Right Markov Processes and Systems of Semilinear Equations with Measure Data
Right Markov Processes and Systems of Semilinear Equations with Measure Data
Tomasz Klimsiak 0
0 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University , Chopina 12/18, 87-100 Torun ́ , Poland
In the paper we prove the existence of probabilistic solutions to systems of the form −Au = F (x, u) + μ, where F satisfies a generalized sign condition and μ is a smooth measure. As for A we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on L1. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer's property (L) of Markov processes and in terms of order compactness of the associated resolvent.
Right Markov processes; Dirichlet forms; Semilinear elliptic systems; Order compactness; Probabilistic potential theory; Measure data; Smooth measure
1 Introduction
Let E be a Radon metrizable topological space, F : E×RN → RN , N ≥ 1, be a measurable
function and let μ = (μ1, . . . , μN ) be a smooth measure on E. In the present paper we
investigate the problem of existence of solutions of the system
Tomasz Klimsiak
Here A is the linear operator associated with a Markov semigroup {Tt , t ≥ 0} on L1(E; m).
Our only assumption on {Tt } is that it is representable by some right Markov process
X = ({Xt , t ≥ 0}, {Px , x ∈ E}) on E, i.e. for every t ≥ 0 and f ∈ L1(E; m),
(Tt f )(x) = Ex f (Xt ) ≡ pt f (x) for m-a.e. x ∈ E,
where Ex denotes the expectation with respect to the measure Px . The class of
operators associated with such semigroups is fairly wide. It includes important local and
nonlocal operators corresponding to quasi-regular Dirichlet forms (see [23, 32, 34]) as
well as interesting operators which are not in the variational form, like some classes of
Ornstein-Uhlenbeck processes (see Example 5.7).
As for F = (f1, . . . , fN ) we assume that it is continuous with respect to u and satisfies
the following sign condition:
≤ G(x)|y|,
x ∈ E, y ∈ RN
for some appropriately integrable positive function G (see hypotheses (H1)–(H4) in
Section 3).
The first problem we encounter when dealing with systems of the form (1.1) is to give
suitable definition of a solution. The problem occurs even in the case of one linear
equation with local operator of the form A = id,j =1 ∂∂xj (aij ∂∂xi ), whose study goes back to the
papers of Serrin [38] and Stampacchia [40]. Serrin [38] constructed an example of
(discontinuous) coefficients aij and nontrivial function u having the property that u ∈ W 1,q (D) for
0
every q < d/(d − 1) and u is the distributional solution of Eq. 1.1 with data μ = 0, F = 0.
Since it was known that in general one can not expect that a solution to Eq. 1.1 belongs to
the space W 1,q (D) with q ≥ d/(d − 1), the problem of the definition of a solution to Eq.
0
1.1 ensuring uniqueness arose. Stampacchia [40] solved this problem by introducing the
socalled definition by duality. Since his work the theory of scalar equations with measure data
and local operators (linear and nonlinear of of Leray-Lions type) have attracted
considerable attention (see [4, 12, 13, 16, 18] for results for equations with smooth measures μ; a
nice account of the theory for equations with general measures has been given in [3]).
The case of nonlocal operators is much more involved. To our knowledge there were
only few attempts to investigate scalar linear equation (1.1) with operator A = α with
α ∈ (0, 1] by analytical methods (see [1, 26]). To encompass broader class of operators and
semilinear equations in [28] (see also [29]) a probabilistic definition of a solution of scalar
problem (1.1) is proposed. The basic idea in [28] is to define a solution via a nonlinear
Feynman-Kac formula. Namely, a solution of Eq. 1.1 is a measurable function u : E → R
such that
u(x) = Ex
F (Xt , u(Xt )) dt + Ex
0 0
for m-a.e. x ∈ E, where Aμ is a continuous additive functional of the process X
corresponding to the measure μ in the Revuz sense (see [19, 23, 32, 35]). In [28] it is proved that in case
N = 1 if F is nonincreasing with respect to u then under mild integrability assumptions on
the data there exists a unique solution to Eq. 1.1. In fact, if A is a uniformly divergence form
operator then the probabilistic solution of Eq. 1.1 coincides with Stampacchia’s solution by
duality.
When studying systems (1.1) with F satisfying merely sign condition (1.3) we encounter
new difficulties, which roughly speaking pertain to weaker regularity of solution of Eq. 1.1
than in the scalar case and to “compactness properties”. In [27] we have studied systems of
the form (1.1) on bounded domain D ⊂ Rd with A = subject to homogeneous (...truncated)