The Onsager-Machlup function for diffusion processes

J. Math. Kyoto Univ. (JMKYAZ)

22-1 (1982) 115-130

The Onsager-Machlup function for diffusion processes By Takahiko FUJITA and Shin-ichi KOTANI (Received Nov. 28, 1980)

O . Introduction L e t M b e a sm ooth d-dim ensional connected Riem annian m a n ifo ld a n d d m be th e L aplace-B eltram i perator. L e t (X e , P,), E m b e the m inim al diffusion processes o n M generated by th e operator — d m +b w h e r e b i s a sm ooth vector 2

f ie ld o n M . F o r a given sm ooth curve 0=(çbt)o,tsr on M s ta r tin g a t 0(0)=q, th e sojourn probability around 0 up to tim e T is defined by (0.1)

ifi(0 )= -P q fp(X t, 0 e ) (t, 0) and som e neighborhood V i n [0, T ] x M of th e curve t (t, 9 5 (t)) by

117

Onsager-Illachlup function 0(t, (x l, x , ••• , x ))=-(t, exp (OW, x e k (t)) 2

k

where e k(t)(e k(0) =e k ) is obtained as the parallel translate of e k along the curve 0 . It is clear that x = (x l, x , ••• , x a) is a normal coordinate of p --0 (t, (x , x , x ä )) with center q =0 (t) for each fixed tE [0, T ] . T he components of the ••• , vector field b, the metric tensor, Christoffel symbol, etc, in this normal coordin ate 0 ( t , • ) fo r each fixed tE [0, T ], are denoted by b '(t, x ), g „(t, x )n ,(t, x ), etc. For a differential operator aon V, we denote by 5 the differential operator on U transformed by the above diffeomorphism : -

2

1

2

5f(t, x)=a(f00 1)(0(t, x )). -

We shall calculate the operotors 5, 3, and 'a/ a t in the following lemma. Lemma 1.2. 5-=b (t, x) i

a

. ax t

a2 3 m = g ii(t, x )

a x ia x i

5a at

g ij( t, x ) rt( t, x

k

a axi

{ 9i(t)+Ei(t,

at

ax

W(u)—çbi(t) ( (u ) is the i-th component o f 0 ( u ) in th e local u— t coordinate 0 ( t, • ) .) and s (t, x ) is a smooth function satisfying where ç1)'(t)=lim .

i

max Iei(t, x)1 =0(1 x1 ) and Og

(1.3)

2

T

max ostT

(1.4) P r o o f.

axi k(

s

t ,

x )

=0(1x 1) f o r any i and k.

Define a function x k (t, p )(tE [0 , T ], p E M ) by an equation

(1.5)

exp (OW, x q t

k(0)= 10

•

Then we have the i-th component of 5 in the base = 5(x )1 t =to , , -

j

x

,

(xi : i-th coordinate function in Rd) (OW, xe k(to)))

=b(x (i, 1))) I( P o = e x p j

(by the definition).

=b (t o , x o ) i

Here it is evident that the fo r

M

a axk

a at -component of 5

is identically zero. The proof

is sim ilar to the above. Next set

at Then we see easily

a

.

c(t, x) a t d-cP(t, x )

a axi

.

T . Fujita and S . Kotani

118

c(t, x )=

5a at (0= at

(0=1.

Moreover note

(1.6)

di(t, x )=

a at a

. t=to

at (x '(tt p))1t=to (P = exP (95(t0), x ek(to))) k

In th e n orm al coordinate 0(t 0 , •), a geodesic ( x ( 0 ) f r o m ( ai) w it h a velocity (p ') satisfies th e following ordinary differential equations d xi(u) (u) d u —

(1.7)

d y u) = 1 du

1 1

(x(u))y (u)y (u) k

xi(0)= a' , y '(0)=

I

.

W e denote by x"(u, a , 13) and y '(u, a, p) the above solutions. T hen the equation (1.5) is equivalent to th e following

y = L(1, qi(t), x

(1.8)

.

k

(t , P)e k(t))

O n th e other hand, e (t) satisfies in th e n o rm a l coordinate 0(t 0, •) k

del(t)

dt

= rim (95(t))0,(t)

d g5m(t)

dt

eik (t o )= d e I(t)

Since n (6 (t0 ))=o , w e have

dt

(t = ô , O (t 0) It=t o = 0 . Hence noting also eRt0o )e i

= 0 a n d x k(to , p)=x k, w e get (1.9)

0=

aai

(1, 0, (x

(t

ay

(1, 0, (x k))

p)

by differentiating both sides o f (1.8) b y t a n d se ttin g t= t o . S ince rwo(to)>=o, it is e a sy to d e d u c e fro m (1.7) that

(1.10)

ax

,

aa;

(1, 0, (x )) =55±0(1x1 ) k

2

axi. (1, o, (x k))=65+0(1

x 12)

Consequently

di(to, x)=

—

$(t0)+o( x1 ) , 2

w hich im plies (1.1). w e can see by tracing th e above calculation carefully that (1.2) is also valid.

Onsager-Machlup function

2.

119

A reduction of the theorem

We now consider a space-time process (t, X e ). Its generator is -1

(t, X e ) EEV } a n d (t A cr, 'Z A,)=0 (tAcr, Xtno). + b . L et u=inf Lemma 1.2 the local generator of )Z becomes ,

t

a+

at

4

Then by

t

a at

1. a a2 g 'i(t x ) . +b (t, x ) a 2 ' axzax.' xi -

+

,

where

1 b k ( t, x ) =P ( t, x ) - - g i(t, x )rt i (t, x)— Sk(t)+st(t, x)

(2.1)

i

2

L e t W(x)=exp (0(0), x e k (0)) and ii ( x ) = P sb ( s ) I p ( x t , Then we have by a property of normal coordinate k

u ,(x ).=

13°

Po ,x

' f ossliPTIX t

I

O t).

-

6,

fo r an y tc[0, T ]} .

6 1

is the distribution of k' in C([0, T ]x U ) starting from x at time 0. L et ugt, x ) be the solution of the following initial boundary problem :

where

a14

t

1 . i)(T — t, x ) 2g

{

at

a

a2 a x ia x i

+ (T t, x ) a

i

} u,1 o n [0, T]X flx161

ug(t, • ) Iaiy=0, ug(0, x ) = 1 f o r x e {Ix I u (t, 0) as s 0 f o r any 0 < t_ T . i

P ro o f . Decompose Z " into tw o parts;

a +C(t, r, 0, E)+(a '') (1) 2

14-=A(t, r, 0, s)

r

*

ar

D P= LP

—

'

(Lier

' ' )

*

(

1)

F u rth e r let Gff(t, x)= U 1 , (

t

Gbf(t, x)= N f i (

t

s

;

x, y)(Df.lf(s, y)dyds , , x, y)(DP)f(s, y)dyds

fo r any sm ooth function f . T h e n th e equation (3.1) c a n b e w ritte n as (3.6)

.

S tep 1. For any smooth function f, Gbf(t, r)= 0 identically. Indeed, Gb f(t, r)=f dt9/ 1t d 4 1 jo ( ap

0

t-

\

6

t— s =

s

2

fi l ( 2 1ot c is .fiaDtaD

, r '0 1,r 0 )(D P f)(s ,r 0 )r 'd r d 0

' r'0', r0)cl0 1}(DVf)(s, rO)rn -1 -drde . t -9 s

Since fi(t, x, y ) is rotation invariant,

ap

13(

6"

r '0 ', r O ) d 0 ' is independent of

8 . M oreover, by th e definition o f th e (*)-operation, w e see al)

DVf(s, r0)d0= Lier•cf(s, re)d0 1

( ar'e)*(1)f(s, r0)cl0=0 .

ED

T h erefore w e have Gbf(t, r)= .çto d s

M a ID

13

(

t - -2s

E ' r '0

1

r0 )d O 'd 0 }1 Db.Ef(s, r0)d0}-rn-idr

=0. Let 2 ) *denotes the adjoint operator with respect to dO.

D

125

Onsager-Machlup function 1-1 (0=1

if

=0

if

0 t< 1 ,

12 (0=0

if

t

=1

if

0 t 0 o n M , satisfying a s sI0

C,(p)=1-Fa(p)6 2 +0(6 3 )

uniformly o n each com pact set in M including every its derivative o f at least second order. Then we h a v e th e following asymptotic behavior ; Pq IP(Xt, çbt) eQ ç b t ) fo r a n y t < 0 , T J I : +

=1.1(0) fi(Y )dy exp

21

T

2

D

t(

3

, S )d s+0 (1 )) a s r 1 0 , 8

where "L(p, v)=1 (P, 0+221a(P). T h is can be proved by th e same method a s in th e c a s e o f a(p)= 0 , i f we ,

-

1 1 u se th e fundamental solution for t y —s2 a(g5(T-0)} 4 in p la c e o f that of

-

V,(9)

If, in particular, C(x ) is chosen so that V ( p ) = 1 + 0 ( 0 at each point j p, q , M, where V ,(x) denotes th e Riemannian v o lu m e o f th e ball around x with radius sC,(x), then a(p)--

R (p).

1

Therefore this procedure cannot

6 d (d + 1 )

make th e scalar ourvature term vanish. Finally, we remark that if X , starts with a smooth initial distribution )2(x)dx (dx : th e Riemannian measure o f M), th e following estimate holds 2

tq(0)--- s a Yi(g5(0)){

D

T

fi(y)dy} exp (f L (0 3 , k c It)e x p ( o

2,T

c 2

) .

T h is estimate is also easily obtained by th e same method.

DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY

References [ 1 1 D . D a rr a n d A . B a c h , T h e Onsager-Machlup function a s Lagrangian fo r th e most probable path of a diffusion process, C om m . M ath. Phys., 60 (1978), 153-170.

130

T . Fujita and S . Kotani

[ 2 ] R . G r a h a m , Path integral formulation o f general diffusion processes, Z. Physik B, 26 (1977), 281-290. 3 N . Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, [ Kodansha, N orth H olland, 1981. H. Ito, Probabilistic construction of Lagrangian of diffusion processes a n d its ap[4 plication, Prg. Theoretical P h y s., 59 (1978), 725-741. [5 1 S . I to , P a r tia l differential equations, Baifukan, 1966, C hap. 2 (in Japanese). [ 6 ] S . M izohata, T h e theory o f p a r tia l differential equations, Cam bridge University P re ss, 1973. [ 7 1 L. Onsager a n d S . M achlup, Fluctuations and irreversible processess, I, II, P h y s. Rev., 9 1 (1953), 1505-1512, 1512-1515. [8 1 M . S p iv a k , A comprehensive introduction to differential geometry, Brandeis University, 1970. [ 9 1 R. L. Stratonovich, O n th e probability functional of diffusion processes, Select. in M ath. Stat. Rrob., 1 0 (1971), 273-286. [ 1 0 ] Y . T akahashi a n d S . W atan ab e, T h e probability functionals (Onsager-Machlup functions) o f d iffu s io n processes, Proc. LM S Sym p. o n Stochastic Integrals at D urham , 1980, L ecture N otes in M ath. 851, Springer.