Obstacle problem for semilinear parabolic equations with measure data

We consider the obstacle problem with two irregular barriers for the Cauchy–Dirichlet problem for semilinear parabolic equations with measure data. We prove the existence and uniqueness of renormalized solutions of the problem as well as results on approximation of the solutions by the penalization method. In the proofs, we use probabilistic methods of the theory of Markov processes and the theory of backward stochastic differential equations.


Introduction
(1.1) Here, f u (t, x) = f (t, x, u(t, x)), (t, x) ∈ D T , and Problem (1.1) with regular barriers and L 2 data is quite well investigated (see, e.g., the classical monograph [2,Section 3.2]). There are only few papers devoted to problem (1.1) with one irregular time-dependent barrier, and to our knowledge, there is no paper on problem (1.1) with two irregular barriers and general soft measure on the right-hand side. In [20,26], the linear problem with one barrier and L 2 data is considered. A semilinear problem (1.1) with L 2 data and f satisfying a Lipschitz and a linear growth condition is considered in [11] in the case of one barrier and in [13] in the case of two barriers. Note, however, that unlike the present paper, in [11,13], the function f may depend on the solution of (1.1) as well as its gradient.
The main goal of the paper is to prove the existence and uniqueness of solutions of (1.1) in the case where the barriers h 1 , h 2 are merely measurable and satisfy some kind of separation condition, ϕ ∈ L 1 (D), f satisfies a monotonicity condition in u and some mild growth condition considered earlier in the theory of PDEs with measure data (see, e.g., [4]). We are also interested in the approximation of solutions of (1.1) by the penalization method. As in [11,13], to study these problems, we adopt a stochastic approach based on the theory of backward stochastic differential equations.
The problem of existence and uniqueness of solutions of (1.1) with measure data is delicate even for regular barriers. If the barriers are irregular, then the additional serious problem is to define solutions of (1.1) in a way that ensures their uniqueness. The classical approach via variational inequalities is not suitable even in the case of one barrier and L 2 data, because even in that case, the solution of the variational inequality with irregular barrier is in general not unique.
To overcome the difficulty with uniqueness of solutions, one could try to adapt to the parabolic case the method of minimal solutions, which was applied successfully in [17] to one-sided elliptic obstacle problems with general Radon measure on the righthand side. Unfortunately, the concept of minimal solutions is not directly applicable to the two-sided obstacle problem.
To address the nonuniqueness problem, one can also try to adopt the approach from the paper [26] devoted to linear parabolic one-sided obstacle problems with L 2 data and define a solution of (1.1) as a pair (u, ν) consisting of a measurable function u : D T → R and a soft measure ν on D T such that is satisfied in some weak sense, h 1 ≤ u ≤ h 2 a.e. and ν satisfies some minimality condition. In case h 1 , h 2 and u are regular, the natural minimality condition says that (1.5) absolutely continuous with respect to the Lebesgue measure. A further complication arises from the fact that in general u is not quasi-continuous. It is, however, quasil.s.c., so determined q.e. Therefore, one can hope that when the barriers are quasi-l.s.c. or quasi-u.s.c., and hence determined q.e., then the minimality condition holds (the integrals in (1.5) are then well defined since ν is soft). Unfortunately, even in that case, the integrals in (1.5) may be strictly positive (a simple example is to be found in [11]). Our definition of a solution of (1.1) is based on a nonlinear Feynman-Kac formula. It may be viewed as a probabilistic extension of the definition described above, because in the linear case with L 2 data, our probabilistic solution (u, ν) coincides with the solution considered in [26]. Let us also note that in the case of one-sided problem, the first component u of the probabilistic solution is a minimal solution in the sense that u = quasi-essinf{v ≥ h 1 , m 1 -a.e. : v is a supersolution of problem (1.7)}, (1.6) where m 1 is the Lebesgue measure on R + × R d and ∂u ∂t + A t u = − f u − μ in D, u(T, ·) = ϕ, u(t, ·) |∂ D = 0, t ∈ (0, T ). (1.7) Thus, our probabilistic approach leads to a parabolic analogue of a solution considered in [17]. In the case of two-sided problem, it leads to some variant of (1.6). Let X = (X, P s,x ) be a Markov family associated with the operator A t , and for a soft measure γ on D T , let A γ denote the additive functional of X associated with γ in the Revuz sense (see Sect. 2). Set By a solution of (1.1), we mean a pair (u, ν) consisting of a function u on D T and a bounded soft measure ν on D T such that for q.e. (s, x) ∈ D T and ν satisfies a minimality condition introduced in [11]. The minimality condition says that for any measurable functions h * 1 , h * 2 on D T having the property that h 1 ≤ h * 1 ≤ u ≤ h * 2 ≤ h 2 a.e. and the processes [s, T ] t → h * i (t, X t ), i = 1, 2, are càdlàg under P s,x for q.e. (s, x) ∈ D T , the following equalities Vol. 15 (2015) Semilinear parabolic equations with measure data 461 where T s denotes the set of all stopping times with values in [s, T ] with respect to the completion of the filtration generated by X .
In the present paper, we are mostly interested in the investigation of renormalized (or entropy) solutions of (1.1). But it is worth mentioning that as a by-product of our proofs, we obtain new results on stochastic representation of solutions of (1.1) and the Cauchy-Dirichlet problem (1.7), which can be regarded as problem (1.1) with h 1 ≡ −∞, h 2 ≡ +∞. Some of these results seem to be new even in the case of problem (1.7) with L 2 data. To our knowledge in all the existing results on stochastic representation of solutions to that problem, some regularity of the boundary of the domain is assumed. In the present paper, we do not require any regularity of D. Let us also note that in the recent paper [25], the existence and uniqueness of renormalized solutions of the Cauchy-Dirichlet problem with more general than (1.2) divergence form operator (for instance p-Laplace operator) and f not depending on x and satisfying the so-called sign condition is proved. In our paper, we consider equations with A given by (1.2), but we allow f to depend on x.

Preliminaries
In this section, we have compiled some basic facts on diffusions associated with the operator A t defined by (1.2) and their additive functionals associated with soft measures on R + × R d . Here, and in the next sections, we assume that a satisfies (1.3). By putting a i j = δ i j outside D T , we can and will assume that a is defined and satisfy (1.3) in all R + × R d .

Time-inhomogeneous diffusions and additive functionals
Let for any Γ ∈ B(R d ). The process X admits the so-called strict Fukushima decomposition, i.e., for every T > 0, } is a two parameter continuous additive functional (CAF) of X (with respect to the filtration {G s t }) of zero energy and {M s,t , 0 ≤ s ≤ t ≤ T } is a two parameter martingale additive functional (MAF) of X (with respect to {G s t }) of finite energy (see [29]). In particular, (see [29]). It follows that under P s,x , the process B s,· defined as where σ · σ * = a and σ −1 is the inverse matrix of σ , is a Brownian motion on [s, T ]. It is also known (see [14]) that X admits the so-called Lyons-Zheng decomposition, i.e., for any T > 0, For a measurable R d -valued function f and s ≤ r ≤ t ≤ T , we put The usefulness of the backward integral defined above comes from the fact that for regular f, where a −1 is the inverse matrix of a (see, e.g., [30]). Let us recall that for every Φ ∈ L 2 (0, T ; Vol. 15 (2015) Semilinear parabolic equations with measure data 463 i.e., for every η ∈ L 2 (0, T ; H 1 (R d )), where m 1 denote the Lebesgue measure on R + × R d .
The following lemma will be needed in the next section to prove regularity of parabolic potentials. For the meaning of the notion of "quasi-every" used below, see Sect. 2.3.
it follows from the above equality that In [14], it is proved that from which the desired result follows, because ξ s,ε ξ s , P s,x -a.s. as ε 0.

Time-homogeneous diffusions
In the next sections, it will be advantageous to consider a certain time-homogeneous diffusion determined by X. The standard construction of it is as follows. We set and consider the process X on Ω defined as for ω = (s, ω). Then, Since the decomposition (2.1) is unique, we may assume that the two parameter AFs A, M of (2.1) are defined for all 0 ≤ s ≤ t and that for every (s, x) ∈ R + × R d the decomposition (2.1) holds true for all t ≥ s. Set (By convention, if ξ is a random variable defined on Ω, we set ξ(ω ) = ξ(ω) for any ω = (s, ω) ∈ Ω . Thus, in particular, Vol. 15 (2015) Semilinear parabolic equations with measure data 465 By (2.2) and (2.6), In fact, by [16,Theorem 12], it is an {G t }-Brownian motion.

Capacity and soft measures
Let W be the space of u ∈ L 2 (R + ; . We define the parabolic capacity of an open set U ⊂ R + × R d as The parabolic capacity of a Borel set B ⊂ R + × R d is defined as where P s,m (·) = R d P s,x (·) dx and m denotes the Lebesgue measure on R d (see the argument following Eq. (5.2) in [22]).
We will say that some property is satisfied quasi-everywhere (q.e. for short) if it is satisfied except for some Borel subset of R + × R d of capacity zero.
Let μ be a signed Borel measure on R + × R d , and let |μ| denote the total variation of μ. We say that μ is smooth if |μ|(B) = 0 for every Borel set B ⊂ R + × R d such that cap(B) = 0, and there exists an ascending sequence {F n } of closed subsets of In what follows by S, we will denote the set of all positive smooth measures on R + × R d and by S 0 the subset of S consisting of all measures of finite energy integral (see, e.g., [21,Section 5] for the definition). By M b (D T ), we denote the set of all signed Borel measures on R + × R d having bounded variation and such that μ( Let us note that in the literature soft measures are usually defined on (0, T ) × D. In the present paper, we consider soft measures on (0, T ] × D because in general the obstacle reaction measure is concentrated on that set.

Additive functionals and soft measures
Let E s,x (resp. E s,x ) denote the expectation with respect to P s,x (resp. P s,x ). Let us recall that a positive additive functional A of X and a positive soft measure μ on If μ, 1 < +∞, then A is called integrable. By Theorems 6.4.7 and 6.4.9 in [23], for every μ ∈ S 0 , there exists a unique positive natural additive functional (NAF) A of X in the Revuz correspondence with μ (see also (5.9), (5.10) in [21]). Since each measure μ ∈ S may be approximated by measures in S 0 (see [ Let us also note that if μ belongs to the subset of S consisting of smooth measures with respect to the capacity determined by the coercive part of the time-dependent form associated with X (see [31] for details), then by [31, Theorem 2.2], the AF A μ is continuous. In fact, in this case, an explicit representation of A μ is known (see Theorem 2.1 and Corollary 4.4 in [14]). Let and let p D denote the transition density of the process X killed at first exit from R + × D, i.e., where p is the transition density of X . It is known that A μ corresponds to μ iff for quasi-every (s, for every f ∈ B + (D T ) (see [19]). In [11], a correspondence between two parameter additive functionals of X and soft measures on D T is considered. An AF {A μ s,t , 0 ≤ s ≤ t ≤ T } of X corresponds to a bounded soft measure μ on D T in the sense of [11] if for q.e. (s, x) ∈ D T , Vol. 15 (2015) Semilinear parabolic equations with measure data 467 where ξ s is defined by (1.8) and p D is the transition density of the process X killed at the first exit from D, i.e., s,t } of X corresponds to μ in the sense of [11] iff the one-parameter AF A of X defined by (2.11) corresponds to μ in the sense of (2.8).
2 We say that a measurable function f : The set of all quasi-integrable functions on D T will be denoted by q L 1 (D T ).
REMARK 2.4. Analysis similar to that in [15,Remark 4.4] shows that if for every Then, extending f on R + × R d by putting zero outside D T , we get that A t = t 0 f (X r ) dr is finite for every t ≥ 0, which implies that it is a positive continuous AF of X . Directly from the definition of the Revuz correspondence, it follows that f · m 1 corresponds to the AF A. This implies in particular that f · m 1 is a smooth measure. Therefore, for every compact Proof. By the assumptions, for q.e. (s, [20], for every k, there exists a minimal solution e k ∈ L 2 (0, T ; H 1 (D)), in the variational sense, of the obstacle problem (2.13) From (2.12), (2.13) and the fact that 0 ≤ e k ≤ 1 q.e. on D T , it follows that 14) The fact that 0 ≤ e k ≤ 1 q.e. follows from the fact that e k is the minimal potential majorizing h k (see [2, Theorem 3.2.28]). Indeed, it is an elementary check that e k ∧ 1 is a potential majorizing h k . Therefore, e k ≤ e k ∧ 1, which implies that e k ≤ 1. Of course, e k ≥ 0 since e k ≥ h k . From this, we get in particular that e k (s, x) = 1 for (s, x) ∈ D k T . Therefore, e k (s, x) 1 q.e. on D T . Consequently, letting k → ∞ in (2.14) and using Fatou's lemma, we get the desired result.

Backward stochastic differential equations
Results of this section together with some results on BSDEs proved in [12] form the basis for our main theorems on the obstacle problem (1.1).

General BSDEs
Suppose we are given a filtered probability space (Ω, G, {G t } t≥0 , P). We will need the following spaces of processes.
V (resp. V + ) is the space of all progressively measurable càdlàg processes V of finite variation (resp. increasing processes) such that D q , q > 0, is the space all progressively measurable càdlag processes η such that E sup 0≤t≤T |η t | q < +∞ for every T > 0.
M q , q > 0, is the space of all progressively measurable processes η such that E( T 0 |η t | 2 dt) q/2 < +∞ for every T > 0. Vol. 15 (2015) Semilinear parabolic equations with measure data 469 Let us also recall that a càdlàg adapted process η is said to be of Doob's class (D) if the collection {η τ : τ is a finite {G t } t≥0 -stopping time} is uniformly integrable.
Let B be a {G t }-Brownian motion, σ be a bounded stopping time, ξ be a G σmeasurable random variable, A ∈ V and let f : Ω × R + × R × R d → R be a function such that f (·, y, z) is progressively measurable for y ∈ R, z ∈ R d (for brevity, in our notation, we omit the dependence of f on ω ∈ Ω). Let us recall that a pair (Y, Let us consider the following assumptions. and every y ∈ R, z, z ∈ R d , P-a.s., (A3) There is κ ∈ R such that ( f (t, y, z) − f (t, y , z))(y − y ) ≤ κ|y − y | 2 for a.e.

Markov-type BSDEs
Let T > 0, f : D T × R × R d → R be a measurable function, ϕ ∈ L 1 (D). By putting ϕ(t, x) = ϕ(x) for (t, x) ∈ D T , we will regard ϕ as defined on D T . Let μ be a soft measure on D T and B be the Brownian motion defined by (2.7). Let and hence, , so the desired result follows from uniqueness of solutions of (3.1). 2 In the sequel, we say that a Borel set N ⊂ D T is properly exceptional if m 1 (N ) = 0 and P s, By the assumption and the definition of N , E s,x P X h (Λ c ) = 0 for every (s, x) ∈ N c , which completes the proof.
2 In the case of Markov-type equations, the following hypotheses are analogues of (A1)-(A5) and (AZ).

If f does not depend on z, then there is C depending only on κ, T such that
for q.e. (s, x) ∈ D T .
for q.e. (s, x) ∈ D T . Put C t = t 0 Z r dr , t ≥ 0. Since Z 1 [0,ζ τ ] = Z , using the property of Z proved in Proposition 3.2, one can check that for every h ≥ 0, Accordingly, C is a continuous AF of the process X killed at first exit from D T . Therefore, it follows from [32, Theorem 66.2] that there exists a Borel measurable function ψ on D T such that Z = ψ(X), dt ⊗ P s,x -a.s. on [0, ζ τ ] × Ω for q.e. (s, x) ∈ D T . Finally, by Tanaka's formula, 2 In what follows, we denote by T k , k ≥ 0, the truncation operator, i.e., LEMMA 3.6. Assume that ψ n , ψ are Borel measurable functions on D T such that ψ n (X) → ψ(X), dt ⊗ P s,x -a.e. on [0, ζ τ ] × Ω for q.e. (s, x) ∈ D T . Then, for some subsequence (still denoted by n), ψ n → ψ, m 1 -a.e. on D T .
Proof. Let (s, x) ∈ D T be such that ψ n (X) → ψ(X), dt ⊗ P s,x -a.s. on [0, ζ τ ] × Ω . Then,  (s, x) can be chosen so that s is arbitrary close to zero, one can choose a further subsequence (still denoted by n) such that ψ n → ψ, m 1 -a.e. on D T .
Proof. By Proposition 3.5, there exists a pair of processes (Y, Z ) such that Y is of class (D), (Y, Z ) ∈ D q ⊗ M q for q ∈ (0, 1), and for q.e. (s, x) ∈ D T , the pair (Y, Z ) is a solution of BSDE s, x (ϕ, D, dμ), i.e., By Proposition 3.5 again, for n ∈ N, there is a pair of processes (Y n , Z n ) such that Y n is of class (D), (Y n , Z n ) ∈ D q ⊗ M q for q ∈ (0, 1), and (Y n , Z n ) is a solution of the following BSDE s,x Let A n t = t 0 n(Y n r − Y r ) − dr . By Proposition 3.5, Y t = v(X t ), P s,x -a.s. for t ∈ [0, T − τ (0)] and Z = ψ(X), dt ⊗ P s,x -a.e. on [0, ζ ] × Ω for some measurable function ψ on D T . Therefore, From this, (3.7) and Proposition 3.5, and Z n = ψ n (X), dt ⊗ P s,x -a.e. on [0, ζ τ ] × Ω , (3.11) where v n (s, x) = E s,x 1 {ζ >T −τ (0)} ϕ(X T −τ (0) ) + (Y m , Z m ) ∈ D q ⊗ M q for q ∈ (0, 1) (it is known that in fact Y m is continuous). By [12,Theorem 5.7], We divide the proof that v has the desired regularity properties into three steps. Step 1. We assume that ϕ ∈ L 2 (D) and μ = g · m 1 for some g ∈ L 2 (D T ), g ≥ 0. Let w ∈ W(D T ) be a weak solution of the problem Then,w ∈ W(S T ) and hence ∂w ∂t By [14,Theorem 4.3],w has a quasicontinuous m 1 version (still denoted byw) such thatw ∈ F S 2 , ∇w ∈ F M 2 and for q.e. (s, x) ∈ S T , Hence, by Lemma 2.1, (3.14) Therefore, v = w satisfies all assertions of the proposition except from (3.6). To prove (3.6), let us fix k > 0 and z ∈ R. Using the convention of Remark 3.1, by Tanaka's formula and taking expectations, gn(w(X r ) − z)g(X r ) dr, Vol. 15 (2015) Semilinear parabolic equations with measure data 477 where {L z t (w(X)), t ≥ 0} denotes the (symmetric) local time ofw(X) at z and sgn(x) = 1 x =0 x |x| . Hence, Multiplying the above inequality by the function i(z) = 1 [−k,k] (z), integrating with respect to z and applying the occupation time formula and Fubini's theorem, we get which when combined with (3.14) yields (3.6).
Step 2. We are going to show that v n satisfies all the assertions of the proposition. To shorten notation, we write v (resp. ψ) instead of v n (resp. ψ n ). Since ϕ m ∈ L 2 (D) and φ m ∈ L 2 (D T ), it follows from Step 1 that there exists v m ∈ FS 2 such that v m ∈ W(D T ) and for q.e. (s, Put v (s, x) = lim sup m→∞ v m (s, x), (s, x) ∈ D T . Then, by (3.12), v (X t ) = Y t , t ∈ [0, ζ τ ], P s,x -a.s. for q.e. (s, x) ∈ D T . This implies that v ∈ FD q for q ∈ (0, 1), v is of class (FD) and v = v q.e. The last statement implies that v belongs to the same spaces of functions as v . We have proved that (3.6) is true if μ = g · m 1 for some g ∈ L 2 (D T ), g ≥ 0, and ϕ ∈ L 2 (D). Therefore, for q.e. (s, x) ∈ D T , From this and Lemma 2.5, Due to [24], from (3.16), it follows that v ∈ T 0,1 2 ∩ L q (0, T ; W 1,q 0 (D)) and for some subsequence (still denoted by n), σ ∇v m → σ ∇v weakly in L q (D T ) for q ∈ [1, d+2 d+1 ). On the other hand, by (3.13) and Lemma 3.6, σ ∇v m → ψ, m 1 -a.e. on D T . Hence, ψ = σ ∇v, m 1 -a.e. on D T . Summarizing, v satisfies all the assertions of the proposition.
Step 3. Using (3.8)- (3.11), one can show in the same manner as in Step 2 that v satisfies all the assertions of the proposition. The only difference from Step 2 is that in estimates of σ ∇T k (v n ) of the form (3.15), (3.16) the functionφ m is replaced by φ n , so to obtain the estimates for σ ∇T k (v n ) which do not depend on n we have to show that for q.e. (s, x) ∈ D T and that φ n T V ≤ μ T V . But (3.17) follows from the fact that v n (s, x) ≤ v(s, x) for q.e. (s, x) ∈ D T and the estimate for φ n T V follows from (3.17) and Lemma 2.5.

Cauchy-Dirichlet problem
It is convenient to begin the study of the obstacle problem (1.1) with the study of the Cauchy-Dirichlet problem which can be regarded as problem (1.1) with h 1 ≡ −∞, h 2 ≡ +∞. Let us recall that every functional Φ ∈ W (D T ) admits decomposition of the form where g ∈ L 2 (0, T ; H 1 0 (D)), G = (G 1 , . . . , G d ), f ∈ L 2 (D T ), i.e., for every η ∈ W(D T ), where ·, · denotes the duality between L 2 (0, T ; H 1 0 (D)) and L 2 (0, T ; H −1 (D)). It is also known that every measure μ ∈ M 0,b (D T ) admits decomposition of the form Accordingly, μ ∈ M 0,b (D T ) can be written in the form for some g ∈ L 2 (0, T ; H 1 0 (D)), G ∈ L 2 (D T ), and f ∈ L 1 (D T ). Let us stress that in general, Φ, f of the decomposition (4.3) cannot be taken nonnegative even if μ is nonnegative. Vol. 15 (2015) Semilinear parabolic equations with measure data 479 We say that a triple (g, G, f ) is the decomposition of μ ∈ M 0,b (D T ) if (4.4) is satisfied.
DEFINITION. We say that a measurable function u : D T → R is a renormalized solution of the problem (4.1) if Note that a different but equivalent definition of renormalized solution of (4.1) is given in [25,Definition 4.1]. Set DEFINITION. We say that a measurable function u : D T → R is an entropy solution of (4.1) (4.5) (4.1) in the sense that u, ∇u ∈ L 1 (D T ) and for any η ∈ C ∞ 0 (D T ), Proposition 4.5 and Theorem 4.11 in [25]).

REMARK 4.1. (i) From [7, Theorem 3.1], it follows that u is a renormalized solution of (4.1) iff it is an entropy solution of (4.1). (ii) If u is a renormalized solution of (4.1), then it is a distributional solution of
and Proof. From the proof of [8,Theorem 2.7], it follows that each μ ∈ M 0,b (D T ) admits a decomposition of the form (4. . Therefore, repeating arguments from the proof of [17, Corollary 3.2], we get the desired result. , ϕ ∈ L 1 (D) and let u n (resp. u) be a renormalized solution of (4.1) with terminal condition ϕ n (resp. ϕ), f ≡ 0 and with −μ n (resp. −μ) on the right-hand side. If μ n − μ T V → 0 and ϕ n − ϕ L 1 → 0, then u n → u, m 1 -a.e.
Proof. By Lemma 4.2, we may assume that μ n , μ are given by (4.7) and (4.8) is satisfied. But then, the lemma follows from [24,Proposition 4]. 2 and let v be defined by (3.5). (4.9) Proof. Without loss of generality, we may assume that ϕ ≥ 0 and μ ≥ 0. Assume for a moment that ϕ ∈ L 2 (D T ) and μ ∈ M + 0,b ∩ W (D T ). From the proof of Proposition 3.7, it follows that v is the q.e. limit of v n , where v n is a weak solution of ∂v n ∂t + A t v n = −n(v n − v) − , v n (T, ·) = ϕ, v n (t, ·) |∂ D = 0, t ∈ (0, T ). Vol. 15 (2015) Semilinear parabolic equations with measure data 481 But it is known (see, e.g., [20, Theorem 1.1]) that {v n } converges in L 2 (D T ) to a unique weak solution of (4.9). Therefore, v is a weak solution of (4.9). Since a weak solution of (4.9) is a renormalized solution, this proves the proposition under the additional assumptions on ϕ, μ. Assume now that ϕ ∈ L 1 (D) is nonnegative and μ ∈ M + 0,b (D T ). By [21,Theorem 5.6], there exists a generalized nest, i.e., an ascending sequence {F n } of compact subsets of D T such that cap(K \F n ) → 0 for every compact K ⊂ D T , with the property that 1 F n · μ ∈ M + 0,b (D T ) ∩ W (D T ) and μ(D T \ n F n ) = 0. Let ϕ n = ϕ ∧ n. By what has already been proved, v n defined as is a renormalized solution of (4.9) with ϕ, μ replaced by ϕ n and μ n , respectively. Since ϕ n − ϕ L 1 → 0 and {F n } is a generalized nest, we conclude from (4.10) that v n (s, x) → v(s, x) for q.e. (s, x) ∈ D T . This completes the proof because by Lemma 4.3, {v n } converges to the renormalized solution of (4.9). 2 THEOREM 4.5. Assume (H1)-(H5). Then, there exists a unique renormalized solution u of (4.1). Moreover, u ∈ FD, u ∈ L q (0, T ; W for q.e. (s, x) ∈ D T . Finally, there exists C > 0 depending only on κ, T such that because from Remark 4.1(ii) and results proved in [5,Section 3] (see also [6]), it follows that if u is a solution of (4.1) then u L q (0,T ;W 1,q 0 (D)) ≤ c f u · m 1 + μ γ T V for some c, γ > 0.

Obstacle problem
We begin with a probabilistic definition of a solution of the obstacle problem.
DEFINITION. Assume (H1), (H4) and let h 1 , h 2 be measurable functions on D T such that h 1 ≤ h 2 , m 1 -a.e. We say that a pair (u, ν) consisting of a measurable function u : D T → R and a measure ν on D T is a solution of the obstacle problem with terminal condition ϕ, right-hand side f + dμ and obstacles h 1 , h 2 (OP(ϕ, f + dμ, h 1 , h 2 for q.e. (s, x) ∈ D T , where g − (X r ) = lim t<r,t→r g(X t ) for g := u, h * 1 , h * 2 . We say that (u, ν) is a solution of the obstacle problem with one lower (resp. upper) barrier h (OP(ϕ, f + dμ, h) (resp. OP(ϕ, f + dμ, h)) for short) if (u, ν) satisfies the conditions of the above definition with h 1 = h, h 2 = +∞ (resp. h 1 = −∞, h 2 = h) and ν ∈ M + 0,b (D T ) (resp. −ν ∈ M + 0,b (D T )). REMARK 5.1. Let us note that in view of Proposition 4.4, condition (b) in the above definition says that u is a renormalized solution of (4.1) with μ replaced by μ + ν. Condition (c) provides a probabilistic formulation of minimality of ν. In [26], minimality of ν was expressed by using the notion of precise versions of functions introduced in [27]. In fact, in the linear case with L 2 data, condition (c) coincides with that introduced in [26], because for every parabolic potential h,ĥ(X t ) = h − (X t ), t ∈ [0, ζ τ ], whereĥ is the precise version of h (see [11] for details).
As in the case of the Cauchy-Dirichlet problem considered in the previous section, the proof of the existence of a solution of the obstacle problem relies heavily on the results on BSDEs proved in [12].
Suppose we are given a filtered probability space and A, B, σ, ξ as in Sect. 3.1. Moreover, suppose that we are given two progressively measurable processes U, L such that U ≤ L and f : Ω × R + × R → R such that f (·, y) is progressively measurable ( f does not depend on z).
Let M q , q > 0, denote the space of continuous martingales such that E([M] T ) q/2 < +∞ for every T > 0.
We will need the following general growth conditions for f .
2 We first prove the comparison and uniqueness results for solutions of the obstacle problem.