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Fine topologies on the boundary. In 1996, Kuznetsov observed that the Euclid-
ian topology is not appropriate for the definition of the trace and that it should be
replaced by a finer topology (see [DK98c], Section 4). Such a topology is defined
by Dynkin in [Dy98]. For this fine topology which we denote by r, a Borel subset
r of dD is closed if and only if SG(iir) C T. However, proving that SG(tt) is closed
for r for every u £ U has remained an open problem. For this reason, Dynkin and
Kuznetsov introduce in [DK98b] in the general setting of equation (2) a second
fine topology on the boundary which we denote by r . The definition of r relies on
cr-moderate solutions. For every Borel set T C dD, set
ur = suplujy ; v £ A/i, i/(dD\T) = 0, uv ^ u).
In [DK98b], it is proved that ur ^ ur and that ur is cr-moderate. (If T is compact,
ur is actually the greatest r-moderate solution dominated by ur)- By definition, a
Borel set T C dD is closed for f if and only if SG(2
) C T (see Section 2.4.2).
The definition of the fine trace. In [DK98b], the fine trace of a solution u of
(3) is defined as the pair
tr(u) = ( I »
r =
and v is the cr-finite measure on dD characterized by
v(B) = sup{/i(£) ; lie Mi, /i(r) = 0, u^ ^ u}
for every Borel set B C dD. (See [DK98b] and Section 2.4.3). Note that this
definition is given in [DK98b] in the general context of equation (2). It coincides
with the rough trace in the subcritical case. If r , r ; C dD and v, v' G Mo, we say
that (r , v) and (r ; , v') are equivalent and write (T, v) ~ (r' , v') if and only if v v'
and TAT' is boundary polar.
If u G U and %T(U) = (T, z/), Dynkin and Kuznetsov prove in [DK98b] that
(A) r C dD is closed for the topology f (i.e. SG(2
) C T).
(B) v is a cr-finite measure of class Mo such that u(dD\T) = 0 and S G ^ ) C I \
Conversely, if a pair (I\ v) satisfies (A) and (B), then
t r ( ^ r S ^ ) ^ (r,i/).
Moreover, u? 0 uv is the minimal solution with a trace equivalent to (r , v) and the
only one that is a-moderate.
Key problems. In the survey [DK98c], Section 7, Dynkin and Kuznetsov present
the key remaining problems (see also [Dy02]. The first one can be stated as follows:
P R O B L E M 1. Is every solution a-moderate ?
By the results presented above, a positive answer to this problem would com-
pletely solve the problem of the classification of the solutions of (3). The following
intermediate question is suggested by Dynkin and Kuznetsov.
P R O B L E M 2. Is ur cr-moderate for every Borel subset T of dD ?
In fact, this is equivalent to saying that u-p UY for every Borel set Y C dD. A
positive answer to this problem would in turn imply that the fine topologies r and
r coincide. Note that it has already been proved that ^ r is cr-moderate, provided
T is relatively open in dD ([Dy02], Theorem 8.1).
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