On the Asymptotic Behavior of the Fundamental Solution of the Heat Equation on Certain Manifolds

Taniguchi Symp. SA Katata 1982, pp. 169-195

On the Asymptotic Behavior of the Fundamental Solution of the Heat Equation on Certain Manifolds Nobuyuki IKEDA') Dedicated to the memory of Dr. Hitoshi Kumano-go.

0 1. Introduction Let M + be an open d-dimensional manifold with (d- 1)-dimensionaI smooth boundary N endowed with a smooth Riemannian metric g +. We consider the case where there exists a symmetric double M = M +U N U M - of the Riemannian manifold ( M + ,g + ) . Let g be the Riemannian metric of M and A be the associated Laplace-Beltrami operator, i.e., in local coordinates (2,x2,. . .,xd)

where g&) = g,((a/ax,),, @/ax,),), G = det k i j ) and (g") = (gtj)V'. certain cases, the coefficients G-'/2a(gtjG'/2)/ax~ occurring in A jump as x crosses the submanifold N, but the heat equation

au ~

at

=

1 -Au, 2

( t , x) E (0, w ) x M ,

still has the continuous minimal fundamental solution p of (1. l), (see [25] and Section 3). In this paper, we are going to study short time asymptotics of the minimal fundamental solution p of (1.1). Following Varadhan [25],we have the following asymptotic formula: if the Riemannian manifold ( M , g) is complete, then for fixed x,y E M ,

x # Y,

where p(x, y ) is the distance between x and y induced by the Riemannian metric g . In [3], Buslaev took up the matter of finding corrections to (1.2) related to the "short-wave asymptotic behavior of diffraction" on This work was supported in part by the Grant-in Aid for Scientific Research.

170

N. IKEDA

smooth convex surfaces.') The problem is to find how the asymptotic behavior of the minimal fundamental solution p of (1.1) reflects the shape of the hypersurface N of M (also, see [17]). To illustrate the situation, we first introduce some geometric notions. We first impose the following : Assumption (i). N endowed with the induced Riemannian structure is a totally geodesic hypersurface of M , i.e., every geodesic of N is also a geodesic of M. If the metric tensor is smooth, it is well known that N is a totally geodesic hypersurface of M if and only if its second fundamental form a vanishes identically (e.g., see [l] and [16]). However, in case where the Riemannian metric g is not smooth, although N is a totally geodesic hypersurface of M , we may assume the following: Assumption (ii). For every V E T,(N) such that g,(V, V) # 0, a,( V , V ) < 0 for every x E N. As usual, a geodesic joining x and y is called an enveloping ray if it is a curve having more than one point in common with boundary N of M ' , (see Buslaev [3]). Under the assumptions (i) and (ii), Buslaev [3] was led to the following conjecture: for a wide class of manifolds it holds that if the geodesic #,,*(t), 0 5 t 5 1, joining x and y , (x,y E M' U N ) , is an enveloping ray, then as t 4 0 , where C is a non-negative constant depending on the shape of the hypersurface N , the arc-length of #z,u([O,11) n N and p(x, y). Although the intuitive background of his considerations is transparent, it is heuristic. Buslaev [3] used the concept of continuum product integral to show (nonrigorously) the asymptotic formula (1.3). Our main aim is to give a rigorous proof of the asymptotic formula (1.3) for a special class of manifolds, (see Theorem 1 in Section 3). The constant C in the right hand side of (1.3) will also be calculated explicitly. We restrict ourselves to the case where x, y E N and x # y . Under the assumptions mentioned in the following sections, by using skew products of diffusion processes, we will reduce the problem to the Laplace asymptotic formula for Wiener functionals similar to Schilder's one in [21] (also see Donsker-Varadhan [5], [6] and Dubrovskii [7], [S]). For details, see Lemma 4.5. As a result, we finally arrive at the calculation of the Wiener *) For a deep connection between the various types of second order partial differenal equations, see [IS].

On the Asymptotic Behavior of the Fundamental Solution

171

integral E"[exp

{- z j :

Iw(s)lds)/w(l)

for

= 01

z

>0 ,

where E W [ . / w ( l )= 01 denotes the expectation with respect to the Wiener measure with the fixed endpoint 0. As proved by Kac [13], the explicit expression for this integral could be obtained in terms of Bessel functions of order 1/3 (or the Airy function), (see also [14], [I51 and [23]). The organization of the paper is as follows. In Section 2 we prepare some geometric notions which will be needed later. The assumptions and main results are stated in Section 3. In Section 4, by using skew products of diffusion processes and the Feynman-Kac formula, we give a sketch of the proof of main results. Section 5 contains the details of the proof. In Section 6, we will give some comments.

3 2.

Preliminaries

First we will give some notations and notions which will be needed later. Throughout this paper we assume that manifolds are connected and a-compact. In general, for every mapping we denote by the Let $ be a curve in a manifold. Then we denote by 4 differential of the tangent vector field of the curve $, i.e., 4 = &(d/ds) where d/ds denotes the standard vector field on the real numbers. Let a : R' -+ R' be a positive continuous function satisfying the conditions :

+

+.

+*

(A.1): ( i ) a([) = a(lE]) for s' E R'. of the function a to [0, co) is a non(ii) The restriction increasing C"-function. (iii) a(0) = 1 and a+(O)< 0 where a + ( [ )is the right derivative of a at [. Let us fix a positive integer d 2 2. Let S be a ( d - 1)-dimensional smooth complete Riemannian manifold with metric g. Consider the product differentiable manifold M = R' X S with its projections rl : R' X s-+ R' and z2:R' x S -+ S. From now on we use the following notations x' = n,(x) and

X = z,(x)

for x E M .

We define a Riemannian metric g on M by gzv, Y)=

(2.1)

szl((d*(x), (rJ*(Y)) + &'>-'&((d*(m,(x*)*(Y)) for X , Y E T z ( M ) and

x = (x', X) E M

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where g is the standard flat Riemannian metric on the 1-dimensional Euclidean space R'. The Riemannian manifold (M, g) is called the warped product of R' and S by the function a-' (cf. [2]). Then it is easy to see that the Laplace-Beltrami operator A on M has the form

A =L

(2.2)

+ a(x')iI

where

L

=

a ax'

A(xl)-l-

(A(x')-a

3

,

and 0 is the Laplace-Beltrami operator on S. Let us consider the submanifolds M " and N given by

M + = {x;x = ( X I , x) E M , x' > O}, M - = {x;x = (XI, x) E M , x' < 0 ) and N = {x;x = ( X I , x) E M , x' = 0 ) . Let g + and g - be the restrictions of g to M + and M - respectively. It is easy to see that ( M , g ) is a symmetric double of the Riemannian manifold ( M + g, + ) . Although the Riemannian metric g is not C', we can still define the notions of the arc-length Z(q5) of a piecewise smooth3) curve q5: [a, b] + M and the global Riemannian distance p(x, y ) between two points x and y in M as in case of smooth Riemannian manifolds. As well known it is easy to see that the arc-length I(#) is independent of the parametrization of the curve. For details, see Varadhan [25], [26] and Milnor [18].

Lemma 2.1. Take any two points x,y E N . Then, for every piecewise smooth curve T,!P : [0, 11 + A4 joining x and y , there exists a piecewise smooth curve q5: [0, 11 -+ N joining these points such that

I(+) L I(#) where the equality can hold only if+([O, 11) c N. Furthermore

p(x, y ) = p(X, 7 )

for

x = (0, x)

and y

=

(0,p)

where p is the global Riemannian distance on (S, g). By piecewise smooth we mean 4 is continuous and piecewise Cm.

On the Asymptotic Behavior of the Fundamental Solution

173

This is an immediate consequence of (2.1) and the assumption (A.l) and so the proof is omitted. Take two points x and y in M. As usual, we can also define the notion of geodesics joining x and y. Take a local coordinate (x2,x3,. . .,x") in 5'. Let Ttj be the coefficients of the Riemannian connection P in (S, g) with respect to the local coordinate (2,xs,. . ., x"). Now, keeping Lemma 2.1 in mind, we can show the following.

-

Lemma 2.2. Let us consider a local coordinate (2,x2, , x") such that x' E R' and X = (x2,x3, . .,xd) E S. Let # ( I ) = ( @ ( t ) ,$ ( t ) ) , 0 5 t 1, be a geodesic. Then (i) for every t such that # ( t ) E M\N,

for i

=

2, 3, . . ., d ,

where $ ( t ) = (#'(t), #'(t), . ., #"(t)) E S and a'(t) = da(E))/dCfort # 0, (ii) for every t E (a, b) such that #((a, b)) c N , #'(t) = 0

(2.5)

3 % dt2

-

Now let # ( t )

=

2

for i = 2, 3, . ., d

f:,($(t))&t)$X(t)

+

j,x=2

(#'(t), $ (t)), 0

5 t 5 1, be a geodesic of M and set

where k ( t ) is the inverse function of h ( t ) defined by h(t) = c

j' a(#'(s))ds ,

c =

([

a(#'(s))ds)-I

.

Then the curve $ ( t ) , 0 t 5 1, is a geodesic of S such that $(O) = $(O) and $(1) = $(1). This is an easy consequence of Lemma 2.2 and the proof is omitted. Before closing this section we give some remarks on assumptions (i) and (ii) in Section 1. We note that the Riemannian connection V + on M' can be naturally extended to the Riemannian connection V; on p.

N. IKEDA

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Let a:T ( N ) x T ( N ) + T ( N ) I be the second fundamental form of the hypersurface N in ( Fg,) , i.e., for every X , Y ET J N ) and x E N , a,(X, Y ) is the normal component of (P:Y), (see [16]). Since, under the assumptions mentioned above, it is possible to choose a field of unit normal vectors globally on N , we can regard a as a mapping from T ( N )x T ( N ) to the space of C"-functions on N ([16]). Combining the considerations mentioned above we have the following: Proposition 2.1. 1) The hypersurface N in ( M , g ) is totally geodesic, i.e., every geodesic of N is a geodesic of M . 2) For every X , Y ET J N ) and x E N ,

where K

=

-a+(O).

The proof is straightforward and is omitted. Throughout this paper we denote by K the positive constant -a+(O).

0 3.

Main results

We consider the heat equation

with the initial condition (3.2)

u(O+,x)=f(x),

XEM.

Let p : (0, m) x M XM -+ (0, m) be the minimal fundamental solution of (3.1) with respect to the Riemannian volume m(dx), i.e., p is the fundamental solution satisfying the condition: Let f be a non-negative continuous function with compact support and u be a solution of (3.1) with (3.2). Then setting

we have

On the Asymptotic Behavior of the Frrndamental Solution

I75

From now on, throughout this paper, we assume the following: (A.2): There exists a bounded non-negative function b defind on [0, w) such that

(3.3)

1 - a(S) = KS - b(E)Ez

for

E E [O,

m)

I

Roughly speaking, the use of the stationary phase method combines the contributions to the asymptotic behavior of p from the various parts of the geodesic in a inultiplicative manner. Then, using the considerations of Varadhan [25], [26] and Molchanov [20], we can restrict ourselves to the calculations in a neighbourhood of the hypersurface N. Hence the assumption (A.2) plays a similar role to the following assumption: the function aIco,.,,is convex in a neighbourhood of the origin. Let 2, be the first eigenvalue of the eigenvalue problem (3.4) with the boundary condition (3.5)

d -u(A, dx

Of)

= 0.

It is well known that 1, > 0 (see 1121, 1141 and [23]). Let Q(3,p) be the set of all minimal geodesics of S joining X and 7,(n, 7 E S ) and set n(x, p) = #Sa(x,y ) , i.e., the number of elements of Q(X, 7). We are now ready to state the main results of this paper. Theorem 1. Let us assume (A.l) and (A.2). Take two points x = (0, X) and y = ( 0 , ~ )in N . V X and 7 are non-conjugate points along each element o f s Z ( ~7) , and I 2 n(X. 7)< CO, then

as t 5.0. Theorem 2. Let us assume (A.l) and (A.2). For every compact subset D of S, there exist positive numbers pt, i = 1,2, 0 < p, < pa < 00 such that

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176

uniformly in X, J

E

D with pl 5 p(x, J )

p2.

Remark 3.1. If M is a smooth Riemannian manifold, the second term in the right hand side of (3.6) does not appear in the asymptotic behavior of p . For details, see Molchanov [18] and MinakshisundaramPleijel [19]. Therefore the second term is a correction depending on how smooth the Riemannian metric g is. Remark 3.2. Let us consider the heat equation with the Neumann boundary condition

where ajan denotes the differention in the direction of the normal of N . Let p + be the minimal fundamental solution of (3.8). Since ( M , g ) is a symmetric double of the Riemannian manifold ( M + ,g'), we have

__

where x* = (-XI, X) for x = (XI, x) E M + . Combining this with Theorem 1, we can obtain the asymptotic formula o f p + ( t x, , y ) as t J 0. Before turning to the proof of Theorem 1 we give a typical example. Example 3.1. Let us consider the Euclidean space Rd endowed the standard flat Riemannian metric g and we denote by Sd-' the (d - 1)sphere of radius 1 endowed the induced Riemannian metric g from Rd. Let M + be the exterior domain of Sd-' in R d . Let (r, 01,02,. . ., t 9 - l ) be the standard polar coordinate in Rd and we define a local coordinate (XI,x2, . ., xd) in M' by

Then, in the local coordinate (x',x2, . . -,xd),the metric tensor g + = (g;) of M is expressed in the following form : +

On the Asymptotic Behavior of the Fundamental Solution

177

g:, = 1, g&(x) = u(x1)-1g&)

for x

=

(XI,X) E M' , and

for j

g&(x) = 0

where u(E) = (1

=

i, j = 2, 3,

. . ., d

2, 3, . . . , d

+ lE1)-z and

g&)

=

g3(a/axf,a/axj) ,

i,j = 2, 3,

. . -,d .

It is easy to see that the function u satisfies (A.l) and (A.2) with K = 2 and b(E) = (3 + 2E)/(1 + E)z. Let p' be the fundamental solution of (3.8). Now, by (3.9), we arrive at the following asymptotic formula: if X and J are not antipodal points on the Sd-' and x = (0, X), y = (0,J ) ,

as

t40.

We will again return to this example in the final section.

9 4.

Skew products of diffusion processes and the Feynman-Kac formula

Let W' be the space of all real continuous functions defined on [0, w). There exists a minimal one-dimensional diffusion measure { Q E ;E E R'} generated by L/2 (see [lo] and [ll]). We denote by Ef the expectation with respect to the measure Q , on W'. We also denote by E f [ - / w ( t )= 71 the conditional expectation with respect to the measure Q , on W' under the condition w ( t ) = 7 . Roughly speaking, this is defined by the usual formula

In Molchanov [20], a rigorous justification of various definitions of conditional processes with fixed endpoints was discussed in detail (cf. [22]). Let p be the minimal fundamental solution of

_a _u _- 1- - 2-4 , at

2

(tyX)€(O,w)XS

with respect to the Riemannian volume FFZ on S. We set

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178

where $;'(w) is the inverse function of q 5 r ( ~ ) . Then, letting + ; I ( W ) inverse function of +,(w), we have

be the

Combining this with the formula (2.2) and using skew products of diffusion processes, we have

where k(t, E, 7 ) is the minimal fundamental solution of

(4.3)

-au - - L-u ,I at 2

(t,X)E(O,W)XR1

with respect to the measure A(C))df, (see [lo] and [ll]). Combining (4.2) with Molchanov's results (Theorem 2.1 and Theorem 3.1 in [20]) we have the following lemmas.

Lemma 4.1. Assume (A. 1) and (A.2). For every compact set D there exists a positive constant po such that for every p, with 0 pl as t -1 0,

c S,

< < po,

where Hl(f):[0, w) -+ R' is a positive continuous function with Hl(0) = 1.

<

Proof. Since 0 #,(w) 2 t, this is an immediate consequence of Theorem 2.1 of Molchanov [20] and (4.2). Also see the related remarks in Molchanov [20]. The details are omitted. Lemma 4.2. Assume (A.l) and (A.2). Take two points x = (0, x), 1 5 n(X, 7)< w and X and p are non-conjugate points y = (0,jj) in M . along each element of Q(X, y ) , then, as t -1 0,

On the Asymptotic Behavior of the Fundamental Solution

179

where H2(E):[0, co) + R’ is a positive continuous function with H,(O) = 1. Proof. This is also an easy consequence of Theorem 3.1 of Molchanov [20] and (4.2). The details are omitted.

Remark 4.1. Let us consider the case where S is a (d- 1)-dimensional simple space form with constant curvature K , ie., sphere, Euclidean space, hyperbolic space (cf. [l]). Then it follows from Molchanov’s result that Lemma 4.1 holds with p, given by

For details, see Molchanov [20]. In case where K 5 0, the explicit formula of p is also well known (cf. [4]). Now it should be mentioned that if x, y E N, the asymptotic behavior of p(t, x, y), as t 4 0, does not feel the Riemannian metric outside a neighbourhood of N . For details, see Varadhan [24], [25], [26] and Molchanov [20]. Hence it is sufficient to prove Theorems 1 and 2 under the following assumption (A.3): There exists a positive constant f , such that

(4.7)

a(E) = a,

>0

for

t 2 to.

To avoid non-essential complexities, from now on, we assume (A.3). For the simplicity of notations, we set

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180

We now consider the l-dimensional Wiener measure P y on W' starting at E and denote by EY the expectation with respect to P y . Then, setting

for p

>0,

K(d - l)Jt .t(l, w;O)}/w(I)

=

X k(t, 0, 0)

we obtain the following Lemma 4.3.

For every p

> 0,

(4.9)

+

4

01

where t(t, w;z) is the local time at z of the l-dimensional Brownian motion { w ( t ) , t 2 0 ) and E r [./w(l) = 01 denotes the conditional expectation with respect to P r under the condition w(1) = 0. Proof. By using the transformation of drift by C and (2.3) it holds that for every bounded continuous function j

Setting u ( t ) = (log A(E))/2 and using Tanaka's formula (see III-(4.1) in [lo]), we can rewrite (4.10) in the following form

On the Asymptotic Behavior of the Fundamental Solution

181

qs'

- 2t

0

(1 - a(w(ts)))ds)z(s' a(w(ts))ds)-l}

Using the scaling property of the Brownian motion, we can again rewrite this in the equivalent form

(f)

(d-l)/Z

exp { - p ' / 2 t } E , W [ ( A 1 ( J f w ) ) - ( d - 1 ) / i

{

x exp - $ ~ ( 1 / tw)

+ a ( ~ , ( ~ ~ I) t . ) ) ) j

which completes the proof. The Feynman-Kac formula allows one to obtain the asymptotic formula related to the Wiener integral in the right hand side of (4.9). Let {An}, 0 < 2, < A, < . . . and {q5n} be the eigenvalues and the normalized eigenfunctions of the eigenvalue problem (3.4) with (3.5) respectively. Then we have

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182

For details, see [9], [12], [13], [14] and [23]. The following is an easy consequence of (4.1 1).

Lemma 4.4. As t

4 0 ,for every p > 0,

Furthermore, for every 0 in p with C, 2 p 2 C,.

< C, < C, <

(4,12) holds iiniformly

03,

Before we further proceed with the proof of Theorems 1 and 2 we will try to explain the idea behind the proof. We first set, for every p > 0,

Since u and C are bounded functions, it is easy to show that for every p > 0, as t .10, Er[exp { -p2A,(d tw)/2t}Gj1)(,lt w ; p)/w(l) (4.14)

=

0]

= E,W[exp { -pzA2(d 7 w)/2t}Gt2)(47w;p)/w(l) = 0](l

+ O(t))

, uniformly in p. In the following section, by using direct asymptotic evaluations of (4.15)

E,W[exp { -p2A,(dJt w)/2t}GjZ)(z/fw ;p)/w(l) = 01 ,

we will prove the following:

Lemma 4.5. As t J 0,fur everyp

> 0,

(4.16)

E,WMP {-pZA,(J7w)/2t}Gt2’(J -~ t w ; P)/W(I) = 01 E,W[exp { -P‘A,(Jt w ) / 2 t } ~ t ~ ywd;tp)/w(l) = 01

(4.17)

Er[exp { - p z A ~ ( ~ t ~ w ) / 2 t } G ~ 3p)/w(l) ~ ( ~ / t= w ;01 +I, E,W[exp { - p 2 A , ( d 7 w ) / 2 t } / w ( l ) = 01

~

~

1,

On the Asymptotic Behavior of the Fundamental Solution

183

and

(4.18)

where

(4.19) Furthermore, for every 0 < C, < C,< 00, (4.16), (4.17) and (4.18) hold uniformly in p with C, p 5 C,.

On the other hand, it follows from Lemmas 4.2 and 4.3 that under the assumptions of Theorem 1

(4.20)

x EY[exp (-p(x, y)2A2(Jfw)/2t}GI’)(J~i w ;p(x, r))/w(l) = 01 x Hz(P(4JMl + o(1)) *

The proof of the theorems. Combining (4.14), ‘(4.20) and Lemmas 4.4 and 4.5 we can easily complete the proof of Theorem 1. Furthermore, by using Lemmas 4.1, 4.4 and 4.5 and (4.14) we arrive at the conclusion of Theorem 2.

It should be noted that it is sufficient for the proof of (3.6) to show

<-. Although the rigorous proof of (4.16), (4.17) and (4.18) are complicated, the intuitive background of these asymptotic relations is clear and closely related to considerations of Schilder [21], (also, see DonskerVaradhan [5], [6]). In fact (4.17) and (4.18) are similar to the Laplace asymptotic formula for integrals in one dimension. The functional

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184

has a proper minimum at 0 over H where h ; h : LO, 11 +R' is absolutely continuous, h(0) = 0 and

[I h ( s ) r d ~< m} . 0

On the other hand

and

are bounded in t . Hence roughly speaking, it is reasonable to expect that the asymptotic fomulas (4.17) and (4.18) hold. Roughly speaking, it is also easy to show (4.16).

9 5.

The proof of Lemma 4.5.

We will prove Lemma 4.5 by establishing several lemmas. Setting, for any positive p, Z,(t; p) = E,W[exp { - p ? A , ( J t w ) / 2 t } G 1 3 ) ( J t w ; p)/w(l) = 01

we have the following.

Lemma 5.1. For every E Z2(t; p)

(591)

> 0,

5 E,W[exp {-p2A,(JJw)/2t}G1,)(J~tw; p ) / w ( l ) = 01 5 (1 + e)12(t; p) + e- sllt for 0 < t S t , ,

where (5.2)

I, =

8,

=

( ( A - 1)K)Z {L 2 A (log (1

+ $))}z.

Proof. Since Kt(1, w ;0) 2 0, the first inequality of (5.2) is clear. Setting

1

8, = 2 A (log (1

we have

+ +)),

On the Asymptotic Behavior of the Fundamental Solution

Since 1 - a(c) 2 0, by (5.3), we have J,(t, 6,; p)

5 E,W[exp{(d - I ) K J t t ( l , w ;0)/4}; (d - l ) K J t t ( ( l , W ;0)/4 2 S2/w(l)= 01

=4Jm 4 h / ( d - l ) K 4:

On the other hand, for 0 < t 4Jrn 46a/(d-l)KJT

p exp { J t ( d - I)Kp/4}e-21zddp.

< tl

7 exp {,/Jt(d - l)K7/4}e-'qad7

Prn

Hence, for 0

< t < t,, J,(t, 8,; p )

S e-"lt

which completes the proof. We define 13(t;p) by (5.4)

13(t;p) = E,W[exp {-p'A2(J

Lemma 5.2.

For every positive

E

t w ) / 2 t } / w ( l )= 01 .

with 0

< < 1, E

185

N. IKEDA

186

where

(

(

6 , = 2p- a log 1 -

(5.6)

Proof.

(5-7)

2

-

>

;)-')'/?

We rewrite I,(t; p ) in the following form

1 3 0 ; p ) - E Y b P { - p2A,(J t w)/2t} x (1 - exp { - p 2 ~ ( A , ( J t ~ ) ) / 2 r } ) / ~ = ( 1 01 )

+ EY[{(A1(Jt x

w))-(d-')'Z

1

- 1

exp { -pzA,(J t w)/2t - pZa(A,(J t w))/2t}/w(l)= 01 .

Since, by (5.6), (1 - a 3 J & ) - ( d - ' ) ' 2 it holds that for 0

(5.8)

- 1

<

E ,

< t < t,,

E,W[{(A,(Jt w ) ) - ( d - l ) / z - 11 x exp { -pzA2(d t w)/2t - p2a(A,(J t w ) ) / 2 t } / w ( l )= 01 - 1) exp { -PZ&/2J t 1 * 5 EI,(r; p )

+

It is also easy to see that for 0

< t < t,,

E$[exp {-p2A,(Jt w)/2t}(l - exp {-p?a(A,(J t w))/2t})/w(l)= 01 = Eiv[exp { - p 2 A , ( J t w)/2t}(l - exp { -p2a(A,(Jt w))/2t});

(5.9)

+ E,W[exp { -p*A,(J

0

< A,(J

t w)

< 6,Jt

/ w ( l ) = 01

t w ) / 2 t } ( l - exp { - p2a(A,(J t w ) ) / 2 t } ) ;

A*(J t w) 6,J t /wq) = 01 - exp { - 6,p2/2J t }EA"[exp { - p2AL(J t ~ ) / 2 t } / w I 1)( = 01

5 exp { - 6,pZ/2J t }

+ &J3(t;p ) ,

x (1

- exp { -pZ6X2a,})

(by (5.6)) .

It is a small step to conclude (5.5) from (5.7), (5.8) and (5.9).

On the Asymptotic Behavior of the Fundamental Solution

Lemma 5.3. Take two any positive constants A , and A,. every t with 0 < t 5 t3,

where b*

=

sup,,,,,

187

Then, for

b(E), t3 = (K/2A,b*)3and

Proof. Since 1 - a(E) = K(EI - b(lEI)Ez and b((E1)2 0, the lower estimate on 4 ( t ; p) is easily obtained. Using A,(w) 2 0 we obtain,

I&; p)

5 PY[M,(w) 1 Alt-'/e/W(l)= 01

+ E , W k p {-pZA,(J-tw)/2t}; (5.12)

M,(w) < Alt-'/',

J: ~ w ( s ) ~ d5s A,t'/'/W(l) = oI

+ Er[exp { - - p ~ ~ ,t(w)/2t); d ~ , ( w< ) ~~t-'/',

J: I w(s) I ds > A,t'/'/w(l) = 0 On the other hand, by the famous formula of Levy, (5.13)

PAv[M,(w) > E/w(s) = 01 = exp

[-$I

, 5

>0 ,

( [ l l ] ) . Since if Ml(w) 5 Alt-"',

by combining (5.13) with (5.12), we get (5.10). We should now note that Lemma 4.4, 5.1, 5.2 and 5.3 imply (4.21) and so, as stated in Section 4, we arrive at (3.6).

N. IKEDA

188

We will next prove Lemma 4.5. To do this we first show the following. Lemma 5.4. Take any positive constants A,, i = 1 , 2 , 3 and choose Then, for every t with 0 t apositive constant E such that 0 E < A,.

<

<

<

t4r

Er[exp { - p Z A , ( J T w ) / 2 t } ;

f I w ( S ) ( h 2 A2t1/',

1

M,(w) 5 A,t-'/'/w(l) = 0

s'

5 E,W[exp { - p 2 A , ( J f w ) / 2 t } ;

Iw(s)Ih

1+

M,(w) 5 Et-l/e/w(l)= 0

5 Aztl/',

exp [ - A 3 / P ] ,

where

and 6 is a positive number such that 0

< 6 < 1/6.

Proof. Set (Y = 1/6 - 6 and a,(w) = inf {s;I w(s)I = E t

.

Define a,(w) by a,(w) = inf { s ;a,(w) 5 s, 1 w(s)I 2 t a }

and set g ( t , E, 17) =

1

~~

1/Eexp [- IE

- g/2/2tl,

E, 4 E R',t

> 0.

By the Markovian property of the Brownian motion, we have Pow"; Iw(s)lds 5 Azt1/6,€t-'/6 5 M,(w) 5 A,t-'/"lw(l)

5 Pow[aaZ(w)- a,(w) (= A2P, a,(w) < l / w ( l ) = 01 =

(5.14)

J-; PF[a,(w)

E

=

1

0

d s ] P L [ o , ( w )I (A#) A (1 - s)/w(l - s)

=

01

On the Asymptotic Behavior of the Fundamental Solution

x exp

= Jz(t, a),

I89

[- 2(1 -(t")2 ]dud% s - u)

(say).

E;

Since 3Aztd

< ( ~ t - "-~ t'>z

for every

t, 0

< t < tr ,

we have

Combining this with the inequality (t"Y

4 2 4 s - u) exp

[- 2 ( 1 - s - T ) I % z F for u

1

1

< (Aptd)A (1

- s),

we have

Hence, for every t with 0 equal to the following

< =

< t < t,,

8A2 d2nA,E

t3d/2

the last term of (5.14) is less than or

exp [ -A,/t 'I3]

5 exp [ -A s / t 1 / 3.]

This completes the proof. We now turn to the proof of Lemma 4.5. Take two any positive numbers 0 < C, < C, < 00 and fix an any positive number E . Take sufficient large constants A,, i = 1 , 2 , 3. Then, by combining Lemma

N. IKEDA

190

5.3 with Lemma 5.4, we can conclude that there exists a positive number t, independent of p with C, p 5 C, such that for every t, 0 < t < t,,

+ exp [ -A 3 / t 1 q By combining this with Lemma 4.4, we get

Since e is any positive number, this implies (4.18). Similarly, using Lemmas 4.4, 5.1, 5.2, 5.3 and 5.4, we obtain (4.16) and (4.17).

0 6.

Remarks

In this section, we give some remarks on the asymptotic behavior of p ( t , x,y) in the case where x = (0, X) E N a n d y = (y’, p) 4 N . For the sake of simplicity, we assume that there exists a unique minimal geodesic #(t) = ($(t), &t)), 0 5 t 5 1 , joining x and y. We also assume that the geodesic 4 is an enveloping ray. Then, as stated in the section 2, there exists tl, 0 t , 1 such that

< <

$“O, 41) c N

and

#((tl, 11)

c M\N

*

We set z = $(t,). Roughly speaking, the use of stationary phase method allows one to get that as t J 0, (6.1)

logp(t9 x , Y )

- log P(ttl, x, 4 + 1%

P(t(l - t A = , Y ) *

Hence, by Theorem 1 and Molchanov’s result ([20]),it is reasonable to expect that as t J 0

On the Asymptotic Behavior o f the Fundamental Solution

191

However, since the Riemannian metric g is not smooth in a neighbourhood of N, it seems hard to give a rigorous proof of (6.1) (or (6.2)). Although we will not go into this, we show that the problem can again be reduced to finding of asymptotic behavior of a Wiener integral. By Lemma 4.2 and using the transformation of drift by C, we can show that as t $ 0 ,

where

Now using the scaling property of Brownian motion, we have

1 92

N. IKEDA

On the other hand, since $(t), 0 4 t 4 1, is an enveloping ray, by using the considerations in the section 2, we have

where aye)

=

E#O

,

[=O

and @(O, y') is the space of all monotone piecewise smooth functions such that +(O) = 0 and +(I) = yl. By using (6.6), (6.5) can be rewritten in the equivalent form

On the Asymptotic Behavior of the Fundamental Solution

193

We set

Finally we again return to the example 3.3. Let n and s be the north and south poles of Sd-' respectively. Then, as stated in Example 3.1 fo Molchanov [20],

where C d - ]is the area of Sd-'. In this case, letting ?' be a great circle arc

N. IKEDA

194

from n and s, n and s are conjugate along 'i with multiplicity d - 1 ([18]). Hence #Sl(n, s) = 03. However, by repeating once again the similar proof to one of Theorem 1, we can show that (3.11) still holds, i.e., --logp+(t, (0, n), (0, s)) =

=2 ~~

2t

+

],22/3*4/3 ~~

2 t 113

+ 0(t-'l3),

as

t

.10 .

Acknowledgements. The author would like to thank S. Watanabe for many discussions of the material. He is also indebted to S.A. Molchanov for bringing the reference [3] to his attention.

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On the Asymptotic Behavior of the Fundamental Solution

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S. Kobayashi and K. Nomizu, Foundations of differential geometry, I, 11, Interscience Publ., New York, 1969. H. P. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, Jour. Diff. Geometry, 1 (1967), 43-69. J. Milnor. Morse theory, _ . Annals. Math. Studies, 51, Princeton Univ. Press, Princeton, 1963. S. Minakshisundaram and A. Pleiiel. Some DroDerties of the eiaenfunctions of the Laplace operators on Rikmannian manifolds, Can. Jo&. Math., 1 (1949), 242-256. S. A. Molchanov, Diffusion processes and Riemannian geometry, Russian Math. Survey, 30 (1975), 1-63. M. Schilder, Some asymptotic formulas for Wiener integrals, Trans. Amer. Math. SOC.,125 (1966), 63-85. D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979. E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations, 1946. S. R.S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure. Appl. Math., 19 (1966), 261-186. -, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., 20 (1967), 431-455. -, Diffusion processes in a small time interval, Comm. Pure Appl. Math., 20 (1967), 659-685. DEPARTMENT O F MATHEMATICS OSAKAUNIVERSITY TOYONAKA, OSAKA 560, JAPAN