ANALYTICAL AND NUMERICAL RESULTS ON ESCAPE OF BROWNIAN PARTICLES. Carey Caginalp. Submitted to the Graduate Faculty of

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1 ANALYTICAL AND NUMERICAL RESULTS ON ESCAPE OF BROWNIAN PARTICLES by Carey Caginalp Submitted to the Graduate Faculty of the Department of Mathematics in partial fulfillment of the requirements for the degree of Bachelor of Philosophy University of Pittsburgh

2 UNIVERSITY OF PITTSBURGH DEPARTMENT OF MATHEMATICS This dissertation was presented by Carey Caginalp It was defended on January 8, and approved by Xinfu Chen, Mathematics Department, University of Pittsburgh Huiqiang Jiang, Mathematics Department, University of Pittsburgh Dehua Wang, Mathematics Department, University of Pittsburgh Marshall Slemrod, Mathematics Department, University of Wisconsin Dissertation Director: Xinfu Chen, Mathematics Department, University of Pittsburgh ii

3 Copyright c by Carey Caginalp iii

4 ANALYTICAL AND NUMERICAL RESULTS ON ESCAPE OF BROWNIAN PARTICLES Carey Caginalp, B-Phil University of Pittsburgh Abstract. A particle moves with Brownian motion in a unit disc with reflection from the boundaries except for a portion (called "window" or "gate") in which it is absorbed. The main problems are to determine the first hitting time and spatial distribution. A closed formula for the mean first hitting time is given for a gate of any size. Also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed. Numerical simulations of the stochastic process with finite step size and sufficient amount of sample paths are compared with the exact solution to the Brownian motion (the limit of zero step size), providing an empirical formula for the divergence. Histograms of first hitting times are also generated. iv

5 TABLE OF CONTENTS. INTRODUCTION MAIN RESULTS PROOF OF THE MAIN RESULTS MONTE CARLO SIMULATIONS BIBLIOGRAPHY v

6 LIST OF FIGURES Movement of a particle subject to Brownian motion Histograms for the escape times of, sample paths for various gate and step sizes Relative and absolute error for various gate and step sizes Representation of relative and absolute error for various gate and step sizes using a three-dimensional plot Cdf (cumulative distribution function) and ecdf (empirical cumulative distribution function) of the exit position along the gate vi

7 . INTRODUCTION Many physical, chemical, biological, and ecological processes can be formulated in terms of a Brownian motion with re ection at most of the domain boundary and absorption from a small part. In chemical processes [], particle A may move around randomly while B is essentially stationary, and a reaction occurs when A enters an attraction basin of B. In cell biology, an ion drifts about within a cell, is re ected when it hits the membrane, which is most of the boundary of the cell, and escapes when it hits a small pore, thereby altering the electrostatic balance in and out of the cell []. A prey moving randomly in a con ned territory dies when it encounters a predator hiding at the entrance to the territory [7]. An epidemic con ned in one region may spread through small unsecured boundary of the region. These applications thus lead to a pure mathematical problem of nding the expected life time (mean rst passage time or MFPT) of a Brownian particle (or a particle subject to Browian motion) in an n-dimensional domain in which the particle is re ected and dies (or escapes or is absorbed) once it We de ne a Brownian Motion as a real-valued stochastic process w (t;!) on R + that satis es the following properties: () w (;!) = with probability. () w (t;!) is a continuous function of t almost everywhere. (3) For every t; s ; the increment w (s) = w (t + s;!) w (t;!) is independent of events prior to t, and is a zero mean Gaussian random variable with variance V ariance := E jw (s)j = s: This problem has been studied by several authors ([, 3, 5, 6, 7, 9] and references therein). Most recently, Chen and Friedman [] proved asymptotic expansions for MFPT when the size of the gate,, is small.

8 In this paper, we start with the equation d! x =! b dt + d! w ((.)) where ~w = [w (t) ; :::; w n (t)] T is Brownian motion in n dimensions. Here is the n n covariance matrix, whose entries determine the changes in the x ; :::; x n directions based on a speci cation of the noise d~w. For example, if = I nn ; then the change in x i depends only on the change in w i. ~x is the position of the particle, so that d~x is the change in the particle s position, and ~ b is the drift velocity. We will assume that the drift velocity is zero and that the covariance matrix is the identity I, so that the motion of the particle along Later, this will be approximated in the numerics by ~x (t + t) = ~x (t) +! N ~ (; ) t. We present a closed formula for MFPT for the fundamental case when is the unit disc and is a connected arc on the boundary. From an analytical point of view this formula provides the exact size of O() in several asymptotic expansions derived in the past [,,, 5, 9]. Singer, Schuss, and Holcman [9] obtained an approximation for the mean escape time up to an O () term from the center of a disc E : T (z) = jj log D + O () ; z E: " where jj is the area contained in. For the case of the unit disc (denoted by B), D = and jj =. Recently, Chen and Friedman [] obtained the approximation T (z) = jzj + j zj ln " + O (") + O (" ) dist (z; ") for z B for the mean escape time, and by using the mean value theorem for harmonic functions, they obtain the average mean escape time T = 8 + ln + O (") ; " both of which are accurate up to an O (") term. Using rigorous asymptotic analysis, they show that these estimates are accurate under the condition dist(z; ") < 3" (i.e., the particle does not start too close to the gate). We call the case fundamental since upon which asymptotic expansions for general domains with multiple small windows can be derived []. From a numerical point of view, this formula provides a reliable test for any numerical algorithm

9 designed to tackle the case when the gate size is very small (so the numerical problem is very sti ). In general without knowing the exact size of O()j j or even O()j j where j j is the length of, an asymptotic expansion is quite often very hard to verify or use, since one cannot easily determine whether the di erence between the numerical solution and the asymptotic expansion is due to the error of discretization or due to the error, say, O()j j log j j, of the underlying asymptotic expansion. Our closed formula provides a concrete criterion here. As a consequence of the relationship between the stochastic problem and elliptic equations (see Theorem ) one can compute numerical solutions on MFPT using Poisson s equation. Here we shall use a Monte Carlo method directly simulating the di usion process and computing related statistics. A simple Monte-Carlo simulation can produce tremendous amount of useful information. For example, from a reasonable amount (> ) of sample paths, we can construct a histogram of exit times (c.f. Figure ) from which we can compute the mean (i.e. MPFT), the variance, the probability density (and its exponential tail), etc., of the rst passage times; also from the locations of exits, we can construct histograms as well as empirical cumulative distribution functions (c.f Figure 5) to nd where the particle exits. A single GHz processor can produce 6 sample paths per hour, if the time step, t, and size of the gate are not too small, say t > 5 and j j >. In applications, this will be su cient to provide needed information for actions of optimal controls. Our Monte-Carlo simulation is based on the following stochastic process fx t ; Y t g tt de ned by Y t+t = X t + p t t ; X t+t = Y t+t maxf; jy t+t j g [ 8 t T := fktg; (.) k= where X = Y are given and f t g tt are i.i.d random variables with (normal) N(; I) distribution. The rst passage time associated with the process is de ned by n = min t T Y t maxf; jy t jg o : (.) In simulation, rst of all, there is a random error due to the statistical sampling. Trusting the quality of common random number (CRN) generator that we use, this error can be controlled by taking an appropriate large number of samples, as a consequence of the Central Limit Theorem. There is also a discretization error due to the approximation of (.) to the 3

10 Figure : Movement of a particle subject to Brownian motion.

11 di usion process, fw t g t>, of the Brownian motion con ned in the unit disc with pure re ection on the boundary. With an exact solution for the particular case of the circle, one can then obtain empirical results as step size and gate size vary. Thus, a careful comparison of the exact solution with the numerical solution illuminates the distinction between nite step and continuous di usion. We would like to point out that replacing the N(; I) distribution of t by a certain transition distribution that depends on X t will diminish the discretization error. We hope that one can either nd a convenient way to use the transition density from W t to W t+t to eliminate the discretization error or nd the asymptotic behavior, as t!, of the distribution di erence between fx t g tt and fw t g t> and also the distribution di erence between the stopping time de ned in (.) and the rst passage time := infft > j W t g to produce a correction for the discretization error. Our simulation with X = (; ) = W suggests that log sin j j = E[] E[ ] p t ; std[ ] E[ ]; j j where E stands for expectation and std the standard deviation; see Figures and where " = j j=, x = p t, and MCT is the sample mean E[ ] + N(; std[ ] =n) with n = ;. The rest of the paper is organized as follows. We present our theoretical results for E[] and std[] in Section, their proofs in Section 3, and results of Monte-Carlo simulations in Section. 5

12 . MAIN RESULTS Our starting point is the formulation for the statistics of the stochastic process as solutions of di erential equations. The following equation for the mean rst passage time is a standard result [8], while the expression for its variance is new. Theorem. Let be a bounded domain with smooth and be a closed subset For each x, let x be the rst time of a particle hitting, assuming that the particle starts from x, is subject to the Brownian motion in, and re ects Then, the mean rst passage time, T (x) := E[ x ], and its variance, v(x) := E[( x T (x)) ], are solutions of the following boundary value problems: T = in ; T = on n T = n ; (.) v = jrt j in ; v = on n v = n : (.) n := n r is the derivative in the direction n, the exterior normal Moreover, the average of the variance can be calculated from the formula v := v(x)dx = T (x)dx =: T jj jj : (.3) The main contribution of this paper is the following closed formula for the mean rst passage time in a special case that has attracted much attention and been the subject of many theoretical investigations in the past; see [,, 5, 9] and the references therein. 6

13 Theorem. (A Closed Formula) In -D, with points identi ed by complex numbers, let := fre i j 6 r < ; " 6 6 "g; := fe i j jj 6 "g: (.) Then the mean rst passage time T (z), for z, is given by T (z) = jzj z + p ( ze + log i" )( ze i" ) sin " : (.5) For the rest of this paper, we will assume and are as in (.). This exact formula allows us to improve the result of Theorem 5. in [] by the following. Theorem 3. The mean rst passage time T has the following properties: T () = log sin " ; T := T (x)dx = T () jj T (e i ) = arccosh maxfsin "; j sin jg sin " 8 R: In addition, setting ^" = sin " = jei" j, we have, when < ^" 6 j zj and z, T (z) = jzj j zj z^" + log + < + O()z^" ^" ( z) j zj where < stands for the real part and O() is a certain function satisfying jo()j < :887 : Finally we consider the location of a particle when it exits. ; Theorem. The probability density of the location of a particle at time of its exit is given by j(e i T (ei ) = 8 >< >: if " < < "; cos q sin " sin if jj < ": That is, for any (Borel the probability that a particle, starting either at the origin or uniformly distributed in, making Brownian motion in, re ecting when it n, and escaping once it hits, ends up escaping from is P () = j(y)ds y where ds y is the surface element at These Theorems will be proven in the next section. 7

14 3. PROOF OF THE MAIN RESULTS Proof of Theorem. For convenience, we call a particle dead as soon as it hits ; otherwise survived. For x, we denote by (x; ; t) the survival probability density at time t of the particle starting from x; that is, (x; y; t)dy is the probability that a particle starting from x lands in the region y + dy at time t before it hits. Then by the Kolmogrov equation, 8 t = in (; ); >< >: = on (; n = on (@ n ) (; ); (x; ; ) = (x ) on fg (3.) where (x ) is the Dirac mass concentrated at x. If we denote by the principal eigenvalue of the operator subject to the mixed Neumann(zero n ) Dirichlet(zero on ) boundary condition, then 6 (x; y; t) 6 Ce t 8 y ; t > ; x where C is some positive constant. Since is positive, we see that decays in time exponentially fast. Next we introduce the function G(x; y) := (x; y; t)dt 8 x ; y : 8

15 Then it is easy to see that G(x; ) = on n G(x; ) = n. In addition G(x; ) = (x; ; t)dt = t (x; ; t)dt = (x; ; ) = (x ): Thus, G is indeed the Green s function of the Laplace operator associated with the Neumann-Dirichlet boundary condition; that is, for every x, G(x; ) is the solution of G(x; ) = (x ) in ; G(x; ) = on n G(x; ) = n : Note that the probability that a particle starting from x survives at time t is p(x; t) := (x; y; t)dy: Consequently, the probability that a particle dies in the time interval [t; t + dt) is p(x; t) p(x; t + dt). Hence, the expected life time (mean rst passage time) is given by n T (x) := E[ x ] = t p(x; t) = p(x; t)dt = o p(x; t + dt) = (x; y; t)dydt = where we have used integration by parts in the third equation. tp t (x; t) dt G(x; y)dy Since G is the Green s function, T is therefore the solution of the mixed Neumann-Dirichlet boundary value problem (.). In Monte Carlo simulations, con dence intervals are estimated in terms of the standard deviation, (x), of the exit time x, de ned by (x) = p v(x); v(x) = E[( x T (x)) ] = E[ x] T (x) : To calculate v, we introduce (x; y; t) = t (x; y; s)ds: 9

16 Then for xed x and t >, (x; ; t) satis es (x; ; t) = on n (x; ; t) = on n ; (x; ; ) = G(x; ) on : Also, (x; ; t) = (x; ; s)ds = t Thus, E[ x] = = = = = o t np(x; t) p(x; t + dt) t s (x; ; s) ds = (x; ; t) = t (x; ; t) in : = t p t (x; t)dt = tp(x; t)dt t (x; y; t)dydt = t t (x; y; t)dtdy (x; y; t)dtdy = (x; y; t)t (y)dydt T (y) y (x; y; t)dydt = T (y) t (x; y; t)dtdy T (y) (x; y; )dy = T (y)g(x; y)dy: This means that M(x) := E[ x] satis es M = T in ; M = on n M = n : Consequently, v = M T is the solution of (.). Finally, we prove (.3) as follows: v := v(x)dx = v(x)t (x)dx = T (x)v(x)dx jj jj jj = T (x)jrt (x)j dx = rt (x) rt (x)dx jj jj = T (x)t (x)dx = T (x)ds: jj jj This completes the proof of Theorem.

17 Remark 5. The transition probability density from W t = x to W t+t = y for the di usion process of Brownian motion con ned in with re ection boundary n and absorption boundary is (x; y; t) where is the solution of (3.). Thus, the exact discretization of the di usion process is W t+t = W t + k (W t ) 8 t T := [ k=fktg where f k g k= are independent random variables and k(x) has probability density (x; ; t). Our Monte-Carlo process de ned in (.) with t N(; I) is just a convenient approximation for t [; ] \ T. Proof of Theorem. We need only show that T given in (.5) satis es (.). For this, we use <(z) and =(z) to denote the real and imaginary part of a complex number z. Notice that in default, for z we have < z + p ( ze i" )( ze i" ) = <( z) + < p ( ze i" )( ze i" ) > : Hence, taking a real value at z =, the function f(z) := log z + p ( ze i" )( ze i" ) sin " is analytic in n fe i" ; e i" g and continuous on. harmonic in. Hence, Consequently, its real part, <(f), is T (z) = jzj + <(f(z)) = jzj = 8 z : Next, for z, we can write z = e i with jj 6 ". Then f(e i ) := lim log p rei + ( re i[ "] )( re i[+"] ) r% sin " q h sin " sin i sin i = log e i= sin " = i arcsin sin sin " 8 [ "; "]: Thus, <(f(z)) = when z. Consequently, T (z) = on.

18 Similarly, for n, we write z = e i with ("; ") to obtain lim q( arg re i[ "] ))( re i[+"] ) = r% + " " + = : Hence, q f(e i ) := lim f(re i ) = log he sin + sin sin " i( )= r% sin " i = ( )i + arccosh sin sin " 8 ["; "]: Here arccosh : [; )! [; ) is de ned by arccosh z := log[z + p z follows from the Cauchy-Riemann equation that in the polar coordinates (r; ), Hence, for n, ] for z >. <(f(ei =(f(ei )) = 8 ("; "): n T (z) <(f(z)) = Therefore, by the uniqueness of the solution of problem (.), T is given by the formula (.5). This completes the proof of Theorem. Proof of Theorem 3. The formula T () and T (e i ) follows directly from (.5) and the calculation of f(e i ) in the proof of Theorem. harmonic functions, T = jj ( jxj ) dx + <f() = + log sin " Also, by the Mean Value Theorem for = T () For the asymptotic (indeed Taylor) expansion, we can write T in (.5) as : T (z) = jzj + log j zj sin " + < log + p + b ; b := p z sin " z = p z^" z : When < ^" 6 j analytic nction flog +p + log + p + zj and z, we have jbj <. Applying the maximum principle for the g on the unit disc we nd that 6 The stated expansion for T thus follows. ln jj 6 :3jj 8 jj 6 :

19 Remark 6. (). It is clear from (.5) that the function T is smooth (C ) in n fe i ; e i g and Hölder continuous with exponent on. (). The constant T is the average of the mean exit time. It was derived in [, 9] in the case of one absorbing window, and in [5] for the case of a cluster of small absorbing windows that T = log " + O(): In [, Theorem 5.], it was rigorously shown that T = log sin " + O()": Clearly, our formula shows that the above O() term is indeed exactly zero. Proof of Theorem. The ux j is calculated T (ei ) =(f(ei )) and the expression of f(e i ). Note that for x, the quantity n(y) r y(x; y; t)ds y dt is the probability that a particle starting from x ends up escaping from ds y in time interval [t; t + dt). Hence, the probability that a particle starting from x ends up escaping from ds y is j(x; y)ds y where j(x; y) = n(y) r y (x; y; t)dt = n(y) r y G(x; y): () First we consider the case that initially the particle is at x =. Then direct computation show that Consequently, G(; z) = log jzj + jzj + T (z) : j(; e i G(; rei )j T (ei ) = j(e i ): Thus, j is the escaping probability density of particles starting from the origin. 3

20 () Next assume that the initial position of a particle is uniformly distributed in. Then the probability that the particle exits from ds y is j(x; y)ds y dx jj = ds y n(y) r y G(x; y) dx jj = ds y jj n(y) r y G(y; x)dx = ds y jj n(y) r yt (y) = j(y)ds y 8 Thus, j is also the escaping probability density of particles initially uniformly distributed in. This completes the proof.

21 . MONTE CARLO SIMULATIONS The Dicretization. We simulate the con ned Brownian motion by n sample paths, P ; ; P n. The sample path P i is described by fxi kt g k= where Xkt i represents the position of the particle at time t = kt. Here fxi kt g are random variables sequentially de ned by (k )t p Xi = i ; Yi k := X i + ik t; X kt i = ; ; n; k = ; ; i = Y k i maxf; jy k i j g ; where f ; ; n g are the starting positions related to the initial distribution of the particle, and f ik ji = ; ; n; k = ; ; g are i.i.d random variables with mean vector (; ) and covariance matrix equal to identity. Note that Y k i is the position at time kt of the discretized Brownian particle if it were not bouncing In our computations, we represent bouncing from the by the Kelvin transformation z! z=jzj for jzj >. Other re ection principles can also be used; for example, one can return the particle to its position before it took the nal step causing re ection, or re ect the particle geometrically from the boundary without signi cant change in the results provided the step size is small. If we are simulating particles starting from a xed point z, we simply take i = z for i = ; :::n. In the computations below we take all particles starting from the center, z =. For exit times from random points, one can generate random starting points from common random numbers (CRNs). As long as f ik g are i.i.d random variables with mean vector (; ), covariance matrix I and nite fourth order momentum, the limit, as t &, of the piecewise linear curve connected by points f(kt; Xi kt )g k= choices of the i.i.d random variables f ik g: is a Brownian motion path. There are two standard 5

22 number of exits per unit time number of exits per unit time number of exits per unit time number of exits per unit time x 5 8 sample paths, x = π / 8 x 5 sample paths, x = π / 8 ε = π / ε = π / 6 max = 6. mean =.58 3 max = 7.65 mean =.9 std =.3637 std = time of exits 6 8 time of exits x sample paths, x = π / 8 x sample paths, x = π / 8 3 ε = π / 8 max = ε = π / 8 max = mean =.537 std =.9.5 mean =.89 std = time of exits 6 8 time of exits Figure : Histograms for the escape times of, sample paths for various gate and step sizes. (i) f ik j i = ; ; n; k = ; ; g are i.i.d random variables with a probability of moving either southeast, southwest, northeast, or northwest, with distance p. (ii) f ik j i = ; ; n; k = ; ; g are i.i.d random variables with N((; ); I) Gaussian distribution. This is the method we use. It has the advantage that we can run statistics for any t >. Numerically, it may be preferable to use x := p t as a parameter to address the accuracy of the approximation of the Brownian motion by discretization. The First Hitting Time. For the ith particle path, fxi kt g k=, its time of rst hitting is de ned as T i := k i t; k i := minfk N j arg(x kt i ) [ "; "]; jy k i j > g: We terminate the simulation for the ith particle when we reach the time T i. Figure shows four histograms of the rst exit times ft i g n i=, in Monte-Carlo simulations with n = ;, x = =56, and gate size " = ; =; =8, and =8, respectively. 6

23 The Randomness Error. In a given Monte-Carlo simulation, f ik g are generated from CRNs according the needed distribution. The sample mean and sample standard deviation are calculated by ^T = n ( nx T i ; ^ = n i= = nx (T i ^T ) ) : i= Denote by T t = lim n! ^T. Then by the Central Limit Theorem, for n >, we can present our conclusion from a Monte-Carlo simulation as ^T T t + N(; ^ =n), or simply T t = ^T ^ p n with 65% con dence; T t = ^T 3^ p n with 99% con dence. In our simulations, we take n = ; particles, so the Central Limit Theorem can be reasonably applied (recalling from the proof of Theorem that the probability distribution density of x has an exponential tail). Our simulation (starting from (; )) shows that ^ is proportional to ^T with proportional constant almost equal to (cf. Figure ). Discretization Error. Using the approximation T t ^T ^= p n, we display the absolute error T t T (> ) and relative error T t =T in Figures 3 and. From the Figures, one sees that log (T t that T t T ) is almost linear in log " and in log x. This suggests T x " ; T t T x "( + j ln sin " j) : Distribution of Exits. Using the positions of the sample exits fx T i i g n i=, we can nd the sample distribution of the exits along the gate and compare it with the theoretical density j(y); y from Theorem. For xed ", we compare the empirical cumulative distribution function ecdf() := n X arg(x T i i )[ ";] for various di erent x, with the true cdf() = P arg(w ) ( "; ) = j(s)ds = " + arcsin sin sin " 7 8 [ "; "];

24 log ( MCT T ) Log (π/ε) Log ( MCT/T ) Log ( MCT T ) A bsolute E rror, paths A bsolute E rror, paths 5 5 MC mean plus std minus std 5 From top down, 5 From bottom up, x=π, π/, π/,..., π/ 9 ε=π, π/, π/,..., π/ Log (π/ε) Relative E rror, paths Log (π/ x) Relative E rror, paths 6 6 MC mean plus std minus std 6 From top to bottom, x=π, π/, π/,..., π/ 9 6 ε=π, π/, π/,..., π/ Log (π/ε) 6 8 Log (π/ x) Figure 3: Relative and absolute error for various gate and step sizes. ε, x, and error in Log 7 Level Set of Log ( MCT T ) log (π/ε) 5 log (π/dx) Log (π/ x) 7 Figure : Representation of relative and absolute error for various gate and step sizes using a three-dimensional plot. 8

25 cumulative distribution empirical cdf cdf and empirical cdfs ( ε = π/ =.578 ) cdf x=π x=π/ x=π/ x=π/8 x=π/6 x=π/3 x=π/6 x=π/8 x=π/56 x=π/ x=π/5=.6359, samples When ε/ x=56, Max Error= θ/ε (scaled exit position e iθ ) ε=π.3 ε=π/ ε=π/. ε=π/8 ε=π/6 ε=π/3. ε=π/6 ε=π/ cdf of exit position Figure 5: Cdf (cumulative distribution function) and ecdf (empirical cumulative distribution function) of the exit position along the gate. where is the rst hitting time of the con ned Brownian motion fw t g that starts from the origin and bouncing from the unit circle. For xed x and various " (noting that cdf depends on "), we consider the uniformly distributed random variable cdf(arg(w )). This is equivalent to plot cdf ecdf graphs. The results are shown in Figure 5. 9

26 BIBLIOGRAPHY [] Xinfu Chen & Avner Friedman, Asymptotic analysis for the narrow escape problem, preprint. [] A.F. Cheviakov, M.J. Ward, & R. Straube, An asymptotic analysis of the mean rst passage time for narrow escape problems, Part II: The sphere, to appear in SIAM Multiscale Modeling and Simulation, 3 pages [3] I.V. Grigoriev, Y.A. Makhnovskii, A.M. Berezhkovskii, & V.Y. itserman, Kinetics of escape through a small hole, J. Chem. Phys. 6 (), [] D. Holcman &. Schuss, Escape through a small opening: receptor tra cking in a synaptic membrane, J. Stat. Phys., 7 (), 975. [5] D. Holcman &. Schuss, Di usion escape through a cluster of small absorbing windows, J. Phys. A: Math Theory, (8), 55. [6] S. Pillay, M.J. Ward, A. Peirce, & T. Kolokolnikov, An asymptotic analysis of the mean rst passage time for narrow escape problems, Part I: Two dimensional domains, to appear in SIAM Multiscale Modeling and Simulation, 8 pages. [7] S. Redner, A Guide to First Passage Time Processes, Cambridge Univ. Press,. [8]. Schuss, Theory and Applications of Stochastic Differential Equations: An Analytical Approach, Springer. New York,. [9] A. Singer,. Schuss & D. Holeman, Narrow escape, Part II: the circular disk, J. Stat, Phys, (6), []. wanzig, A rate process with an entropy barrier, J. Chem. Phys. 9 (99),