Asymptotic behavior of the first exit times of randomly perturbed dynamical systems with unstable equilibrium points

Nonlinear

Analysis,

Themy,

Methods

Pergamon

& Applications, Vol. 30, No. Proc. 2nd World Congress

7, pp. 4077-4087, of Nonlinear

1997 Analysts

0 1997 Elsevier Science Ltd in Great Britain. All tights reserved

Printed

0362-546X/97$17.00+0.00

PII: SO362-546X(96)00242-8

ASYMPTOTIC BEHAVIOR OF THE FIRST EXIT TIMES OF RANDOMLY PERTURBED DYNAMICAL SYSTEMS WITH UNSTABLE EQUILIBRIUM POINTS TOSHIO MIKAMI Department Key words and phrses: systems,

of Mathematics, first exit time,

Hokkaido

unstable

University,

equilibrium

Sapporo

point,

060, Japan

randomly

perturbed

dynamical

large deviations. 1. INTRODUCTION

Let X’(t, 2) (t 2 0, z E Rd, E > 0) be the solution of the following stochastic differential equation: dX’(t, XT)= b(XE(t, I))& + E”2a(Xe(t, z))dW(t), xyo, zc)= z,

(1.1).

where b(.) = (bi(.))$l : Rd H Rd is globally Lipschitz continuous, where a(.) = (aij(.))t,,,,i : Rd H Md(R) is bounded, Lipschitz continuous, and uniformly nondegenerate, and where W(.) is a ddimensionalWiener process(cf. [15, 231). Since b(.) is globally Lipschitz continuous, {XO(t, .)}tE~ is a dynamical system on Rd. Moreover the following is known (cf. [lo, 11, 29, 301);for any z E Rd, T > 0 and 6 > 0, llm

Ed0

P(

sup O
(X’(t,

z) - X”(t,

x)1 < 6) = 1.

In this sense,X”(t, x) can be consideredas the small random perturbations of X”(t, Z) for sufficiently small E > 0. Let D(c Rd) be a bounded domain which contains the origin o, with a C2-boundary aD, and supposethat b(s) = o if and only if 2: = o. The asymptotic behavior, as E --) 0, of the first exit time 75(z) of Xs(t,z) from D defined by T;(X)

E inf{t

> 0; X’(t,

cc) $ D}

(1.2).

has been studied by many authors. The first result on the asymptotic behavior of T;(Z) as E-+ 0 was given by M. I. F’reidlin and A. D. Wentzell (cf. [lo, 11, 301). THEOREM 1.1 (cf. [ll], p. 127, Theorem 4.2). Supposethat X0(&z) ED = o for all z E B. Then the following holds; for any z E D and 6 > 0,

(t

> 0) and limt+~Xo(t,x)

(1.3). where we put

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J

VD= imf{0 t I~P(s))-‘(+(s)/~~ - b(ds)))12W2; cp(O) = 0,v(t)EaD,

(1.4).

{v(s); 0 5 s < t} c D, t > 0}(> 0).

As a refinement V. Day [2].

of THEOREM

1 .l, the weak convergence

result for 75(z)

was obtained

by M.

THEOREM 1.2 (cf. [2], p. 322, Corollary 2). Suppose that 8D is of class Cs, b and g is of class C’, and the eigenvalues of (abi(o)/axJ)&l E D (t > 0) and h ave negative real parts, and that X0(&z) XO(t, Z) = o for all x E n. Then the following holds; for any 3: E D and t > 0, lim,,, jz+iP(Tfj(x)/E[T&(x)]

> t) = ezp(-t).

(1.5).

[12] gives a similar result to [2]; the weak convergence result for the First exit time of X’(t,x) from the stable manifold of one asymptotically stable equilibrium point of XO(t,x) to another (cf. also [20-22, 281). Slight generalization of [12] and its application to geophysics is given in [16]. The asymptotic expansions of the distribution functions of T&(Z) was obtained by Fleming and James [8] (cf. also [7]). In case the origin D is not asymptotically stable, the asymptotic behavior of oh as E -+ 0 was first studied by Y. Kifer (cf. [18]). To explain his result, let us give some notation. Put Al = {Z E n; there exists s = S(Z) < 0 such that XO(t,x) q! a for t < s and such that XO(t,x) E D for t > S. X0(&x) -+ o as t -+ w}; A2 z {Z E D; there exists s = S(X) 2 0 such that X0(&x) $ D for t > s and such that XO(t,z) E D for t < s. XO(t,x) --) o as t -+ --a~}; AJ = {Z E D’; there exist s1 = si(~) 2 0 2 s2 = ~~(1) such that XO(t,x) @ib fort E (-00,s~) u(s~,oo), and such that X0(&x) E D for s2
1.3 (cf. [18], Theorems

2.1 and 2.2). Suppose that (A.D)

holds.

Then for any 6 > 0 and

x E AI u {o}\aD, ~~oJYl~~(x)lhdlii{E[T;(x)]/ and for any 6 > 0 and x E A2

U

l’(2X)) - 11 < 6) = 1, log(E-“@q}

= 1,

(1.6).

As\aD, liiP(lT&(x)/T;(x)

- 11 < 6) = 1, liiE[T;)(X)]

= T;(x).

(1.7).

The condition that B = {o} u A1 u A2 u AS is a topological one, and it does not imply that the eigenvalues of (3ba(o)/axj)$,l have non-zero real parts. In fact, in THEOREM 1.1, they do not

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suppose it. For instance, in THEOREM 1.1, (@(o)/drj)& origin is not exponentially stable (cf. [13]). A s a typical and b(l) = -z3 for IX] 5 1. Then db(o)/dz: = o and X0(&r)

Analysts

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can be a zero matrix in which case the example, suppose that d = 1, D = (-1,l) = ~(1 + 2&-i/2 satisfies the assumption

in THEOREM 1.1. Therefore the following problem comes out natually; study the asymptotic behavior of T;(X) as E -+ 0 when n = {o} u Al u A2 u A2 with A2 u A3 # 0, and when (8bi(o)/&j)f,j,l is a zero matrix. In section 2 we study the asymptotic behavior of TV when (%i(~)/&j)&,l is a zero matrix. In section 3, we discuss the refinement of Kifer’s result (cf. [18]); the large deviations results for TE(Z)/ log(E-l’(2x) ) (cf. (251); the weak convergence results for TV -log(E--11(2x)) (cf. [5]). In section 4, we explain the idea of proof of the results in sections 2-3. In section 5 we discuss the not very and super large deviations for T;(Z). The reader can find results related to those in this paper in [3, 4, 6, 9, 17, 19, 24) and references therein. 2. CASE

THE

FIRST

DERIVATIVES

OF b VANISH

AT THE

ORIGIN

In this section we study the asymptotic behavior of T&(X) when D = {o} u A2 in which case the origin o is a repulsive equilibrium point, and when (SJ’(O)/&~)$, is a zero matrix. In particular, we consider the case when b(z) N ]zle+1 as 3: -+ o for some P > 0. Let us first introduce our assumptions. (A.0). D(c Rd) is a bounded domain which contains O. D = {0} u A2 (A.1). There exist positive constants e and Cl such that for I E D

lb(x)15 Cllq+l. (A.2).

There exist positive

constants

(2.1).

6,, e and C2 such that for z for which < z,b(z)

>2

]z] < 6,,

Czlzlet2

(2.2).

THEOREMS 2.1-2.2 below implies that F(t,o) exit D at time of order F-~/(~+~) under (A.O)(A.2). This is a big difference between the case I = 0 (cf. THEOREM 1.3) and the case e > O. T;(O) is, as E 4 0, of order log(l/e) when I = 0, but it is, as E -+ 0, of polynomial order of ~-l when e > O. Moreover the asymptotic behavior of ~g(o) as E -+ 0 depends on the first derivatives of b when e = 0, but it does not depend on the derivatives of b when e > 0. THEOREM 2.1 (cf. [26], Theorem for any 6 > 0, lim E’O (II).

Suppose that D = {o}

u

1.1 and Corollary

P(,&-6)~/(~+2)

< T;(o)

1.2). (I). Suppose that (A.O)-(A.2)

<

AZ. Then for any z E D\(o)

E-(1+w(e+2)

and 6 > 0,

) = 1.

hold.

Then

(2.3).

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THEOREM 2.2 (cf. [26], Theorem following holds. (I). For any 6 > 0,

of Nonlinear

1.3 and Corollary

linsz~p{~~(“~)~(~+‘)E[~~(o)]}

Analysts

1.4). Suppose that (A.O)-(A.2)

hold.

Then the

5 1 < ilmknf{Ee(1-6)l(L+2)E[7~(o)]}.

E’O

(2.5).

E--t0

(II). For any x E D\(o),

fzi E[7;;(x)]= $(x). Remark 2.1. Only in a special case of D = (0)

u

Al

(2.6).

AZ u AJ, the asymptotic

u

behavior

of T&(O) is

studied under (A.0) and (2.1)-(2.2) (cf. [26], Theorem 4.1). Next we show that E~/(~+~)T;(~) weakly converge, as E --) 0, to the life time T(O) of an explosive diffusion process and that E[E~/(~+~)T;(o)]converge, as E -+ 0, to E[T(o)], under the stronger assumption than (A.O)-(A.2). Let us state the assumptions (H.O)=(A.O).

that are stronger

than (A.O)-(A.2).

(H.l). There exists e > 0 such that b(x) can be written as follows; b(x) = B(Z) + R(X) B(Z) and R(z) are locally Lipschitz continuous functions which satisfy the following;

B(tx) = tt+‘B(x)

for

x

E D. Here

for all t > 0 and x E Rd,

(2.7).

and there exist Ci and yi E (0, l] such that R(x)

for all

5 Cllx(e+1+71

x

E Rd.

(2.8).

(H.2). inf lzi=l

Remark 2.2. (H.l)

holds with

< B(x),

x >>

(2.9).

0.

y1 = 1 when P E N, b E Cf+2(Rd;Rd), @b”(o)/i3x:

.3x?

for all i = 1, ‘, d, k = 0, , e and (j,, . , jdj for which also holds if b(x) = Ix[~x(@ Cf+l (Rd; Rd)) for e E N. Let Y’(t, Z) (t 2 0, z E Rd, E > 0) be the solution up to the life time (cf. [14, 15, 231): dYE(t,

x) = B(Y’(t,

x);dt

and

= 0

jr +

+ j,j = Ic and j,

of the following

stochastic

2 0 (m = 1,. differential

, d).

equation,

+ &2a(o)dW(t), (2.10).

Y’(0,

x) = x,

and denote by T(Z) the life time of {Y’(t,z)}~<~:

It

Second World Congress of Nonlinear Analysts T(1)

Then the following THEOREM

which

3

inf{t

is stronger

2.3 (cf. [27], Theorem

3).

> o;o~uqPJt IYl(s,

x)1 = es}

than THEOREMS Suppose

that

4081 (2.11).

2.1-2.2 is true.

(H.O)-(H.2)

hold.

Then

E~~(L+~)T&(o) weakly

converges to ~(0) as E -+ 0, and

hi E[@+*‘~&(o)] = E[T(o)] < 00. As a corollary COROLLARY

to THEOREM

(2.12).

2.3, we get the following.

2.4 (cf. [27], Corollary

hold. Then ~&(o)/E[~f,(o)] weakly

1). Suppose that (H.O)-(H.2)

converges to ~(o)/E[~(o)l as E+ 0. 3. REFINEMENT

OF KIFER’S RESULT

Let us go back to Kifer’s case (A.D). We first consider the large deviations for T&(Z)/ log(E-11(2x)) for z E Al u {o}\ao. The large deviations for ~~(z)/~~(z) for x E A2U As\aD can be obtained by the routine argument Put

on large deviations

(cf. Ill]).

f(E) = log(E-“@q. Then the following THEOREM A1 u {o}\aD

is a generalization

of Kifer

3.1 (cf. [25], p. 493, Theorem and 0 < T < 1,

[18]. 1.2).

h{log(-logP(TS,(x)/f(e) THEOREM anyT>l,

3.2 (cf. [25], p. 493, Theorem

PROPOSITION x E A1 u {o}\G’o

Suppose

that

(A.D)

> T)/loge} = (T -

case, we only have the following

3.3 (cf. [25], p. 493, Proposition and T 2 1,

holds.

Then

for any

< T))/logc} = T - 1.

1.3). Suppose that (A.D)

h{logP(T&(o)/f(E) In multi-dimensional

(3.1).

1.4).

speaking,

(3.2)-(3.4)

E

(3.2).

holds and that d = 1. Then for

1)/2.

(3.3).

result when T > 1. Suppose

that

(A.D)

holds.

Then

li;~~p{log P(T&(x)/~(E) > T)/ logc} < 00. Remark 3.1. Roughly

x

means the following;

(3.2) implies,

for any

(3.4).

as E -+ 0,

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JY6(x)lf(,)

and (3.3) implies,

Analysts

- d-ET-‘),

(3.5).

> T) N E(T-1)‘2,

(3.6).

< 0

as E -+ 0, P(T;(o)/f(E)

and (3.4) implies

of Nonlinear

that there exists a positive

constant

C such that for sufficiently

small E > 0,

P(TfJ(X)/f(E) > T) > EC.

(3.7).

We would like to point out that the following is known (cf. [18]); for any 2’ > 1 and z E A1 u there exists a positive constant C1 such that for sufficiently small E > 0,

P(Tg(x)/f(E) > T) < PI.

(3.8).

Hence (3.7)-(3.8) imply that P(FD(z)/f(&) > T) is of polynomial From THEOREM 3.2, we have the following conjecture which U(Z) =Identity matrix (cf. [25], section 5). CONJECTURE

3.4. Suppose

that (A.D)

>

lii{lOgP(Th(X)/f(E)

holds.

T)/lOg&}

order of E, as E + 0, for T > 1. is true when b(z) = Db(o)r and

Then for any I E A1 u {0}\6’o

= ~f?l~X(O,

{o}\aD,

(Re(Xi)T/X

and 2’ 2 1,

- 1)/2).

(3.9).

I=1

Next we give the limit theorem for T;(X) - log(E-1/(2x) ) (z E AI u (0)) which V. Day (cf. [5]). The following is the additional assumption; (H). d = 2 and Db(o) have two real eigenvalues y1 > 0 > ~2. Let K and v be mutually independent random variables such that

P(K P(u

Then the following

<- t) = 2exp[-(t =

+ e-2’)]/(7r)“2

1) = P(Y = -1) = l/2.

was obtained

by M.

(t E RI, (3.10).

is known.

THEOREM 3.5 (cf. [5], Theorem 4.2). Suppose that (A.D) and (H) hold. Then there exist constants C+l and Cl such that for any x E Al U {o}, nFD(x) - log(E-1/2) converges to K + C,, as E 4 0, in distribution. 4. IDEA

OF PROOF

In this section we briefly explain the idea of proof of THEOREMS 2.3, 3.1, 3.2 and 3.5. Let us first consider THEOREM 2.3. In one word, we can say that Xc(t,o) is close to Y’(t,o) until it exit a small neighborhood of the origin o. Once it exit a small neighborhood of the origin 0, it

SecondWorld Congress of NonlinearAnalysts

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exit D within a fmite time which is independent of E. This is a fundamental fact in Freidlin-Wentzell theory (cf. [lo, 11, 301). More precisely speaking, for R > 0 and sufficiently small E > 0,

lI(~+2)XE(~-L/(e+2)t, N

p(

I,-l/(e+2)yE(E--1/(1+2)

sup

(4.1).

,,)I 2 R)

4 0112 RI = J”,;v’~ P% o)l > 9,

O
since {,-ll(e+2)y&(,-r/(L+2)t, o)} 05~has the sameprobability law as that of

{Y1(t, O))OS~

from

(2.7)

and (2.10); and P(

sup

IY’(t,o)J

2 R) + P(T(o)

as R

5 T)

+ 03.

(4.2).

O
Next let us consider THEOREMS 3.1, 3.2 and 3.5. Supposethat d = 1, D a(~) = Xx (A > 0) and o(x) = 1. Then D = {o} u A2 and

= (-a,@

(a,

p >

0),

t

X’(t)

0) = C2

exp(Xt)

exp(-Xs)dW(s).

(4.3).

s0

For

T E (0,

l),

P(T~(o) I IT)

o
I P(E”2exdWE)T)

--

(4.4).

E

for someconstant C1 > 0 which dependson D (cf. (3.1) for notation), since for t E [0,f(~)Tj,

I-WC 011I &2

exp(-Xs)dW(s)l

=P(V(E)T) OS&!&T

I Jd

from (4.3); and

P(T;(o) I f(~)Tl 2 P(lX’(f(~)T,o)l = d’2 e4ME)T)I 1”“’

evcp(-J+skWs)I2 Cd

(4.5).

exp( -Xs)dW(.s))

(4.6).

for someconstant CZ> 0 which dependson D. ForT>l, as~+O,

P(T~(o)

2 f(e)T)

,-u P(IX’(~(E)T,O)I

=

E”’

edAf(E

i’(e)T

< C,)

for someconstant Cs > 0 which dependson D. From (4.4)-(4.6), using (4.7) below, we get THEOREMS 3.1-3.2; ~‘/~exp(Xf(~)T)

Finally we consider THEOREM 3.5. Putting exp(-WE)), we get

= ~(l-~)/~. t

(4.7).

= 75(o) in (4.3) and using the fact that $I2 =

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Analysts (4.8).

The first and the second part on the right respectively. Remark 4,1. From (4.3), X’(75(0),

o)/ lT’(”

hand side of (4.8) converge

exp(-Xs)dW(s)

5.

= cl/*

ezzp(X76(0))

to K and C,, as E + 0,

>

0.

DISCUSSION

In this section we consider the “not very large deviations” and “super Let us first introduce H. Cram&r’s result [l] to explain the terminology and “super large deviations”. Let {X,,}~!, be a sequence of i.i.d random variables on some probability that the following holds;E[X1] = 0, E[(XI)~] = 1, and there exists 6 > 0 for all ~(1~1 < 6), and P(X1 E do) has a non-zero absolutely continuous

large deviations” for pD(z). “not very large deviations” space (0, B, P). Suppose such that E[ezp(zXl)] < oc part.

Then

for T > 0 and

a E (0, l/2),

P(k

X,&“+‘/2

2 7.)- ezp(-lnar12/2)

(5.1).

k=l

as n -+ 00 (cf. [l], Theorem 5). But (5.1) is not true when o = 5). On the other hand, as n+ co,

l/2

(cf. [I], the last part of Chapter

(5.2). k=l

in Prob. for LY> 0 by way of the central limit theorem. Therefore (5.1) means that the probabilities on *+1/2 can be approximated by a normal distribution. Such phenominon large deviations for CL=, Xk/n is called “not very large deviations”. When a = l/2, it is called “(very) large deviations”. When a > l/2, it is called “super large deviations” (cf. [30]). Let us go back to THEOREM 1.2. From the proof one can show that for any t > 0, z E D and g(E) > 0 for which

lim,,og(~)

= 0, ~(9(~)G(~)l-w~~(~ll

> t) - =P(--tl!AE)),

(5.3).

as E 4 0. This means that in the case considered in THEOREM 1.2, there is no difference between not very, very and super large deviations. The same thing can be mentioned for results [12, 20-22, 281.

Let us next consider

Kifer’s

case THEOREM

1.3.

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Large deviations for ~b(z)/f(~) are consideredin THEOREMS 3.1 and 3.2 and PROPOSITION 3.3.

One can prove a result on super large deviations for oh in the sameway as in the proof of THEOREM 3.2, and the proof is omitted. Take F(E) > 0 such that

)~7$4If(E) l&F(E)

= -J,

(5.4).

= 0

(5.5).

for any 6 > 0. Then we get the following. PROPOSITION 5.1. Supposethat (A.D) holds and that d = 1. Then for any r > 0, iiiF(E)-llOgP([Tgo) Remark

- f(E)]/F(E)

> 7) = -Xr.

(5.6).

5.1. Prom (5.4), for any r > 0,

P([TE(O) - f(E)IIF(E) < -7) = 0,

(5.7).

for sufficiently small E > 0. This is true, since

--TF(E) < --f(E)5 qw - f(E) for sufficiently small E > 0, from (5.4). Our approach in [25] is not useful to consider not very large deviations for 75(z). But we can showa example. Take F(E) > 0 such that P_m,[l/F(E)+ &E)/f(E)] = 0.

(5.8).

Then we get the following. PROPOSITION 5.2. Supposethat (A.D) holds, and that d = 1 and that b(z) = Xz. Then for any r > 0. liioF(E)-’

logP([T;(o)

-f(E)]/&)

> r) = -A?-,

(5.9).

lim P(E)-l log{- logP([Tc(o) - ~(E)]/F(E) < -r)} = -Xr, (5.10). E--t0 Since (5.9) can be shown in the sameway as in the proof of THEOREM 3.2, we only sketch the proof of (5.10).

4086

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where p(t) is a one-dimensional Wiener process, by way of the time change. This is true, since X”(t, 0) = eq(A[t - f(c)]) lt eq(-As)dW(s). Remark 5.2. PROPOSITIONS 5.1-5.2 implies that there is no difference between not very, very and super large deviations for T;J(x) when (A.D) holds. We have no idea about the not very, very and super large deviations for T;(Z) when (A.O)-(A.2) in section 2 holds. REFERENCES 1. Cram&

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