Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form

We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous function $u$ is a renormalized solution to an elliptic (or parabolic) equation in the sense of our definition iff $u$ is its probabilistic solution, i.e. $u$ can be represented by a suitable nonlinear Feynman-Kac formula. This implies in particular that for a broad class of local and nonlocal semilinear equations there exists a unique renormalized solution.


Introduction
The aim of this paper is to extend the notion of renormalized solution to encompass semilinear elliptic and parabolic equations involving measure data and operators associated with Dirichlet forms. The paper consists of two parts. In the first one we are concerned with elliptic equations of the form − Lu = f (x, u) + µ. (1.1) In (1.1), L is the operator associated with a regular Dirichlet form (E, D(E)) on L 2 (E; m) and f : E × R → R is a measurable function. As for µ we assume that it is a bounded smooth measure on E, i.e. a measure of bounded total variation on E which charges no set of zero capacity associated with the form (E, D(E)). Note that the class of operators L we consider is quite large. It contains many local as well as nonlocal operators. The model examples are Laplacian and fractional Laplacian (many other examples are to be found for instance in [9,13,15,17]). An important problem one encounters when dealing with equations of the form (1.1) is to define properly a solution. In case L is local and (1.1) is linear, i.e. f does not depend on u, some definition, now called Stampacchia's definition by duality, was proposed in [25]. To deal with semilinear equations the definitions of entropy solution (see [2]) and of renormalized solution (see [6]) have been introduced. For a comparison of different forms of these definitions as well as remarks on other concepts of solutions see [6]. In case E = D ⊂ R d is a bounded domain and L is a uniformly elliptic operator in divergence form with Dirichlet boundary conditions associated with the classical form E(ϕ, ψ) = D (a∇ϕ, ∇ψ) dx, ϕ, ψ ∈ H 1 0 (D) one of the equivalent definitions of a solution of (1.1) given in [6] says that a quasicontinuous u : D → R is a renormalized solution of (1.1) if f (·, u) ∈ L 1 (D), T k u ∈ H 1 0 (D) for k > 0, where T k u = ((−k) ∨ u) ∧ k, and there exists a sequence {ν k } of bounded smooth measures on D such that ν k T V → 0 as k → ∞ and for any bounded quasi-continuous v ∈ H 1 0 (D) and k ∈ N, In fact, the notion of entropy or renormalized solution can be applied to deal with more general then (1.1) equations in which L is a Leray-Lions type operator and µ is not necessarily smooth. Another approach to (1.1), covering both local and nonlocal operators, have been proposed in [13,15]. In this probabilistic in nature approach, a quasi-continuous (with respect to the form E) function u : E → R is a solution of (1.1) if the following nonlinear Feynman-Kac formula is satisfied for quasi-every x ∈ E. Here M = (X, P x ) is a Markov process with life time ζ associated with E, E x denotes the expectation with respect to P x and A µ is the additive functional of M associated with µ in the Revuz sense (see Section 2). In (1.3) we only assume that u is quasi-continuous and the integrals make sense. In fact, if (1.3) holds and f (·, u) ∈ L 1 (E; m) then using the probabilistic potential theory and the theory of Dirichlet forms one can show that u has some additional regularity properties. Namely, T k u belongs to the extended Dirichlet space D e (E) for every k > 0.
In [13,15] also a purely analytical definition of a solution of (1.1) resembling Stampacchia's definition is proposed (see also [11,16] for another approach in case of linear equation with L being a fractional Laplacian). We call it a solution in the sense of duality. In [13,15] it is shown that under quite general assumptions on E, f, µ a function u is a solution in the sense of duality if and only if it is a probabilistic solution defined by (1.3). However, the definition in the sense of duality seems to be not particularly handy tool for investigating (1.1).
The natural question arises whether the concept of renormalized solution can be carried over to general (possibly nonlocal) operators corresponding to E (for some partial results in this direction see [1]). An obvious related question to ask is what is the relation between (1.2) and (1.3), i.e. between renormalized and probabilistic solutions? It appears that (1.2) is the right form of the definition to be generalized to encompass wider class of operators. In the paper, under the assumption that E is transient, we define renormalized solution of (1.1) as a quasi-continuous function u : E → R such that f (·, u) ∈ L 1 (E; m), T k u ∈ D e (E) for k > 0 and there is a sequence {ν k } of bounded smooth measures on E such that ν k T V → 0 and for every k ∈ N and every bounded quasi-continuous v ∈ D e (E . What is perhaps more important, it also says that in the study of (1.1) one can use both probabilistic and analytical methods from the theory of PDEs. Let us point out once again, that contrary to [6], in our theorem we assume that the measure µ is smooth. An interesting open problem is how to define renormalized solutions for general bounded measures, at least for some classes of nonlocal operators. Finally, let us note that in case L = ∆ the equivalence between probabilistic and renormalized solutions to (1.1) was observed in [14].
In the second part of the paper we consider parabolic equation of the form where ϕ : E → R, f : [0, T ] × E → R, the operators ∂ ∂t + L t correspond to some time dependent regular Dirichlet form E 0,T and µ is a bounded measure on (0, T ] × E which is smooth with respect to the capacity associated with E 0,T . In case L t are local, a definition of a renormalized solution of equations of the form (1.5) involving more general nonlinear local operators L t of Leray-Lions type but with f not depending on u have been introduced in [7] (see also [5] for earlier existence results for equations with general bounded measure µ and [3] for uniqueness results in the case where µ is a function in L 1 ). In [8,20] definitions of renormalized solutions to (1.5) with Leray-Lions type operators and f depending on u have been proposed (in [20] equations with general, not necessarily smooth measures are considered). Another definition of a renormalized solution, which is suitable for handling equations with local operators and nonlinear f , have been introduced in [21]. It may be viewed as parabolic analogue of (1.2). Existence and uniqueness results for weak solutions to linear equations with fractional Laplacian and µ being a function in L 1 are proved in [16]. A probabilistic approach to (1.5) has been developed in [12]. A probabilistic solution of (1.5) is defined similarly to (1.3), but with M replaced by a time-space Markov process associated with E 0,T . In [12] the existence, uniqueness and regularity of probabilistic solutions of (1.5) is proved for f satisfying some natural conditions (monotonicity together with mild integrability conditions) and general operators associated with E 0,T .
Similarly to the elliptic case, in the paper we generalize the notion of a renormalized solution of [21] to the case of general operators corresponding to E 0,T . Then we show that the proposed definition is equivalent to the probabilistic definition considered in [12]. As in elliptic case, this shows that the renormalized solutions are properly defined and gives new information on the structure of solutions. We illustrate the utility of our result by stating some theorems on existence and uniqueness of renormalized solutions of parabolic equations with f satisfying the monotonicity condition and mild integrability conditions.
For simplicity, in the paper we confine ourselves to equations with operators corresponding to regular forms, but our results can be generalized to quasi-regular forms (see remarks at the end of Sections 3 and 4).

Preliminaries
In the paper we assume that E is a locally compact separable metric space and m is an everywhere dense Radon measure on E, i.e. m is a non-negative Borel measure on E finite on compact sets and strictly positive on non-empty open sets.
We set are defined analogously. We set H = L 2 (E; m) and H 0,T = L 2 (0, T ; H). The last space we identify with L 2 (E T ; m 1 ), where m 1 = dt⊗m. By (·, ·) H , (·, ·) H 0,T we denote the usual inner products in H and H 0,T , respectively.

Dirichlet forms
In what follows we assume that (E, D(E)) is a (non-symmetric) Dirichlet form on H, i.e. positive definite closed form satisfying the weak sector condition and such that (E, D(E)) has both the sub-Markov and the dual sub-Markov property. For the definitions we refer the reader to [17]. Here let us only recall hat (E, D(E)) satisfies the weak sector condition if there is K > 0 (called the sector constant) such that In fact, from the sub-Markov and the dual sub-Markov property of (E, D(E) it follows that (G α ) α>0 , (Ĝ α ) α>0 may be extended to sub-Markov resolvents on L ∞ (E; m) and on L 1 (E; m), respectively (see [19,Section 1.1]).
Let f ∈ L ∞ (E; m) be a non-negative function. Since G 1/l f increases as l ↑ ∞, the potential operator Gf = lim l→∞ G 1/l f is m-a.e. well defined but may take the value ∞. We say that E is transient if Gf < ∞ m-a.e. for every non-negative f ∈ L ∞ (E; m). LetẼ denote the symmetric part of E, i.e.Ẽ(u, v) = 1 2 (E(u, v) + E(v, u)). The extended Dirichlet space D e (E) associated with a Dirichlet form (E, D(E)) is the family of measurable functions u : E → R such that |u| < ∞ m-a.e. and there exists añ E-Cauchy sequence {u n } ⊂ D(E) such that u n → u m-a.e. The sequence {u n } is called an approximating sequence for u ∈ D e (E).
For u ∈ D e (E) we set E(u, u) = lim n→∞ E(u n , u n ), where {u n } is an approximating sequence for u. If moreover E satisfies the strong sector condition (2.1) then we may extend E to D e (E) by putting E(u, v) = lim n→∞ E(u n , v n ) with approximating sequences {u n } and {v n } for u ∈ D e (E) and v ∈ D e (E), respectively (see [19,Section 1.3]). This extension satisfies again the strong sector condition. By [19,Theorem 1.3.9], if (E, D(E)) is transient then (D e (E),Ẽ ) is a Hilbert space.
Given a Dirichlet form (E, D(E)) we define quasi notions with respect to E (exceptional sets, nests and quasi-continuity) as in [17, Chapter III] (see also [19, Sections 2.1, 2.2]). We will say that a property of points in E holds quasi everywhere (q.e. for short) if it holds outside some exceptional set.
In the paper we assume that (E, D(E)) is regular (see [17,Section IV.4] or [19, Section 1.2] for the definition). By [17,Proposition IV.3.3], if (E, D(E)) is a regular Dirichlet form then each element u ∈ D(E) admits an quasi-continuous m-version, which we denote byũ, andũ is q.e. unique for every u ∈ D(E). If moreover (E, D(E)) is transient then such a unique m-versionũ exists for every u ∈ D e (E). This follows from [9, Theorem 2.1.7] and the fact that D e (E) and the notion of quasi-continuity only depend on the symmetric part of E.
A positive measure µ on B(E) is said to be smooth (µ ∈ S(E) in notation) if µ(B) = 0 for all exceptional sets B ∈ B(E) and there exists an nest {F k } k∈N of compact sets such that µ(F k ) < ∞ for k ∈ N.
Given a transient form (E, D(E)) satisfying the strong sector condition we will denote by S If ν ∈ S (0) 0 (E) then from the Lax-Milgram theorem it follows that there is a unique elementÛ ν (called a copotential of ν) such that

Time dependent Dirichlet forms
We assume that we are given a family {B (t) , t ∈ [0, T ]} of Dirichlet forms on H with common domain V and sector constant K independent of t. We also assume that By putting B (t) = B for t ∈ R \ [0, T ] we may and will assume that B (t) is defined and satisfies (a), (b) for t ∈ R.
By the definition of a Dirichlet form V is a dense subspace of H and (B, V ) is closed. Therefore V is a real Hilbert space with respect toB 1 (·, ·), which is densely 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 u ∈ V we denote by ∂u ∂t the derivative in the distribution sense of the function t → u(t) ∈ V and we set We will consider time dependent Dirichlet forms E and E 0,T associated with the where ·, · is the duality pairing between V ′ and V and Note that E can be identified with some generalized Dirichlet form (see [ , and then using it we define quasi-notions (exceptional sets, nests and quasi-continuity) as in [19,Section 6.2]. Note that by [19, Theorem 6.2.11] each element u of W has a quasi-continuous m 1 -version. We will denote it byũ.
To define E 0,T , we set H 0, In what follows we adopt the convention that any element of W 0,T is already in C([0, T ]; H). With this convention we may define the spaces By the definition of W 0,T , ∂/∂t : Its adjoint is defined on W T and is given by −∂/∂t (see, e.g., [27]). Finally, we set where now ·, · denote the duality pairing between V ′ 0,T and V 0,T . As in the case of E, the form E 0,T can be identified with some generalized Dirichlet form (see [

Markov processes and additive functionals
In what follows ∆ is a one-point compactification of E. If E is already compact then we adjoin ∆ to E as an isolated point.
In the case of Dirichlet forms (and elliptic equations) we adopt the convention that every function f on E is extended to E ∪ {∆} by setting f (∆) = 0.
In the case of time dependent Dirichlet forms (and parabolic equations) we adopt the convention that every function ϕ on E is extended to Dirichlet forms Let (E, D(E)) be a regular Dirichlet form on H. Then there exists a unique Hunt process M = (Ω, (F t ) t≥0 , (X t ) t≥0 , ζ, (P x ) x∈E∪{∆} ) with state space E, life time ζ and cemetery state ∆ properly associated with (E, D(E)) (see Theorems IV.3.5, IV.6.4 and V.2.13 in [17]). The last statement means that for every α > 0 and f ∈ B b (E) ∩ H the resolvent of M, that is the function It is known (see, e.g., [17,Theorem VI.2.4]) that there is a one to one correspondence (called Revuz correspondence) between smooth measures µ and positive continuous additive functionals (positive CAFs) A of M. It is given by the following relation where E m denotes the expectation with respect to the measure P m (·) In what follows the positive CAF of M corresponding to µ ∈ S(E) will be denoted by where |µ| denotes the total variation of µ. Note that by [15,Lemma 2.3], in the above definition of the class R(E) one can replace m-a.e. by q.e. By M 0,b (E) we denote the space of all signed measures µ on E such that |µ| ∈ S(E) and |µ|(E) < ∞. By [15, , (X t ) t≥0 , ζ, (P z ) z∈E 1 ∪{∆} ) with state space E 1 , life time ζ and cemetery state ∆ properly associated with E in the sense that for every α > 0 and f ∈ B b ∩ L 2 (E 1 ; m 1 ) the resolvent of M defined as is a quasi-continuous version of the resolvent G α f associated with E. Moreover, by [19, where τ (t) is the uniform motion to the right, i.e. τ (t) = τ (0) + t, τ (0) = s, P z -a.s. for z = (s, x). Let S(E 1 ) denote the set of smooth measures on E 1 (with respect to E), which we define analogously to S(E) (see, e.g., [12,26] for details). We say that a positive AF where E m 1 denotes the expectation with respect to P m 1 (·) = E 1 P z (·) m 1 (dz) (see [18,24]). It is known (see [12,Section 2]) that for every µ ∈ S(E 1 ) there exists a unique positive natural AF A of M, i.e. a positive AF of M such that A and M have no common discontinuities, such that A is in the Revuz correspondence with µ. In what follows we will denote it by A µ . In fact, A µ is a predictable process (see [10]). On the contrary, if A is a positive natural AF of M then by Proposition in Section II.1 of [24] and [18,Theorem 5.6] there exists a smooth measure µ on E 1 such that A is in the Revuz correspondence with µ.
Let S(E 0,T ) denote the set of all µ ∈ S(E 1 ) with support in E 0,T and for µ ∈ S(E 0,T ) let

Elliptic equations
Let (E, D(E)) be a regular Dirichlet form on H. We consider the problem In what follows M is the Markov process of Section 2 associated with (E, D(E)). Let us recall that a càdlàg adapted (with respect to (F t )) process Y is said to be of class (D) if the collection {Y τ , τ is a finite (F t )-stopping time} is uniformly integrable.
Definition 3.1. Let f : E × R → R be a measurable function and let A µ be a CAF of M corresponding to some µ ∈ R(E). We say that a pair (Y x , M x ) is a solution of the backward stochastic differential equation under the measure P x if (a) Y x is an (F t )-progressively measurable càdlàg process such that Y x t∧ζ → 0, P x -a.s. as t → ∞, Y x is of class (D) under P x and M x is a càdlàg (F t )-local martingale under P x , The following definition is taken from [13,15].
Definition 3.2. Let µ ∈ R(E). We say that a quasi-continuous function u : E → R is a probabilistic solution to (1.1) if f u · m ∈ R(E) and for q.e. x ∈ E, Set By [9, Lemma A.3.6] there exists a MAF M od M such that M x t = M t , t ≥ 0, P x -a.s. for q.e. x ∈ E. Therefore under P x for q.e. x ∈ E. By the strong Markov property, under P x we have which shows (3.3). Taking t = 0 in the above equality and using (3.5) we get It follows that for q.e. x ∈ E, u(X T ∧ζ ) → 0, P x -a.s. as T → ∞.
In what follows we assume that (E, D(E)) is transient and satisfies the strong sector condition. For a measure µ on E and a function u : E → R we use the notation whenever the integral is well defined.
We adopt the following definition of renormalized solution of (1.1). In the case of local operators, this is essentially [6, Definition 2.29].
Definition 3.4. Let µ ∈ M 0,b (E). We say that u : E → R is a renormalized solution of (1.1) if (a) u is quasi-continuous, f u ∈ L 1 (E; m) and T k u ∈ D e (E) for every k > 0, (b) there exists a sequence {ν k } ⊂ M 0,b (E) such that ν k T V → 0 as k → ∞ and for every k ∈ N and every bounded v ∈ D e (E), Theorem 3.5. Assume that (E, D(E)) is transient, satisfies the strong sector condition and that µ ∈ M 0,b (E).
(i) If u is a probabilistic solution of (1.1) and f u ∈ L 1 (E; m) then u is a renormalized solution of (1.1).
(ii) If u is a renormalized solution of (1.1) then u is a probabilistic solution of (1.1).
Proof. (i) Let u be a probabilistic solution of (1.1) and let M be the martingale of Remark 3.3(ii). For k > 0 put From the fact that (Y, M ) is a solution of (3.3) it follows that By the Meyer-Itô formula (see, e.g., [23, Theorem IV.70]), for some increasing processes A 1,k , A 2,k . Since T k y = y ∧ k − ((y + k) ∧ 0) for y ∈ R, it follows from (3.8) and (3.9) that From (3.8), it follows immediately that A 1,k , A 2,k are AFs of M. Since u is a probabilistic solution, u(X t ) → 0, P x -a.s. as t → ∞ for q.e. x ∈ E. Therefore from (3.8) and continuity of A µ we conclude that for q.e. x ∈ E, Since E x (u ∧ k)(X 0 ) = (u ∧ k)(x) ≤ u(x) and u is a probabilistic solution of (1.1), it follows from the above that Similarly, by (3.9) we have It follows that for q.e. x ∈ E, E x (A 1,k ζ + A 2,k ζ ) < ∞. Therefore by [9, Theorem A.3.16] there exists AFs B 1,k , B 2,k of M such that B i,k is a compensator of A i,k , i = 1, 2 under P x for q.e. x ∈ E. Since A 1,k , A 2,k are increasing, B 1,k , B 2,k are increasing, too. Furthermore, since by [9,Theorem A.3.2] the process X has no predictable jumps, it follows from [9, Theorem A.3.5] that B 1,k , B 2,k are continuous. Thus B 1,k , B 2,k are increasing CAFs of M such that A i,k − B i,k , i = 1, 2, are martingales under P x for q.e.
x ∈ E. Let b i k , i = 1, 2, denote the Revuz measure of B i,k . Since for q.e. x ∈ E, E x A i,k t = E x B i,k t for t ≥ 0, from (3.11), (3.12) and [13,Lemma 5.4] we conclude that From (3.13) and the fact that u(X t ) → 0, P x -a.s. as t → ∞ for q.e. x ∈ E it follows that for q.e. x ∈ E, with Clearly ν k ∈ M 0,b (E) and ν k T V → 0 as k → ∞. By [15,Theorem 4.2] (see also [13,Proposition 5.9] in the case of regular symmetric forms), T k u ∈ D e (E) for every k > 0.
x ∈ E. By Fubini's theorem, for q.e. x ∈ E we have By the Markov property the right-hand side of the above equality equals for q.e. x ∈ E. By (3.14), T k u = Rλ k . Therefore from the above generalized resolvent equation it follows that for every bounded v ∈ D(E) we have By [17,Theorem 2.13(iii)] the left-hand side of (3.15) converges to E(T k u, v) as α → ∞.
The right-hand side is equal to λ k , αR α v , whereR α denotes the resolvent of a Hunt process associated with the form (Ê, D(E)) defined asÊ(u, v) = E(v, u), u, v ∈ D(E).
Since the functions αR α v are bounded uniformly in α > 0 and by Propositions I.2.13(ii) and III.3.5 in [17] we may assume that αR α v →ṽ q.e. as α → ∞, the right-hand side of (3.15) converges to λ k ,ṽ as α → ∞. Thus (3.6) is satisfied for bounded v ∈ D(E). Now assume that v is bounded, say by l, and v ∈ D e (E). Let v n be an approximating sequence for v. Then T l (v n ) → v m-a.e. and, by [9, Corollary 1.6.3], in (D e (E),Ẽ) as n → ∞. Taking a subsequence if necessary we may assume that T l v n →ṽ q.e. By what has already been proved, for n ∈ N we have Letting n → ∞ we get (3.6), which completes the proof of (i).
(ii) If u is a renormalized solution of (1.1) then u is quasi-continuous and (3.6) is satisfied for all functions v of the form v =Û ν with ν ∈Ŝ Therefore T k u is a solution in the sense of duality (see [15,Section 3.3] or [13,Section 5] for the definition) of the linear problem By [15,Proposition 3.9] (or [13,Proposition 5.3] in the case of symmetric forms) T k u is a probabilistic solution of (3.16). In particular (3.14) (with the measure ν k of (3.16)) is satisfied. Since ν k T V → 0 as k → ∞, there is a subsequence (still denoted by k) such that for m-a.e. x ∈ E. To see this, let us first observe that if µ ∈ S (0) 0 (E) andũ ≤ c µ-a.e., where u =Û µ, then u ≤ c m-a.e. Indeed, we have 0 (E) for n ∈ N and m(E \ ∞ n=1 F n ) = 0. Let F n,N = {x ∈ F n : Û µ n (x) ≤ N } and µ n,N = 1 F n,N ·µ n = 1 F n,N ·m. Then µ n,N ∈ S (0) 0 (E) and Û µ n,N ≤ Û µ n ≤ N µ n -a.e. Therefore by the observation made above,Û µ n,N ≤ N m-a.e., and hence Û µ n,N ≤ N q.e. Moreover, Since Û µ n is q.e. finite, m(F n \ ∞ N =1 F n,N ) = m({x ∈ F n : Û µ n (x) = ∞} = 0. Therefore from (3.19) one can deduce that (3.17) holds for m-a.e. x ∈ F n for each n ∈ N. Since m(E \ ∞ n=1 F n ) = 0, we see that (3.17) holds for m-a.e. x ∈ E. Letting k → ∞ in (3.14) and using (3.17) we conclude that (3.4) holds true for m-a.e. x ∈ E. In fact, since u and the right-hand side of (3.4) are quasi-continuous, (3.4) holds for q.e. x ∈ E, which completes the proof.
To illustrate the utility of Theorem 3.5 we now give some results on existence and uniqueness of renormalized solutions of (1.1) with f satisfying the monotonicity condition and mild integrability conditions. To state the results we will need the following hypotheses.
In what follows we assume that (E, D(E)) satisfies the assumptions of Theorem 3.5.
We close this section with some general remarks on possible generalization of our results and on their applicability.
An inspection of the proof of Theorem 3.5 reveals that it only makes use of some general results from the theory of stochastic processes that are valid for general semimartingals, some results from [15], which are proved for quasi-regular forms and the fact that M associated with E in the resolvent sense is a standard process (the fact that M a Hunt process is not needed). Therefore the proof of Theorem 3.5 carries over to quasi-regular Dirichlet forms.
By using probabilistic methods one can prove that for many interesting equations there exists a unique probabilistic solution u such that f u ∈ L 1 (E; m). This can be done for instance for f satisfying (E1)-(E3). For specific examples of local and nonlocal operators A satisfying our assumptions see, e.g., [9,13,15,17]. Then as in Corollary 3.7 and Theorems 3.8 above, a direct consequence of Theorem 3.5 is that u is a renormalized solution to (1.1), i.e. has clear analytical meaning, and that u is the unique renormalized solution. On the other hand, renormalized solutions to (1.1), which are obtained by analytical methods, automatically have stochastic representation of the form (3.4). We may then use (3.4) (and the theory of BSDEs; see Remark 3.3) to study further properties of the solution by probabilistic methods. For instance, probabilistic methods are quite effective in proving comparison results and hence uniqueness.
It would be desirable to define renormalized solutions for equations with general bounded measures µ, at least for some classes of nonlocal operators. Another interesting open problem is to give other equivalent to Definition 3.4 analytical definitions of a solution (like in the local case considered in [6]).

Parabolic equations
In this section we assume that the family {B (t) , t ∈ [0, T ]} satisfies the assumptions of Section 2.2 and E 0,T is the time dependent Dirichlet form defined by (2.6). By L t we denote the operator associated with B (t) in the sense of (3.2) and by ∂u ∂t + L t the operator corresponding to E 0,T , i.e. the generator of the strongly continuous contraction semigroup corresponding to (G 0,T α ) α>0 .
We consider the Cauchy problem In what follows we maintain the notation of Sections 2.2 and 2.3 concerning time dependent forms and associated Markov processes. In particular, M is a Markov process associated with E and ζ τ is defined by (2.8). By abuse of notation, in this section for u : E 0,T → R and µ ∈ R(E 0,T ).
We will say that a Borel measurable function u on E 0,T is quasi-càdlàg if for q.e. z ∈ E 0,T the process t → u(X t ) is càdlàg on [0, T − τ (0)], P z -a.s.
. We say that a quasi-càdlàg function u : E 0,T → R is a probabilistic solution to (4.1) if f u · m 1 ∈ R(E 0,T ) and for q.e. z ∈ E 0,T , Let ϕ, f u , µ be as in the above definition. Then by [12,Proposition 3.4] the right-hand side of (4.2) is a quasi-càdlàg function of z. Therefore the quasi-càdlàg requirement on u in the above definition can be omitted.
(ii) From the proof of [12,Theorem 5.8] it follows that if u is a probabilistic solution to (4.1) then there is an adapted process M such that M is a martingale under P z for q.e. z ∈ E 0,T and , t ≥ 0, for ν defined in (i) (see the beginning of the proof of [12,Proposition 3.4]). Hence Our definition of a renormalized solution is similar to [21,Definition 4.1].
(i) If u is a probabilistic solution of (4.1) and f u ∈ L 1 (E 0,T ; m 1 ) then u is a renormalized solution of (4.1).
(ii) If u is a renormalized solution of (4.1) then u is a probabilistic solution of (4.1).
Proof. (i) Let u be a probabilistic solution of (4.1). For k > 0 put By Remark 4.2 there is a martingale M such that (4.3) is satisfied. As in the proof of Theorem 3.5, applying the Meyer-Itô formula we show that for k > 0, for some increasing processes A 1,k , A 2,k such that and for q.e. z ∈ E 1 . Hence E z A i,k ζτ < ∞ for q.e. z ∈ E 0,T . From this and [9, Theorem A. 3.16] it follows that there is a positive increasing AF B i,k of M such that B i,k is the compensator of A i,k under P z for q.e. z ∈ E 0,T . In particular,  [9]. On the other hand, by [10,Theorem 16.8], there exists a Borel function a : E 0,T → R such that ∆A µ t = a(X t− ), t > 0, P z -a.s. for q.e. z ∈ E 0,T . From this and the fact that X is quasi-left continuous it follows that ∆u(X Tn ) = a(X Tn ). By this and (4.9), Combining (4.6)-(4.8) with (4.10), (4.11) we get By the above and the definition of the measures b 1 k , b 2 k we have From this and (4.10), (4.11) we obtain with By what has already been proved, ν k ∈ M 0,b (E 0,T ) and ν k T V → 0 as k → ∞. Moreover, by by [12,Theorem 3.12], T k u ∈ V 0,T , so what is left is to show that (4.4) is satisfied. We shall show that (4.4) follows from (4.14) by the same method as in elliptic case (see the proof of the fact that (3.6) follows from (3.14)). Let λ k = δ {T } ⊗ ϕ · m + f u · m + µ + ν k and let A = A λ k . By Fubini's theorem, for q.e. z ∈ E 0,T . Since by Remark 4.3 and (4.14), T k u = R 0,T λ k , it follows from the above equation that for every bounded v ∈ W 0 . Since the left-hand side of (4.15) equals E 0,T (αG 0,T α T k u, v), it converges to E 0,T (T k u, v) as α → ∞. LetR 0,T α denote the resolvent associated with the dual formÊ 0,T . By [26,Corollary III.3.8] applied to the functions αR 0,T α v we may assume that αR 0,T α v converges toṽ q.e. as α → ∞. It follows that the right-hand side of (4.15) converges to λ k ,ṽ as α → ∞. Therefore letting α → ∞ in (4.15) we obtain (4.4), which completes the proof of (i).
From this and (4.4) it follows that T k u is a solution in the sense of duality (see [12,Section 4] for the definition) of the linear problem (− ∂ ∂t − L t )T k u = f u + µ + ν k , T k u(T ) = T k ϕ, (4.16) so by [12,Corollary 4.2] T k u is a probabilistic solution of the above equation. Therefore (4.14) (with the measure ν k of (4.16)) is satisfied. Since ν k T V → 0 and for every Borel set F ⊂ E 0,T such that m(F ) < ∞ we have (R 0,T |ν k |, 1 F ) H 0,T = |ν k |, Ĝ0,T 1 F ≤ T ν k T V , one can find a subsequence (still denoted by k) such that R 0,T ν k (z) → 0 as k → ∞ for m 1 -a.e. z ∈ E 0,T . Therefore letting k → ∞ in (4.14) we show that (4.2) holds true for m 1 -a.e. z ∈ E 0,T , and hence for q.e. z ∈ E 0,T , because u and the right-hand side of (4.2) are quasi-continuous.
Remark 4.6. One can show that the function u + a appearing in (4.12) and (4.13) is equal quasi everywhere to the precise version of u (For the notion of a precise version of a parabolic potential see [22]).
We now illustrate the applicability of Theorem 4.5. Let us consider the following hypotheses.
Proof. Follows from Theorem 4.5 and [12, Corollary 5.9].  The results of [12] used in the proof of Theorem 4.5 can be generalized to quasiregular time dependent Dirichlet forms (see [12,Remark 4.4]). Moreover, if the forms B (t) , t ∈ [0, T ], are quasi-regular, then by [26,Theorem IV.2.2] there exists a special standard process M properly associated in the resolvent sense with the time dependent form determined by {B (t) , t ∈ [0, T ]}. Since one can check that the results from the theory of stochastic processes used in the proof of Theorem 4.5 hold true for such process M, Theorem 4.5 can be extended to quasi-regular time dependent Dirichlet forms.