Diffusion processes with generators of generalized divergence form

J. M ath. K yoto Univ. (JMKYAZ) 27-4 (1987) 597-619

Diffusion processes with generators of generalized divergence form By

Hirofumi OSADA § O . Introduction.

In this paper we shall discuss a c e rta in c la ss of diffusion processes with singular coefficients. Our class contains th e class o f d iv erg en c e fo rm . Hence we call it generalized divergence form (G. D. F . in abbreviation). G. D. F . have some applications to problems arising from fluid mechanics. Indeed, t h e B urgers process ([1 2 ]) and the vortex process are of generalized divergence form . Especially the vortex process is a n example o f generalized divergence form w hich is not of divergence form ([3 ], [1 1 ], [1 3 ]). Since the coefficients of these diffusion processes are singular, the construction is not easy. Our results are useful to construct such diffusion processes a n d also applicable fo r a homogenization problem ([8], [9]).

a

is sa id to b e a diveraz ' gence form if a i i (t, x) a re measurable functions satisfying A differentiable operator A =

Via

i

(0.1)

2-1Ie12 i, .1

e=(e1)

1

Fundamental solutions of parabolic equations in a divergence form have been researched by many mathematicians. Especially important theoretical improvements were done by Nash [7] and Moser [ 6 ] , T h e most remarkable result was obtained by Aronson [ 1 ] . He showed the following estimate : L et p(s, x, t, y) be a fundamental solution of 7 — A . Then 2

(0.2)

C1t-n/2exP —C11x 3'1 — s )] P (s , x , t, y ) 2

[

_ C t n exp[ 3

-

12

—

C41x

—

Y1 /(t s)] 2

—

for all 0 —1 and w hen q < -1 . Step 1. q > - 1 ( q # 0 ) . W e d e fin e U(s)=((1 —s)/6, (1+s)/6)x Q (s ), where 1/3 s 1/2. Let 1 and l' be real numbers satisfying 1/3/2 t+constant,

x—>lx+ constant.

We set h(t)=4' if t 13and = 0 if t r

w it h C1 8 = v exp [C1 7 ]. F o r (s, y ) w it h 0 < s t and 41 3 1 t, it is clear that h(s, y) - - C /4, w hile f o r (s, y ) with O < s t a n d 1yl > r , w e h a v e h(s, 3)) —C,,r /(4t)—C ,/4. Then ,

2

.

2

16

2

1

maxf u(s, y) d y C i s exP E— Cur /LItillvd L 2( R . ) • 0r

C„(t a) -

- 1 1 ,4

exp [

1/2

Ci5r7(t an0 tz- i>r p (a , z, t, y) d z ) . y

—

2

—

Hence (3.19)

P(a, z, t, yrin(z)dz

C 4 v(t—o. ) - 4 /1 exp [

2C15r /(t

-

2

—

an •

Similarly we have (3.20)

tz-vi>r

P(or, z, t, Y Y m (Y )d y-C LI)(t— a)

/2 exp [

-1 1

—

2C15r 2 /(t -

Now, w e se t r= x — 3/1/2 and r = ( s + 0 / 2 . W e first assume t— s5 r 2

By the semigroup property o f p w e have p(s, x, t, y )= 1 i n p(s, x, r , z)p(r, z, t, y)m(z)dz . .

Split the integral over o v e r lz—x IG r . T h e n Ji q

lz - x 1 2 r

Rn

into a n integral J , over lz—xl

p(s, x , r,

and an integral

p(r, z, t, y) m(z)dzY . 2

z)2m(z)dz} 1/2 {, iz-xl1r

J2

/2

Using (3.20) to estimate th e first integral on the right and (3.5) to estimate the second and noting a r = s + t and 2 r = lx —y , we obtain (3.21)

./1 5.C15(t—s)-11/2exp [— C J x

—

Y1 2 /(t — s)]

w ith positive constants C a n d C 1 depending only on 2, p , ).) and n. Next we estimate J2 . W e observe that I < r = x — y1/2 implies that Hence z— 15

I z -y l2 r

6

p (s, x, r ,

p ( 7 ,

z ) 2 m ( z ) d z } 1 " { .Ç I ,-

z ,

t ,

y ) 2

m(z)dz}- 1 / 2 .

,

Using (3.19) to estim ate the second integral on the right a n d (3.5) to estimate th e first, w e o b ta in (3.21) for 1 2 . T h u s w e h a v e derived (3.3) in the case We finally discuss the case t — s> r . By means o f (3.5), w e have 2

p (s, x , t, y)_ C„(t—s) ni -

C (t —s) n/ exp 17

-

2

2

x— y1 2 / 4(t—s)]

w ith a positive constant C 17 depending only on 2, p, u a n d n . T his completes the proof o f (3.3). Q. E. D. Proof o f (3.4). ( 3 . 4 ) follows from (3.3) and Proposition 2 . 6 . Since this step

616

Hirofunzi Osada

is sam e as Lemma 4.1 in Ichihara [4], w e omit it. § 4. Proof o f theorems.

I n t h i s section w e s h a ll c o m p le te t h e p ro o f o f T h e o re m s. W e suppose =in

-1

{

7i G

z,j=1

( n

;

2 , p , v ) .

Proof o f Theorem 1. L e t {A } b e an approximation sequence o f A : k

Ak

(4.1)

G (n ; 22, 2p, 2y). o

Assuming the coefficients f in", s a t i s f y limMk=771

strongly in L

k -Yoo

(4.2)

strongly in L ((0 , 0 0 )x R ) ,

lim 4 7 =a i , = Ci

)

n

g

k

lir n k -Yoo

lo c (R n ) y

s t r o n g l y in Lg1 0 c ((0, 00)X Rn)

for a ll 2 _ q < 0 0 . L e t P (s, x, t, y ) be a fundam ental solution of 7 —A . Since the coefficients are sm ooth, p " is u n iq u e . M oreover it follows from Proposition 2 .5 a n d 3 .1 t h a t {p (s, x, t, Y)}k=i,2.. i s precompact i n C (S ), w h e r e S = { (s, x , t, y): x , y e in Rn O s