2

KAZUAKI TAIRA

2) b{ e C°°(RN).

3) c G C°°(R iV ) and c 0 in D.

4) The integral kernel s(x, y) is the distribution kernel of a properly supported,

pseudo-differential operator S € L\~0K(Ti. ), « 0, which has the transmission

property with respect to the boundary dD (cf. Section 2.2), and s(x,y) 0 off the

diagonal {(x,x);x € R } i n R x R . The measure dy is the Lebesgue measure

o n R ^ .

5) The function cr(x,y) is a C°° function on D x D such that a(x,y) = 1 in

a neighborhood of the diagonal {(x,x)\x £ D} in D x D. The function r(x,y)

depends on the shape of the domain D. More precisely, it depends on a family of

local charts on D in each of which the Taylor expansion is valid for functions u.

For example, if D is convex, one may take a(x, y) = 1 on D x D.

6) Wl(x) = c(x) + JD s(x, y)[l - r(x, y)]dy 0'mD.

The operator W is called a second-order Waldenfels operator (cf. [1]), The

differential operator P is called a diffusion operator which describes analytically

a strong Markov process with continuous paths (diffusion process) in the interior

D. The operator Sr is called a second-order Levy operator which is supposed to

correspond to the jump phenomenon in the interior D\ a Markovian particle moves

by jumps to a random point, chosen with kernel s(x,y) and function o~(x,y), in

the interior D. Therefore, the Waldenfels operator W is supposed to correspond to

such a diffusion phenomenon that a Markovian particle moves both by jumps and

continuously in the state space D.

The intuitive meaning of condition 6) is that the jump phenomenon from a point

x € D to the outside of a neighborhood of x in D is "dominated" by the absorption

phenomenon at x. We remark that in the case when cr(z, y) = 1 on D x 5 , condition

6) is reduced to the following simple one:

6') Wl(x) = c(x) 0 in D.

Let L be a second-order boundary condition such that in local coordinates

(si,- • • ,XN-I)

(0.2)

Zu(ar') = Qu(x') + n{x')^(x') - S(x')Wu{x') + Tu(x')

,N-l n? JV-1

s

f E «°V)ir7r-(*')+ £/?V)|V)+7(*(*'))

+ M*')fj(*') - 6(x')Wu(x') + Ux')u{x') + ]T CV)f^0O

+ / r(x',y') [«(2/') - r(x', y') [u{x') + ^ ( » - *i)^L(»'))] tf

+ Jt(x\y)\u(y)-T(x',y)^u(x')+^(yj-xj)^(x')jyyj,

where: