largest among thosecompatible with relativity. 2he equations governing this process are: d x ( t ) = c ( t ) dt,d c ( t ) = − ω c ( t ) dt + ωc d w ( t ) (1)where, according to Ito, d c ( t ) = c ( t + dt ) − c ( t ) and d w ( t ) = w ( t + dt ) − w ( t ) is a two component Wienerincrement on the plane perpendicular to c ( t ) such that E [ | d w ( t ) | ] = 2 dt .The velocity d c ( t ) remains on the surface of a sphere i.e, | c ( t ) | = c that we assume to be the speed of light.In fact, it is straightforward to verify that according to Ito, d c ( t ) = 0 (in next section this equality is explicitlyproven for the general class of processes generated by Lorentz boosts).Indeed, the second equation describes the uniform Wiener process on a sphere which was studied for thefirst time at least 70 years ago [27, 28]. We already mentioned that d w ( t ) = w ( t + dt ) − w ( t ) is a twocomponent Wiener increment on the plane perpendicular to c ( t ), nevertheless, the recipe for the constructionof the underlying process is not univocal.In the following pages we will omit the time as an explicit argument when it is not strictly necessary. Forexample, we will simply write c , d c , w and d w for c ( t ), d c ( t ), w ( t ) and d w ( t ).In the early seventies Strook and then Ito [29, 30] constructed the increment d w in the second of the equations(1) as d w = (cid:0) I − nn T (cid:1) d B (2)where n ( t ) = c ( t ) /c is a time dependent unitary vector, B is a standard three dimensional Brownian motion, I is the 3 × n T is the transposed of the column vector n . One gets d c = − ω c dt + ˆ σ d B where ˆ σ = ωc (cid:0) I − nn T (cid:1) is a 3 × d w = n × d B , (3)which leads to d c = − ω c dt + ˆ σ ′ d B where ˆ σ ′ = ωc [ n ] with [ n ] being the 3 × n .It is easy to check that ˆ σ ˆ σ T = ˆ σ ′ ˆ σ ′ T = ω c (cid:0) I − nn T (cid:1) which implies that the Fokker-Planck equation isthe same for choices (2) and (3). See [33, 34] for properties and applications.Both implementations of the bi-dimensional increment d w are made by a three dimensional Wiener process B ( t ), which is somehow redundant for the construction of a bi-dimensional Brownian motion. We proposehere to use in place of the standard three dimensional Wiener process B ( t ) = B ( t ) , B ( t ) , B ( t ) a standardbi-dimensional Wiener process w ( t ) , w ( t ) (we write w ( t ) , w ( t ) in place of B ( t ) , B ( t ) to avoid confusion).Our choice is d w = n dw + n dw (4)where n ( t ) and n ( t ) are two unitary vectors perpendicular each other and also perpendicular to n ( t ). TheWiener increments are independent which implies E [ dw ( t ) dw ( t )] = 0 and they are standard which means E [( d w ( t )) ] = 2 dt . The orientation of n ( t ) and n ( t ) can be arbitrarily chosen on the plane perpendicular to n ( t ). For example, given a constant vector v , one can chose n = v × n | v × n | , n = n × n , (5)so that v , n = c /c and n are on the same plane and n is perpendicular to it. To fix the ideas one can put thenorth pole in the v direction, so that n is tangent to a meridian and points to north, while n is tangent toa parallel and points to est. In this way it is simple to pass to spherical coordinates. At the poles (where n equals ± v / | v | ) the unitary vectors n and n can be arbitrarily chosen perpendicularly to v .According to our representation the second equation in (1) rewrites as: d c = − ω c dt + ωc ( n dw + n dw ) , (6)the advantage being that we use only a two component Wiener process in place of a three component one,moreover this stochastic equation is straightforwardly associated to the velocity spherical laplacian in the Fokker-Plank equation when it is expressed in terms of longitude and latitude.3 Lorentz boosts and stochastic equations in a generic inertial frame
In the frame where the process is ’at rest’ the velocity c ( t ) of the particle evolves according to equation (6)where n and n are defined by (5). Then assume that this ’rest frame’ moves at constant velocity u (withoutrotating) with respect to a second inertial frame. Since the choice of v in (5) is arbitrary, we can leave it tocoincide with u . In the next we will only use v to indicate both the velocity in (5) and the velocity of the ’restframe’.According to special relativity, the velocity c ′ ( t ) of the particle in the second frame is c ′ ( t ) = 11 + v · c ( t ) c (cid:20) α c ( t ) + v + (1 − α ) v · c ( t ) v v (cid:21) (7)where v , c and α = (cid:16) − v c (cid:17) are constant. By special relativity the velocity in this second inertial frame isalso luminal ( | c ′ | = c ) (at the end of this section we will show that indeed ( c ′ ) = c = c ). If one also takesinto account that the time increment dt ′ in the second frame satisfies dtdt ′ = 1 α (cid:18) − v · c ′ c (cid:19) , (8)he should be able to write from (7) and (8) a stochastic equation for c ′ ( t ′ ) analogous to (6) and (5). Notice thatthe argument of c ′ is now t ′ and that the new equation should express the increment d c ′ = c ′ ( t ′ + dt ′ ) − c ′ ( t ′ )in terms of c ′ ( t ′ ), dt ′ and of the increments dw ′ = w ( t ′ + dt ′ ) − w ( t ′ ) and dw ′ = w ( t ′ + dt ′ ) − w ( t ′ ).In the second frame the particle will still instantaneously move at light speed but, contrarily to the case ofthe process in the ’rest frame’, its average velocity will not vanish at large times but it will equal v .Define δ c ′ = c ′ ( t + dt ) − c ′ ( t ) (notice the difference with d c ′ ), then a long and tedious application of Itocalculus (see the Appendix) leads to δ c ′ = − ω α (cid:20) − v · c ′ c (cid:21) c ′ dt + ω cα (cid:18) − v · c ′ c (cid:19) ( n ′ dw + n ′ dw ) (9)where n ′ and n ′ are the two unitary vectors perpendicular to c ′ defined as in (5) (with n , n and n replaced by n ′ = c ′ /c , n ′ and n ′ ). Then, taking into account (8) and remembering that dw /dw ′ = dw /dw ′ = ( dt/dt ′ ) one gets d c ′ = − ω α (cid:20) − v · c ′ c (cid:21) c ′ dt ′ + ω c (cid:20) α (cid:18) − v · c ′ c (cid:19)(cid:21) d w ′ (10)where d w ′ = n ′ dw ′ + n ′ dw ′ is a two component increment perpendicular to c ′ in complete analogy with theprocess in the ’rest frame’. Notice that by (7) the three vectors v , c and c ′ are on the same plane so that n , n ′ also are on the same plane. As a consequence n and n ′ are both perpendicular to that plane so that n = n ′ .One can easily prove from (6) that the speed of the particle remains constantly luminal i.e, d | c ′ ( t ) | = 0, infact by Ito calculus d | c ′ | = − ω α (cid:20) − v · c ′ c (cid:21) | c ′ | dt ′ + 2 c ω α (cid:20) − v · c ′ c (cid:21) dt ′ + 2 c ω α (cid:20) − v · c ′ c (cid:21) c ′ · d w ′ (11)where the second term at the right comes from the second order contribution to Ito differential. Since c ′ and d w ′ are perpendicular the equation reduces to d | c ′ | = − ω α (cid:20) − v · c ′ c (cid:21) (cid:0) | c ′ | − c (cid:1) dt ′ = 0 (12)where he last equality holds if the initial velocity is luminal i.e, | c (0) | = c .4he fact that the process remains luminal is not astonishing since a particle moving at the speed of lightalso moves at the speed of light in any other inertial frame. Therefore, the equation (10) defines a class of lightspeed processes which transform one in the other by Lorentz boost.Notice that in a generic inertial frame, although the particle still only moves at the speed of light, it has anaverage velocity v for large times. This is a consequence of the fact that, according to (10), the diffusion of c ′ slows down the more v · c ′ is large. This means that the particle spends more time with values of c ′ alignedwith v and less time when it is anti-aligned. This is exactly the same one has with the (1+1)-dimensionalKac process associated to the telegrapher equation. In this simpler case a non vanishing average velocity isdetermined by the fact that unbalanced rates of inversions lead to a longer permanence of the velocity in oneof the two directions.We also would like to stress again that the Kac process could be generalized to 3+1 dimensions in a differentway considering a velocity which performs jumps in place of having continuous trajectories, but we think theprocess presented here, having as a constitutive ingredient the Wiener process, is the most natural choice forthis generalization. In this section we only consider the process in the ’rest frame’, all results concerning averages can be eventuallyLorentz transformed for the processes in a generic inertial frame.The stochastic equation (1) can be recast in an integral equation: x ( t ) = x (0) + Z t c ( s ) ds, c ( t ) = e − ω t (cid:20) c (0) + ω c Z t e ω s d w ( s ) (cid:21) . (13)This is not a solution because d w ( s ), according to (4) and (5), depends on c ( t ). Let us mention that the proofthat the particle velocity remains constantly luminal i.e, | c ( t ) | = c can be eventually also obtained by the secondintegral equation in (13).Starting from second integral equation in (13) one can easily find out that the following averages hold for t ≥ s ≥ E [ c ( t )] = e − ω t c (0) ,E [ c ( t ) · c ( s )] = c e − ω ( t + s ) + 2 ω c e − ω ( t + s ) Z s e ω u du = c e − ω ( t − s ) . (14)Notices that the first of the equalities above says that the average velocity E [ c ( t )] vanishes for large t however,for a generic inertial frame, E [ c ′ ( t ′ )] → v for large t ′ . Then, using these averages and the first of the integralequations in (13), one also obtains E [ x ( t )] = x (0) + 1 − e − ω t ω c (0) ,E [( x ( t ) − x (0)) ] = 2 c Z t Z s e − ω ( s − u ) du ds = 2 c ω t − c ω (1 − e − ω t ) . (15)The above averages imply, for large times, a diffusive behaviour with coefficient c ω , in this limit one has infact E [( x ( t ) − x (0)) ] ∼ c ω t . On the contrary, for short times E [( x ( t ) − x (0)) ] ∼ c t which means ballisticbehavior at the speed of light.It is interesting to note that it is possible to find out a pair of autonomous variables, i.e, two variableswhose stochastic equations can be written and eventually solved independently from the equations of the othervariables. Consider first the equation for the variable x · n : d ( x · n ) = x · d n + n · d x = cdt − ω ( x · n ) dt + ω x d w (16)5here the second equality is derived from (1) taking into account that n = c c . On the other side: x d w ( t ) = x · n dw + x · n dw = (cid:2) ( x · n ) + ( x · n ) (cid:3) dw = (cid:2) | x | − ( x · n ) (cid:3) dw where the second equality only usethe composition rules of independent Gaussian increments ( dw also is a unitary Gaussian increment) and thelast equality only use the identity ( x · n ) + ( x · n ) = | x | − ( x · n ) . Moreover d | x | = 2 x · d x = 2 c ( x · n ) dt (17)where the second equality also simply derives from (1) . In conclusion the pair of variables x · n , | x | satisfy theautonomous system d ( x · n ) = cdt − ω ( x · n ) dt + ω (cid:2) | x | − ( x · n ) (cid:3) dw,d | x | = 2 c ( x · n ) dt. (18)Taking an average ad solving the resulting system one finds out E [ x ( t ) · n ( t )] = cω − cω e − ω t + e − ω t x (0) · n (0) ,E [ | x ( t ) | ] = 2 c ω t − c ω (1 − e − ω t ) + 2 1 − e − ω t ω x (0) · c (0) . + | x (0) | . (19)While the first equality is new, the second can be obtained combining the two equalities in (15).The fact that these two variables can be isolated obviously does not mean that the probability densitydepends only on these two. Nevertheless, in case the process starts from the origin position ( x (0) = 0) and theinitial velocity distribution is uniform on the surface of the sphere, it is easy to realize that the density onlydepends on these two variables at any time.The condition x (0)=0 can be eventually replaced by the more generic x (0)= x , to return to the previouscase it is in fact sufficient to consider the variables ( x ( t ) − x ) · c ( t ) and | x ( t ) − x | which also satisfy (18). In conclusion we have found that equation (10) describes a class of light speed processes which transform onein the other by Lorentz boost. Their main characteristics can be resumed as follows: • the particle only moves at light speed and the velocity performs a Brownian motion on a sphere of radius c = light speed. For large times it has an average velocity v , this is is a consequence of the fact thataccording to (10) the diffusion of the velocity slows down the more v · c ′ is large. In turn, this means thatthe particle spends more time with values of c ′ aligned with v and less time when it is anti-aligned. This isexactly the same situation one has with the (1+1)-dimensional Kac process associated to the telegrapherequation since the probability rate of inversion of velocity can be different for left/right and right/leftinversions; • the class of processes that we propose generalizes to 3+1 dimensions the 1956 idea of the Polish physicistand mathematician Marek Kac which considered a (1+1)-dimensional process where the particle travels atlight speed (left or right) and randomly inverts its velocity. Although the Kac process could be generalizedto 3+1 dimensions in a different way, for example considering a velocity which performs jumps in place ofhaving continuous trajectories, we think the process presented here, having as a constitutive ingredient theWiener process, is the most natural choice for this generalization. Moreover, since the speed is always themaximum possible it is posses the trajectories which better mimics the (infinite speed) Wiener trajectoriesgiven the relativistic constraint; • for large times the behavior of the position is diffusive with coefficient c ω , one has in fact E [( x ( t ) − x (0)) ] ∼ c ω t . On the contrary, for short times E [( x ( t ) − x (0)) ] ∼ c t which means ballistic behavior at the speedof light. 6his process is the natural candidate for modeling the diffusion of mass-less particles, nevertheless, its useshould not limited to this case. The situation is similar in the non-relativistic realm; thought that a particlewith infinite speed is unphysical, the Wiener process is largely used to model its erratic movement.Another point that deserves investigation and which contributed to prompt this work is the possible connec-tion of the Fokker-Plank backward equation associated to this process with relativistic equations as Klein-Gordonand Dirac. The goal would be to find a generalization of the Gaveau et al. approach to the (3+1)-dimensionalcase. This topic is presently under study. Appendix: Ito calculus
In this appendix we apply Ito calculus, in order to obtain equation (9) from equation (7). Since we defined δ c ′ = c ′ ( t + dt ) − c ′ ( t ), then from (7) we immediately get δ c ′ = 11 + v · ( c + d c ) c (cid:20) α ( c + d c ) + v + (1 − α ) v · ( c + d c ) v v (cid:21) −
11 + v · c c h α c + v + (1 − α ) v · c v v i (20)where c + d c = c ( t + dt ) and c = c ( t ). This is still not a Ito increment, but it is the trivial application of thedefinition δ c ′ = c ′ ( t + dt ) − c ′ ( t ). This equation can be exactly rewritten as δ c ′ = dǫ dǫ (cid:20) (1 − α ) c v v −
11 + v · c c (cid:16) α c + v + (1 − α ) v · c v v (cid:17)(cid:21) + 11 + dǫ αd c v · c c (21)where d c is given by (6) and where dǫ = 11 + v · c c v · d c c = − ω c v · c v · c c dt + ωc v · n v · c c dw . (22)The next step is to rewrite the right side of the equation (21) in terms of the new variables c ′ , n ′ and n ′ .First of all, using (7) we immediately rewrite the equation (21) as δ c ′ = 11 + dǫ " dǫ (cid:18) (1 − α ) c v v − c ′ (cid:19) + 1 − v · c ′ c α d c , (23)but we also need to rewrite dǫ and d c in terms of the new coordinates. In order to reach this goal we need torecall that c = 11 − v · c ′ c (cid:20) α c ′ − v + (1 − α ) v · c ′ v v (cid:21) , → v · c = 11 − v · c ′ c (cid:2) v · c ′ − v (cid:3) , (24)moreover c n = 11 − v · c ′ c (cid:20) αc n ′ − v × n ′ + (1 − α ) v · c ′ v v × n ′ (cid:21) , → v · n = α v · n − v · c ′ c . (25)From these two last equations one easily realize that the three vectors c , v and c ′ are co-planar. As well, n and n ′ lie on the same plane. Moreover, n ′ = n is perpendicular to that plane. We get dǫ = − ω c v · c ′ − v α dt + ωc v · n ′ α dw , (26)where d c is given by (6) with n = n ′ and c and n given respectively by the first equation in (24) and the firstequation in (25).We are now ready compute the Ito differential i.e, we are ready to rewrite the differential δ c ′ keeping onlyterms of order dt . To obtain this result we have first of all to expand (23) to the second order with respect tothe differentials: 7 c ′ ≃ " dǫ (cid:18) (1 − α ) c v v − c ′ (cid:19) + 1 − v · c ′ c α d c − " ( dǫ ) (cid:18) (1 − α ) c v v − c ′ (cid:19) + dǫd c ] 1 − v · c ′ c α , (27)then, we have replace the second order differentials ( dǫ ) and dǫ d c by the terms proportional to dt of theiraverages: ( dǫ ) ≃ (cid:16) ωc (cid:17) ( v · n ′ ) α dt , dǫd c ≃ ω α ( v · n ′ ) n dt (28)where n must be expressed in terms of the new variables by the first equation in (25).Let us rewrite equation (27) as δ c ′ = δ A + δ B + δ C (29)where δA is the term proportional to dt which comes from the first order differentials (the deterministic partof the first term between square parenthesis in (27)), δB is the term proportional to dt wich comes from thesecond order differential (the second term between square parenthesis in (27)) and δC is the random term (therandom part of the first term between square parenthesis in (27)). After having expressed all the old variablesin therms of the new ones (except n ), we have δ A = − (cid:20) ω c v · c ′ − v α (cid:18) (1 − α ) c v v − c ′ (cid:19) + ω α (cid:18) α c ′ − v + (1 − α ) v · c ′ v v (cid:19) (cid:21) dt,δ B = − ω α ( v · n ′ ) (cid:20) v · n ′ c (cid:18) (1 − α ) c v v − c ′ (cid:19) + (cid:18) − v · c ′ c (cid:19) n (cid:21) dt,δ C = " ωc v · n ′ α (cid:18) (1 − α ) c v v − c ′ (cid:19) dw + 1 − v · c ′ c α ωc ( n ′ dw + n dw ) (30)with n given by (25) in terms of the new variables. After some rearrangement of terms we get: δ A = − (cid:20) ω c v · c ′ α − ω α (cid:21) v dt + (cid:20) ω c v · c ′ − c α (cid:21) c ′ dt = ω α (cid:20) − v · c ′ c (cid:21) ( v − c ′ ) dt. (31)This differential lies in the plane of c ′ and n ′ and can be decomposed along these two vectors: δ A = − ω α (cid:20) − v · c ′ c (cid:21) c ′ dt + ω α (cid:20) − v · c ′ c (cid:21) ( v · n ′ ) n ′ dt. (32)Analogously, the term δ B , after decomposition along c ′ and n ′ , can be rewritten as δ B = − ω α ( v · n ′ ) (cid:18) − v · c ′ c (cid:19) " v · n ′ − v · c ′ c (cid:18) (1 − α ) v · c ′ v − (cid:19) + n · c ′ c ′ dt − ω α ( v · n ′ ) (cid:18) − v · c ′ c (cid:19) " v · n ′ (cid:0) − v · c ′ c (cid:1) (cid:18) (1 − α ) v · n ′ v (cid:19) + n · n ′ n ′ dt. (33)Since c ′ · n = − c · n ′ , from (24) by scalar product with n ′ , one gets c ′ · n = 11 − v · c ′ c (cid:20) − (1 − α ) v · c ′ v (cid:21) v · n ′ , (34)therefore the first term at the right side of equality (33) vanishes, moreover from (24) and by the definitions of n and n ′ one has that n · n ′ = c · c ′ c = 11 − v · c ′ c (cid:20) αc − v · c ′ + (1 − α ) ( v · c ′ ) v (cid:21) , (35)8hich, can be substituted in the second term of (33) in order to obtain δ B = − ω α ( v · n ′ ) (cid:18) − v · c ′ c (cid:19) n ′ dt. (36)A scalar product of the third of (30) with c ′ gives c ′ · δ C = ωcα (cid:20) v · n ′ (cid:18) (1 − α ) v · c ′ v − (cid:19) + (cid:18) − v · c ′ c (cid:19) ( c ′ · n ) (cid:21) dw = 0 (37)where the equality is obtained using (34). Construction is coherent since the Wiener increment of c ′ has nocomponent parallel to c ′ itself. Therefore, by decomposition along n ′ = n ′ and n ′ we have δ C = ωcα (cid:20) − v · c ′ c (cid:21) n ′ dw + ωcα (cid:20) (1 − α ) ( v · n ′ ) v + (cid:18) − v · c ′ c (cid:19) c ′ · c c (cid:21) n ′ dw , (38)which given (35) can be rewritten as δ C = ω cα (cid:18) − v · c ′ c (cid:19) ( n ′ dw + n ′ dw ) . (39)Finally, by δ c ′ = δ A + δ B + δ C we finally obtain δ c ′ = − ω α (cid:20) − v · c ′ c (cid:21) c ′ dt + ω cα (cid:18) − v · c ′ c (cid:19) ( n ′ dw + n ′ dw ) , (40)which is the equation (9) that we use in section 3 and which allows to find out the equation (10) whichcharacterizes the general class of the light speed processes. References [1] R. M. Dudley,
Lorentz-invariant Markov processes in relativistic phase space. , Arkiv f¨or Matematik ,241268, (1965).[2] R. M. Dudley, A note on Lorentz-invariant Markov processes. , Arkiv f¨or Matematik , 575581, (1967).[3] R. M. Dudley, Asymptotics of some relativistic Markov processes. , Procedings of the National Academy ofSciiences USA, , 35513555, (1973).[4] F. Debbasch, K. Mallick, and J. P. Rivet, Relativistic Ornstein-Uhlenbeck Process.
Journal of StatisticalPhysics , 945966, (1997).[5] C. Barbachoux, F. Debbasch, and J. P. Rivet, The spatially one-dimensional relativistic Ornstein-Uhlenbeckprocess in a n arbitrary inertial frame.
European Physical Journal B, , 3747, (2001).[6] J. Dunkel and P. H¨anggi, Theory of the relativistic Brownian motion: The (1+1)-dimensional case.
PhysicaReviev E , 016124, (2005).[7] J. Dunkel and P. H¨anggi, Theory of the relativistic Brownian motion: The (1+3)-dimensional case.
PhysicaReviev E , 036106, (2005).[8] J. Dunkel, P. Talkner, and P. H¨anggi., Relativistic diffusion processes and random walk models.
PhysicaReviev E , 043001, (2007).[9] J. Franchi and Y. Le Jan, it Relativistic diffusions and Schwarzschild geometry. Communications on Pureand Applied Mathematics , 187251, (2006). 910] J. Angst and J. Franchi, Central limit theorem for a class of relativistic diffusions.
Journal of MathematicalPhysics , 083101, (2007).[11] J. Franchi, Relativistic diffusion in G¨odels universe.
Communications in Mathematical Physics , 523555,(2009).[12] C. Chevalier and F. Debbasch,
Relativistic diffusions: a unifying approach.
Journal of Mathematical Physics , 043303, (2008).[13] J. Herrmann, Diffusion in the special theory of relativity.
Physical Review E , 051110, (2009).[14] Z. Haba, Relativistic diffusion.
Physical Reviev E , 021128, (2009).[15] Z. Haba, Relativistic diffusion with friction on a pseudo-Riemannian manifold.
Classical and QuantumGravity , 095021, (2010).[16] Z. Haba, Non-linear relativistic diffusions.
Physica A: Statistical Mechanics and its Applications ,2776-2786, (2011).[17] I. Bailleul,
A stochastic approach to relativistic diffusions.
Annales de l’Institut Henri Poincar´e, Probabilit´eset Statistiques , 760-795, (2010).[18] J. A. A. F´elix, A relativistic diffusion model in kinetic theory.
PhD Thesis, Editor: Universidad de Granada,1-133, (2015).[19] T. Ichinose and H. Tamura,
Imaginary-Time Path Integral for a Relativistic Spinless Particle in an Elec-tromagnetic Field.
Communications in Mathematical Physisics bf 105, 239-257, (1986).[20] G. F. De Angelis and M. Serva,
On the relativistic Feynman-Kac-Ito formula.
Journal of Physics A: Math-ematical and General , L965-L968, (1990).[21] G. F. De Angelis, A. Rinaldi and M. Serva, Imaginary-time path integral for a relativistic spin-( 1/2)particle in a magnetic field.
Europhysics Letters , 95-100, (1991).[22] M. Serva, Relativistic stochastic processes associated to Klein-Gordon equation.
Annales de l’Institut HenriPoincar´e, Phisique th´eorique , 415-432, (1988).[23] M. Kac, A stochastic model related to the telegrapher’s equation.
Rocky Mountain Journal of Mathemat-ics , 497-510, (1974). Reprinted from Some Stochastic Problems in Physics and Mathematics.
MagnoliaPetroleum Company Colloquium Lectures in the Pure and Applied Sciences , (1956).[24] B. Gaveau, T. Jacobson, M. Kac and L. S. Schulman, Relativistic Extension of the Analogy between Quan-tum Mechanics and Brownian Motion.
Physical Review Letters , 419-422, (1984).[25] G. F. De Angelis, G. Jona-Lasinio, M. Serva and Nino Zanghi, Stochastic mechanics of a Dirac particle intwo spacetime dimensions.
Journal of Physics A: Mathematical and General , 865-871, (1986).[26] E. Nelson, Dynamical Theories of Brownian Motion.
Princeton University Press, Princeton, New Jersey.(1967).[27] K. Yosida,
Brownian motion on the surface of the 3-sphere.
Annals of Mathematical Statistics , 292296,(1949).[28] K. Yosida, On brownian motion in a homogeneous Riemannian space.
Pacific Journal of Mathematics ,263270, (1952).[29] D. Stroock, On the growth of stochastic integrals.
Z. Wahrsch. verw. Gebiete, , 340-344, (1971).1030] K. Itˆo, Stochastic calculus . International Symposium on Mathematical Problems in Theoretical Physics(Kyoto), Lecture Notes in Physics , 218-223, Springer (1975).[31] G. C. Price and D. Williams, Rolling with ’Slipping’.
S´eminaire de Probabilit´es XVII (Paris), Lecture Notesin Mathematics , 194-297, Springer-Verlag, (1983).[32] M. van den Berg and J. T. Lewis,
Brownian Motion on a Hypersurface.
Bulletin of the London MathematicalSociety , 144-150, (1985).[33] D. R. Brillinger, A Particle Migrating Randomly on a Sphere.
Journal of Theoretical Probability , 429-443, (1997).[34] M. M. G. Krishna, Joseph Samuel and Supurna Sinha, Brownian motion on a sphere: distribution of solidangles.
Journal of Physics A: Mathematical and General33