On uniqueness and structure of renormalized solutions to integro-differential equations with general measure data

We propose a new definition of renormalized solution to linear equation with self-adjoint operator generating a Markov semigroup and bounded Borel measure on the right-hand side. We give a uniqueness result and study the structure of solutions to truncated equations.


Introduction
In the paper, E is a locally compact separable metric space and m is a Radon measure on E with full support. Let (A, D(A)) be a non-positive definite self-adjoint operator on L 2 (E; m) associated with some Dirichlet form (E, D(E)) on L 2 (E; m). The main goal of the present paper is to give a new definition of renormalized solution to the linear equation − Au = µ (1.1) with general (possibly nonsmooth in the Dirichlet forms theory sense) bounded Borel measure µ on E. It is known that such a measure admits unique decomposition into the absolutely continuous, with respect to the capacity Cap generated by (E, D(E)), part µ d (so-called diffuse part or smooth part of µ) and the orthogonal, with respect to Cap, part µ c (so-called concentrated part). The problem of right definition to (1.1) is rather subtle if we require that the solution u be unique.
In the paper, we assume that the resolvent (R α ) α>0 generated by A is Fellerian, i.e. R α (C b (E)) ⊂ C b (E) for some (and hence for all) α > 0, and there exists a Green function G for A (see Section 2.2).
Our new definition reads as follows: u ∈ B(E) is a renormalized solution to (1.1) if (i) T k (u) := (u ∧ k) ∨ (−k) ∈ D e (E), k ≥ 0, where D e (E) is the extended Dirichlet space, i.e. an extension of D(E) such that D e (E) with inner product E is a Hilbert space.
(ii) There exists a family of bounded smooth measures (ν k ) k≥0 on E such that E(T k (u), η) = µ d , η + ν k , η , η ∈ D e (E) ∩ B b (E), k ≥ 0, (iii) ν k → µ c in the narrow topology, i.e. for every η ∈ C b (E), A similar definition of a solution to (1.1), also guaranteeing uniqueness, was introduced recently in my joint paper with Rozkosz [10]. In that paper by a solution we mean u ∈ B(E) satisfying (i) and (ii), and the following condition (iii') lim k→∞ E G(x, y) ν k (dy) = E G(x, y) µ c (dy) for m-a.e. x ∈ E.
Condition (iii) is much simpler than (iii') because it does not involves the notion of the Green function. One of the main results of the present paper says that (i)-(iii) still ensure uniqueness for solutions to (1.1). In the second part of the paper we show interesting properties of the family (ν k ) k≥0 : a structure theorem, so-called reconstruction property and the narrow convergence of variations.
The above definition (i)-(iii) is a counterpart to the definition introduced by Dal Maso, Murat, Orsina and Prignet [3] for equations with local nonlinear operators of Leray-Lions type of the form A(u) = div(a(·, ∇u)).
For such operators, E appearing in (i), (ii) is replaced by E(u, v) := E a(·, ∇u)∇v dm, and the domain of E is the natural energy space in which E(u, u) is finite. As a matter of fact, this modified definition (i)-(iii) is one of the four equivalent definitions of renormalized solutions considered in [3].
The concept of renormalized solutions was a crucial step in the development of the theory of elliptic and parabolic equations with (nonlinear) local operators and measure data since it gives partial uniqueness results. The complete uniqueness result in nonlinear case is still an open problem.
It is worth noting here that among the defintions considered in [3] the defintion presented above have some remarkable feature. Namely, in condition (ii) the term E(T k (u), η) is well defined since both T k (u) and η are in the domain of E. In the other definitions considered in [3], a different variational formulas (counterparts to (ii)) are considered. In these formulas the term E(u, η) always appears which of course requires an extension of the form (E, D(E)) in such a way that E(u, η) makes sense for rich enough class of test functions η. All the used extensions of E in [3] are based on the property which is true only for the forms associated with local operators. For that reason only definition of type (i)-(iii) can be directly adopted to the nonlocal case. In case E is nonlocal of the form for some symmetric positive measure J on E × E \ d, Alibaud, Andreianov and Bendahmane [1] proposed the following extension of E: for bounded η and u such that A careful analysis shows that thanks to (1.5) and regularity of h both integrals in (1.4) are well defined. However, the crucial assumption that h has compact support makes this approach applicable only to equations with smooth (diffuse) measure data. In [1], imitating one of the definition considered in [3], the authors introduced the following definition of a solution to (1.1) with A generated by (1.3), E = R d and smooth measure data (in fact for µ ∈ L 1 (R d ) but it naturally extends to smooth measure data): a measurable function u satisfying (1.5) is a renormalized solution to (1.1) if We extend this definition to general forms E considered here (and smooth measure data) and show that if u is a renormalized solution to (1.1) in the sense of definition (i)-(iii), then u is a renormalized solution to (1.1) is the sense of (1.6) and (1.7). In order to get existence and uniqueness result for solutions to (1.1) with general measure data, we propose definition (i)-(iii) which seems to be the more suitable formulation of the definition of renormalized solution to (1.1) since it is applicable not only to general measure data but also to wide class of operators associated with local and non-local Dirichlet forms.
We prove that for bounded Borel measure µ there exists a unique renormalized solution to (1.1). Moreover, if u is renormalized solution to (1.1) then and even stronger convergence of {ν k } holds. Namely From this it follows in particular that u is a renormalized solution to (1.1) in the sense of (i)-(iii) if and only if u is a duality solution to (1.1) in the sense of Stampacchia. The notion of duality solutions for linear equations with uniformly elliptic divergence form operators and general measure data was introduced by Stampacchia in [18]. His approach was adapted to fractional Laplacian in [5,15]. The general formulation for operators A generated by Markov semigroups was introduced in [6] (see also [7] for the case of smooth measure data).
In the second part of the paper, we give a complete characterization of the family (ν k ) k≥0 . Recall that each regular symmetric Dirichlet form (E, D(E)) admits the following (unique) Beurling-Deny decomposition Here E (c) is the local part of E, J is a symmetric positive Radon measure outside the diagonal d of E × E and κ is a smooth Radon measure on E, called the killing measure. We show that and {l k (u), k ∈ Z} characterized as follows where µ c u is a positive smooth Radon measure measure on E given by From (1.10) it follows in particular that if E is local (i.e. J ≡ 0), then for all sequences {b n }, {c n } of positive numbers such that b n < c n , n ≥ 1 and b n , c n → ∞ as n → ∞ (the so-called reconstruction property). Observe also that j k (u) may be concentrated on the whole E. Hence, contrary to the local case, the measures ν k need not be concentrated on the set {|u| = k}.
In the present paper we focus our attention on linear equation (1.1). However, our results also apply to semilinear equations of the form with f being a measurable function on E × R. By a renormalized solution to (1.12) we mean a measurable function u on E such that f (·, u) ∈ L 1 (E; m) and (i)-(iii) hold when we replace µ d by f (·, u) + µ d in condition (ii) (since (f (·, u) + µ) d = f (·, u) + µ d ).
From our results it follows that if f is nonincreasing with respect to the second variable, then there exists at most one renormalized solution to (1.12).

Notation and standing assumptions
In the paper, E is a locally compact separable metric space and ∂ is a one-point compactification of E. If E is already compact, the ∂ is an isolated point. We adopt the convention that each function f on E is extended to E ∪ {∂} by setting f (∂) = 0.

Dirichlet forms and potential theory
In the whole paper, (E, D(E)) is a symmetric Dirichlet form on We also assume that it is transient, that is there exists a strictly positive function g on E such that E |u|g dm ≤ c E(u, u), u ∈ D(E). We denote by (T t ) t≥0 the semigroup of contractions on L 2 (E; m) generated by (A, D(A)) and by Cap the capacity on E defined as follows: for and open U ⊂ E, and for an arbitrary B ⊂ E, We say that a property P holds quasi everywhere (q.e. in abbreviation) if it holds outside a set N with Cap(N ) = 0. We say that a measurable function u on E is quasicontinuous if for every ε > 0 there exists a closed set F ε ⊂ E such that Cap(E \ F ε ) < ε and u |Fε is continuous. By [4, Theorem 2.1.7], every function u ∈ D e (E) has an mversionũ which is quasi-continuous.
For a Borel measure on E, |µ| stands for its total variation. We say that a Borel measure µ on E is smooth if |µ|(B) = 0 for every Borel set B ⊂ E such that Cap(B) = 0, and there exists a strictly positive quasi-continuous function η on E such that |µ|, η < ∞. The set of all positive smooth measures on E will be denoted by S. We denote by M 0,b (E) the set of bounded smooth measures, and by M + 0,b (E) the set of positive bounded smooth measures. Each µ ∈ M(E) admits unique decomposition of the form (1.2), where µ d is a smooth measure and µ c is concentrated on the set B ⊂ E such that Cap(B) = 0.

Probabilistic potential theory
By [4,Section 7], there exists a Hunt process In the paper, we assume that X satisfies absolute continuity condition, i.e. there exists a positive Borel function p on R + ×E ×E such that for all x ∈ E, t > 0 and f ∈ B + (E) , We denote by (P t ) t≥0 (resp. (R α ) α>0 ) the semigroup (resp. resolvent) associated with X. Recall that for all t, α ≥ 0 and f ∈ B + (E), Here G is the Green function defined by In the sequel, we write that some property P holds q.a.s. (resp. a.s.) if it holds outside a set B ∈ F ∞ such that P x (B) = 0 for q.e. x ∈ E (resp. for every x ∈ E). Recall The set of all excessive functions will be denoted by Exc. It is well known that for any η, ψ ∈ Exc the mapping is nonincreasing. Moreover, if ψ = Rµ for some positive Borel measure µ and ψ < ∞ m-a.e., then Let A + c denote the set of positive continuous additive functionals of X (see [4, Section 5.1]). It is well known that there exists an isomorphism (the so-called Revuz duality) defined as follows: for every A ∈ A + c and every continuous positive η on E, dA r is nonincreasing for any η ∈ Exc and A ∈ A + c , and the above equality also holds for η ∈ Exc. Moreover, if η is excessive or is a positive continuous function η on E, then In the sequel, for given µ ∈ S we set A µ : For a given càdlàg special semimartingale Y and k ∈ R, we denote by L k (Y ) the local time of Y at k (see [17, page 212]). We also put where ϕ ′ is the left derivative of ϕ and µ = (ϕ ′ ) ′ with second derivative taken in distributional sense. Since Y is a special semimartingale, there exists process p J k (Y ) which is the dual predictable projection of J k (Y ). In case Y = u(X) q.a.s. it is easy to observe that L k (u(X)) and J k (u(X)) are positive additive functionals. By the definition of local times, L k (u(X)) is continuous, and since X is a Hunt process, p J k (u(X)) is continuous too (and it is still a positive additive functional, see [4,Theorem A.3.16]). We set 3 Integral, duality, probabilistic and very weak solutions By [6, Proposition 3.2], Cap(N ) = 0, and by [6,Theorem 3.7],ũ defined as is a quasi-continuous m-version of u. Therefore we may assume that any integral solution u to (1.1) is quasi-continuous and (3.1) is satisfied for q.e.
x ∈ E.
The next definition introduced in [6] is a generalization, to the class of operators considered in the present paper, of Stampacchia's definition introduced in [18] in case A is a uniformly elliptic diffusion operator in divergence form. In case A is a fractional Laplacian, duality solutions were considered in [5] (on R d ) and in [15] (on bounded domains in R d ).
It is worth noting here that in case µ is a smooth measure it is possible to define duality solutions for general operators corresponding to transient regular Dirichlet forms, i.e. without the additional assumption that there exists the Green function for A (see [7]).
The following definition of probabilistic solution to (1.1) was introduced in [6]. To formulate it, we first recall that M is called a local martingale additive functional (local MAF for short) of X if M is an additive functional of X and M is a local martingale under the measure P x for q.e. x ∈ E (see [6] for details).
x ∈ E and τ k ր ζ q.a.s. we have Any sequence {τ k } of F-stopping times such that E x sup t≤τ k |u(X t )| < ∞ for q.e. x ∈ E and τ k ր ζ q.a.s. is called the reducing sequence for u.  (i) If u is a probabilistic solution to (1.1), then u is an integral solution to (1.1).
From now on, we always consider quasi-continuous versions of solutions to (1.1) (no matter which one of the definition we consider).

3)
and for every k > 0, for some local MAF M of X and reducing sequence {τ n } for u(X). By the Tanaka-Meyer formula, Therefore taking the expectation of both sides of the above equation and applying (2.2) yields for q.e.
x ∈ E, which shows (3.4). Since u + ∧ k ≤ u + , we have By this and [6, Lemma 4.6] again, λ k (u) ∈ M b (E) for k > 0. Using the same argument, but with the function ϕ(x) = x − ∧ k, we show that λ k (u) ∈ M b (E) for k < 0 and (3.5) is satisfied.
Proof. Follows from Proposition 3.7 and the fact that T k (u) = u + ∧ k − u − ∧ k.

Renormalized solutions for general measure data
In this section, we consider two equivalent definitions of renormalized solution to (1.1) and we study their relations to other concepts of solutions considered in Section 3. The first definition was introduced in [10]. The second is new. Its advantage over the first one is that it is simpler because it does not involve using the notion of the potential of the measure.

First definition
Remark 4.2. In [8] it is shown that if µ ∈ M 0,b (E) and u is a renormalized solution to (1.1), then ν k T V → 0 as k → ∞, where ν k T V = |ν k |(E).  Our goal is to show that condition (iii) in Definition 4.1 may be replaced by the following condition (iii'): ν k → µ c in the narrow topology.
Proof. We will prove the first assertion. The proof of the second one is analogous.
Let η be a bounded excessive function. From the above equation, Proposition 4.3 and Revuz duality we conclude that Since η ∈ Exc, we deduce from the above equation that ν 1 k , η is nondecreasing. Therefore we may pass to the limit in the above equation as k → ∞. We then have It is clear that R(1 {0<u≤k} · µ − d ) + 1 2 R(λ 0 (u)) is an excessive function. By (3.7), is also an excessive function. Therefore both limits with respect to t on the right-hand side of (4.1) are nondecreasing. It is clear that this is also true for limits with respect to k. Therefore we may change the order of the limits in (4.1). By Proposition 3.7, we then have This proves the first assertion.
Proof. Since (E, D(E)) is transient, there exists a strictly positive bounded function g 0 such that Rg 0 is bounded (see [14, Corollary 1.3.6.]). Set g := R 1 g 0 . Then Rg = RR 1 g 0 = R 1 Rg 0 ≤ Rg 0 = g. It is clear that g is bounded, finely-continuous and strictly positive. For n ≥ 1, we set where T is the set of all F-stopping times. We have so −h 2 n is defined as h 1 n but with h replaced by −h. Therefore it is enough to prove that {h 1 n } has the desired properties. Observe that Hence, by [13], h 1 n + nRg is an excessive function. It is clear that nRg is also excessive. Thus h 1 n ∈ Exc − Exc. By the definition, {h 1 n } is nonincreasing. Moreover, with τ 0 = 0, we have Since Rg is an excessive function and h is continuous, the process (nRg + h)(X) is càdlàg under the measure P x for every x ∈ E. For every ε > 0 there exists τ x n,ε ∈ T such that From this we conclude that Assume for a moment that we know that the above inequality implies that τ x n,ε → 0 in probability P x as n → ∞. Then, by continuity of h and (4.2), This implies that lim n→∞ h 1 n (x) ≤ h(x). Since, h 1 n ≥ h, we get the desired result. What is left is to show that τ x n,ε → 0 in probability P x as n → ∞. Aiming for a contradiction, suppose that there exist ε 1 , ε 2 > 0 and a subsequence (still denoted by (n)) such that Since g is strictly positive, there exists δ > 0 such that g(x) ≥ 2δ. Set Since g is finely-continuous, for any sequence t n ց 0, lim n→∞ P x (σ x > t n ) = 1. Let in contradiction with (4.3).
Theorem 4.6. Let u be a renormalized solution to (1.1). Then as k → ∞ in the narrow topology.

Second definition
(iii) ν k → µ c in the narrow topology. (ii) Assume that either R α (C b (E)) ⊂ C b (E) for some (hence for all) α > 0 and u ∈ L 1 (E; m) or R α (B b (E)) ⊂ C b (E) for some (hence for all) α > 0. If u is a renormalized solution to (1.1) in the sense of Definition 4.8, then u is a renormalized solution to (1.1) in the sense of Definition 4.1.
Proof. Assertion (i) follows from Proposition 4.3, Corollary 3.8 and Theorem 4.6 (since u is quasi-continuous it is finite q.e.). To prove (ii), assume that u is a renormalized solution to (1.1) in the sense of Definition 4.8. By Definition 4.8(ii), Therefore {Rν k } converges q.e. as k → ∞. Let v denote its limit. Let η ∈ B(E) be a bounded positive function such that Rη is bounded. Observe that Since ν k is narrowly convergent, sup k≥1 |ν k |, η < ∞. Therefore |u|, η < ∞ for every η ∈ B + (E) such that Rη is bounded. We also have has the strong Feller property. Then, by (4.5) and the Lebesgue dominated convergence theorem, The third equation follows from strong Feller property. Letting α → ∞ gives v, η = Rµ c , η .
Since η ∈ B(E) was an arbitrary positive function such that Rη is bounded, v = Rµ c . Therefore, by (4.4), u = Rµ d + Rµ c = Rµ, q.e., so by Proposition 4.3, u is a renormalized solution to (1.1) in the sense of Definition 4.1. Assume now that Letting k → ∞ and then α ց 0 shows that u = Rµ. By Proposition 4.3 again, u is a renormalized solution to (1.1) in the sense of Definition 4.1. Remark 4.11. Even in the case of local operators Definition 4.8 of renormalized solutions to (1.1) is in some cases more convenient in applications then the other definitions considered in [3]. For instance, Petitta, Ponce and Porretta [16] applied formulation of this type to solve evolution equations with smooth measure data and absorption on the right-hand side.
Let f : E × R → R be a measurable function. In [10] we have proved a uniqueness result for solutions, in the sense of Definition 4.1, to semilinear equations (1.12). Thanks to the equivalence proved in Theorem 4.9 we have the uniqueness result for solutions to (1.12) in the sense of Definition 4.8. Let us also note here that the existence of renormalized solutions to semilinear equations (1.12) with smooth measure data and f satisfying merely the sign condition with respect to the second variable was proved in [9]. In the case of general measure data the existence problem for (1.12) is a very subtle matter. Its investigation requires introducing the notion of reduced measures (see [6]). Definition 4.12. Let µ ∈ M b (E). We say that u ∈ B(E) is a renormalized solution to (1.12) if (i) T k (u) ∈ D e (E) for every k ≥ 0, and f (·, u) ∈ L 1 (E; m), Theorem 4.13. Let µ ∈ M b (E) and f be non-increasing with respect to the second variable.
where ν k is a bounded smooth measure. By Corollary 3.8, and by the definition of the measure λ k (u) (see (2.4)), In this section, we study the structure of the measures l k (u) and j k (u). We show that j k (u) has an explicit formula via u and the kernel of the nonlocal part of the operator A. As for the measure l k (u), we show the so-called reconstruction formula which is well known for equations with measure data and local (nonlinear) Leray-Lions type operators (see, e.g., [3]).
We adopt the notation introduced in Introduction. For the Beurling-Deny decomposition of (E, D(E)) we defer the reader to [4]. Note that for every u ∈ D e (E) there exists a unique smooth Radon measure µ c u such that (see [4, (3.2.20)]) By [4,Lemma 5.3.3],

By this and (5.3),
In general, µ c u is not a Radon measure. However, by (5.3) and the fact that µ c is bounded, µ c u , |h(u)η| is finite for all η ∈ B b (E) and h ∈ B b (E) such that h has compact support.
where a ij ∈ L 1 loc (D; m) and a = [a ij ] i,j=1,...,d is a non-negative definite symmetric matrix. To give a precise definition of the operator A, we assume that the form is closable. This is satisfied for instance if a ij ∈ H 1 loc (D) for i, j = 1, . . . , d or a ≥ λI for some λ > 0 (see, e.g., [4, page 111]). Let (E, D(E)) denote the closure of (E 0 , C ∞ c (D)). Then there exists a unique self-adjoint operator (A, D(A)) such that D(A) ⊂ D(E), and It is clear that where σ is such that σ · σ T = a. By |x − y| d+2α dy dx.

Renormalized solutions for smooth measure data
In [1] a definition of renormalized solutions to (1.1) with purely jumping operator on R d and µ ∈ L 1 (R d ) was introduced. In this section, we show that this definition can be extended to general smooth measure data and the class of operators considered in the present paper, so in particular to the class of operators considered in [5]. We also show that if µ ∈ M 0,b , then renormalized solutions considered in the previous sections are renormalized solutions in the sense of the new definition formulated below.
Letting k → ∞ and applying Fatou's lemma and Corollary 4.7 we get E×E (u(x) − u(y))(T l (u)(x) − T l (u)(y))J(dx, dy) ≤ l µ T V , l > 0. Since |T k (u)(x) − T k (u)(y)| ≤ |u(x) − u(y)|, applying the Lebesgue dominated convergence theorem shows that the left-hand side of the above equality tends to E(u, h(u)η) as k → ∞. On the other hand, by Remark 4.2, lim k→∞ ν k T V = 0, which shows that condition (ii) of Definition 6.1 is satisfied.