4
KAZUAKI TAIRA
are supposed to correspond to the diffusion along the boundary, the absorption
phenomenon, the reflection phenomenon, the viscosity phenomenon and the jump
phenomenon on the boundary and the inward jump phenomenon from the bound-
ary, respectively.
The intuitive meaning of condition 9) is that the jump phenomenon from a
point x* G dD to the outside of a neighborhood of x' in D is "dominated" by the
absorption phenomenon at x'.
This paper is devoted to the functional analytic approach to the problem of con-
struction of Feller semigroups with VentceP boundary conditions. More precisely,
we consider the following problem:
Problem. Conversely, given analytic data (W, L), can we construct a Feller semi-
group {Tt}to whose infinitesimal generator 21 is characterized by (W,L) ?
We say that the boundary condition L is transversal on the boundary dD if it
satisfies the condition:
/ t{x',y)dy = +oo if n{x') =
Six1)
= 0.
JD
Intuitively, the transversality condition implies that a Markovian particle jumps
away "instantaneously" from the points x' G dD where neither reflection nor vis-
cosity phenomenon occurs (which is similar to the reflection phenomenon). Prob-
abilistically, this means that every Markov process on the boundary dD is the
"trace" on dD of trajectories of some Markov process on the closure D = D U dD.
The next theorem asserts that there exists a Feller semigroup on D corresponding
to such a diffusion phenomenon that one of the reflection phenomenon, the viscosity
phenomenon and the inward jump phenomenon from the boundary occurs at each
point of the boundary dD:
Theorem 1. We define a linear operator 21 from the space C(D) into itself as
follows:
(a) The domain of definition D(21) of 21 is the set
(0.3) D(21) = {u G C(D); Wu G C(D), Lu = 0} .
(b) 21u = Wu, wGD(21).
Here Wu and Lu are taken in the sense of distributions.
Assume that the boundary condition L is transversal on the boundary dD. Then
the operator 21 generates a Feller semigroup {Tt}to on D.
We remark that Theorem 1 was proved before by Taira [12] under some additional
conditions (cf. [12, Theorem 10.1.3]), and also by Cancelier [3] (cf. [3, Theoreme
3.2]). Takanobu and Watanabe [14] proved a probabilistic version of Theorem 1 in
the case when the domain D is the half space R ^ (cf. [14, Corollary]).
The problem of construction of Feller semigroups has never before, to the au-
thor's knowledge, been studied in the non-transversal case. In this paper we con-
sider the following case:
(A) ii(x') + 6(x') - 7(z ; ) 0 on dD.
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