RECTILINEAR AND BROWNIAN MOTION FROM A RANDOM POINT IN A CONVEX REGION

A particle is projected from a point P in a subset E of a convex region H to a point Q in a uniformly random direction. The probability that Q lies in the interior of H at time t is obtained for two types of motion of the particle, rectilinear (i.e. straight-line) and Brownian. In the case of rectilinear motion, the first passage time through the boundary of H is considered. Results are obtained in terms of the generalized overlap function for embedded bodies.


INTRODUCTION
Consider a particle moving from point P at time t = 0 to point Q at t > 0. P is taken to lie in a subset E of a compact convex body n H R ⊂ .The set E may be nonconvex.It may be disconnected and may have lower dimensionality than H ; indeed, E may consist of a single point (see Fig. 1).In the next section we relate the probability that Q lies in the interior of H to the generalized overlap function for embedded bodies introduced by Enns and Ehlers (1988).Thereafter we obtain this probability first for the case of motion along a straight path and then for the case of Brownian motion.Depending on its equations of motion, the particle may leave and re-enter the region H any number of times.We are concerned only with its presence or absence inside H at time t .Fig. 1.A particle moves from P to Q in time t.
There is a considerable literature on escape processes and first exit times, especially as related to the example of Brownian motion; see, for example, Getoor (1979) or Wendel (1980).For more general books, see Gihman and Skorohod (1975) or Knight (1981).

Let ( )
Q t denote the location of the particle at time 0 t ≥ .Let (0) P Q = be a uniformly random starting point in E .Denote the distance between (0) Q and ( ) Q t by ( ) X t .Assuming that motion is isotropic, it is possible to relate the probability of finding ( ) Q t inside H to the overlap function which is defined by .Averaging with respect to net distance X and initial point P , one obtains the unconditional probability that ( ) = Ω E (Enns and Ehlers, 1988) relating ( ) h t and the overlap function.
It was shown by Enns and Ehlers (1988) that is the distribution function of the length R of a random ray generated by selecting a point in E and a direction, independently uniformly distributed and with terminal point in the boundary where the R -expectation is with respect to ν - measure.
We now turn to specific types of particle motion.

RECTILINEAR MOTION
For a particle undergoing straight-line motion with position function ( ) u t along the directed half- line originating at P (direction equal to the direction of motion so that ( ) where ( ) .Moments of T may then be obtained by evaluating For the case of motion at constant velocity, where ( ) Obviously, this case corresponds to where R is the ν -random ray length from a uniformly random point in E to the boundary of H.A shape-independent moment for n-dimensional H is ( ) Enns andEhlers (1988, 1993) give the overlap functions for the case of concentric balls.Let This situation models the physically important cases of particles generated in a linear or circular region inside the 3-ball.We have The integrations for the moments are elementary.In each case, the moments may be written in the form ( ( , ) ( , )) 1: Clearly, more complicated position functions ( ) u t may be substituted in Eq. 3, leading to relatively straight-forward tedious integrations.Note that, depending on the form of ( ) u t , a particle may leave and re-enter H in (0, ) t .
One modification of practical interest for constant-speed motion is the case of radioactive particles with short lifetimes.Such particles may decay before reaching the boundary of H . Let d T denote the random lifetime of the particle (time to decay, given birth at point P ) and let the random time to reach the boundary in the absence of decay be 0 T .If the decay process is independent of the particle's motion, then the particle vanishes unless One application is the situation where an experimenter can make observations only in a limited field of view (under a microscope, say).An identifiable cell might be observed to be in a certain part of the field of view at time zero and then might be observed again later.
If we expand ( , ) X F R t as used in Eq. 2 in a Taylor series in R , we find which gives ( ) h t in terms of moments of the length of a ν -random ray.

Remark. It seems physically curious that, since
( 1) (0, ) 0 ( ) Fig.2shows how ( ) h t decreases with time.For the case of a b < and 3 d n = = we find ( ) ( ; , ) ( ; , ) h t q t a b q t a b = + −

Fig. 3 .
Fig. 3.The probability h(t) for concentric spherical regions E and H. .