ie - Arizona Math - University of Arizona
LIMITING EXIT LOCATION DISTRIBUTIONS IN THE STOCHASTIC EXIT PROBLEM ROBERT S. MAIER AND DANIEL L. STEIN
Abstract. Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point . If the deterministic dynamics are perturbed by white noise (random perturbations) of strength , the system state , the exit location on will eventually leave the domain of attraction of . We analyse the case when, as the boundary is increasingly concentrated near a saddle point of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter , equal to the ratio of the stable and unstable eigenvalues of the linearized deterministic flow at . If then the exit location distribution is generically asymptotic as to a Weibull distribution with shape parameter , on the lengthscale near . If it is generically asymptotic to a distribution on the lengthscale, whose moments we compute. Our treatment employs both matched asymptotic expansions and stochastic analysis. As a byproduct of our treatment, we clarify the limitations of the traditional Eyring formula for the weak-noise exit time asymptotics.
#$% &'(*)+-,.
!"
0/1
&'(-23+,
Key words. Stochastic exit problem, large fluctuations, large deviations, Wentzell-Freidlin theory, exit location, saddle point avoidance, first passage time, matched asymptotic expansions, singular perturbation theory, stochastic analysis, Ackerberg-O’Malley resonance. AMS subject classifications. 60J60, 35B25, 34E20
1. Introduction. We consider the problem of noise-activated escape from a planar domain 4 with smooth boundary, in the limit of weak noise. If 57698;:.?69@ A%B is a smooth vector field on a neighborhood of the closure 4 C , we define the random process DFE 8HG=6I8KJ <E 8KG== , >L6I@ AB by the Itˆo stochastic differential equation (1.1)
W M <E M MZY W J H8 G=F6N: < 8 D E K8 G== GPORQ$S-T-ULVW"X < 8 D E K8 G== 8KG= Y W
and an appropriate initial condition. Here 8KG= , [?6\@ AB , are independent Wiener processes W and X16I83X < = is a B -by- B noise matrix, like 5 a function of position D 678HJ < = , >]6^@ZA%B . The associated diffusion tensor _`6I8Ka @8A? 9B;=
= L8? Y 6[Z 6 8?A\^] _a`Vb $ X$< 6>= L8? 8 J" 8?S(N*PO)Q)RcIWT - . / Y 6 8Md`e] _f\gb - /. $ 8 6LZ h"i8?S(N*PO#QSR@IjT 8 - /. Yk ln/ m ohp ] 8 A s q r 6 6 = L8?P5 = "8 = t 6H= q t 6H= 8? Z L*P(+CD t 6H=+L8?uY 6 Y = o "9Y 6Sv rw6 = 8? 53=@"98A=B oyxzp ]{F (5.2) Assume for the moment that the appropriate lengthscale on which the inner approximation should be defined is the |}*PO#QSR~ lengthscale. If so, we can employ the ‘stretched’ variable 6 p 5 6 "f8 6 S(N* O#QSR , in terms of which (5.2) becomes bt YzR o Y (5.3) C 6H= 8? Y 6 Y = "fr 6 = L8? Y 6 v = oyxzp ]{F ? 6 p 6 = = 8? O L8? 8 8? 78?
p
We can change variables to reduce this covariant equation to a noncovariant, but more understandable form. Under a linear change of variables , i.e., , the matrices and transform to and respectively. Choosing transforms to the identity matrix. But since is a saddle point, irrespective of coordinate transformations the linearized drift will have one positive eigenvalue ( ) and one negative eigenvalue ( ). By a further change of variables (a rotation) we can set . So we can choose
c8? L8? Vp J8M O)QSR J8M N8M rhO R 8M p ] 9L8? ps 8M L] 8?} (5.4) for some real constant . The constant is not determined by z8M and N8M . Since JL8? p is preserved under rotations, with respect to the new system of coordinates equa-
tion (5.3) becomes
(5.5)
b zY R o C Y O R "czL8? Y Y
b zY R o C Y R R Y Y vO O oDx "cN8? Y R v R oyx " Y R v O oyxzp ]{F
17
LIMITING EXIT LOCATION DISTRIBUTIONS
fN#?
we may take the region to be the rightIn terms of the transformed coordinates half plane , and its boundary to be the -axis. This was the convention of Figs. 4.1 and 4.2. This system of coordinates is computationally easy to work with. Suppose for simplicity that ; this is an innocuous normalization condition that can be absorbed into a redefinition of time (and noise strength ). Then
f ¡
£7¤z¥?§¦s¨
(5.6)
©
¢
M
ª « ¥M¬¦® ¯W ¨ ²° ¡ ± ³
± µ ¦ ´ £¶N¥M ´ ·+£ ¤ L¥? , as usual. The Kolmogorov equation (5.5) reduces to (5.7) ¸¨ ¢¢ ¹ º ¸¨ ¢¢z P¹ ° ¢ ¢ z» ¹D¼ º ± ¢ ¢ ½» ¹y¼ °c¯ ¢ ¢ ½» ¹y¼z¦ ¡{¾ « Also, a bit of matrix computation, using the form (5.6) for L¥? and ¿ ¥?¦ À , yields (5.8a) Á ¶ )¡y'¦Â¡ ¨DS¡{S¡y (5.8b) LÁ{¤)¡y'¦Â ± º ¨D ¯ )¡{S¸ ¡y ¸ (5.8c) SÁ à ¶ +Ä Ã ¶ ¦Â ±Å¸ ° ¨D ¯ ° ± º )¡ ¸ (5.8d) SÁ{à ¤DÄ Ã ¤ ¦Â ° ±½¯ ± ° ¨ °f¯ ° ±½¯ ±A° ¨B, ¸ ± ± ° ¨~# for the four eigenvectors of the linearized Hamiltonian flow Æ2¥M at the point L¥Ç)¡ in phase space, as given by (4.5). (Normalization is irrelevant here; the negatives of these vectors could equally well have been chosen.) That Á{¶È¦e¡ ¨~ agrees with the convention of Figs. 4.1 and 4.2. We note in passing that the formulæ (5.8) explain the positioning of the rays Á{¤ , Á à ¶ , and Á{à ¤ in Figs. 4.2(a), 4.2(b), and 4.2(c). Recall that in those figures the MPEP was taken to approach from the first quadrant; if ± É ¨ , it is generically tangent to Á ¶ , and hence to the positive M -axis. By examination of (5.8), if ±É ¨ and ¯É ¡ then Á{¤ will lie between the positive ? -axis and Á{à ¤ , while if ¯ Ê¡ then Á{à ¤ (taken to point into the right half-plane) will lie between the positive ? -axis and Á3¤ . So the Subcases A and B of Figs. 4.2(a) and 4.2(b) are simply the subcases ¯2É ¡ and ¯ ¡ . This correspondence assumes of course that the MPEP is tangent to the positive M -axis; if it approached ¥ from the fourth quadrant, and were tangent to the negative ? -axis instead, then the interpretation in terms of sgn ¯ would be reversed. Formulæ (5.8) justify the positioning of the rays in Fig. 4.2(c), as well. If ± ˨ , we know by the arguments of the last section that the approaching MPEP is generically tangent to Á à ¶ . By (5.8c), Á à ¶nÌ ±° ¨D ¯ , and our convention that the MPEP approaches from the first quadrant mandates that ¯iÍ ¡ . It is easy to verify, by examining (5.8), that in the right-half plane, when ± ¨ and ¯ Ë¡ the Á{¤ ray necessarily lies between the the Á à ¶ ray and the the Á{à ¤ ray. This was the positioning of Fig. 4.2(c). In our new coordinate system, it is easy to study quantitatively the quadratic behavior of the action Î near the saddle point. Substituting (5.6) into the Riccati equation (4.3), and solving, gives Ï ¸ ±Å° ¨~ ¯ ¨ °M± °@¯,± (5.9) ¥?Ц ±Å° ¨B ¯ Ï º °@¯P± ± °f± ³ º ¸ Ï matrix ¥?Ȧ » ÎÑ ÒHÓy¥?Ô¼¬¦ » ¢Õ7Ò;·N¢Ö Ó ¥M;¼ . Recall as the formula for the rank- Hessian that generically this limiting Hessian matrix arises (when ×ØÙ¥ in the complement of the wedge) only when ± g¨ . If ±MÉ ¨ , L׬ØÛÚ as ×9Ø®¥ along the MPEP. where
def
18
R. S. MAIER AND D. L. STEIN
Ü
âAá ãLä?å
in the wedge (to the extent that it is quadratic) can be The quadratic behavior of computed similarly. In Subcase A of , the limiting Hessian matrix in the wedge exists, and equals by Table 4.1 the rank-1 matrix corresponding to the tangent space sp . The formula
ÝVÞàß
æ ãç{èéSêyå,éBãSç{ë è7éNì ë è åîí ñ ò å öãñ ãÝ ó åß å A â ã M ä ¬ å ï ð Ý ã ã å á (5.10) Pñ òôó Ý ó ß ò9õ ö ñ Ý ó ß Ý ó ß òK÷ follows from (5.8) elementary In Subcase B, and when ÝJøùß , the ú ûÙmanipulations. ä will not be quadratic. ãLúå inbythesomewedge behavior of Ü as it follows from úûüä along ýzþ (from the However, the formula (4.10) that its behavior as wedge side) is quadratic, with limiting second derivative ÿ á ¬ãä?å¬ï ð Ý (5.11) ä This limiting second derivative has a simple interpretation: on the wedge side of , the cost (action) of the most probable trajectory leading to any point on ýzþ arises from the drift on ýþ itself. (By (5.6), the drift along ýzþ is proportional to Ý .) ã
û å Gaussian (or inThe formulæ (5.9)–(5.11) make quite precise the far-field ã Bé ò å verted Gaussian) asymptotics that must be imposed on the solution of the transformed Kolmogorov equation (5.7). First, to leading order we must have
ã é ò å ö )âAãLä?å ð é û! outside the wedge. Also, in Subcase A of ÝfÞgß we must have "ã #é ò å $%'& ö âAá ãLä? å ( ð*) é Ùû+ inside the wedge. (5.13) In Subcase B of ÝMÞËß , or when ÝMøgß , we must have "ã #é ò å & ö ÿ ,ãLä?åã ò å ò ð ) é Ùû+ along ýzþ , on the wedge side, á (5.14) ò ãê{é- å!ïµê must since the boundary ýþ is the -axis. The Dirichlet boundary condition also be imposed, since the quasistationary density is absorbed on the boundary. to solve the partial differential equation (5.7) on the half-plane /In ê , subjectit istonottheseeasyboundary . general conditions. Our treatments of the generic ÝÂÞsß and generic Ýeøüß cases, in Sections 6 and 7 respectively, will be crafted to circumvent this ã 1 2 ò å problem. For the case ÝJÞß we shall expand on a larger lengthscale than 0 ; for the case Ýcøß , on which we have less information, we shall use stochastic analysis. In advance of our detailed treatments, we observe that the far-field behavior of the exit location density 354 6ã 1 #2 ò å ä (on the 0 lengthscale near ) follows from (5.12)–(5.14). By (2.3) the exit location density is simply proportional to the normal derivative of the inner approximation 7948 . If we âAãä?å , âAá ãäMå and ÿ á ,¬ãLä?å into (5.12)–(5.14), we obtain substitute the above expressions for (5.12)
far-field asymptotics (5.15a)
ã6:~å; 354
Ý9Þ ß
(Subcase A), and
if
(5.15b)
%@?>êA-ã6: ò *1,åCBé % ö Ý ã Ý ã ó ß
ã6:~åGF 354
åò õ ñ ò ó Ý ó ß å ò 6ã : ò D1,å ÷
:D D1 2 ò éE:D D1 2 ò
@?HêA-yãH: ò D1,åIB éJ:D D1 2 ò ? ö ã6: ò D1,å B éED: D1 2 ò Ý
û ó
û ö
û ó
û ö ;
;
19
LIMITING EXIT LOCATION DISTRIBUTIONS
if KL(M (Subcase B), and (5.15c)
W XZY%[]\_^C` K Q K ` ` M S a Q6R,aDdDefSCgihERDdDej#klanm p c o NPO%Q6RTSU :A@ Kolmogorov .B @1C @
87 , as follows. In the linear approximation we take 9$:
87 when 0ji 7 , for both Subcase A and Subcase B. set 5 A family of solutions of equation (6.8), each with far-field asymptotics that are Gaussian in + and 0 , can be found by inspection. Each solution is of the form #]VRon6p1qr +[0 C > 0 $ if 0js 7 ; 5 + 0 kml 70 (6.9) 7 if 0ji , > for some constant . Actually we shall use antisymmetrized (odd) versions of these solutions, 7 t ?7 on account of absorption of since 5 must satisfy the Dirichlet boundary condition 5 def
24
R. S. MAIER AND D. L. STEIN
u,v . Antisymmetrizing under wyxz|{4w yields }~ w$ m ] N ~
Abstract. Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point . If the deterministic dynamics are perturbed by white noise (random perturbations) of strength , the system state , the exit location on will eventually leave the domain of attraction of . We analyse the case when, as the boundary is increasingly concentrated near a saddle point of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter , equal to the ratio of the stable and unstable eigenvalues of the linearized deterministic flow at . If then the exit location distribution is generically asymptotic as to a Weibull distribution with shape parameter , on the lengthscale near . If it is generically asymptotic to a distribution on the lengthscale, whose moments we compute. Our treatment employs both matched asymptotic expansions and stochastic analysis. As a byproduct of our treatment, we clarify the limitations of the traditional Eyring formula for the weak-noise exit time asymptotics.
#$% &'(*)+-,.
!"
0/1
&'(-23+,
Key words. Stochastic exit problem, large fluctuations, large deviations, Wentzell-Freidlin theory, exit location, saddle point avoidance, first passage time, matched asymptotic expansions, singular perturbation theory, stochastic analysis, Ackerberg-O’Malley resonance. AMS subject classifications. 60J60, 35B25, 34E20
1. Introduction. We consider the problem of noise-activated escape from a planar domain 4 with smooth boundary, in the limit of weak noise. If 57698;:.?69@ A%B is a smooth vector field on a neighborhood of the closure 4 C , we define the random process DFE 8HG=6I8KJ <E 8KG== , >L6I@ AB by the Itˆo stochastic differential equation (1.1)
W M <E M MZY W J H8 G=F6N: < 8 D E K8 G== GPORQ$S-T-ULVW"X < 8 D E K8 G== 8KG= Y W
and an appropriate initial condition. Here 8KG= , [?6\@ AB , are independent Wiener processes W and X16I83X < = is a B -by- B noise matrix, like 5 a function of position D 678HJ < = , >]6^@ZA%B . The associated diffusion tensor _`6I8Ka @8A? 9B;=
= L8? Y 6[Z 6 8?A\^] _a`Vb $ X$< 6>= L8? 8 J" 8?S(N*PO)Q)RcIWT - . / Y 6 8Md`e] _f\gb - /. $ 8 6LZ h"i8?S(N*PO#QSR@IjT 8 - /. Yk ln/ m ohp ] 8 A s q r 6 6 = L8?P5 = "8 = t 6H= q t 6H= 8? Z L*P(+CD t 6H=+L8?uY 6 Y = o "9Y 6Sv rw6 = 8? 53=@"98A=B oyxzp ]{F (5.2) Assume for the moment that the appropriate lengthscale on which the inner approximation should be defined is the |}*PO#QSR~ lengthscale. If so, we can employ the ‘stretched’ variable 6 p 5 6 "f8 6 S(N* O#QSR , in terms of which (5.2) becomes bt YzR o Y (5.3) C 6H= 8? Y 6 Y = "fr 6 = L8? Y 6 v = oyxzp ]{F ? 6 p 6 = = 8? O L8? 8 8? 78?
p
We can change variables to reduce this covariant equation to a noncovariant, but more understandable form. Under a linear change of variables , i.e., , the matrices and transform to and respectively. Choosing transforms to the identity matrix. But since is a saddle point, irrespective of coordinate transformations the linearized drift will have one positive eigenvalue ( ) and one negative eigenvalue ( ). By a further change of variables (a rotation) we can set . So we can choose
c8? L8? Vp J8M O)QSR J8M N8M rhO R 8M p ] 9L8? ps 8M L] 8?} (5.4) for some real constant . The constant is not determined by z8M and N8M . Since JL8? p is preserved under rotations, with respect to the new system of coordinates equa-
tion (5.3) becomes
(5.5)
b zY R o C Y O R "czL8? Y Y
b zY R o C Y R R Y Y vO O oDx "cN8? Y R v R oyx " Y R v O oyxzp ]{F
17
LIMITING EXIT LOCATION DISTRIBUTIONS
fN#?
we may take the region to be the rightIn terms of the transformed coordinates half plane , and its boundary to be the -axis. This was the convention of Figs. 4.1 and 4.2. This system of coordinates is computationally easy to work with. Suppose for simplicity that ; this is an innocuous normalization condition that can be absorbed into a redefinition of time (and noise strength ). Then
f ¡
£7¤z¥?§¦s¨
(5.6)
©
¢
M
ª « ¥M¬¦® ¯W ¨ ²° ¡ ± ³
± µ ¦ ´ £¶N¥M ´ ·+£ ¤ L¥? , as usual. The Kolmogorov equation (5.5) reduces to (5.7) ¸¨ ¢¢ ¹ º ¸¨ ¢¢z P¹ ° ¢ ¢ z» ¹D¼ º ± ¢ ¢ ½» ¹y¼ °c¯ ¢ ¢ ½» ¹y¼z¦ ¡{¾ « Also, a bit of matrix computation, using the form (5.6) for L¥? and ¿ ¥?¦ À , yields (5.8a) Á ¶ )¡y'¦Â¡ ¨DS¡{S¡y (5.8b) LÁ{¤)¡y'¦Â ± º ¨D ¯ )¡{S¸ ¡y ¸ (5.8c) SÁ à ¶ +Ä Ã ¶ ¦Â ±Å¸ ° ¨D ¯ ° ± º )¡ ¸ (5.8d) SÁ{à ¤DÄ Ã ¤ ¦Â ° ±½¯ ± ° ¨ °f¯ ° ±½¯ ±A° ¨B, ¸ ± ± ° ¨~# for the four eigenvectors of the linearized Hamiltonian flow Æ2¥M at the point L¥Ç)¡ in phase space, as given by (4.5). (Normalization is irrelevant here; the negatives of these vectors could equally well have been chosen.) That Á{¶È¦e¡ ¨~ agrees with the convention of Figs. 4.1 and 4.2. We note in passing that the formulæ (5.8) explain the positioning of the rays Á{¤ , Á à ¶ , and Á{à ¤ in Figs. 4.2(a), 4.2(b), and 4.2(c). Recall that in those figures the MPEP was taken to approach from the first quadrant; if ± É ¨ , it is generically tangent to Á ¶ , and hence to the positive M -axis. By examination of (5.8), if ±É ¨ and ¯É ¡ then Á{¤ will lie between the positive ? -axis and Á{à ¤ , while if ¯ Ê¡ then Á{à ¤ (taken to point into the right half-plane) will lie between the positive ? -axis and Á3¤ . So the Subcases A and B of Figs. 4.2(a) and 4.2(b) are simply the subcases ¯2É ¡ and ¯ ¡ . This correspondence assumes of course that the MPEP is tangent to the positive M -axis; if it approached ¥ from the fourth quadrant, and were tangent to the negative ? -axis instead, then the interpretation in terms of sgn ¯ would be reversed. Formulæ (5.8) justify the positioning of the rays in Fig. 4.2(c), as well. If ± ˨ , we know by the arguments of the last section that the approaching MPEP is generically tangent to Á à ¶ . By (5.8c), Á à ¶nÌ ±° ¨D ¯ , and our convention that the MPEP approaches from the first quadrant mandates that ¯iÍ ¡ . It is easy to verify, by examining (5.8), that in the right-half plane, when ± ¨ and ¯ Ë¡ the Á{¤ ray necessarily lies between the the Á à ¶ ray and the the Á{à ¤ ray. This was the positioning of Fig. 4.2(c). In our new coordinate system, it is easy to study quantitatively the quadratic behavior of the action Î near the saddle point. Substituting (5.6) into the Riccati equation (4.3), and solving, gives Ï ¸ ±Å° ¨~ ¯ ¨ °M± °@¯,± (5.9) ¥?Ц ±Å° ¨B ¯ Ï º °@¯P± ± °f± ³ º ¸ Ï matrix ¥?Ȧ » ÎÑ ÒHÓy¥?Ô¼¬¦ » ¢Õ7Ò;·N¢Ö Ó ¥M;¼ . Recall as the formula for the rank- Hessian that generically this limiting Hessian matrix arises (when ×ØÙ¥ in the complement of the wedge) only when ± g¨ . If ±MÉ ¨ , L׬ØÛÚ as ×9Ø®¥ along the MPEP. where
def
18
R. S. MAIER AND D. L. STEIN
Ü
âAá ãLä?å
in the wedge (to the extent that it is quadratic) can be The quadratic behavior of computed similarly. In Subcase A of , the limiting Hessian matrix in the wedge exists, and equals by Table 4.1 the rank-1 matrix corresponding to the tangent space sp . The formula
ÝVÞàß
æ ãç{èéSêyå,éBãSç{ë è7éNì ë è åîí ñ ò å öãñ ãÝ ó åß å A â ã M ä ¬ å ï ð Ý ã ã å á (5.10) Pñ òôó Ý ó ß ò9õ ö ñ Ý ó ß Ý ó ß òK÷ follows from (5.8) elementary In Subcase B, and when ÝJøùß , the ú ûÙmanipulations. ä will not be quadratic. ãLúå inbythesomewedge behavior of Ü as it follows from úûüä along ýzþ (from the However, the formula (4.10) that its behavior as wedge side) is quadratic, with limiting second derivative ÿ á ¬ãä?å¬ï ð Ý (5.11) ä This limiting second derivative has a simple interpretation: on the wedge side of , the cost (action) of the most probable trajectory leading to any point on ýzþ arises from the drift on ýþ itself. (By (5.6), the drift along ýzþ is proportional to Ý .) ã
û å Gaussian (or inThe formulæ (5.9)–(5.11) make quite precise the far-field ã Bé ò å verted Gaussian) asymptotics that must be imposed on the solution of the transformed Kolmogorov equation (5.7). First, to leading order we must have
ã é ò å ö )âAãLä?å ð é û! outside the wedge. Also, in Subcase A of ÝfÞgß we must have "ã #é ò å $%'& ö âAá ãLä? å ( ð*) é Ùû+ inside the wedge. (5.13) In Subcase B of ÝMÞËß , or when ÝMøgß , we must have "ã #é ò å & ö ÿ ,ãLä?åã ò å ò ð ) é Ùû+ along ýzþ , on the wedge side, á (5.14) ò ãê{é- å!ïµê must since the boundary ýþ is the -axis. The Dirichlet boundary condition also be imposed, since the quasistationary density is absorbed on the boundary. to solve the partial differential equation (5.7) on the half-plane /In ê , subjectit istonottheseeasyboundary . general conditions. Our treatments of the generic ÝÂÞsß and generic Ýeøüß cases, in Sections 6 and 7 respectively, will be crafted to circumvent this ã 1 2 ò å problem. For the case ÝJÞß we shall expand on a larger lengthscale than 0 ; for the case Ýcøß , on which we have less information, we shall use stochastic analysis. In advance of our detailed treatments, we observe that the far-field behavior of the exit location density 354 6ã 1 #2 ò å ä (on the 0 lengthscale near ) follows from (5.12)–(5.14). By (2.3) the exit location density is simply proportional to the normal derivative of the inner approximation 7948 . If we âAãä?å , âAá ãäMå and ÿ á ,¬ãLä?å into (5.12)–(5.14), we obtain substitute the above expressions for (5.12)
far-field asymptotics (5.15a)
ã6:~å; 354
Ý9Þ ß
(Subcase A), and
if
(5.15b)
%@?>êA-ã6: ò *1,åCBé % ö Ý ã Ý ã ó ß
ã6:~åGF 354
åò õ ñ ò ó Ý ó ß å ò 6ã : ò D1,å ÷
:D D1 2 ò éE:D D1 2 ò
@?HêA-yãH: ò D1,åIB éJ:D D1 2 ò ? ö ã6: ò D1,å B éED: D1 2 ò Ý
û ó
û ö
û ó
û ö ;
;
19
LIMITING EXIT LOCATION DISTRIBUTIONS
if KL(M (Subcase B), and (5.15c)
W XZY%[]\_^C` K Q K ` ` M S a Q6R,aDdDefSCgihERDdDej#klanm p c o NPO%Q6RTSU :A@ Kolmogorov .B @1C @
87 , as follows. In the linear approximation we take 9$:
87 when 0ji 7 , for both Subcase A and Subcase B. set 5 A family of solutions of equation (6.8), each with far-field asymptotics that are Gaussian in + and 0 , can be found by inspection. Each solution is of the form #]VRon6p1qr +[0 C > 0 $ if 0js 7 ; 5 + 0 kml 70 (6.9) 7 if 0ji , > for some constant . Actually we shall use antisymmetrized (odd) versions of these solutions, 7 t ?7 on account of absorption of since 5 must satisfy the Dirichlet boundary condition 5 def
24
R. S. MAIER AND D. L. STEIN
u,v . Antisymmetrizing under wyxz|{4w yields }~ w$ m ] N ~
