Smooth measures and capacities associated with nonlocal parabolic operators

We consider a family $\{L_t,\, t\in [0,T]\}$ of closed operators generated by a family of regular (non-symmetric) Dirichlet forms $\{(B^{(t)},V),t\in[0,T]\}$ on $L^2(E;m)$. We show that a bounded (signed) measure $\mu$ on $(0,T)\times E$ is smooth, i.e. charges no set of zero parabolic capacity associated with $\frac{\partial}{\partial t}+L_t$, if and only if $\mu$ is of the form $\mu=f\cdot m_1+g_1+\partial_tg_2$ with $f\in L^1((0,T)\times E;dt\otimes m)$, $g_1\in L^2(0,T;V')$, $g_2\in L^2(0,T;V)$. We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator $\frac{\partial}{\partial t}+L_t$ and a functional from the dual $\mathcal{W}'$ of the space $\mathcal{W}=\{u\in L^2(0,T;V):\partial_t u\in L^2(0,T;V')\}$ on the right-hand side of the equation.


Introduction
In the study of parabolic problems of the form − ∂ t u − ∆ p u = f (·, u) + µ in D, u| (0,T )×∂D = 0, u(T, ·) = ϕ, (1.1) where D is a bounded open set in R d , ∆ p is the usual p-Laplacian, p > 1, and µ is a bounded measure on D 0,T := (0, T ) × D charging no set of zero parabolic pcapacity associated with ∂ ∂t − ∆ p (see below) an important role is played by the result on the decomposition of µ proved by Droniou, Porretta and Prignet [5] (see, e.g., [5,23,24]; note that in these papers more general than ∆ p operators of the form A(u) = div a(t, x, ∇u) are considered). The decomposition proved in [5] says that each such measure µ (we call it diffuse) is of the form µ = f + div(G) + ∂ t g (1. 2) for some f ∈ L 1 (D 0,T ), G ∈ (L p ′ (D 0,T )) d and g ∈ L p (0, T ; W 1,p 0 (D) ∩ L 2 (D)). Recently, in [15], the converse to this result was proved. The decomposition (1.2) is a counterpart to the decomposition of diffuse measures proved in the stationary case by Boccardo, Gallouët and Orsina [3]. The decomposition of [3] was extended to the Dirichlet forms setting in [14].
There has recently been increasing interest in semilinear evolution problems of the form − ∂u ∂t − L t u = f (·, u) + µ, u(T, ·) = ϕ, (1.3) involving operators L t associated with a (possibly nonlocal) Dirichlet form and bounded measure that do not charge the sets of zero parabolic capacity associated with ∂ t + L t (see [12,11,13] and the references therein). Motivated by possible applications to problems of the form (1.3), in the present paper we investigate the structure of such measures. We extend the results of [14] to the parabolic setting and at the same time the results of [5] with p = 2 to more general parabolic operators. As a by-product, we obtain some results on the existence of solutions to equations of the form (1.3) with µ ∈ W ′ 0 and on the structure of additive functionals associated in the Revuz sense with bounded smooth measures.
Let E be a locally compact separable metric space, E 0,T := (0, T ) × E for some T > 0, and let m be a Radon measure on E such that supp[m] = E. In the paper we consider smooth measures with respect to parabolic capacities associated with a family {L t , t ∈ [0, T ]} of closed operators generated by a family {(B (t) , V ), t ∈ [0, T ]} of regular (non-symmetric) Dirichlet forms on L 2 (E; m), with common domain V , satisfying some mild regularity assumptions. Our general Dirichlet forms setting allows us to treat both local and nonlocal operators. The results of the paper are new even for local operators. However, in our opinion, the most interesting fact is that we are able to describe the structure of smooth measures (and related additive functionals) for capacities associated with quite large class of parabolic nonlocal operators.
The model example of the family of local operators satisfying our assumptions is the family of divergence form operators  In case of problem (1.1), the natural capacity is the p-parabolic capacity cap p defined for open set U ⊂ D 0,T by cap p (U ) = inf{ ∂ t u L p ′ (0,T ;V ′ p ) + u L p (0,T ;Vp) : u ≥ 1 U dt ⊗ dx-a.e.}, where V p = W 1,p 0 ∩ L 2 (D) and V ′ p is the dual of V p (see [5,27]). To study evolution problems with the operator ∂ t + L t , Pierre [27] introduced the capacity c 2 defined for open U ⊂ E 0,T by where m 1 = dt ⊗ m. In the potential theory, Borel measures on E 0,T which do not charge sets of zero capacity c 2 and satisfy some quasi-finitness condition are called smooth measures. In particular, each bounded Borel measure "absolutely continuous" with respect to c 2 is smooth. Of course, in case of operators of the form (1.4) the classes of diffuse measures and bounded smooth measures coincide.
In our main theorem we extend (1.2) to the Dirichlet form setting. Let M 0,b (E 0,T ) denote the set of all bounded smooth measures on E 0,T . We show that each µ ∈ M 0,b (E 0,T ) admits decomposition of the form µ = f · m 1 + g 1 + ∂ t g 2 (1.7) with f ∈ L 1 (E 0,T ; m 1 ), g 1 ∈ V ′ = L 2 (0, T ; V ′ ), g 2 ∈ V = L 2 (0, T ; V ), i.e. for every bounded quasi-continuous η ∈ W 0 = {u ∈ V : ∂ t u ∈ V ′ , u(0) = 0}, we have where ·, · denotes the duality between V ′ and V. We also show the converse of this theorem. Namely, each µ ∈ M b (E 0,T ) having decomposition (1.7) is smooth. Note that the converse is an extension of the result proved by Fukushima [6] to time dependent Dirichlet forms. The proof of the fact that µ ∈ M 0,b (E 0,T ) can be written in the form (1.7) is purely analytic. Essential to the proof of the converse part are probabilistic methods.
In applications to (1.1), the analysis of additional properties of the term g appearing in the decomposition (1.2) proved to be important. For instance, crucial to the definition and the existence result of a solution u of (1.1) is the fact that g regularizes u with respect to time in the sense that u − g ∈ W ⊂ C([0, T ], L p (D)), where W = {u ∈ L p (0, T ; V p ) : ∂ t u ∈ L p ′ (0, T ; V ′ p ). This property together with some other useful properties of g has been proved in [5]. In the present paper, applying the potential theory tools, we show that g 2 from (1.7) enjoys similar properties. We also show some new results on the regularity of g 2 . We show the following useful properties.
-g 2 has an m 1 -versiong 2 which is quasi-càdlàg (i.e quasi-right-continuous with left limits; this notion generalizes the notion of qusi-continuity; see Section 9) and g 2 is a difference of c 2 -quasi-l.s.c. functions.
It is worth pointing out here that the proofs of the above results in the general setting requires us to use quite different methods then those used in [5] for the Leray-Lions type operators, which are strongly based on the regularization of the measure µ by a convolution operator.
In the proof of our main decomposition theorem, we apply some deep results from the potential theory for evolution operators proved by Pierre [25,26,27], as well as from the probabilistic potential theory for time dependent or generalized Dirichlet forms developed in the papers by Oshima [19,20,21,22], Stannat [29,30] and Trutnau [31,32]. In these papers the definitions of the capacity (and hence some quasi-notions) are different. In Section 3, which is technical but important for us, we show that all these capacities are in fact equivalent on E 0,T . This allows us to apply freely the results from the papers mentioned above.
One of the most important ingredient of the proof that each µ ∈ M b (E 0,T ) having decomposition (1.7) is smooth is an existence result for the Cauchy problem (1.3) with µ ∈ W ′ 0 and f not depending on u. If µ ∈ L 2 (0, T ; V ′ ), then the existence of a solution to (1.3) follows from the classical theory of variational inequalities (see [16]). However, if µ ∈ W ′ 0 , then the situation is more difficult. To prove the existence of a solution, a decomposition similar to (1.7), but for functionals from W ′ 0 is needed. In case of (1.1), such a decomposition and an existence result were proved in [5]. In Section 5, we prove a similar decomposition result for µ ∈ W ′ 0 , and then, in Section 6, we deal with the existence of a solution of (1.3) with µ ∈ W ′ 0 . Though for our applications we only need the existence result for liner equations, in the paper we show that there exist a solution for semiliner problem (1.3) under the assumption that ϕ ∈ L 2 (E; m) and u → f (·, u) is continuous, nonincreasing and satisfies the linear growth condition. We think that this result may be of independent interest. Finally, note here that in the proof that µ given by (1.7) is smooth we use a very recent result from the paper by Beznea and Cîmpean [1] on the characterization of quasimartingale functions.
It is well known that there is a one to one correspondence, called Revuz correspondence, between positive smooth, with respect to the Dirichlet form (B, V ), measures on E and positive continuous additive functionals of the Hunt processes associated with (B, V ) (see [7,18]). In the parabolic case the situation is more subtle. Let X denote a Hunt process associated with a generalized Dirichlet form E in the resolvent sense (see Section 2) which is generated by the operator ∂ ∂t + L t . In the paper we first show that with each smooth measure µ on E 0,T with respect the form E one can associate uniquely a positive additive functional A µ of X which is natural, i.e. has no common discontinuities with the Hunt process associated with E. This is a counterpart to the known result concerning smooth measures on R × E (see [20,22]). Then we analyse more carefully the nature of jumps of A µ in the case where µ ∈ M 0,b (E 0,T ). Roughly speaking, our main result says that for µ ∈ M 0,b (E 0,T ) the jumps of A µ are related to g 2 from decomposition (1.7). We show that g 2 has always a quasi-càdlàg modificatioñ g 2 , i.e. an m 1 -versiong 2 such that the process t →g 2 (X t ) is right-continuous with left limits, and for every predictable stopping time τ , In other words, A µ has jumps that coincide with the jumps of the processg 2 (X) in predictable stopping times. This implies thatg 2 is quasi-continuous if and only if A µ is continuous.

Preliminaries
In this paper, E is a locally compact separable metric space and m is an everywhere dense Radon measure on E, i.e. m is a positive Borel measure on E, which is finite on compact sets and strictly positive on nonempty open sets.
We set E 1 = R × E, m 1 = dt ⊗ m, and for T > 0, we set E 0,T = (0, T ) × E. We denote by B(E 1 ) (resp. B(E 0,T )) the set of all Borel measurable subsets of E 1 (resp. E 0,T ). With the customary abuse of notation, the same symbols are used to denote the sets of real Borel measurable functions on E 1 (resp. E 0,T ). B b (E 1 ) is the set of all real bounded Borel measurable functions on

Time dependent Dirichlet forms
Let H = L 2 (E; m) and (·, ·) H denote the usual inner product in H. In this paper, we assume that we are given a family {(B (t) , V ), t ∈ [0, T ]} of regular (non-symmetric) Dirichlet forms on H with common domain V ⊂ H (see [18, Chapter I] for the definitions) satisfying the following conditions.
(a) There is K ≥ 0 such that |B and ϕ ∈ V .
To shorten notation, we continue to write B for B (0) . By putting B (t) = B for t / ∈ [0, T ], we may and will assume that B (t) is defined and satisfies (c) for all t ∈ R. We denote byB (t) the symmetric part of B (t) , i.e.B (t) (ϕ, ψ) = 1 2 (B (t) (ϕ, ψ) + B (t) (ψ, ϕ)). Since V is a dense subspace of H and (B, V ) is closed, V is a real Hilbert space with respect toB 1 (·, ·), which is densely and continuously embedded in H. We equip V with the norm · V defined by ϕ 2 V = B 1 (ϕ, ϕ), ϕ ∈ V . We denote by V ′ the dual space of V , and by · V ′ the corresponding norm. For T > 0, we set and We shall identify H and its dual H ′ . Then V ⊂ H ≃ H ′ ⊂ V ′ continuously and densely, and hence V ⊂ H ≃ H ′ ⊂ V ′ continuously and densely. For given u ∈ V let ∂ t u denote the derivative in the distribution sense of the function t → u(t) ∈ V , and let It is well known that there is a continuous embedding of W into C([0, T ]; H), i.e. for every u ∈ W one can findū ∈ C([0, T ]; H) such that u(t) =ū(t) for a.e. t ∈ [0, T ] (with respect to the Lebesgue measure) and In what follows, we adopt the convention that any element of W is already in C([0, T ]; H). With this convention we may define the spaces Note that W 0 , W T are reflexive spaces as closed linear subspaces of the reflexive space W.
The linear operator ∂ t on H with domain W T will be denoted by Λ. Its adjoint, i.e. the operator −∂ t with domain W 0 , will be denoted byΛ.
We denote by E the generalized Dirichlet form associated with Λ and the family where ·, · is the duality pairing between V ′ and V, and The generalized form associated withΛ and the family {( T ], will be denoted byÊ. In the paper, we denote by (G α ) α>0 (resp.Ĝ α ) α>0 the resolvent (resp. coresolvent) associated with the form E, i.e. (G α ) α>0 , (Ĝ α ) α>0 are strongly continuous resolvents of contractions on H such that (for a construction of the resolvents see, e.g., [30,Chapter I]).
Let ψ ∈ L 1 (E 0,T ; m 1 ) ∩ B(E 0,T ) be a function such that 0 < ψ ≤ 1. We define the capacity Cap ψ associated with E as in [30, Section III.2] (see also [32]), that is for an where (G 1 ψ) U is the 1-reduced function of G 1 ψ, and then for arbitrary so Cap ψ associated with E coincides with the capacity defined as Cap ψ but for the dual formÊ. In particular, Cap ψ coincides with the capacity considered in [31]. By [30, Proposition III.2.8], Cap ψ is a Choquet capacity. Also note that by [30, Proposition III.2.10], the family of sets A ⊂ E 0,T such that Cap ψ (A) = 0 is the same for all ψ as above.
We will denote by Cap 1 the capacity associated with E 1 and defined in [21] (the definition is also given in [20] and [ Some relations between Cap ψ and Cap 1 , as well as relations between Cap ψ and some other notions of parabolic capacity considered in the literature will be studied in Section 3. We say that a set A ⊂ E 0,T (resp. A ⊂ E 1 ) is E-exceptional (resp. E 1 -exceptional) if Cap ψ (A) = 0 (resp. Cap 1 (A) = 0), and we say that a property holds E-quasieverywhere (resp. E 1 -quasi-everywhere) if the set of those x ∈ E 0,T (resp. x ∈ E 1 ) for which it does not hold is E-exceptional (resp. E 1 -exceptional).
Recall that an increasing sequence {F n } of closed subsets of E 0,T (resp. E 1 ) is called an E-nest (resp. E 1 -nest) if Cap ψ (F c n ) → 0 (resp. Cap 1 (F c n ) → 0) as n → ∞. An increasing sequence {F n } of closed subsets of E 1 is called a generalized E 1 -nest if Cap 1 (K \ F n ) → 0 for every compact K ⊂ E 0,T .
A function u : E 0,T → R (resp. u : E 1 → R) is called E-quasi-continuous (resp. E 1 -quasi-continuous) if there exists an E-nest (resp. E-nest) {F n } such that u| Fn is continuous for every n ∈ N.
It is known (see [30,Proposition IV.1.8]) that each u ∈ W 0 has an E-quasicontinuous m 1 -version. Similarly, each u ∈ W T has an E-quasi-continuous m 1 -version. We will denote them byũ.

Dirichlet forms and Markov processes
Let ∆ be adjoint to E 1 as the point at infinity. We adopt the convention that every function f on E 1 (resp. E 0,T ) is extended to E 1 ∪ {∆} (resp. E 0,T ∪ {∆}) by setting f (∆) = 0. Let By [19,Theorem 4.2] (see also [21, with life time ζ 1 and cemetery state ∆ associated with E 1 in the resolvent sense, i.e. for all α > 0 and f ∈ B b (E 1 ) ∩ L 2 (E 1 ; m 1 ) the resolvent of X 1 defined as where π : Similarly, by [30, Section IV.2], there exists a Hunt process X = (Ω, (F t ) t≥0 , (X t ) t≥0 , (P x ) x∈E 0,T ∪{∆} ) with life time ζ associated with E in the resolvent sense, i.e. for all It is clear that Ω ⊂ Ω 1 and X = X 1 on Ω.
It is a Hunt process associated with the dual formÊ, For all α > 0 and whereÊ x denotes the expectation with respect toP x , is an E-quasi-continuous m 1version of the coresolventĜ α f . A modification of the argument used to prove (2.12) and (2.13) shows thatP The relation (2.13) (resp. (2.15)) implies that the operator G α (resp.Ĝ α ) is well defined for α = 0, and that Therefore Rf ∈ G α (H), and by (2.4), In other words, . A similar argument applies toR 0 andĜ 0 .

Smooth measures
The set of all E-smooth (resp. E 1 -smooth) measures will be denoted by Let µ be a positive Borel measure on E 0,T such that µ does not charge sets of zero capacity. We call it a measure of finite energy integral if there is C ≥ 0 such that we call it a measure of finite co-energy integral. In both cases,η denotes an E-quasicontinuous m 1 -version of η. The set of all positive smooth measures on E 0,T of finite energy (resp. co-energy) integral will be denoted by S 0 (E 0,T ) (resp.Ŝ 0 (E 0,T )).
Lemma 2.1. Let µ be a positive Borel measure on E 0,T . Then Proof. We provide the proof of (i). The proof of (ii) is analogous. If (2.18) is satisfied, then of course µ ∈ S 0 (E 0,T ). Suppose that µ is of finite energy integral. Then, by [32,Lemma 4.2] applied to the dual formÊ, for every positive η ∈ W 0 there exists u ∈ such that Let µ ∈ S 0 (E 0,T ). The element u ∈ V defined by (2.18) is uniquely determined. We will denote it by U 1 µ. Similarly, for µ ∈Ŝ 0 (E 0,T ), the element u ∈ V defined by (2.19) is uniquely determined. We will denote it byÛ 1 µ. We also set It is clear that for any α ≥ 0, (2.21)

Parabolic capacity
Our basic capacity associated with E is Cap ψ . Exceptional sets with respect to Cap ψ have nice probabilistic interpretation given by (2.9). However, in the literature devoted to partial differential equations, usually some other notions of capacity are used. In this section, we recall some of them and prove that they are all equivalent to Cap ψ . These results will be needed in the next sections.  .7), Hence, by (2.12), x ∈ E 0,T . From this and (2.6) and (2.12) it follows that Similarly, from (2.14) and (2.15) it follows that where E 1 η·m 1 denotes the expectation with respect to the measure P 1 η·m 1 defined as We now recall the capacity considered in Pierre [27] (see also [25,26]). Let U be a relatively compact open subset of E 0,T . By [30, Proposition III.1.6], for every n ≥ 1 there exists a unique solution e n U ∈ W T of the following problem  [27]), there exists a measure µ U ∈ S 0 (E 0,T ) such that e U = U 1 µ U . By [26, Proposition II.1], supp[µ] ⊂ U . In [27], the capacity of U is defined by As usual, for an arbitrary A ⊂ E 0,T , we define By [27,Proposition 2], c 0 is a Choquet capacity. Proof. Let U be a relatively compact open subset of E 0,T . Let Ĝ 1 ψ be a quasicontinuous modification ofĜ 1 ψ. By the definitions of Cap ψ and U 1 µ U , and the fact thatĜ 1 is Markovian, we have Hence Cap ψ ≤ c 0 . Let K be a compact subset of E 0,T such that Cap ψ (K) = 0. Then there exists a nonincreasing sequence U n of open relatively compact subsets of E 0,T such that Cap ψ (U n ) ց 0 and K ⊂ U n . Let η ∈ W 0 ∩ C c (E 0,T ) be such that η ≥ 1 U 1 (see [26,Lemma II.3]). Then, by (3.4), Therefore there exists a subsequence (still denoted by n) such that {e Un } is weakly convergent in V. Since e Un ց 0 m 1 -a.e., in fact e Un → 0 weakly in V.
Hence c 0 (K)=0. Since Cap ψ and c 0 are Choquet capacities, this implies the desired result. Write where C c (E 0,T ) is the set of all real continuous functions on E 0,T having compact support. For a compact set K ⊂ E 0,T , we set Next, for an open U ⊂ E 0,T , we set CAP(U ) = sup{CAP(K) : K ⊂ U, K-compact}, and finally, for arbitrary A ⊂ E 0,T , we set CAP(A) = inf{CAP(U ) : A ⊂ U, U open}. We will also consider the capacity c 2 (see [27]) defined as for an open set U ⊂ E 0,T , and then by c 2 (A) = inf{CAP 1 (U ) : A ⊂ U, U open} for an arbitrary set A ⊂ E 0,T . By [27,Proposition 2], c 2 is a Choquet capacity. Proof. By virtue of Lemma 3.3, it suffices to prove that CAP is equivalent to c 2 . Let K be a compact subset of E 0,T and let ε > 0, a > 1. Choose η ε ∈ C such that η ε W ≤ CAP(K) + ε and η ε ≥ 1 K . Since η ε is continuous, there exists an open set V a such that aη ε ≥ 1 Va and K ⊂ V a . We have Letting ε ↓ 0 and then a ↓ 1 we see that c 2 (K) ≤ CAP(K) for every compact K ⊂ E 0,T . From this and the fact that c 2 is a Choquet capacity we conclude that c 2 ≤ CAP. Now suppose that c 2 (K) = 0 for some compact set K ⊂ E 0,T . Let U n = {(s, x) ∈ E 0,T : dist((s, x), K) < n −1 }. Since E 0,T is locally compact and K is compact, we may assume that U n are relatively compact. Since c 2 is a Choquet capacity, b n := c 2 (U n ) ց 0. Fix ε > 0 and choose η ε n ∈ W such that η ε n ≥ 1 Un m 1 -a.e. and η ε n W ≤ b n + ε. By putting η ε n (t) = η ε n (T ) for t ≥ T and η ε n (t) = η ε n (0) for t ≤ 0, we may assume that η ε n ∈ W 1 . Write where j is a smooth positive function with support in [0, 1] such that R j(t) dt = 1. It is clear that By regularity of the forms B (t) , for every a > 0 there exists η ε,a n ∈ C such that (see [21,Lemma 1.1]). Let ξ n ∈ C be a positive function such that ξ n = 1 on K and supp[ξ n ] ⊂ U n−1 . The existence of ξ n follows from [21, Lemma 1.1]. We put η ε,a n = η ε,a n + (ξ n − η ε,a n ) + . Observe thatη ε,a n ∈ C andη ε,a n ≥ 1 K . Let a l ↓ 0. By (3.7), up to a subsequence, (ξ n − η ε,a l n ) + → (ξ n − J n (η ε n )) + weakly in W 2 as l → ∞. Hence, up to a subsequence, By (3.6), (ξ n − J n (η ε n )) + = 0, so letting k → ∞ in the above inequality and using (3.7) and (3.8) we get CAP(K) ≤ J n (η ε n ) W ≤ η ε n W ≤ b n + ε. Hence CAP(K) = 0. Since c 2 is a Choquet capacity, this implies that CAP(A) = 0 if c 2 (A) = 0.

Smooth measures and associated additive functionals
The results of this section will be needed in Sections 8 and 9. Recall that a positive additive functional (AF in abbreviation) A 1 of X 1 is said to be in the Revuz correspondence with a positive µ ∈ S 1 (E 1 ) if where E 1 m 1 denotes the expectation with respect to the measure P 1 m 1 defined as P 1 m 1 (·) = E 1 P x (·) m 1 (dx). Similarly, we say that a positive AF A of X is in the Revuz correspondence with a positive µ ∈ S(E 0,T ) if where E m 1 denotes the expectation with respect to P m 1 (·) = E 0,T P x (·) m 1 (dx).
A similar result holds for A 1 .
It is known (see [11,Section 2]) that for every µ ∈ S 1 (E 1 ) there exists a unique positive natural AF A 1 of X 1 (i.e. a positive AF of X 1 such that A 1 and X 1 have no common discontinuities) such that A 1 is in the Revuz correspondence with µ. In what follows we denote it by A 1,µ . In fact, A 1,µ is a predictable process (see [9,Theorem 5.3]). In the proposition below, we show a similar result for positive smooth measures on E 0,T .
whereμ denotes the extension of µ to E 1 such that µ(E 1 \ E 0,T ) = 0 and A 1,μ is the positive natural AF of X 1 in the Revuz correspondence withμ.
Proof. We first assume that µ ∈ S 0 (E 0,T ). Since µ is zero outside E 0,T , by Lemma 3.1 it is clear thatμ ∈ S 0 (E 1 ). By the remark preceding Proposition 4.1, there exists a positive natural AF A 1,μ of X 1 in the Revuz correspondence withμ. Let R 1μ (x) = E 1 x A 1,μ ζ . By (4.1), for x ∈ E 0,T . Hence, by [9, Theorem 9.3] and (2.11) (for the dual process), for every Thus R 1μ = U µ m 1 -a.e. on E 0,T . By (2.12) and (4.5), for all τ ≤ ζ and x ∈ E 0,T , Hence R 1μ is a natural potential with respect to X. By [2,Theorem IV.4.22], there exists a unique positive natural AF A of X such that From this and the fact that R 1μ = U µ m 1 -a.e. on E 0,T we get . Thus µ is a Revuz measure of A. Let A µ := A. From (4.6) we easily get (4.4).
To prove the existence of A µ in the general case, we choose an E-nest {F n } of compact subsets of E 0,T such that µ n = 1 Fn · µ ∈ S 0 (E 0,T ). Such a nest exists by [32,Theorem 4.7]. By what has already been proved, for all α ≥ 0 and for q.e.
x ∈ E 0,T . Since the additive functional in the Revuz correspondence with a smooth measure is uniquely determined, we have dA µn ≤ dA µm for n ≤ m and A 1,μn x ∈ E 0,T , Therefore, letting n → ∞ in (4.7), yields (4.4).

Remark 4.2.
Since the capacities Cap ψ associated with E and the dual formÊ coincide (see Section 2.1), the set of smooth measures S(E 0,T ) associated with E coincides with the set of smooth measures on E 0,T associated with the dual formÊ. Therefore from Proposition 4.1 applied toÊ andX it follows that for a positive µ ∈ S(E 0,T ) there exists a unique predictable AF ofX in the Revuz correspondence with µ. We will denote it byÂ µ .
Proof. First we assume that µ ∈Ŝ 0 (E 0,T ) and η ∈ B + (E 0,T ) ∩ H. Then, by (2.21), the fact that R α η is an m 1 -version of G α η and (2.4), From this and (4.9) it follows that the first equality in (4.11) is satisfied. To show it in the general case, we set η n = ng 1+ng (η ∧ n) for some strictly positive g ∈ H and choose an E-nest {F n } of compact sets such that µ n = 1 Fn · µ ∈Ŝ 0 (E 0,T ). Such a nest exists by [32,Theorem 4.7]. By what has already been proved, E 0,T R α η n dµ n = E 0,T η nRα µ n dm 1 , n ≥ 1.
Letting n → ∞ and using [30, Remark 3.6] yields the first equality in (4.11). The proof of the second equality is similar, so we omit it.
Given a positive smooth measure µ on E 0,T , we denote by R α • µ the Borel measure on E 0,T defined by the formula and byR α • µ we denote the Borel measure on E 0,T defined by By Lemma 4.3, the measures R α • µ,R α • µ are absolutely continuous with respect to m 1 and

Decomposition of smooth measures
In what follows, we denote by ·, · the duality pairing between W ′ 0 and W 0 . In this section, we prove that each µ ∈ M b (E 0,T ) admits decomposition (1.7). The converse statement will be proved in Section 8. We start with a decomposition of elements of the space W ′ 0 . Proposition 5.1. Let g ∈ W ′ 0 . Then there exist g 1 ∈ V ′ , g 2 ∈ V such that Proof. The proof is a slight modification of the proof of [5, Lemma 2.24]. We give it for completeness.
Since T (u) F = (u, ∂ t u) F = u W , T is an isometry. We equip T (W 0 ) with the norm of F and define i.e. Φ g (T (u)) = g, u for u ∈ W 0 . Then Φ g is a continuous linear functional on T (W 0 ). By the Hahn-Banach theorem, Φ g extends to a continuous linear functional on F (still denoted by Φ g ) such that for some g 1 ∈ V ′ , g 2 ∈ V. One can check that g 1 V ′ ≤ Φ g F ′ and g 2 V ≤ Φ g F ′ , which when combined with (5.2) yields (5.1). Hence for v ∈ W 0 , and the lemma is proved.
Then there exist f ∈ L 1 (E 0,T ; m 1 ) and g ∈ W ′ 0 such that µ = f · m 1 + g, (5.5) i.e. for every bounded v ∈ W 0 we have Proof. Let µ = µ + − µ − be the Hahn decomposition of µ. Then µ + , µ − ∈ M 0,b (E 0,T ). Therefore, without loss of generality, we can assume that µ is positive. By [32,Theorem 4.7], there exists a nest {F n } such that 1 Fn · |µ| ∈ S 0 (E 0,T ) for every n ∈ N. Set µ n = 1 F n+1 \Fn · µ. Then |µ n | ∈ S 0 (E 0,T ) and Hence Let j : V → W ′ 0 be the mapping defined as From (5.8) and reflexivity of the space W 0 it follows that the sequence {j(U 1 µ α n )} converges to j(U 1 µ n ) weakly in W ′ 0 as α → ∞. By the Banach-Saks theorem, there is a sequence {α l } such that α l → ∞ as l → ∞ and Therefore one can find a subsequence {k n } such that for every n ∈ N, f kn (µ n ). (5.10) By (4.10) and Lemma 4.3, for every i ∈ N we have Since αR α is a Markov operator, it follows that Letting N → ∞ and using (5.7), (5.10) we obtain (5.6), which completes the proof of the proposition.
Remark 5.4. The decomposition (5.5) also holds for arbitrary µ ∈ S(E 0,T ). In this case, in general, f / ∈ L 1 (E 0,T ; m 1 ) but f is quasi-integrable, i.e. there exists a nest {F n } of compact subsets of E 0,T such that 1 Fn · f ∈ L 1 (E 0,T ; m 1 ), n ≥ 1. The proof of the decomposition (5.5) in case µ ∈ S(E 0,T ) runs as the proof of Proposition 5.3 with the only difference that in (5.11) we replace V i by F i defined in the proof of Proposition 5.3. We then get Combining Proposition 5.1 with Proposition 5.3 we get the following theorem.
Note that f, g 1 , g 2 in Theorem 5.5 are not uniquely determined. In what follows, any triplet of functions having the same properties as the triplet (f, g 1 , g 2 ) appearing in Theorem 5.5 will be called a decomposition of µ.
Remark 5.6. From (5.10) it follows that if µ ∈ M 0,b (E 0,T ) is positive, then the L 1 part f of the decomposition of µ can be chosen to be positive.
6 Semilinear equations with right-hand side in W ′ 0 In this section, using the decomposition of elements of W ′ 0 given in Proposition 5.1, we will show an existence and uniqueness result for the following Cauchy problem In (6.1), g ∈ W ′ 0 , f : E 0,T × R → R and ϕ ∈ L 2 (E; m). We will need the following assumption: there exist λ ∈ R, M ≥ 0 and a positive ̺ ∈ H such for all x ∈ E 0,T and y, y ′ ∈ R, Definition. We say that u ∈ V is a solution of (6.1) if In what follows, for a function η on E 0,T and ε > 0, t ∈ [0, T ], we set η t = η1 [t,T ]×E and It is clear that for every η ∈ V , η t ε → η t strongly in V as ε ց 0.
Theorem 6.1. Assume that f is a measurable function such that (6.2) is satisfied. Then there exists a unique solution of (6.1).
In the proposition below, we provide a stochastic representation of the solution of (6.1) with f = 0 and g ∈ W ′ 0 ∩ M 0,b (E 0,T ). It will be needed in Section 9.

Further properties of g 2
We know from Theorem 5.5 that each µ ∈ M 0,b (E 0,T ) admits decomposition of the form µ = f · m 1 + g 1 + ∂ t g 2 (7.1) with f ∈ L 1 (E 0,T ; m 1 ), g 1 ∈ V ′ and g 2 ∈ V. In this section, we prove some further regularity results for g 2 .
x ∈ E 0,T we have The following proposition is a counterpart to [24, Theorem 1.1].
It is known (see [24,Example 3.1]) that not every bounded smooth measure can be written in the form (7.1) with bounded g 2 . We are going to show that g 2 is always quasi-bounded with respect to the capacity c 2 . To this end, we first recall the definition of a parabolic potential (see, e.g., [25]).

Definition.
A measurable function u ∈ V ∩ L ∞ (0, T ; H) is called a parabolic potential if for every positive v ∈ W 0 , The set of parabolic potentials will be denoted by P 2 . By [26, Proposition I.1], for every u ∈ P 2 there exists a unique Borel measure µ u such that for every η ∈ W 0 ∩ C b (E 0,T ), Let µ ∈ S 0 be positive. Then, by [12,Proposition 3.1], µ ∈ P 2 and µ Rµ = µ. Following [27], for u ∈ P 2 we set For an open set open U ⊂ E 0,T , we define By [27,Theorem 1], there exists α > 0 such that For k ≥ 0, we set T k (s) = max{−k, min{k, s}}, s ∈ R.
It is clear that T k (Rµ) satisfies (7.6), which when combined with (7.8) implies that T k (Rµ) ∈ P 2 and the desired inequality holds true.
where A ν is a natural AF of X in the Revuz correspondence with ν. By Proposition 6.2, v is a solution to (6.1) with f = 0, ϕ = 0 and g replaced by −ν. Observe that w = v − g 2 is a solution to the Cauchy problem where χ is an element of V ′ such that χ, η = B(g 2 , η), η ∈ V. Since g 1 + χ ∈ V ′ , we have w ∈ W T . Therefore (7.9) for some w ∈ W T . By [26,Theorem III.1], w has an m 1 -versionw which is c 0 -quasicontinuous, so c 0 -quasi-bounded. Therefore we only need to show that v + (x) : We will show this for v + . The proof for v − is analogous. Let v k + = T k (v + ). By Lemma 7.5, v k + ∈ P 2 , so by [26,Corollary III.3], there exists an m 1 -versionṽ k + of v k + which is c 0 -quasi-l.s.c. Henceṽ + := sup k≥1ṽ k + is also c 0 -quasi-l.s.c., and of course, it is an m 1 -version of v + . By Lemma 7.5 and (7.7), Let {F 1 n } be a c 0 -nest such that (ṽ + ) |Fn is l.s.c. for each n ≥ 1. Set F n = F 1 n ∩{ṽ + ≤ n}. Then F n is closed. Moreover, as n → ∞, which proves thatṽ + is c 0 -quasi-bounded.

0
We already know that each bounded smooth measures admit decomposition of the form (1.7). The problem whether a bounded measure admitting decomposition (1.7) is smooth is more delicate. In this section, we give positive answer to this question. In the proof of our result, we will make use of Fukushima's decomposition, which we now recall. For an AF A of X its energy is defined by t =ũ(X t ) −ũ(X 0 ), t ≥ 0. We have On the other hand, by (8.1), e(A [u] ) = e(M [u] ), which proves the proposition.
Lemma 8.2. Assume that A, A n , n ≥ 1, are additive functionals of X.
(ii) If E η·m 1 sup t≥0 |A n t − A m t | p → 0 as n, m → ∞ for some p ≥ 0 and strictly positive η ∈ B(E 0,T ), then there exists a subsequence (still denoted by n) such that E x sup t≥0 |A n t − A m t | p → 0 as n, m → ∞ for q.e. x ∈ E 0,T .
Proof. We will prove (ii). The proof of (i) is analogous. We may assume that and E 0,T η dm 1 = 1. Letm 1 = η · m 1 and It is clear that N is a nearly Borel set. We will show that Cap ψ (N ) = 0. Since Cap ψ is a Choquet capacity, we may and will assume that N is compact. We have By the strong Markov property, Since A is additive, Hence But by (8.2) and the Borel-Cantelli lemma, Pm 1 (B ′ ) = 0, which implies that P x (B ′ ) = 0 for m 1 -a.e.
Since the measure f · m 1 is smooth, without loss of generality we may assume that f = 0. Let Φ be a functional on W 0 defined by the right-hand side of (5.12) (with f = 0). It is clear that Φ ∈ W ′ 0 . Let u ∈ V be a solution to (6.1) with ϕ = 0, f = 0 and g = Φ.
By [30,Proposition I.3.7], u n → u strongly in V. By Fukushima's decomposition, there exists an m 1 -inessential set N such that for every x ∈ E 0,T \ N , Hence Step 3. We will show that there exists a natural AF A of X of finite variation and a martingale AF M of X such that By the definition ofũ, for every x ∈ E 0,T , u n (X t ) →ũ(X t ), t ≥ 0, P x -a.s.

Decomposition of measures and additive functionals
In this section, we study the structure of the additive functional A µ in the Revuz correspondence with µ ∈ M 0,b (E 0,T ). Specifically, we want to get deeper understanding of the nature of jumps of A µ and their relation to the decomposition (f, g 1 , g 2 ) of µ given in Theorem 5.5. Before proceeding, remarks concerning two special cases of µ are in order.
We see that in both special cases considered above the additive functionals corresponding to µ are continuous. This suggests that the jumps of A µ stem from the component g 2 of the decomposition of µ. In what follows we will show that this is indeed true and we will make this statement more precise.
Following [11] we adopt the following definition.
Definition. We say that a Borel measurable function u on E 0,T is quasi-càdlàg if for q.e. x ∈ E 0,T the process t → u(X t ) is càdlàg on [0, T − τ (0)) under the measure P x .
Since A µ is predictable, by [4, Chapter IV, Theorem 88B], there is a sequence {τ n } of predictable stopping times exhausting the jumps of A µ , i.e. In what follows, we denote by A µ,c the continuous part of A µ and by A µ,d the pure jump part of A µ . For a given càdlàg process Y , we write ∆Y t = Y t − Y t− , where Y t− = lim sրt Y s . Theorem 9.1. Let µ ∈ M 0,b (E 0,T ) and (f, g 1 , g 2 ) be a decomposition of µ from Theorem 5.5. Then (i) g 2 has a quasi-càdlàg m 1 -versiong 2 .