A note on the transition from diffusion with the flow to diffusion against the flow , for first passage times in singularly perturbed drift-diffusion models

We consider some singularly perturbed ODEs and PDEs that correspond to the mean first passage time T until a diffusion process exits a domain fl in !Rn. We analyze the limit of small diffusion relative to convection, and assume that in a part of fl the convection field takes the process toward the exit boundary. In the remaining part the flow does not hit the exit boundary, instead taking the process toward a stable equilibrium point inside fl. Thus fl is divided into a part where the diffusion is with the flow and a complementary part where the diffusion is against the flow. We study such first passage problems asymptotically and, in particular, determine how T changes as we go between the two parts of the domain. We shall show that the mean first passage time may be exponentially large even in the part of fl that is with the flow, and where a typical sample path of the process hits the exit boundary on much shorter time scales.


Introduction
Diffusion processes/models have been used in many applied areas, dating back to the work of Einstein on Brownian motion [4].
Subsequent early work on diffusions and stochastic differential equations was done by Smoluchowski, Langevin, Ornstein and Uhlenbeck, and Kramers (see [8,13,17,18] and the early survey in [l]).Such models have found applications in a large number of areas, including chemical kinetics, genetics, signal filtering, and mathematical finance (see the books of Schuss for more discussion of various applications [15,16]).At times diffusion models arise naturally, while at other instances they arise as continuous limits of discrete models, where the limit may be viewed as a type of functional central limit theorem.Such is often the case for problems in mathematical biology and queuing theory, where a discrete model may be approximated by a continuous diffusion model as the size of the (discrete) state space becomes large.
Give a diffusion process Xd(t), whose state space is ltln or a subset thereof, one is often interested in computing the probability, Pr[Xd(t) E SJXd(O) = x], that the process is in some set S at time t given that it started at the point x at time t = 0. Another important quantity is the first passage time until the process Xd(t) reaches some set or region.Typically we would start with Xd(O) = x, for x E n C Itln (dim(n) = n) and ask when the process exits n, and thus reaches the boundary an, which will typically be some ( n -1 )-dimensional manifold.The physical significance of the first passage time depends on the particular application.For example, it may be the time for the crossing of a potential barrier leading to a chemical reaction [8], the time until a cycle slip in a phased-locked loop [20], the time until a queuing system reaches its maximum capacity of customers, or the time until a certain biological species becomes extinct [9].More on first passage times and applications appears in the books [15,16] and [14], with the latter being devoted entirely to first passage processes, including various applications such as fractal networks and reaction-diffusion problems.For a model that incorporates both diffusive and convective (or drift) effects, the mean first passage , time where T = min{t: Xd(t) E an}, satisfies an elliptic PDE of the form L[T] = -1.This is sometimes referred to as the Dyhkin equation [3], and is closely related to the backward Kolmogorov equation of Markov processes.Here L is an elliptic partial differential operator and we have also the boundary condition that T = 0 for x E an.
For some simple models, such as a Brownian motion or an Ornstein-Uhlenbeck process, we can solve explicitly for the mean passage time, at least in one dimension.But for complicated drift/diffusion fields, complicated geometries of an, and problems in dimension n > 1 it is usually very difficult to solve exactly for T, and thus approximate, e.g., asymptotic or numerical, methods must be used.A popular and fruitful limit to consider is where convection dominates diffusion, and then L has a form such as L = E Ll + a(x) • \7, so a small parameter multiplies the highest derivatives in L, and the problem becomes of "singular perturbation" type.Singular perturbation methods for computing the first passage times are developed in [10][11][12], and in the probability literature such an asymptotic limit is often treated using the theory of large deviations (see [5,19] and [2] for some classic references).
The behavior of the first passage time T(x) as E ---+ o+ is highly dependent upon the behavior of the drift field a(x) and how it interacts with the boundary an which the process is to hit.We refer to the subcharacteristics of the PDE as the solutions to the ODE(s) x = a(x), which is an elementary dynamical system, and this corresponds to neglecting diffusion entirely and thus approximating Xd(t) by a deterministic or "fluid" process.If the subcharacteristics hit the boundary we refer to the problem as "diffusion with the flow".Then the diffusive effects may be small.If, however, there is a globally stable equilibrium point x 0 that is inside the domain n, then all subcharacteristics flow toward the equilibrium point and possibly none of them hit the boundary.Then the effects of diffusion become more significant and they are needed for the process to ever reach the exit boundary.This is called "diffusion against the flow", and a singular perturbation method for computing Tis discussed in [10], where it was found that T(x) is asymptotically a constant, say C(c:), that is exponentially large for small E (roughly O(eK/c:) for some K > 0).Thus the passage time is independent of the starting value, since most likely the process will flow along a subcharacteristic toward the.stable equilibrium at x 0 , and spend a long time in a neighborhood of this point before finally undertaking the "large deviation" to exit the domain n.
The purpose of this note is to study the asymptotics of the mean first passage time for problems where some subcharacteristics in n hit the exit boundary while others do not.Thus we study the transition from diffusion with the flow to that of against the flow.By studying some model problems in one and two dimensions, we shall gain a better understanding of this transition, for E ---+ o+.We divide [l into the two parts D+ and D_, where in D+ a subcharacteristic flows towards the stable equilibrium without hitting an while in D_ it hits the boundary.We shall show that while in D+ the solution is asymptotically exponentially large and constant, in D_ it may be exponentially large but highly dependent on the initial value of the process.In other parts of D _ the solution may be 0( 1) as E ---+ 0, and there the deterministic approximation is valid.The analysis carefully studies the boundary region(s) near an, and also the transition curve C that separates D+ from D_. Near C we shall show that there are two nested internal layers that lead to different expansion of T; these correspond to the distance from C being either O(c: 1 1 3 ) or 0( JE"), and we note that C is (a portion of) the unique subcharacteristic that becomes tangent to the boundary an.The main focus here is on the mathematical (singular perturbation) methodology.
The remainder of the paper is organized as follows.In Section 2 we consider one-dimensional models, that may be solved exactly, and make some observations about the structure of the first passage time as E ---+ 0. In Section 3 we analyze in detail a two-dimensional problem, that cannot be analyzed exactly, and that will show the basic asymptotic approach.A brief discussion, and possible generalizations, appears in Section 4.

One-dimensional passage times
We consider a Brownian particle X(t) moving in a potential field U(x), with a small diffusion coefficient c:.Then we define T = min{t: X(t) = x 0 } to be the first passage time to the point x 0 .The mean value satisfies the backward Kolmogorov equation with the "absorbing" boundary condition We assume that the potential U(x) satisfies U'(x) < 0 for x < x 2 or x > x 1 , U'(x) > 0 for x2 < x < x 1 , for some x2 < x1 < xo, U(-oo) = oo, U'(x2) = 0 = U'(x1) and U 11 (x2) > 0, U"(x 1 ) < 0. Hence, x 2 is a local minimum of the potential while x 1 is a local maximum.Equivalently, x 2 (resp., x 1 ) is a stable (resp., unstable) equilibrium point of the flow x = a(x) = -U'(x), which is a deterministic, or "fluid", approximation to the stochastic process X(t).We also assume that U(x) is smooth for all x :( x 0 , and in Fig. 1 we sketch a typical potential.The mean first passage time T(x) depends also on c and the exit point x 0 , and we evaluate it in the limit of small diffusion (c ---+ o+) and various ranges of x and x 0 .The ODE in (2.2) is easily solved to give: Proposition 1.The mean first passage time has the integral representation For c; ---+ 0 and initial conditions x < x 1 the drift takes the process toward the stable equilibrium at where it spends a large amount of time before finally exiting at x 0 .This type of behavior is essentially independent of the starting point x, unless x becomes very close to the unstable equilibrium at x 1 , and is referred to as "small diffusion against the flow".In contrast, if x > x 1 then U'(x) < 0 for x E (x 1 , x 0 ] and the drift a(x) = -U'(x) takes the process toward the exit point x 0 .This is called "small diffusion with the flow", and then we might expect that for c;---+ 0, (2.2) and (2.3) may be approximated by the first order equation whose solution is rxo 1 (2.6) This means that we approximate the stochastic process by its fluid limit and thus neglect diffusion completely.But, we show below that (2.6) may hold only in a certain subset of the interval [x 1 , x 0 ], and in fact it may never hold.Below we summarize several asymptotic results for (2.4) as c; ---+ 0 and thus identify precisely the conditions under which (2.6) holds.

O(E)
x. -U'(ry) rJ + -U'(x*)JU 11 (x2) + . (2.12) These results show that if U(x 0 ) > U(x 2 ), i.e., the value of the potential at the exit boundary exceeds that at the stable equilibrium, the "fluid" behavior in (2.6) is never observed, and the first passage time is always exponentially large in E. In such cases T(x) is exponentially large and independent of x for x < x 1 and undergoes the transition, as x increases through xi, to the behavior in (2.9) which does depend significantly on the starting point x.The expression in (2.10) may be viewed as a "boundary layer" correction to (2.9), for starting points near the exit boundary.Note that in order for T(x) to become roughly 0(1), z = (x 0 -x)/E would need to be exponentially small, of the order O(e[U(xz The results in Proposition 2 may be explained intuitively as follows.For initial conditions x such that x 1 < x < min{xo, x.} a typical sample path of X(t) is taken toward x 0 by the drift, but there is a very small (in fact, exponentially small) fraction of sample paths that reach the range x < x 1 , which is the domain of attraction of the stable equilibrium point at x 2 • While this fraction of paths is extremely small they tend to lead to exponentially large exit times (similarly to the case x < x 1 ).So, on the one hand a fraction close to one leads to the 0(1) exit times in (2.7), but on the other hand a very small fraction leads to exponentially large times.Our analysis shows that the former sample paths dominate the mean exit time for x E (x., xo) (if U(x) = U(x 2 ) has a solution for x E (x 1 , x 0 )) while the latter paths dominate for x E (x 1 , x*), and possibly for all x < x 0 .We also note that neither of the expansions in (2.9) and (2.11) exhibit any singular behavior as x --+ x*, and (2.12) shows that for x ~ x* the two classes of sample paths contribute roughly equally to the expansion of T(x ).
Below we sketch only briefly the derivation of Proposition 2, since it follows from a standard application of the Laplace method to the integral in (2.4).We write T(x) = c-1 J: 0 I(ry; c) dry with is attained at the interior points~ = x 2 and we have x 2 < rJ < Xi. (2.14) (2.15) (2.16) If rJ > x 1 , U(O has local minima both at ~ = x 2 and at the upper limit ~ ry.The relative sizes of these two contributions depend on whether U(ry) ~ U(x 2 ).Thus if U(ry) = U(x 2 ) has the solution rJ = x* (> x,) then for TJ < x* (2.16) holds, while (2.14) applies for rJ > x*.For rJ ~ x* we simply add (2.14) to (2.16).
Now we integrate c-1 I from rJ = x to T/ = x 0 , noting that different approximations (out of (2.13)- (2.16)) may apply in different subintervals.Suppose first that x E (x 1 , x 0 ) and U(x 0 ) > U(x 2 ).Then (2.16) applies over rJ E [x, x 0 ] and the integral over rJ is a Laplace integral with the major contribution coming from the lower limit rJ = x; this leads to the expression in (2.9).
then the interval of integration is small, but we may still approximate the integrand using (2.16).This leads to where we expanded (2.16) about rJ = x 0 .Evaluating the integral in (2.17) leads to the expression in (2.14) applies over the entire range of integration and we thus obtain (2.11).But, if x E (x 1 , x*), (2.16) applies and we again obtain (2.9 and this leads to (2.8).This completes the derivations.

A two-dimensional example
Here we analyze a singularly perturbed two-dimensional diffusion model, which will illustrate some asymptotic phenomena similar to those in Section 2, but also lead to new complications.Since this model does not seem exactly solvable, we shall apply a singular perturbation approach.

Problem statement
Let n be a domain in JR. 2 and let T be the random time until a diffusion process (X(t), Y(t)) exits n by hitting the boundary an, i.e., T = min{t: (X(t), Y(t)) E an}.Taking the drift vector as (a(x, y), b(x, y)), and again assuming a small diffusion coefficient c, then mean first passage time  We shall assume that there is a unique stable equilibrium point at (x, y) = (0, 0) that is inside n and consider the drift field where a is a constant.The boundary will be the straight line x + y = 1 so that For simplicity we shall also take a = 2, but the analysis is virtually unchanged for any a > 1.In Fig. 2 we sketch the domain n.
If c = 0 the problem in (3.2) becomes a deterministic one, and T becomes the time it takes for a "subcharacteristic" curve to reach the line that is an.The subcharacteristics are defined as the solutions to (i:, y) = (-ax, -y) so we get the curves y = Clxl 1 /a and if a= 2 these are the parabolas y 2 = kx, k E R The origin is thus a stable improper node of this simple dynamical system.Through each point (x, y) f-(0, 0) passes a unique subcharacteristic that eventually approaches the origin.But a subcharac- where and The curve C = { (x, y): y = 2yf=i, x < -1} is the unique subcharacteristic that becomes tangent to the boundary (at the point (-1, 2)) before continuing toward the origin.Thus for (x, y) E D+ the drift field takes the process toward the stable equilibrium before hitting the boundary, while for (x, y) E D_ we hit the line x + y = 1.We would hence expect T(x, y) to be, for small c, very large and nearly constant in D+, but be 0(1) for (x, y) ED_.Neglecting diffusion entirely in (3.2) and solving the elementary PDE leads to and we note that the above satisfies T(x, 1x) = 0 for x ::;; -1.As y decreases through C the expression in (3.9) becomes complex and thus clearly invalid.However, we shall show that (3.9) is correct only in a subset of D_, just as (2.11) was only valid for x E (x*, x 0 ) c (x 1, x 0).We shall also obtain the correct form of T(x, y) in the remaining part of D_, and study carefully the asymptotic transition(s) along the critical parabola y 2 + 4x = 0 (y > 2, x < -1 ), and near the "corner point" (2, -1) where the curve is tangent to the exit boundary; it is at this corner where the structure of T(x, y) is the most complicated.

Summary of results
An asymptotic analysis of (3.5) leads to the following results.
Proposition 3. The asymptotic expansions ofT(x, y) as E ---+ 0 are where D+, D_ and an all meet; TJ < 0.  -2t) and (s, t) are related to (x, y) via the mapping The range t > 0 ands < -1 corresponds to (x, y) E D_.Furthermore, is the Jacobian associated with (3.20).The equation x [Ai'(ro)] e-u,Py(x,l-x) c F(x)Ai 2 (vi where rk are again the Airy roots, and v=;r(43x 4 + 18x 2 + 3) The results in Proposition 3 show that the asymptotic structure of T(x, y) is in fact quite intricate, even in the simple case of the linear drift functions in (3.4) and the linear exit boundary.Later we discuss possible generalizations.For now we note that T(x, y) has essentially different behaviors in the three regions D+, D* and Df.In D+ the mean first passage time is exponentially large and independent of the starting point (x, y).In D* it is also exponentially large in 1/c:, with an additional subexponential factor that is exp[O(c:-1 1 3 )] (cf.(3.16)), and now depends intricately upon (x, y).In Df we have T = 0(1) and there (3.9) applies asymptotically, which neglects diffusion entirely.
The remaining five ranges in the (x, y) plane treated in Proposition 3 represent boundary layer and internal layer expansions.Cases (ii) and (v) apply near the exit boundary and (ii), which applies for 1 -x -y = O(c:), gives a boundary layer correction to (3.10), while (v) applies in the thicker range 1 -x -y = O(c: 2 1 3 ) and gives a boundary layer correction to the second term in the right-hand side of (3.6).For values of x < -4.615 ... no boundary layer correction is needed as (3.9) satisfies the boundary condition for all x < -1.
More discussion of the various asymptotic matchings occurs within the derivations in Section 3.3.
In Fig. 3 we sketch the three main regions D+, D* and Df, also indicating the separating curves I' and C (y = 2Fx).

and (3.3). Thus we analyze the PDE
In the range D+, where Tis asymptotically large and independent of (x, y), the method of Matkowsky and Schuss [10,11] applies, which yields the results in (3.10) and (3.11).We merely sketch the main points.If C(c:) ---+ oo and c: ---+ 0, T(x, y) '"" C(c:) is a possible asymptotic solution in D+, as all subcharacteristics reach the origin.But this solution does not satisfy the boundary condition in (3.3).
]} du cv -hnc:. (3.35)The main contribution to the integral comes from u = 2/3, and using the boundary layer expansion We next consider D_, where the subcharacteristics reach the exit boundary x + y = 1, at points that have x > -1, so that (3.11) does not apply.We employ the ray method of geometrical optics, where we seek an asymptotic solution of (3.32) in the form Here v is a constant and for convenience we included the multiplicative factor C(E), to facilitate the asymptotic matching between the regions D_ and D+.As long as C(1:)e7/J/r:: is asymptotically large, which, in view of (3.10), requires that 'lj;(x, y) + 1 /3 > 0, the term -1 in the right-hand side of (3.32) will be negligible compared to the left-hand side.But, if 'lj;(x, y) + 1/3 < 0 then we can construct an expansion of the solution of (3.32) that remains 0(1) as E ---+ 0, and this yields precisely (3.9) as the leading term.By linear superposition, T(x, y) in D_ will be asymptotic to the sum of (3.9) and (3.37), with the condition 'lj;(x, y) + 1/3 = 0 determining a curve I' in the (x, y) plane, above (resp., below) which (3.9) (resp., (3.37)) is the dominant part.Of course, we have yet to determine 'lj;(x, y).Using (3.37) in (3.32) we obtain the eiconal and transport equations In addition, the sub-exponential (O[exp(1:-1 1 3 )]) factor in (3.37) must satisfy The factor involving 'lj; 1 is needed in order to accomplish the asymptotic matching between (3.37) and the boundary layer near x + y = 1, x < -1, as well as the internal layer( s) that we shall construct near the transition curve y = 2-j=x, x < -1.But since 1: 1 / 3 does not appear in (3.32), 'lj; 1 satisfies a fairly simple homogeneous linear PDE, once 'ljJ is computed.Equation (3.38) admits many different solutions, depending on the "initial manifold" on which the values of 'ljJ are given.The characteristic curves, or rays, for the PDE are obtained by solving the system and the solution 'ljJ follows from Here "." denotes the directional derivative along a ray.We choose the initial manifold to be the exit boundary, where (x, y) = (s, 1 -s), s < -1, so a ray hits the exit boundary where x = s.Furthermore, we need the solution where each ray is tangent to the exit boundary.Then the exit boundary will become a caustic of these rays.That this is the appropriate ray family for this problem can be argued by examining the structure of the corner layer, where (x, y) ~ (-1, 2) and (3.12) applies, as this solution involves the Airy functions that are characteristic of caustic curves.By expanding (3.12) as we go away from :hat (-1, 2) into D _, we can also argue that (3.37) is the appropriate form of the expansion, and in particular we that the sub-exponential factor must be included.We shall first analyze D_, then the boundary [)[2 for x < -1, and finally the comer region near ( -1, 2).However, the singular perturbation analysis could also be done by considering the regions in a different order.

52)
It remains to determine the functions K 0 (s) in (3.52), 'ljJ 1 (s) in (3.49), and the constant v in (3.37).This we accomplish below, by asymptotic matching to some boundary and comer layer expansions.Some more properties of the rays and the function 1/J(x, y) are discussed in the Appendix.
We choose k = 0, since the higher roots would lead to exponentially smaller terms in the ray expansion in (3.37).The necessity of taking k = 0 will also follow from asymptotic matching considerations, The boundary condition is S(O, T) = 0 for all T, and we can also obtain asymptotic matching conditions for S(u, T), as T-+ ±oo.For x > -1 the boundary layer expansion in (3.11), noting that with UT fixed.By expanding (3.54) as x-+ -1, noting that and we get a second matching condition: S(u, T) , . .__, c:vi F(l)eT3 /384e-uT/8 x exp [-2 ~~3 T] Ai(2-2 1 3 u + r 0 ), T-+ -oo.We previously encountered very similar PDEs when analyzing Brownian particles moving in a timedependent drift field [7], and problems in financial math dealing with option values evolving under the CEV (constant elasticity of variance) process [6].The only difference was that in [6], TJ had the opposite (3.87) This solution can also be obtained as a limiting case of some more general results in [7], where we analyzed problems similar to (3.84), but on semi-infinite T 1 intervals.We have now completely determined the leading term in (3.37),where v = 1/6, and that in (3.54),where v 1 = 0.
It remains to analyze the vicinity of the transition curve y = 2yf=x, and its analysis can be used to give an alternate derivation of (3.87).We first consider the scale y -2yf=x = c 1 1 3 77, with 77 > 0 and x < -1.On this scale we set  G(x,77) =Go (1-3 x)(-l _ x) .

6 )
applies but only in the range x E [x*, x 0 ], and for x E [x 1 , x,) the first passage time remains exponentially large.The scale x -x* = O(E) represents the transition from T(x) being exponentially large to being 0(1), with the deterministic approximation (2.6) taking hold as O'.---+ + 00.

Fig. 2 .
Fig. 2. A sketch of the domain fl.(Colors are visible in the online version of the article; http://dx.doi.org/10.3233/ASY-141259.) teristic may hit the exit boundary x + y = 1 before reaching the origin.Thus we may naturally divide D into two main parts, writing D= D+UD_ uc,
1 (where exit boundary meets D+) Here v 1 is another constant that we shall determine later.The exponential factors in (3.54) must be included in order for (3.54) to have a chance (as u -+ +oo) of asymptotically matching to (3.37).By using(3.54)in the homogeneous version of (3.32), expanding for small E, and noting that (3.38) holds along x + y = 1, we obtain at the first two orders in E the following equations: 2Bo,uu + [ v; Vl -2x + 5x2 'lj;;(x)