Obstacle problem for semilinear parabolic equations with measure data

We consider the obstacle problem with two irregular reflecting 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 and well as results on approximation of the solutions by the penaliztion method. In the proofs we use probabilistic methods of the theory of Markov processes and the theory of backward stochastic differential equations.


Introduction
Let D ⊂ R d , d ≥ 2, be an open bounded set and let µ be a bounded soft measure on D T ≡ [0, T ] × D (we call a Radon measure µ soft if it does not charge sets of zero parabolic capacity). Suppose we are also given f : D T × R → R, ϕ : D → R and two functions h 1 , h 2 : D T →R such that h 1 ≤ h 2 . In the present paper we investigate the obstacle problem u(T, ·) = ϕ, u(t, ·) |∂D = 0, t ∈ (0, T ). (1.1) Here f u (t, x) = f (t, x, u(t, x)), (t, x) ∈ D T , and where a : D T → R d ⊗ R d is a measurable symmetric matrix-valued function such that for some Λ ≥ 1, for a.e. (t, x) ∈ D T . Problem (1.1) with regular barriers and L 2 data is quite well investigated (see, e.g., the classical monograph [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 righthand side. In [18,22] the linear problem with one barrier and L 2 data is considered. Semilinear problem (1.1) with L 2 data and f satisfying the Lipschitz and the linear growth condition is considered in [9] in the case of one barrier and in [12] in the case of two barriers. Note, however, that unlike the present paper, in [9,12] 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 the monotonicity condition in u and some mild growth condition considered earlier in the theory of PDEs with measure date (see, e.g., [3]). We are also interested in approximation of solutions of (1.1) by the penalization method. As in [9,12] 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 such a way to guarantee 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 when the barrier is irregular then in general a solution of the variational inequality associated with (1.1) is 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 [15] to one-sided elliptic obstacle problem with general Radon measure on the right-hand 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 [22] devoted to linear parabolic one-sided obstacle problem 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 ∂u ∂t + A t u = −f u − µ − ν in D, u(T, ·) = ϕ, u(t, ·) |∂D = 0, t ∈ (0, T ) (1.4) 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 where ν + , ν − denote the positive and the negative part of the Jordan decomposition of ν, i.e. the reaction measure acts only when u touches the barriers. If h 1 , h 2 are merely measurable, the problem is to make sense of (1.5), because in general ν is not 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 then 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 [9]). Our definition of a solution of (1.1) is based on a nonlinear Feynman-Kaca 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 [22]. 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 [15]. 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 Section 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 [9]. The minimality condition says that for any measurable functions h * 1 , h * 2 on D T such that e. and the processes [s, T ] → h * i (t, X t ), i = 1, 2, are càdlàg under P s,x for q.e. (s, x) ∈ D T we have Our main result says that under natural mild assumptions on the data there exists a unique solution (u, ν) of (1.1). Moreover, from the nonlinear Feynman-Kac formula (1.9) we deduce that u is a renormalized (and an entropy as well) solution of problem (1.4). If h 1 , h 2 are quasi-continuous, condition (1.10) has purely analytic interpretation. We show that under this additional assumption u is quasi-continuous and (1.10) reduces to condition (1.5). In the case of irregular barriers an analytical formulation of the minimality condition is possible in the linear case with L 2 data. In that case (1.10) may be expressed by using the notion of precise version of function introduced in [23].
The reason for adopting here probabilistic definition of a solution of (1.1) not only pertains to the difficulties with the analytic formulation of the minimality condition for the reaction measure ν. One major advantage of the probabilistic definition is that it fits well to a general scheme of proving existence of solutions of equations with measure data which was successfully adopted in the paper [13] devoted to semilinear elliptic equations with operators associated with general symmetric regular Dirichlet forms. The scheme comprises two essentially different parts. In the present context the first part consists in using stochastic methods to show existence of a pair (u, ν) satisfying (1.9), (1.10) and such that the process t → u(t, X t ) has some integrability properties. As a matter of fact this part follows rather easily from results on doubly reflected BSDEs proved recently in [10]. The second part consists in using the nonlinear Feynman-Kac formula (1.9) to prove additional regularity properties of u and to show that u is a renormalized solution of (1.4).
Results from [10] are also used to show that the solution of (1.1) can be approximated by the penalization method. For instance, we show that under the same assumptions under which there exists a unique solution (u, ν) of (1.1), if u n is a renormalized solution of the problem then u n → u q.e. on D T and ∇u n → ∇u in the Lebesgue measure. From [10] and the main result of the present paper it also follows that q.e. on D T , u = quasi-essinf{v ≥ h 1 , m 1 -a.e.: v is a supersolution of (1.7) with µ replaced by µ − ν − }. (1.11) If h 2 ≡ +∞ then ν − = 0, and so (1.11) reduces to (1.6). Finally, let us mention that from [10] and our main result it follows that u can be also characterized as a solution of the following stopping time problem (sometimes called Dynkin game): for any h * 1 , h * 2 as in condition (1.10), 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 investigation of renormalized (or entropy) solutions of (1.1). But it is worth mentioning that as a byproduct 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 [21] 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 ij = 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 Ω = C(R + ; R d ) be the space of continuous R d -valued functions on R + = [0, +∞), X be the canonical process on Ω, F s t = σ(X u , u ∈ [s, t]) and for given T > 0 letF s t = σ(X u , u ∈ [T +s−t, T ]). We define G as the completion of F s T with respect to the family P = {P s,µ : µ is a probability measure on B(R d )}, where P s,µ (·) = R d P s,x (·) µ(dx), and then we define G s t (Ḡ s t ) as the completion of F s t (F s t ) in G with respect to P. Let p denote the fundamental solution for the operator A t and let X = {(X, P s,x ) : (s, x) ∈ R + ×R d } be a time-inhomogeneous Markov process for which p is the transition density function, i.e. P s,x (X t = x; 0 ≤ t ≤ s) = 1, P s,x (X t ∈ Γ) = Γ p(s, x, t, y) dy, t > s for any Γ ∈ B(R d ). The process X admits the so-called strict Fukushima decomposition, i.e. for any 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 [25]). In particular, M s,· is a ({G s t }, P s,x )martingale. Moreover, for every (s, (see [25]). It follows that under P s,x the process B s,· defined as where σ · σ * = a, is a Brownian motion on [s, T ]. It is also known (see [11]) 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 , (see, e.g., [26]). Let us recall that for every Φ ∈ L 2 (0, T ; i.e. for every η ∈ L 2 (0, T ; H 1 (R d )), The following lemma will be needed in the next section to prove regularity of parabolic potentials.
, it follows from the above equality that In [11] it is proved that from which the desired result follows, because ξ s,ε ր ξ s , P s,x -a.s. as ε ց 0. ✷

Time-homogeneous diffusions
In the next sections it will be advantageous to consider a time-homogeneous diffusion determined by X. The standard construction of it is as follows. We set and consider the proces X on Ω ′ defined as Let F ′ t = σ(X u , u ≤ t), F ′ ∞ = σ(X u , u < +∞) and let G ′ ∞ denote the completion of F ′ ∞ with respect to the family P ′ = {P ′ µ : µ is a probability measure on R + × R d } and G ′ t denote the completion of F ′ t in G ′ ∞ with respect to P ′ . Then X ′ = {(X t , P ′ s,x ); (s, x) ∈ R + × R d } is a time-homogeneous Markov process with respect to the filtration {G ′ t } with the transition density P ′ (t, (s, x), Γ) = P (s, x, s + t, Γ s+t ), (2.5) where Γ s+t = {x ∈ R d : (s + t, x) ∈ Γ}. For h, t ≥ 0 we define θ h : Ω ′ → Ω ′ and τ (t) : Ω ′ → R + by putting 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 for ω ′ = (s, ω). Since P s,x (A s,t = A s,u + A u,t , 0 ≤ s ≤ u ≤ t) = 1 for every (s, x) ∈ R + × R d and M has the same property, it follows from (2.6) and the fact that the energy of A equals zero and the energy of M is finite that A = {A t , t ≥ 0} is a CAF of X ′ of zero energy and M = {M t , t ≥ 0} is a MAF of X ′ of finite energy. In particular, M is a ({G ′ t }, P ′ s,x )-martingale for every (s, x) ∈ R + × R d . Moreover, from (2.1), (2.4) it follows that for every (s, By (2.2) and (2.6), In fact, by [14,Theorem 12], it is an {G ′ t }-Brownian motion.

Capacity and soft measures
. 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

It is known that for a Borel
where P s,m (·) = R d P s,x (·) dx and m denotes the Lebesgue measure on R d .
From now on we 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 Borel measure on R + × R d . 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 R + × R d such that for every compact In what follows by Let us note that in the literature soft measures are usually defined on (0, T ) × D. In the present paper we define 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 ) and let m 1 denote the Lebesgue measure on R + × R d . Let us recall that a positive CAF A of X ′ and a positive soft measure µ on R + × R d are in the Revuz correspondence if It is known (see [17,24]) that under (2.8) the family of all integrable positive CAFs of X ′ and the family of all bounded positive soft measures on R + × R d are in one-to-one correspondence. In what follows the additive functional corresponding to a positive bounded soft measure µ will be denoted by and let p ′ D denote the transition density of the process X ′ killed at first exit from for every f ∈ B + (D T ) (see [17]).
In [9] 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 [9] if for q.e. (s, x) ∈ D T , 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.

Let us define
Using (2.3)-(2.5) and the fact that ζ τ (ω ′ ) = (ξ s (ω) − s) ∧ (T − s) one can show that (2.10) is satisfied iff (2.9) is satisfied with A µ replaced by A. Therefore a two parameter AF {A µ s,t } of X corresponds to µ in the sense of [9] iff the one-parameter AF A of X ′ defined by (2.11) corresponds to µ in the sense of (2.8).
Proof. Follows from [9]. ✷ We say that a measurable function f : The set of all quasi-integrable functions on D T will be denoted by qL 1 (D T ).
Remark 2.4. Analysis similar to that in [13,Remark 4.4] shows that if for every Directly from the definition of the Revuz duality it follows that f · m 1 corresponds to the PCAF A. This implies in particular that f ·m 1 is a smooth measure. Therefore for every compact K ⊂ R + × R d and for every ε > 0 there exists an open set Proof. By the assumptions, for q.e. (s, [18], 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 .g., [7]). By [1], for q.e. (s, x) ∈ D T , 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]). 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) ր 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 [10] 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. 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 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 : The processes Y, Z in the above definition may be considered as processes on R + . Indeed, if we set Z t = 0 for t ≥ σ and Y t = Y σ for t ≥ σ the then the equation in condition (c) is satisfied for every t ≥ 0 if we admit the convention that b a = 0 for a ≥ b.
Let us consider the following assumptions.

Markov type BSDEs
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 (s, x) ∈ D T . We say that a pair (Y s,x , Z s,x ) consisting of an R-valued process Y s,x and an R d -valued process Z s,x is a solution of BSDE s, Assume that (A1), (A2), (AZ) are satisfied, Z ∈ M q for some q > α, there exist and hence ). 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. ✷ In the case of Markov type equations the following hypotheses are analogues of (A1)-(A5) and (AZ).
Moreover, for q.e. (s, x) ∈ D T and every t ∈ [0, T − τ (0)], If f does not depend on z then there is C depending only on κ, T such that Therefore by [10, Proposition 3.10], for q.e. (s, so from [8, Lemma A.3.5] it follows that there exists a process Y such that Y using once again [8, Lemma A.3.5] we conclude that there exists a process Z such that Z s,x = Z, dt ⊗ P ′ s,x -a.e. on [0, ζ τ ] × Ω ′ for q.e. (s, x) ∈ D T . Thus, for q.e. (s, x) ∈ D T , we deduce existence of a pair (Y, Z) such that (3.2) is satisfied. Let Λ ⊂ Ω ′ be a set of those ω ∈ Ω ′ for which Y is càdlàg and Then Λ ∈ G ′ ∞ and of course P ′ s,x (Λ) = 1 for q.e. (s, x) ∈ D T . Therefore by Proposition 3.2 and Lemma 3.3, for every t ∈ [0, T − s], for q.e. (s, x) ∈ D T . Put C t = t 0 Z r dr, t ≥ 0. Since Z1 [0,ζτ ] = Z, using the property of Z proved in Proposition 3.2 one can check that for every h ≥ 0, Accordingly, C is a CAF of the process X killed at first exit from D T . Therefore it follows from [27, 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 the Itô-Tanaka formula, If κ ≤ 0 then by (H3), −Ŷ r (f (r, X r , Y r )) ≥ |f (r, X r , Y r )|−|f (r, X r , 0)|. On substituting this into (3.4) and then taking expectation on both sides of the inequality we get (3.3) with C = 1. The general case can be reduced to the case κ ≤ 0 by the standard change of variables. ✷ In what follows we denote by T k , k ≥ 0, the truncature operator, i.e.
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 there exists a subsequence (still denoted by n) such that p D (s, x, θ, y)T k (|ψ n − ψ|) → 0, m 1 -a.e. on [s, T ] × D, which by the positivity of p D (s, x, ·, ·) and the definition of T k implies that ψ n → ψ, m 1 -a.e. on [s, T ] × D. Since (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 . ✷ We say that a measurable function u : D T → R is of class (FD) if for q.e. (s, x) ∈ D T the process [s, T ] ∋ t → u(t, X t ) is of Doob's class (D) under the measure P s,x .
We say that a measurable function u : D T → R is quasi-càdlàg if for q.e. (s, x) ∈ D T the process [s, T ] ∋ t → u(t, X t ) is càdlàg under P s,x .
FD is the set of all quasi-càdlàg functions u : D T → R. FS q (resp. FD q ), q > 0, is the set of all quasi-continuous (resp. quasi-càdlàg) functions u : D T → R such that for q.e. (s, x) ∈ D T , E s,x sup s≤t≤T |u(t, X t )| q < +∞.
F M q , q > 0, is the set of all measurable functions u : D T → R such that for q.e. (s, x) ∈ D T , is the set of all measurable functions u on D T such that for every k ≥ 0, T k (u) ∈ L 2 (0, T ; H 1 0 (D T )). From [3] it follows that for every u ∈ T 0,1 0 there exists a unique measurable function v on D T such that ∇T k (u) = 1 {|u|<k} v, m 1 -a.e. We shall set ∇u = v.
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 for q.e. (s, x) ∈ D T . By [10, Theorem 5.7], for q.e. (s, x) ∈ D T , 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 and (3.7), once again by Proposition 3.5, and and ψ n is a measurable function on D T . Now fix n ∈ N and put (Y , Z) = (Y n , Z n ), φ = φ n . Then for q.e. (s, The proof that v has the desired regularity properties we divide into 3 steps. Step 1. We assume that µ = g · m 1 for some g ∈ L 2 (D T ) and ϕ ∈ L 2 (D). Let w ∈ W(D T ) be a weak solution of the problem ∂w ∂t + A t w = −g in D T , w(T, ·) = ϕ, w(t, ·) |∂D = 0, t ∈ (0, T ).
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, s,x -a.s. for q.e. (s, x) ∈ D T . This implies that v ′ ∈ FD q , 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 ) and ϕ ∈ L 2 (D). Therefore for q.e. (s, x) ∈ D T , From this and Lemma 2.5, . (3.16) Due to [20], 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 Φ = (g) t + div(G) + f, (4.2) where g ∈ L 2 (0, T ; H 1 0 (D)), G, 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.
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 in the sense of distributions, Note that different but equivalent definition of a renormalized solution of (4.1) is given in [21].
(ii) If u is a renormalized solution of (4.1) then it is a distributional solution of (4.1) in the sense that u, ∇u ∈ L 1 (D T ) and for any η ∈ C ∞ 0 ((0, T ] × D), (see Proposition 4.5 and Theorem 4.11 in [21]).
Proof. It is enough to use Lemma 4.2, observe that u n satisfies all the properties of Proposition 4 in [20], apply equivalence between renormalized and entropy solutions for parabolic equations proved in [6] and uniqueness of them (see [20]). ✷ Then v ∈ T 0,1 2 , v ∈ L q (0, T ; W 1,q 0 (D)) for q ∈ [1, d+2 d+1 ) and v is a renormalized solution of the problem ∂v ∂t + A t v = −µ, v(T, ·) = ϕ, v(t, ·) |∂D = 0, t ∈ (0, T ). (4.7) 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 But it is known (see [18,23]) that {v n } converges in L 2 (D T ) to a unique weak solution of (4.7). Therefore v is a weak solution of (4.7). Since a weak solution of (4.7) is a renormalized solution, this proves the proposition under the additional assumptions on ϕ, µ. Assume now that ϕ ∈ L 1 (D) is nonegative and µ ∈ M + 0,b (D T ). By [19] 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 Fn ·µ ∈ M + 0,b (D T )∩ W ′ (D T ) and µ(D T \ n F n ). Let ϕ n = ϕ∧ n. By what has already been proved, v n defined as v n (s, is a renormalized solution of (4.7) 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 (4.8) 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 (4.7). ✷ 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 1,q 0 (D)) for q ∈ [1, d+2 d+1 ) and for q.e. (s, x) ∈ D T . Finally, there exists C > 0 depending only on κ, T such that ) there exist C, γ > such that u L q (0,T ;W 1,q 0 (D)) ≤ C( ϕ L 1 (D) + f (·, ·, 0) L 1 (D T ) + µ T V ) γ , because from Remark 4.1(ii) and results proved in [4, Section 3] (see also [5]) 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, the right-hand side f + dµ and obstacles h 1 , 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 [22] minimality of ν was expressed by using the notion of precise versions of functions introduced in [23]. In fact, in the linear case with L 2 data, condition (c) coincides with that introduced in [22], because for every parabolic potential h,ĥ(X t ) = h − (X t ), t ∈ [0, ζ τ ], whereĥ is the precise version of h (see [9] for detailes).
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 [10].
Suppose we are given a filtered probability space and A, B, σ, ξ as in Subsection 3.1, and moreover, 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).
We will need the following general growth conditions for f .
(H6) There exists a measurable function v : There exists a measurable function v : D T → R, a measure λ ∈ M 0,b (D T ) and φ ∈ L 1 (D), φ ≥ ϕ, such that v is a renormalized solution of (5.1) and Proof. Follows immediately from Proposition 5.3.

✷
Before proving our main result on existence and approximation by the penalization method of solutions of the obstacle problem with one barrier let us recall that a function v on D T is a supersolution of PDE(ϕ, f + dµ) if there exists a measure λ ∈ M + 0,b (D T ) such that v is a renormalized solution of the problem ∂v ∂t (i) There exists a solution (u, ν) of OP(ϕ, f + dµ, h 1 ) such that u is of class (FD), u ∈ FD q for q ∈ (0, 1), ∇u ∈ F M q for q ∈ (0, 1), u ∈ T 0,1 2 , u ∈ L q (0, T ; W 1,q 0 (D)) for q ∈ [1, d+2 d+1 ). (ii) For n ∈ N let u n be a renormalized solution of the problem Then u n ր u q.e. on D T , u n → u in L q (0, T ; W 1,q 0 (D)), q ∈ [1, d+2 d+1 ) and ν n → ν weakly, where dν n = n(u n − h 1 ) − dm 1 .
By (5.5) and (5.6), for q.e. (s, x) ∈ D T . It is an elementary check that A is an AF of X ′ . In fact, by (5.7), it is a positive functional. Therefore there exists a positive smooth measure ν on D T such that A = A ν . From this and (5.5) it follows that for q.e. (s, x) ∈ D T . Let v denote the function from condition (H6). By Lemma 5.2, v(X) satisfies (A6) (with L = h 1 (X)) under P ′ s,x for q.e. (s, x) ∈ D T . Therefore arguing as at the beginning of the proof of [10, Theorem 5.7]) and using Theorem 4.5 one can show that there exists a supersolutionv of PDE(ϕ ∨ φ, f + dµ) such that u n ≤v q.e. on D T for n ∈ N. From this, the fact that u 1 ≤ u q.e. on D T and (H3) for q.e. (s, x) ∈ D T . Hence, by Lemma 2.5, ν ∈ M + 0,b (D T ). From (5.8) and Proposition 4.4 it follows now that u is a renormalized solution of the problem (4.1) with µ replaced by µ + ν, u ∈ T 0,1 2 and u ∈ L q (0, T ; W 1,q 0 (D)) for q ∈ [1, d+2 d+1 ). Actually, from the definition of a solution of reflected BSDE and (5.5), (5.7) it follows that the pair (u, ν) is a solution of OP(ϕ, f + dµ, h 1 ). Moreover, from (5.5) and the fact that for q.e. (s, x) ∈ D T the process Y s,x is of class (D) and Y s,x ∈ D q for q ∈ (0, 1) we conclude that u is of class (FD) and u ∈ FD q for q ∈ (0, 1). Furthermore, since (u(X), ψ(X)) is a solution of BSDE s,x (ϕ, D, f u + d(µ + ν)) it follows from Proposition 3.7 that for q.e.