Right Markov processes and systems of semilinear equations with measure data

In the paper we prove the existence of probabilistic solutions to systems of the form $-Au=F(x,u)+\mu$, where $F$ satisfies a generalized sign condition and $\mu$ 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 $L^1$. 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.


Introduction
Let E be a Radon metrizable topological space, F : E × R N → R N , N ≥ 1, be a measurable function and let µ = (µ 1 , . . . , µ N ) be a smooth measure on E N . In the present paper we investigate the problem of existence of solutions of the system −Au = F (x, u) + µ. (1.1) Here A is the linear operator associated with a Markov semigroup {T t , t ≥ 0} on L 1 (E; m). Our only assumption on {T t } is that it is representable by some right Markov process X = ({X t , t ≥ 0}, {P x , x ∈ E}) on E, i.e. for every t ≥ 0 and f ∈ L 1 (E; m), where E x denotes the expectation with respect to the measure P x . 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 [20,28,31] 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 = (f 1 , . . . , f N ) we assume that it is continuous with respect to u and satisfies the following sign condition: F (x, y), y ≤ G(x)|y|, x ∈ E, y ∈ R N (1. 3) 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 = d i,j=1 ∂ ∂x j (a ij ∂ ∂x i ), whose study goes back to the papers of Serrin [30] and Stampacchia [35]. Serrin [30] constructed an example of (discontinuous) coefficients a ij and nontrivial function u having the property that u ∈ W 1,q 0 (D) for every q < d/(d − 1) and u is the distributional solution of (1.1) with data µ = 0, F = 0. Since it was known that in general one can not expect that a solution to (1.1) belongs to the space W 1,q 0 (D) with q ≥ d/(d − 1), the problem of the definition of a solution to (1.1) ensuring uniqueness arose. Stampacchia [35] solved this problem by introducing the so-called 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 [4,9,10,13,15] 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 [1,23]). To encompass broader class of operators and semilinear equations in [25] (see also [26]) a probabilistic definition of a solution of scalar problem (1.1) is proposed. The basic idea in [25] is to define a solution via a nonlinear Feynman-Kac formula. Namely, a solution of (1.1) is a measurable function u : E → R such that for m-a.e. x ∈ E, where A µ is a continuous additive functional of the process X corresponding to the measure µ in the Revuz sense (see [16,20,28,32]). In [25] 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 (1.1). In fact, if A is a uniformly divergence form operator then the probabilistic solution of (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 (1.1) then in the scalar case and to "compactness properties". In [24] we have studied systems of the form (1.1) on bounded domain D ⊂ R d with A = ∆ subject to homogeneous Dirichlet boundary condition. In [24] it is observed that in general, if F only satisfies the sign condition, one cannot expect that F (·, u) ∈ L 1 (D; m). Moreover, it may happen that the first integral on the right-hand side of (1.4) is infinite. This together with the comments given before show that for systems, even in the case of a uniformly elliptic divergence form operator, neither the distributional definition nor the probabilistic via the Feynman-Kac formula (1.4) are applicable. For these reasons in [24] more general than in [25,26] probabilistic definition of a solution of (1.1) is adopted. It uses the representation of u in terms of some backward stochastic differential equation (BSDE) associated with A, F, µ (in case F (·, u) is integrable the representation reduces to (1.4)). This approach via BSDEs only requires quasi-integrability of F (·, u). It turns out that this mild demand is always satisfied for solutions of (1.1). Therefore in the present paper we use some suitable generalization of the definition from [24] (see Section 3).
As for "compactness properties", let us note that in [25] it is shown that if N = 1 and F is nonincreasing then for A associated with a Dirichlet form the function F (·, u) is integrable but in general, u is not integrable (even locally). Since in case N ≥ 2 also the function F (·, u) need not be integrable, it is fairly unclear what type of function space possessing Banach structure to use to get the existence result for (1.1). In [24] we have used the specific structure of the operator A = ∆ to prove that a solution of (1.1) equals locally (i.e. on some finely open sets) to some function from H 1 0 (D), which allowed us to apply the Rellich-Kondrachov theorem on finely open sets (see also [17,18] for the theory of Laplacians on finely open domains). In general, this approach fails. To overcome the difficulty, in the present paper we introduce a notion of compactness property relating the process X to given solid P and positive subadditive set function m on E (not necessarily measure). The compactness property is intended to study ma.e. convergence of sequences of functions defined on E, pointwise convergence (when m is a counting measure) and quasi-everywhere convergence (when m is the capacity determined by A). It appears that such analysis of pointwise behaviour of sequences of functions, in particular sequences of the form {p t f n }, {R α f n }, where p t f is defined by (1.2) and R α f is the probabilistic resolvent defined by is sufficient for the proof of existence of probabilistic solutions to (1.1). Roughly speaking, given a solid P ⊂ B + (E) and a positive subadditive set function m on E we say the triple (X, P, m) has the compactness property if for some α > 0 the probabilistic resolvent (1.5) carries the family P in a relative compact set in the topology of m-a.e. convergence (see Section 2.1). If m is the counting measure then we will omit m in the notation and simply say that (X, P) has the compactness property.
In applications the family P = {u ∈ B + (E); u ≤ 1} ≡ B 1 plays pivotal role. Mokobodzki (see [14, Section XII, Theorem 89]) has observed that if X satisfies hypothesis (L) of Meyer then (X, B 1 ) has the compactness property. Actually, he has observed that R α : equipped with the topology of uniform convergence is equivalent to hypothesis (L) of Meyer. In Section 2 using results of [36,37] we prove that (X, B 1 ) has the compactness property iff X satisfies Meyer's hypothesis (L).
In Section 4 we show that if m is an excessive measure then (X, B 1 , m) has the compactness property iff (1.6) Here by order compactness we mean that for every positive v ∈ L 1 (E; m), R α carries order intervals [0, v] = {u ∈ L 1 (E; m) : 0 ≤ u ≤ v} in relatively compact subsets of L 1 (E; m). We also investigate some stability properties of the compactness property with respect to transformation of the underlying process. The most important result in this direction is Proposition 2.8. It says that for every B ∈ B(E), if (X, P, m) has the compactness property then (X B , P(B), m) has the compactness property, where X B denotes the part of X on B and P(B) = {u ∈ P; u(x) = 0, x ∈ E \ B}. We have already mentioned that it is reasonable to expect that F (·, u) and u are quasiintegrable which roughly speaking means that they are integrable on subsets of E whose complements have small capacity naturally generated by the operator A. The significance of Proposition 2.8 is that it allows to reduce the proof of existence of solutions of (1.1) to the analysis of the system (1.1) on such sets. Let us also note that in some sense Proposition 2.8 resembles results on compactness of positive operators subordinated to compact operator (see [2] and Corollary 2.10).
The second problem that we address in Section 2 is to find conditions on a sequence {u n } of functions on E, which together with the compactness property imply that {u n } is relatively compact in the topology of m-a.e. convergence. Our main result is Theorem 2.2, which says that if (X, P, m) has the compactness property and {u n } ⊂ P satisfies the condition then {u n } has a subsequence convergent m-a.e. Condition (1.7) is satisfied for instance if for m-a.e. x ∈ E the sequence of processes {u n (X)} is tight in the Skorokhod topology J 1 under the measure P x . It is worth noting here that in the paper the notion of compactness of a triple (X, P, m) is defined for general normal processes (i.e. markovianity of the process X is not required) and that Theorem 2.2 is proved for such wide class of processes. In Section 2.3 we show that if X is associated with a transient symmetric regular In Sections 3 and 4 we define a probabilistic solution of (1.1) and give an existence result for system (1.1). The basic space in which solutions are looked for is the space D of measurable functions u on E such that the family {u(X τ ), τ is a stopping time} is uniformly integrable under P x for q.e. x ∈ E. We show that D[E] ⊂ D if X is associated with a semi-Dirichlet form. We call a finely continuous function u ∈ D such that F (·, u) is quasi-integrable a solution of (1.1) if there exists a local martingale additive functional M of X such that for m-a.e. x ∈ E and every T > 0, where ζ is the life-time of X and A µ is the positive co-natural additive functional associated with measure µ.
We first study probabilistic solutions to (1.1) in case X is associated with a semi-Dirichlet form and (X, B 1 , m) has the compactness property. In Section 3 we show that if µ is smooth and satisfies some integrabilty condition, F satisfies the sign condition (1.3), then there exists a solution of (1.1). We also show that if F is monotone, i.e.
then the probabilistic solution to (1.1) is unique.
The case of general right Markov processes is considered in Section 4. We show that if X satisfies Meyer's condition (L) then under the same hypotheses as in Section 3 there exists a solution to (1.1). Using (1.6) one can formulate the existence result in purely analytic terms, without relating to the concept of the compactness property. Namely, if the resolvent of the operator A is order compact on L 1 (E; m), F satisfies the sign condition and the data are appropriately integrable then there exists a solution of (1.1). As a matter of fact we assume some additional regularity condition on the semigroup {T t , t ≥ 0} but we think that it is technical and can be omitted.
In Section 5 we give some examples of operators and processes to which our results apply. Among others we give a simple example of Ornstein-Uhlenbeck semigroup, i.e. semigroup generated by differential operator of the form which is not of variational form (or, equivalently, is not analytic). The Ornstein-Uhlenbeck process with generator L is not associated with a Dirichlet form but satisfies Meyer's hypothesis (L). This shows that the class of processes considered in Section 4 includes important processes that do not belong to the class considered in Section 3.

Normal processes
Let E be a Radon metrizable topological space (see [6]) and B(E) be the set of all numerical Borel measurable functions on E. W adjoin an isolated point ∆ to E and set E ∆ = E ∪ {∆} (in E ∆ we have natural topology in which E is open). We denote by B ∆ (E) the set of all numerical Borel measurable functions on E ∆ . Let (Ω, G) be a measurable space and {X t , t ∈ [0, ∞]} be a stochastic process on E ∆ such that X ∞ = ∆ and if X t 0 = ∆ for some t 0 ∈ [0, ∞] then X t = ∆ for t ≥ t 0 . We denote by ζ the life-time of X, i.e. ζ = inf{t ≥ 0; X t = ∆}.
For x ∈ E ∆ let P x be a probability measure on (Ω, G). Let {G t , t ∈ [0, ∞]} be a filtration in G and let {G 0 t , t ≥ 0} be a natural filtration generated by X. We assume that (a) for every t ≥ 0, In the whole paper for a given Borel set B ⊂ E we denote by Let B + (E) = {f ∈ B(E); f (x) ≥ 0, x ∈ E} and let B r (E) denote the set of u ∈ B(E) such that u(x) ∈ R for x ∈ E. In the whole paper we adopt the convention that f (∆) = 0 for every numerical function f on E. For every t ≥ 0, α ≥ 0 and f ∈ B + (E) we put Unless otherwise stated, in this section m is a nonnegative subadditive set function on E.
Definition. (a) We say that a triple (X, P, m) has the compactness property if for every {u n } ⊂ P there exist a set Λ ⊂ (0, +∞) and a subsequence (n ′ ) ⊂ (n) such that sup Λ = +∞ and for every α ∈ Λ the sequence {R α u n ′ } is m-a.e. convergent and its limit is m-a.e. finite. (b) We say that a pair (X, P) has the compactness property if the triple (X, P, m) has compactness property with m being the counting measure.
In the sequel for given P ⊂ B r,+ (E) we set P * = P − P.
Definition. We say that a sequence {u n } ⊂ P * satisfies Theorem 2.2. Assume that (X, P, m) has the compactness property. If {u n } ⊂ P * satisfies (M 1 ) then there exists a subsequence (n ′ ) ⊂ (n) such that {u n ′ } is m-a.e. convergent and its limit is m-a.e. finite.
Proof. Let Λ ⊂ (0, +∞) be a countable set such that sup Λ = +∞ and let (n ′ ) ⊂ (n) be a subsequence such that for every α ∈ Λ, {R α u n ′ } is m-a.e. convergent and its limit is finite m-a.e. Let A ⊂ E be a set of those x ∈ E for which lim n ′ R α u n ′ (x) does not exist or exists and is infinite for some α ∈ Λ. It is clear that m(A) = 0. Let B be the set of those x ∈ E for which condition (M 1 ) does not hold. We put w = sup n |u n |. By (P1) and (P2), u + n , u − n ∈ P and sup n u + n , sup n u − n ∈ P. Since w ≤ sup n u + n + sup n u − n , we see that w(x) < ∞ for x ∈ E and that without loss of generality we may assume that R α w(x) < ∞ for m-a.e. x ∈ E and every α ∈ Λ. Let C ⊂ E be the set of those Let us fix ε > 0 and let θ ε Let α 0 ∈ Λ be such that α 0 > 1/θ ε x . Then we have Therefore there exists α ε x ∈ Λ such that On the other hand, by (2.3), for n ′ k ≥ n ε x and α ∈ Λ such that α ≥ α ε x , Put ε = β x /9. By the compactness property of the triple (X, P, m) there exists . By (2.5) and (2.6), x, α ε x )}, which contradicts (2.4) and proves the theorem. ✷ Corollary 2.3. If (X, P, m) has the compactness property and {u n (X)} is a sequence of càdlàg processes on some interval [0, T ] tight in the Skorokhod topology J 1 under the measure P x for m-a.e. x ∈ E then {u n } has a subsequence convergent m-a.e.
Proof. If {u n (X)} is tight under P x for m-a.e. x ∈ E then condition (M 0 ) is satisfied for m-a.e. x ∈ E and sup n≥1 |u n (x)| is finite m-a.e. Of course the same is true for Observe that if {u k n } satisfies (M 0 ) then it satisfies (M 1 ). Therefore from Theorem 2.2 it follows that {u k n } converges m-a.e. up to a subsequence for every k ≥ 1. From this we easily deduce that there exists a subsequence (n ′ ) ⊂ (n) such that {u n ′ } converges m-a.e.

Right Markov processes
Let us recall that X satisfies Meyer's hypothesis (L) if there exists a σ-finite Borel measure m on E such that R α (x, dy) ≪ m for every x ∈ E and some (and hence every) The measure m of the above definition will be called a reference measure for the process X or a reference measure for the resolvent {R α , α > 0}. Let Proposition 2.4. Assume that X is a right Markov process. Then (X, B 1 ) has the compactness property iff X satisfies Meyer's hypothesis (L).
Proof. Assume that X has a reference measure m. Then for every α > 0 there exists a measurable function r α : In particular, for every x ∈ E, which shows that (X, B 1 ) has the compactness property. Now assume that (X, B 1 ) has the compactness property. From the resolvent identity it is clear that if R α has a reference measure for some α > 0 then X has a reference measure. Consequently, if X does not have a reference measure then for every α > 0 the resolvent R α does not have a reference measure. In [37] it is proved that if R α does not have a reference measure then there exists a compact perfect set K ⊂ E such that Since K is uncountable, there exists a sequence {v n } ⊂ B + b (K) such that |v n (x)| ≤ 1/γ for x ∈ E, n ≥ 1 and {v n } has no subsequence converging pointwise. Thanks to the properties of the operator 1 K R α , for every n ≥ 1 there exists u n ∈ B + 1 (E) such that 1 K R α u n = v n . This implies that there exists a sequence {u n } ⊂ B + 1 (E) such that {R α u n } has no subsequence converging pointwise. ✷ Remark 2.5. Let X be a Markov process and m be its excessive measure. If for some α > 0, R α carries a family P in a relatively compact set in the topology of ma.e. convergence then R β has the same property for every β > α. To see this, let us suppose that β > α and {u n } ⊂ P * . Then there exists a subsequence (n ′ ) ⊂ (n) such that {R α u n ′ } is m-a.s. convergent and its limit is finite m-a.e. By the resolvent identity, Set v = lim n ′ R α u n ′ and w = sup n ′ |u n ′ |. By (P1) and (P2), w ∈ P. Hence R α w < ∞, m-a.e., and consequently, R β w < ∞ m-a.e. for every β ≥ α. Let g be a positive Borel measurable function on E such that E R α w · g dm < ∞. Since m is an excessive measure, applying the Lebesgue dominated convergence theorem yields is m-a.e. convergent and its limit is finite m-a.e. This when combined with (2.7) implies the convergence of {R β u n ′′ }. Therefore if X is a Markov process then the compactness of a triple (X, P, m) is equivalent to saying that for some α > 0 the operator R α carries the family P in a relatively compact set in the topology of m-a.e. convergence.

Hunt processes associated with Dirichlet forms
In the rest of this section X is a Hunt process associated with a regular semi-Dirichlet form on L 2 (E; m). Let us recall that a semi-Dirichlet form on is the set of all continuous functions on E with compact suppport) such that C is dense in D[E] in the norm determined by E α 0 and in C 0 (E) in the norm of uniform convergence.
It is well known that with every semi-Dirichlet form (E, D[E]) one can associate uniquely a Hunt process X (see [ (·, ·)) is a Hilbert space.
By cap we denote the capacity on subsets of E naturally associated with (E, D[E]) (see [31, Section 2.1]). We say that some property holds quasi everywhere (q.e. for short) if it holds except for a set N ⊂ E such that cap(N ) = 0.
We say that an increasing sequence {F n } of closed subsets of E is a nest if for every compact K ⊂ E, cap(K \ F n ) → 0 as n → ∞.
We say that a Borel measure µ on E is smooth if it charges no set of zero capacity and there exists a nest {F n } such that |µ|(F n ) < ∞, n ≥ 1.
It is well known (see [31,Section 4.1]) that for every smooth measure µ there exists a unique continuous additive functional A µ of X in the Revuz duality with µ.
In the whole paper for a positive smooth measure µ and α ≥ 0 we write Observe Proof. Let {g k } be a sequence of Borel measurable functions on E such that 0 ≤ g k (x) ≤ 1, g k (x) ր 1 for x ∈ E and g k ·w ∈ L 2 (E; m) for every k ≥ 1. Write Therefore to prove the lemma it suffices to show that for every k ≥ 0, {v k n } is convergent q.e. But this follows immediately from the inequality and [31, Theorem 2.2.5]. ✷ Corollary 2.7. A triple (X, P, m) has the compactness property iff (X, P, cap) has the compactness property.
and for α ≥ 0 and u ∈ B + (E) set Proof. Let {u n } ⊂ P + (B). By the assumption there exists a set Λ ⊂ (0, +∞) such that sup Λ = +∞ and a subsequence (n ′ ) ⊂ (n) such that for every α ∈ Λ the sequence {R α u n ′ } is convergent m-a.e. and its limit is finite m-a.e. By Dynkin's formula, Therefore it suffices to show that up to a subsequence, {H α E\B (R α u n ′ )} is m-a.e. convergent and its limit is finite m-a.e. But this follows immediately from Lemma 2.6, because q.e. convergence of {R α u n ′ } implies that {e −ατ B R α u n ′ (X τ B )} is convergent P x -a.s. for m-a.e. x ∈ E, moreover we have |R α u n | ≤ R α w, m-a.e., where w = sup n |u n | ∈ P. Therefore we can apply the Lebesgue dominated convergence theorem to sequence {e −ατ B R α u n ′ (X τ B )} , because {|R α u n | > R α w} as a finely open m-negligible set is exceptional, which in turn implies that |R α u n | ≤ R α w q.e., hence Observe that the assertion of Proposition 2.8 holds true if we replace the process X B killed outside a Borel set B by the process X A killed with rate −dL t , where L t = e −At for some positive continuous additive functional A of X (for notation see [20,Theorem A.2.11]). To see this it suffices to repeat the proof of Proposition 2.8 with τ B replaced by the stopping time where Z is a random variable of exponential distribution with mean 1 independent of X and satisfying Z(θ s (ω)) = (Z(ω) − s) ∨ 0.
Let us recall that a Markov process X 0 on E 0 ∈ B(E) is called a subprocess of X if its semigroup {p 0 t , t ≥ 0} extends naturally to E subordinate to {p t , t ≥ 0}, i.e. for every f ∈ B + (E) and t ≥ 0, Corollary 2.10. Let X 0 be a subprocess of X. If (X, P, m) has the compactness property then (X 0 , P, m) has the compactness property.
Proof. By [8, page 103], X 0 is equivalent to the process X killed with rate −dL t , where L t = e −At for some PCAF A of X. Therefore the desired result follows from Remark 2.9. ✷ Proposition 2.11. Assume that E is positive and (X, P, m) has the compactness property. If {u n } ⊂ F e ∩ P and sup n≥1 E(u n , u n ) < ∞ (2.10) then there exists a subsequence (n ′ ) ⊂ (n) such that {u n ′ } is m-a.e. convergent and its limit is m-a.e. finite.
Proof. Let η ∈ D[E] be such that η > 0, m-a.e. and η ∞ < ∞. By known properties of Dirichlet forms, We can assume that u n ≥ 0, m-a.e. for every n ≥ 1, because from (P1) it follows that u + n ∈ P and it is known that u + n ∈ F e a and E(u + n , u + n ) ≤ E(u n , u n ). Under the assumption of nonnegativity of u n , u k n ≡ T k (u n ) · η ∈ F e ∩ P ∩ L 2 (E; m). By an elementar calculus, αR α u k n − u k n L 2 (E;m) ≤ α −1 E(u k n , u k n ), which when combined with (2.11) gives αR α u k n − u k n L 2 (E;m) ≤ α −1 c(k, η). (2.12) By the assumption there exists a subsequence (n ′ ) ⊂ (n) and a subset Λ ⊂ (0, ∞) such that sup Λ = +∞ and {αR α u k n ′ } is convergent in L 2 (E; m) for every α ∈ Λ. This and (2.12) imply that there exists a further subsequence (n ′′ ) ⊂ (n ′ ) such that {u k n ′′ } is convergent in L 2 (E; m). From this it follows easily that {u n ′′′ } is convergent m-a.e. for some further subsequence (n ′′′ ) ⊂ (n ′′ ). In the sequel by T we denote the set of all stopping times to given filtration F.
Definition. We say that a Borel measurable function u on E is of class (FD) if for m-a.e. x ∈ E the family {u(X τ ), τ ∈ T } is uniformly integrable under the measure P x .
By D we denote the set of all Borel measurable functions on E of class (FD).
From the above formula we easily deduce that e α u ∈ D which implies that u ∈ D. Since , we get the result.
(ii) If we assume additionally that E is positive and transient then in the same manner as in (i) we can show that F e ⊂ D.
For α ≥ 0 and ρ ∈ B(E) such that ρ > 0 let us define the space In the sequel for a given v ∈ B + (E) we write By the assumption, without loss of generality we may assume that for every k ∈ N the sequence {v k n } is m-a.e. convergent as n → ∞. Since v k n ≤ R α (g k · v) ∈ L 2 (E; m), {v k n } converges in L 2 (E; m), and hence, by (2.13), in E α . By [20, Lemma 5.1.1] this implies that there exists a subsequence (still denoted by n) such that for q.e. x ∈ E, lim n,m→∞ for q.e. x ∈ E. By the Lebesgue dominated convergence theorem, v k n − v k m α → 0 as n, m → ∞, so it is enough to show that v k n − v n α ≤ C(k) for some C(Kk such that C(k) → 0 as k → ∞. To this end, let us observe that Since v ∈ D 0 , both integrals on the right-hand side of last inequality are finite. Therefore by the Lebesgue dominated convergence theorem, C(k) → 0 as k → ∞, which shows the "if" part. Now assume that R α : D 0 → D α is order compact. Let {u n } ⊂ B + (E)be such that u n (x) ≤ 1 for x ∈ E. It is clear that 1 ∈ D 0 and {u n } ⊂ [0, 1], so by order compactness of R α : D 0 → D α it follows that there exists a subsequence (still denoted by n) such that lim n,m→∞ In particular R α u n − R α u m L 1 (E;ρ·m) → 0 as n, m → ∞ from which we conclude that (X, B 1 , m) has the compactness property. ✷

Elliptic systems with measure data on Dirichlet space
In this section we assume that (E, D[E]) is a transient regular semi-Dirichlet form on L 2 (E; m). By X we denote a Hunt process associated with (E, D[E]).
In the sequel we adopt the convention that an N -dimensional process Y or function u has some property defined for one-dimensional processes or functions (for instance Y is a MAF or CAF of X, u is of class (FD) etc.) if its each coordinate has this property.
Let F : E × R N → R N be a measurable function and µ = (µ 1 , . . . , µ N ) be a Borel measure on E such that (H1) µ i is a smooth measure such that R|µ i | < ∞ q.e., (H2) for every r ≥ 0 the mapping x → sup |y|≤r |F (x, y)| belongs to qL 1 (E; m), (H3) for every x ∈ E the mapping y → F (x, y) is continuous, (H4) there exists a non-negative function G such that RG < ∞ q.e. and for every x ∈ E and y ∈ R N , F (x, y), y ≤ G(x)|y|.
Definition. We say that a Borel measurable function f on E is quasi-integrable if for q.e. x ∈ E, By qL 1 (E; m) we denote that set of all quasi-integrable functions on E.
Remark 3.1. In the literature one can find another definition of quasi-integrability which we call here quasi-integrability in the analytic sense. According to this definition a measurable function f on E is quasi-integrable if for every ε > 0 there exists an open set U ε ⊂ E such that cap(U ε ) < ε and f | E\Uε ∈ L 1 (E \ U ε ; m). In [25] it is proved that if f is quasi-integrable in the analytic sense then it is quasi-integrable.
We say that a real process M is a local martingale additive functional (local MAF) of X if it is an additive functional of X (see [20,Section 5.1]) and M is a local martingale under P x (with respect to the filtration F) for each x ∈ E\N , where N is an exceptional set of M .
We would like to emphasize that the notion of local MAF differs from the notion of MAF locally of finite energy considered in [20,Section 5.5]. For instance, M having the last property is local AF, i.e. is additive on [0, ζ) only.
Let us consider the following system Definition. We say that a function u : E → R N is a solution of (3.1) if (a) u is quasi-continuous and u ∈ D, x ∈ E and every T > 0, x ∈ E, then u is a solution of (3.1). Indeed, by the Markov property, From this it is easily seen that u ∈ D and u satisfies (b). It is also clear that (c) is satisfied. That u is quasi-continuous it follows from [25,Lemma 4.2]. Now let us put By [20,Lemma A.3.5] there exists a càdlàg process M such that We first show that if F is monotone, i.e. F satisfies the condition then the probabilistic solution of (3.1) is unique. In the sequel for a given x ∈ R N such that x = 0 we writê Proposition 3.3. Assume that (H5) holds. Then there exists at most one solution of (3.1).
Proof. Let u 1 , u 2 be solutions of (3.1) and M 1 , M 2 be local MAFs associated with u 1 , u 2 , respectively. Put u = u 1 − u 2 and M = M 1 − M 2 . Then for every bounded τ ∈ T and q.e. x ∈ E. By the Tanaka-Meyer formula and (H5), for q.e. x ∈ E we have Let {τ k } be a fundamental sequence for the local martingale ·∧ζ 0 ŝ gn(u(X) r− ), dM r . Putting t = 0 in the above inequality with τ replaced by τ k and then taking the expectation with respect to P x we get for q.e. x ∈ E. Since u ∈ D, letting k → ∞ we conclude that |u| = 0 q.e. ✷ Theorem 3.4. Assume that (X, B 1 , m) has the compactness property and (H1)-(H4) are satisfied. Then there exists a solution of (3.1). Proof.
Step 1. We first assume that R|µ| ∞ < ∞ and there exists a strictly positive bounded Borel measurable function g such that |F (x, y)| ≤ g(x) for x ∈ E, y ∈ R N and Rg ∞ < ∞. Let ρ be a strictly positive Borel measurable function on E such that ρ(x)m(dx) < ∞ and let Φ : The mapping Φ is well defined since |R(F (·, u)) + Rµ| ≤ Rg + R|µ| ∈ L 2 (E; ρ · m). By (H3), Φ is continuous. We shall show that Φ is compact. To see this, let us consider {u n } ⊂ L 2 (E; ρ·m). By Remark 3.2, the function v n = Φ(u n ) is a probabilistic solution of the system −Av n = F (x, u n ) + µ.
Let v(x) be equal to the right-hand side of the above equality for x ∈ E such that Rg(x) + R|µ|(x) < ∞ and zero otherwise. Then by [25,Lemma 4.2], v is quasi-continuous and v ∈ D.
Since v = u, m-a.e., we have Thus v is a solution of (3.1) (see Remark 3.2).
Step 2. Now we consider the general case. Let g be a strictly positive bounded Borel measurable function on E such that Rg ∞ < ∞ (for the existence of g see [31, Corollary 1.3.6]) and let {F n } be a generalized nest such that R|µ n | ∞ , where µ n = 1 Fn · µ. Put Then F n satisfies (H2)-(H4) and R|F n | ≤ n 2 Rg, which implies that R|F n | ∞ < ∞. By Step 1, for each n ≥ 1 there exists a solution u n of the system −Au n = F n (x, u n ) + µ n .
Therefore there is a MAF M of X such that u n (X t ) =u n (X T ∧ζ ) + T ∧ζ t F n (X r , u n (X r )) dr + for q.e. x ∈ E. By the Tanaka-Meyer formula, By the above inequality and (H4), x ∈ E. Letting T → ∞ and using the fact that u n ∈ D we conclude that for q.e. x ∈ E, |µ| r if the right-hand side of (3.6) is finite and v(x) = 0 otherwise. By [25], v is quasi-continuous, v ∈ D and v is a probabilistic solution of the equation By (3.9), σ k ր ∞. Let δ k,l = τ k ∧ σ l . By (3.5), (3.8), (3.9) and the construction of δ k,l we have Hence lim t→0 + sup n |u n (x) − E x u n (X t∧δ k,l ∧ζ )| = 0 (3.10) for q.e. x ∈ E. Now we will show that (3.10) holds for x ∈ U k with E x |u n (X t∧δ k,l ∧ζ )| replaced by E x [|u n (X t∧τ k )|1 {t<τ k } ]. To this end, let us first observe that P x (τ k > 0) = 1 for x ∈ U k , because U k is finely open. We have Since lim t→0 + lim l→∞ P x ({t ≥ τ k } ∪ {t ≥ δ k,l }) = 0 for x ∈ U k and v ∈ D, it follows that for x ∈ U k the right-hand side of the above inequality tends to zero as l → +∞ and then t → 0 + . This and (3.10) imply that where {p k t , t ≥ 0} is the semigroup associated with the process X U k . By Proposition 2.8 the triple (X U k , B 1 (U k ), m) has the compactness property. Moreover, X U k is normal since U k is finely open. Therefore it follows from Theorem 2.2 and (3.8) that there exists a subsequence (n ′ ) ⊂ (n) such that {u n ′ 1 U k } is convergent q.e. By using standard argument and the fact that k U k = E q.e. one can now construct a subsequence (m) ⊂ (n) such that {u m } is convergent q.e. on E. Without loss of generality we may assume that (m) = (n). Let us write u = lim sup u n and δ k = δ k,k . By (3.5), , P x -a.s., so applying [7, Lemma 6.1] we can conclude that for every q ∈ (0, 1), Applying the Lebesgue dominated convergence theorem and using (H3), the construction of F n , {δ k } and the convergence of {u n } we conclude that for q.e. x ∈ E the first and second term on the right-hand side of the above inequality converges to zero as n, m → ∞. To show the convergence of the third term, let us observe that Since E x ζ 0 dA |µ| r < ∞ q.e., it is enough to show that lim n,m→∞ P x (∃ t>0 X t ∈ F n ∆F m ) = lim n,m→∞ P x (σ Fn∆Fm < ∞) = 0 for q.e. x ∈ E. But this follows immediately from the fact that {F n } is a nest (see [31,Theorem 3.4.8]). By what has already been proved, uniformly on compacts in probability P x for q.e. x ∈ E. Therefore letting n → ∞ in (3.5) we see that there exists a local MAF M of X such that (3.2) is satisfied for q.e.
x ∈ E. The fact that u ∈ D and u satisfies condition (b) of the definition of a probabilistic solution of (3.1) follows from (3.6) and (3.7). ✷

Systems with operators generated by right Markov processes
In the present section we assume that X is a general transient right Markov process on E satisfying hypothesis (L) of Meyer. Let us fix an excessive (σ-finite) measure m on E, i.e. a Borel measure on We say that a set B ⊂ E is m-polar if there exists an excessive function v such that A ⊂ {v = ∞} and v is finite m-a.e.
In this section we say that a property holds q.e. if it holds except for some m-polar set.
Recall that a set N ∈ B n (E) is m-inessential if it is m-polar and absorbing for X.
Definition. An F-adapted increasing [0, ∞]-valued process {A t , t ≥ 0} is called positive co-natural additive functional (PcNAF) of X if there exist a defining set Ω A ⊂ F ∞ and an m-inessential Borel set N A ⊂ E such that (a) P x (Ω A ) = 1 for x / ∈ N A and θ t Ω A ⊂ Ω A , t ≥ 0, (b) for every ω ∈ Ω A the mapping t → A t (ω) is right continuous on [0, ∞) and finite valued on [0, ζ) with A 0 (ω) = 0, Remark 4.1. It is known (see [21,Proposition 6.12]) that for any m-polar set N there exists a Borel m-inessential set B such that N ⊂ B. Therefore if some property holds q.e. then without loss of generality we may assume that it holds everywhere except for possibly an m-inessential set.
Given a PcNAF A and f ∈ B + (E) set By µ A we denote the Revuz measure associated with A, i.e. the measure defined as In this section by a nest we understand an increasing sequence {B n } of nearly Borel sets such that P m (lim n→∞ τ Bn < ζ) = 0.
Definition. A Borel measure µ on E is called smooth if it charges no m-polar sets and there exists a nest {G n } of finely open nearly Borel sets such that µ(G n ) < ∞, n ≥ 1.
It is known (see [16,Therems 6.15,6.21,6.29]) that for every PcNAF A its Revuz measure µ A is smooth and for every smooth measure µ there exists a unique PcNAF A µ such that its Revuz measure is equal to µ.
Proof. That u is finely continuous follows from [34,Theorems 36.10,49.9]. Since u is finely continuous, F = {u = ∞} is finely closed. Therefore which when combined with [6, Corollary 1.8.6] implies that F is m-polar. ✷ Proof. Sufficiency is obvious. To prove necessity, let us assume that (X, B 1 , m) has the compactness property and for v ∈ D let us choose {u n } ⊂ B + (E) such that u n ≤ v, m-a.e. for n ≥ 1. Write u k n = T k (u n ). Since v ∈ D, R α v is finite m-a.e. for every α > 0. Let g be a strictly positive Borel measurable function on E such that (R α v)g dm < ∞. By the assumption, for every k ≥ 0 there exists a subsequence (n ′ ) ⊂ (n) such that {R α u k n ′ } is convergent in L 1 (E; g · m). Therefore to show the existence of a subsequence (m) ⊂ (n) such that R α u m converges in L 1 (E; g · m) it is enough to prove that R α u k n − R α u n L 1 (E;g·m) ≤ C(k) for some independent of n constants C(k) such that C(k) → 0 as k → +∞. Observe that Since (R α v)g dm < ∞, C(k) → 0 as k → ∞. ✷ Proposition 4.4. Let X be a right Markov process and m be an excessive measure. Then (X, B 1 , m) has the compactness property iff R α : L 1 (E; m) → L 1 (E; m) is order compact for some (and hence for every) α > 0.
Proof. Necessity. Assume that (X, B 1 (E), m) has the compactness property. Let v ∈ L 1 (E; m) and {u n } ⊂ [0, v]. Write v k n = R α u k n , u k n = T k (u n ) for n, k ≥ 1. By the compactness property of (X, B 1 (E), m) there exists a subsequence (still denoted by (n)) such that v k n is m-a.e. convergent.
Sufficiency. Now assume that R α : L 1 (E; m) → L 1 (E; m) is order compact. Let {u n } ⊂ B + (E) be such that u n (x) ≤ 1 for x ∈ E, n ≥ 1. Let {g k } be a sequence of positive functions in L 1 (E; m) such that g k ր 1 and let ρ be a strictly positive function in L 1 (E; m). Write u k n = u n g k , v k n = R α u k n , v n = R α u n . By the assumption, for every k ≥ 1 there exists a subsequence (still denoted by n) such that v k n converges in L 1 (E; m). It follows that for every k ≥ 1 there exists a subsequence (still denoted by n) such that v k n converges in L 1 (E; ρ · m). This when combined with the fact that implies the existence of a subsequence (n ′ ) ⊂ (n) such that v n ′ converges in L 1 (E; ρ·m). Therefore there is a further subsequence (n ′′ ) ⊂ (n ′ ) such that v n ′′ converges m-a.e. ✷ Let us consider the following system where (A, (D(A)) is the operator defined by for some α > 0. Since m is an excessive measure, from which it follows immediately that R α f = 0, m-a.e., if f = 0, m-a.e. Therefore (4.2) makes sense. Also note that by the resolvent equation the definition of (A, D(A)) is independent of α > 0. Proof. We assume that there exists a Borel function g ∈ B + (E) such that |F (x, y)| ≤ g(x) for x ∈ E, y ∈ R N and Rg is finite m-a.e. Let ρ be a strictly positive Borel function on E such that The mapping Φ is well defined since |R(F (·, u)) + Rµ| ≤ Rg ∈ R|µ| ∈ L 2 (E; ρ · m).
In fact Φ : L 2 (E; ρ · m) → B L 2 (E;ρ·m) (0, r), where r = Rg L 2 (E;ρ·m) + R|µ| L 2 (E;ρ·m) . Φ is continuous by (H3). Let {u n } ⊂ L 2 (E; ρ · m). Define v n by putting v n (x) = RF (·, u n )(x) + Rµ(x) for x such that Rg(x) + R|µ|(x) < ∞ and v n (x) = 0 otherwise. By the assumptions and Lemma 4.2, v n is finely continuous and finite q.e. By Remark 4.1 we may assume that it is finite except for an m-inessential set. Then by the strong Markov property formula (3.4) holds. Therefore repeating the arguments following (3.4) and applying Proposition 4.3 we conclude that Φ is compact. The rest of the proof now runs as in Step 2 of the proof of Theorem 3.4 (we use Proposition 4.5 instead of Proposition 2.8). ✷

Applications
In this section we give several examples of processes having the compactness property.
Example 5.1. Let {µ t , t > 0} be a convolution semigroup on R d and let X be a Hunt process with the transition function It is known (see [5]) that if for some ε > 0 lim |x|→∞ |x| −ε |Re ψ|(x) → ∞, (5.1) whereμ t (x) = e −tψ(x) , x ∈ R d (μ t stands for the Fourier transform of µ t ) then the Lebesgue measure m on R d is a reference measure for X. Therefore if X is a Lévy process with the characteristic exponent ψ satisfying (5.1) then (X, B 1 ) has the compactness property. Consequently, our existence and uniqueness results of Section 3 (Theorem 3.4 and Proposition 3.3) apply to systems with operator A of the form ψ(∇) with ψ satisfying (5.1). A model example is ψ of the form ψ(x) = |x| α , x ∈ R d , for some α ∈ (0, 2], which corresponds to the fractional Laplacian ψ(∇) = (∇ 2 ) α/2 = ∆ α/2 . where N (e tA x, Q t ) is the Gaussian measure on H with mean e tA x and covariance operator Q t is representable by the Ornstein-Uhlenbeck process being a solution of the SDE dX(t, x) = AX(t, x) dt + Q 1/2 dW (t) (see [12] for details). By the Cameron-Martin formula (see, e.g., [11]), X satisfies Meyer's hypothesis (L), which implies that (X, B 1 ) has the compactness property. Therefore results of Section 3 apply to systems with Ornstein-Uhlenbeck operator being a generator of the semigroup {T t }. By [20], if the following Sobolev type inequality holds for some c > 0, p 0 > 2, λ 0 ≥ 0, then m is a reference measure for X associated with (E, D[E]). Consequently, (X, B 1 ) has the compactness property. It is known that (E µ , D[E µ ]) is a quasi-regular Dirichlet form and that the associated standard special process X µ is a subprocess of X (see [20,Section 6.4]). Therefore (X µ , P, m) has the compactness property. , where Z t = (τ (t), X τ (t) ), t ≥ 0 and τ is the uniform motion to the right i.e. τ (t) = τ (0) + t, P s,x (τ (0) = s) = 1. Then Z is a Markov process with reference measurem = dt ⊗ m. Indeed, we have Then (X R d \A , B 1 (R d \ A)), where A = {x ∈ R d : x i = 0 for some i = 1, . . . , d}, has the compactness property.
We close this section with an example of a right Markov process X which is not associated with a Dirichlet form, so that the results of Section 3 can not be applied to systems with operator associated with X. However, X satisfies Meyer's hypothesis (L), so that results of Section 4 are applicable. It is clear that Ker Q t = {0} and Q t > 0 for every t > 0. Let {P t , t ≥ 0} be the semigroup generated by the operator L on L 2 (R 2 ; µ), where µ = N (0, Q ∞ ). It is well known that where X is a unique solution of the SDE dX t = AX t dt + Q 1/2 dW t , X 0 = x.