Limit theorems for diffusion processes on compact manifolds

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63

m e a s u r e m ( d x ) on M. A c o n f o r m a l diffusion X is called a conformal Brownian motion if the g e n e r a t o r o f X is the h a l f L a p l a c e - B e l t r a m i o p e r a t o r ½A(g) c o r r e s p o n d ing to the H e r m i t i a n m e t r i c g. F i n a l l y a c o n f o r m a l diffusion X is called a K~ihler diffusion if the H e r m i t i a n metric g c o r r e s p o n d i n g to X is a K~ihler metric on M. If n = 1, as is well k n o w n , these classes all c o i n c i d e . I f n = 2, the last two classes c o i n c i d e a n d are s m a l l e r t h a n the classes o f s y m m e t r i c c o n f o r m a l diffusions a n d there are m a n i f o l d s on w h i c h no s y m m e t r i c c o n f o r m a l diffusions exist (for e x a m p l e , H o p f m a n i f o l d ) . I f n / > 3 , these classes are different in g e n e r a l a n d there exist m a n i f o l d s (for e x a m p l e , I w a s a w a m a n i f o l d ) w h i c h a d m i t a c o n f o r m a l B r o w n i a n m o t i o n b u t no K~ihler diffusion.

References [1] M. Fukushima and M. Okada, On conformal martingale diffusions and pluripolar sets, J. of Funct. Anal. 55(3) (1984), 377-388. [2] L. Schwartz, Semi-martingales sur des varietes, Lect. Notes in Math. 780 (1980).

A Remark on One-Dimensional

Diffusion

Processes with Discontinuous

Scale

Functions K i y o s h i K a w a z u , * Yamaguchi University, Japan Y u k i o O g u r a , Saga University, Japan Y o i c h i r o T a k a h a s h i , University of Tokyo, Japan In the s t u d y o f b i r t h - d e a t h processes in r a n d o m e n v i r o n m e n t s , several a u t h o r s d e a l with o n e - d i m e n s i o n a l diffusion p r o c e s s e s with strictly i n c r e a s i n g d i s c o n t i n u o u s scale functions. A m o n g them, K a w a z u a n d K e s t e n (1985) o b t a i n e d J~-convergence for a s e q u e n c e o f these processes. In this talk we first note that those processes are M a r k o v i a n b u t not strictly M a r k o v i a n in general, a n d then we give a sufficient c o n d i t i o n for J ~ - c o n v e r g e n c e o f a s e q u e n c e o f such processes. The c o n d i t i o n is satisfied by a r a t h e r w i d e r class o f sequences t h a n that d e a l t with b y K a w a z u a n d Kesten. We also talk a b o u t o n e - d i m e n s i o n a l diffusion processes w h o s e scale functions are c o n t i n u o u s b u t not strictly increasing. The s a m p l e p a t h s o f these p r o c e s s e s are not in the s p a c e D in g e n e r a l , a n d we can not expect J ~ - c o n v e r g e n c e for a s e q u e n c e o f those processes. In this case we c h a n g e the metric o f the real line so that we can o b t a i n a J ~ - c o n v e r g e n c e t h e o r e m for a s e q u e n c e o f such processes.

Limit Theorems for Diffusion Processes on Compact Manifolds Y o k o Ochi, Osaka University, Japan Let (M, g ) be a c o m p a c t , c o n n e c t e d R i e m a n n i a n m a n i f o l d with s m o o t h b o u n d a r y 0M. W e c o n s i d e r a diffusion process {x(t)} g e n e r a t e d b y A / 2 + b with b o u n d a r y

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64

c o n d i t i o n yu = 0, w h e r e b a n d 3' are a s m o o t h v e c t o r field on M a n d the unit n o r m a l v e c t o r field on a M respectively. T h e n , there exists a c o n t i n u o u s D'~-valued p r o c e s s X such that for every c~ ~ D1, X,(ct) = ~[o,,1 ct, t>~0, a.s., w h e r e D~ a n d D~ are the s p a c e o f all s m o o t h 1-forms on M e n d o w e d with the S c h w a r t z t o p o l o g y a n d its d u a l space, respectively. W e c o n s i d e r the D ~ - v a l u e d c o n t i n u o u s p r o c e s s e s yA = (A-~/2 Y~)t>~o a n d Z ~ = ( A - ~ / 2 ( X ~ , - AtE)),>~o, where Y is the m a r t i n g a l e p a r t o f X a n d E is an e l e m e n t o f D~ such that, for ct c Dl, E ( a ) - - l i m r ~ X r ( a ) / t a.s. O u r p u r p o s e is to show t h a t yA a n d Z A c o n v e r g e w e a k l y in C ( [ 0 , oo)--> D ' 0 to the D'~-valued W i e n e r p r o c e s s e s ~ a n d "0 with zero m e a n a n d the c o v a r i a n c e f u n c t i o n a l s ((a,/3))(t A S) a n d ((a - du~,/3 - du~))(t ^ s), respectively. H e r e ((a,/3)) = ~M ( a , / 3 ) ( x ) m ( d x ) a n d m is the i n v a r i a n t p r o b a b i l i t y m e a s u r e o f {x(t)}. Also for every a ~ D1, the exact 1-form du~ is d e f i n e d u n i q u e l y by a differential e q u a t i o n r e l a t e d with g, b a n d Y- F o r details, see Y. Ochi, Limit t h e o r e m s for a class o f diffusion p r o c e s s e s (to a p p e a r in Stochastics) a n d N. I k e d a - Y . Ochi, C e n t r a l limit t h e o r e m s a n d r a n d o m currents (to a p p e a r ) .

Asymptotic Behavior of Moments of One-Dimensional Diffusion Processes Yukio Ogura* and Matsuyo Tomisaki, Saga University, Saga, Japan Let [ X ( t ) , Pa], a ~ I = (l, + ~ ) , -oo<~ I < o o , b e a diffusion p r o c e s s with the gene r a t o r qd= ( d / d m ) ( d / d x ) a n d the G r e e n f u n c t i o n G(s, x, y). W e a s s u m e that, for some

a c I,

G(s, a, a ) ~ s

~Lo(1/s)

(s~O),

m ( x ) = m((a, x ] ) ~ xt3Lm(x)

(x~oo).

w h e r e 0<~ a <~ 1,/3 > O; L~, L~ are s.v. (slowly varying) at c~; a(t) ~ b(t), t+O [tl'oo] s t a n d s for lim,;ot,~o~] a ( t ) / b ( t ) = 1. Let k(t) be the inverse f u n c t i o n o f t ~ tin(t). T h e n k(t) is regularly v a r y i n g at oo with i n d e x 1/(/3 + 1): k ( t ) ~ tl/(~÷~)Lk(t)(t~oo) for an Lk S.V. at oo.

Theorem. I f a function f on I satisfies f ~ L~oc(m)\Ll(m) and f ( x ) ~ xVLs(x) (x~oo) with fl + y >~O, c ~ + / 3 + 7 > 0 and Lf s.v. at oo, then Ea[f" X~.~)(Xr)] ~ t~+~v-l)/(t3+l)K~v(t)

as t~oo,

for a K,~v s.v. at oo such that 2

(~+)')/(fl+l)

+7

xc~(t)G(k(t))L~(t)"

1

/3+2

'

/~ -~- ")/

-~

if~+~,>o,

"Y