Large Time Behavior of Solutions to Parabolic Equations with Dirichlet Operators and Nonlinear Dependence on Measure Data

We study large time behavior of renormalized solutions of the Cauchy problem for equations of the form ∂tu − Lu + λu = f(x, u) + g(x, u) ⋅ μ, where L is the operator associated with a regular lower bounded semi-Dirichlet form and μ is a nonnegative bounded smooth measure with respect to the capacity determined by . We show that under the monotonicity and some integrability assumptions on f, g as well as some assumptions on the form , u(t, x) → v(x) as t →∞ for quasi-every x, where v is a solution of some elliptic equation associated with our parabolic equation. We also provide the rate convergence. Some examples illustrating the utility of our general results are given.


Introduction
Let E be a locally compact separable metric space, m an everywhere dense Borel measure on E and let L be the operator associated with a regular lower bounded semi-Dirichlet form (B, V ) on L 2 (E; m). The main purpose of the paper is to study large time behavior of solutions of the Cauchy problem ∂ t u − Lu + λu = f (x, u) + g(x, u) · μ in (0, ∞) × E, u(0, ·) = ϕ on E. (1.1) In Eq. 1.1, ϕ : E → R, f, g : E × R → R are Borel measurable functions, μ is a smooth measure with respect to the parabolic capacity determined by (B, V ). The class of operators corresponding to regular lower bounded Dirichlet forms is quite large. It contains both local operators whose model example is the Laplace operator or Laplace operator perturbed by the first order operator, as well as nonlocal operators whose model example is the α-Laplace operator α/2 with α ∈ (0, 2) or α-Laplace operator with variable exponent α satisfying some regularity conditions. Many interesting examples of operators associated with regular semi-Dirichlet forms are to be found in [11,15,18,22,26]. In fact, our methods also allow to treat equations with operators associated with quasi-regular forms (see remarks at the end of Section 5).
As for the data ϕ, f, g, we assume that ϕ ∈ L 1 (E; m), f, g are continuous and monotone in the second variable u and satisfy mild integrability conditions. Our basic assumption on μ is that it is a smooth measure (with respect to the capacity associated with (B, V )) of class R + (E), i.e. a positive smooth measure such that E x A μ ∞ < ∞ for quasi-every (q.e. for short) x ∈ E, where A μ is the additive functional of the Hunt process associated with (B, V ) in the Revuz correspondence with μ. Equivalently, our condition imposed on μ means that the potential (associated with (B, V )) of μ is m-a.e. finite. It is known that if (B, V ) is a non-symmetric form, and moreover, it is transient or λ > 0, then R + (E) contains the class M + 0,b (E) of positive bounded smooth measures on E (see Section 2). In general, the inclusion M + 0,b (E) ⊂ R + (E) is strict (see Section 2). Elliptic equations with unbounded measures of class R + (E) are considered for instance in the monograph [23]; see also Section 6.
Let v be a solution of the elliptic equation Our main result says that under the assumptions on ϕ, f, g mentioned before and some additional mild assumptions on the semigroup (P t ) and the resolvent (R α ) associated with (B, V ), lim t→∞ u(t, x) = v(x) (1.3) for q.e. x ∈ E. We also estimate the rate of convergence. Our main estimate says that for every q ∈ (0, 1) there is C(q) > 0 such that for q.e. x ∈ E, The quantities on the right hand-side of Eq. 1.4 can be estimated for concrete operators L. We give some examples in Section 6. To our knowledge, in case L is a nonlocal operator, our results (1.3), (1.4) are entirely new. In case L is local, we generalize the results obtained in the paper [12] in which g ≡ 1 and L is a uniformly elliptic divergence form operator. Note, however, that in [12] systems of equations are treated. We also strengthen slightly the results of [20] concerning asymptotic behavior of nonnegative solutions of equations involving Laplace operator and absorbing term of the form h(u)|∇u| 2 with h satisfying the "sign condition". Some other results on asymptotic behavior, which are not covered by our approach, are to be found in [28][29][30]. In [29,30] equations involving Leray-Lions type operators and smooth measure data are considered while [28] deals with linear equations with general, possibly singular, bounded measure μ. Note that the methods used in [28][29][30] do not provide estimates between the parabolic solution and the corresponding stationary solution.
In order to prove (1.3) and (1.4), we develop the probabilistic approach initiated in [12]. We find interesting that it provides a unified way of treating a wide variety of seemingly disparate examples (see Section 6).
Although in the paper we deal mainly with the asymptotic behavior for solutions of Eq. 1.1, the first question we treat is the existence and uniqueness of solutions of problems (1.1) and (1.2). Here our results are also new, but our proofs rely on our earlier results proved in [15,18] in case g ≡ 1. In fact, in the parabolic case we prove the existence and uniqueness of solutions to problems involving operators L t and data f, g, μ depending on time, i.e. more general then problem (1.1). Finally, let us note that in the paper we consider probabilistic solutions of Eqs. 1.2 and 1.3 (see Section 3 for the definitions). It is worth pointing out, however, that in the case where (B (t) , V ) are (non-symmetric) Dirichlet forms, the probabilistic solutions coincide with the renormalized solutions defined in [17] (in the elliptic case under the additional assumption that (B, V ) satisfies the strong sector condition and either (B, V ) is transient or λ > 0). For local operators these renormalized solutions coincide with the usual renormalized solutions (see [9,31] and also [16]).

Preliminaries
In the paper E is a locally compact separable metric space, E 1 = R×E, m is an everywhere dense Borel measure on E and m 1

Dirichlet Forms
Let H = L 2 (E; m) and let (·, ·) denote the usual inner product in H . We assume that we are given a family {B (t) , t ∈ [0, T ]} of regular semi-Dirichlet forms on H with common domain V ⊂ H (see [26,Section 1.1]). We assume that the forms B (t) are lower bounded and satisfy the sector condition with constants α 0 ≥ 0, K ≥ 1 independent of t ∈ [0, T ]. Let us recall that this means that Without loss of generality, we assume α 0 < 1. We also assume that where B(ϕ, ϕ) = B (0) (ϕ, ϕ). By putting B (t) = B for t / ∈ [0, T ], we may and will assume that B (t) is defined and satisfies (2.1) for all t ∈ R. As usual, we denote byB (t) the symmetric part of B (t) , i.e.B (t) (ϕ, ψ) = 1 2 (B (t) (ϕ, ψ) + B (t) (ψ, ϕ)). Note that by the assumption, V is a dense subspace of H and the form (B, V ) is closed, i.e. V is a real Hilbert space with respect toB 1 (·, ·), which is densely and continuously embedded in H . We denote by · V the norm in V , i.e. ϕ 2 V = B 1 (ϕ, ϕ), ϕ ∈ V . We denote by V the dual space of V , and by · V the corresponding norm. We set H = L 2 (R; H ), V = L 2 (R; V ), V = L 2 (R; V ) 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 denote by E the time dependent Dirichlet form associated with the family where ·, · is the duality pairing between V and V, and Note that E can be identified with some generalized Dirichlet form (see [ We denote by S(E) the set of all smooth measures on E with respect to the form (B, V ) (see, e.g., [26,Section 4.1] for the definition). S(E 1 ) is the set of all smooth measures on E 1 with respect to E (see [14]), and S(E 0,T ) is the set of all smooth measures on E 1 with support in E 0,T . We denote by M b (E 0,T ) the set of all signed Borel measures on E 1 with support in E 0,T such that |μ|(E 1 ) < ∞, where |μ| stand for the total variation of μ.
By the above inequality and Eq. 2.1, Let f be a continuous function on R with compact support such that f ≥ 1 on [−T , 2T ] and let η ε = f ψ ε . Then η ε ∈ W and by [26, (6.2.21)] and Eq. 2.7, where C > 0 depends only on c and α. Since ε > 0 was arbitrary, the desired result follows.

Markov Processes and Additive Functionals
In what follows E ∪ {∂} is a one-point compactification of E. If E is already compact then we adjoin ∂ to E as an isolated point. When considering Dirichlet forms, we adopt the convention that every function f on E is extended to E ∪ {∂} by setting f (∂) = 0. When considering time dependent Dirichlet forms, we adopt the convention that every function is an E-quasi-continuous m 1 -version of the resolvent associated with the form E. By [26,Theorem 6.3 is a decomposition of X into the process on R and on E, then τ is the uniform motion to the right, i.e.
is a Hunt process with life time ξ = inf{t ≥ 0 : X t ∈ ∂} associated with the form (B (0) , V ). Let us recall that an additive functional (AF for short) of M is called natural if A and M have no common discontinuities. It is known (see [14,Section 2]) that for every μ ∈ S(E 1 ) there exists a unique positive natural AF A of M such that A is in the Revuz correspondence with μ, i.e. for every m 1 -integrable α-coexcessive function h with α > 0, where E h·m 1 denotes the expectation with respect to P h·m 1 (·) = E 1 P z (·)h(z) m 1 (dz). In what follows we will denote it by A μ . Conversely, if A is a positive natural AF of M then modifying the proof of [11, Lemma 5.1.7] (we replace quasi-notions and facts used in the proof in [11] by the corresponding quasi-notions and facts from [26,; for the case of (non-symmetric) Dirichlet form see also [25,Theorem 5.6]) one can show that there exists a smooth measure on E 1 such that A is in the Revuz correspondence with μ. We set By [14,Proposition 3.4], in the definition of R(E 0,T ) one can replace m 1 -a.e. by q.e. (with respect to E). By [14,Proposition 3.8], if (B, V ) is a (non-symmetric) Dirichlet form or, more generally, a semi-Dirichlet form satisfying the duality condition (see [14] for the definition), then We set (i) For every s ≥ 0 the distribution of (X • θ τ (0) , A 0,μ • θ τ (0) ) under P s,x is equal to the distribution of (X, dr. Therefore (i) follows from the fact that the distribution of X under P 0,x is equal to the distribution of X • θ τ (0) under P s,x . Now assume that μ belongs to the set S 0 (E) of smooth measures of finite energy. . From this and the first part we deduce that (i) is satisfied for everyμ ∈ S 0 (E). By [26, Lemma 4.1.14], there exists a nest {F n } such that 1 F n ·μ ∈ S 0 (E) for each n ∈ N. Since we already know that (i) holds forμ replaced by 1 F n ·μ, applying the monotone convergence theorem we conclude that it holds forμ replaced by 1 ∞ n=1 F n ·μ, and hence forμ because the set E \ ∞ n=1 F n is exceptional.
. Under (2.9) the distribution of A under P s,x is equal to the distribution of A 0,μ under P 0,x . Hence One can check that A is a CAF of M. Let ν denote its Revuz measure. Then for every f of Hence ν = dt ⊗μ = μ. Since additive functionals are uniquely determined by their Revuz measures, this proves (ii).

Parabolic PDEs and Generalized BSDEs
For t ∈ [0, T ] let L t denote the operator associated with the form ( (see [22,Proposition I.2.16]). Suppose we are given measurable functions ϕ : E → R, f, g : E T × R → R and μ ∈ R(E 0,T ). In this section we consider the following Cauchy problems with terminal and initial conditions: and Definition Let z ∈ E T . We say that a pair (Y z , M z ) is a solution of the BSDE Here |A μ | t denotes the total variation of the process A μ on [0, t]), (c) Eq. 3.4 is satisfied P z -a.s.
Let us recall that a càdlàg (F t )-adapted process Y is of Doob's class D under P z if the collection {Y τ : τ ∈ T }, where T is the set of all finite valued (F t )-stopping times, is uniformy integrable under P z . Let L 1 (P z ) denote the space of càdlàg (F t )-adapted processes Y with finite norm It is known that L 1 (P z ) is complete (see [10, p. 90]). Moreover, if processes Y n are of class D and Y n → Y in L 1 (P z ), then Y is of class D. To see this, let us fix ε > 0 and choose n so that Y n − Y z,1 ≤ ε/2. Since the family {Y n τ } is of class D, there exists δ > 0 such that if P z (A) < δ, then A |Y n τ | dP z < ε/2. It follows that if P z (A) < δ then for every finite which shows that {Y τ } is uniformly integrable (see [34,Theorem I.11]). To simplify notation, in what follows we write Definition (a) We say that u : is a solution of the Cauchy problem with terminal condition of the form Remark 3.1 If Eq. 3.6 has the uniqueness property (i.e. has a unique solution v T for every To see this, let us write Of course, Eq. 3.7 will be proved once we show that It is known (see [14, p. 1213]) that there exists a generalized nest {F n } on E 0,T +a such that n,T +a : denote the solution of the linear equation By [14,Theorem 3.7], v n T +a is a weak solution of Eq. 3.10. Therefore making a simple change of variables shows that v n T +a,a is a weak solution of the linear equation ∂v n Using the probabilistic representation of the solution of Eq. 3.10 and the fact that {F n } is a nest, one can easily show that v n T +a → v T +a pointwise as n → ∞. Similarly, using the probabilistic representation of the solution of Eq. 3.12 one can show that v n T +a,a converges pointwise as n → ∞ to the solution of Eq. 3.8, that is to v T . This and Eq. 3.11 imply (3.9).
In the rest of this section we say that some property is satisfied quasi-everywhere (q.e. for brevity) if the set of those z ∈ E 1 for which it does not hold is exceptional with respect to the form E.
In what follows we say that a Borel measurable F : Let us remark that if μ = m 1 , then A μ t = t, t ≥ 0, so m 1 -quasi-integrability coincides with the notion of quasi-integrability considered in [14,Section 5]) (see also [13,Section 2]).
Our basic assumptions on the data are the following.
(P2) f (·, ·, y), g(·, ·, y) are measurable for every y ∈ R and f (t, x, ·), g(t, x, ·) are continuous for every In what follows we denote by D q (P z ), q > 0, the space of all Then there exists a càdlàg (F t )-adapted process M such that M t = M z t , t ∈ [0, T ], P z -a.s. for q.e. z ∈ E 0,T , and for q.e z ∈ E 0,T the pair (u(X), M) is a unique solution of Eq. 3.4 on the space ( , F, P z ). Moreover, u(X) ∈ D q (P z ) for q ∈ (0, 1) and M is a uniformly integrable martingale under P z for q.e. z ∈ E 0,T . Finally, for q.e. z ∈ E 0,T , Proof By using the standard change of variables (see, e.g., the beginning of the proof of [4, Lemma 3.1]), without loss of generality we may and will assume that α ≤ 0 in condition (P3).
We first prove (ii). The uniqueness of a solution of BSDE (3.4) follows from (P3), (P5) and the fact that μ is nonnegative. The proof is standard. We may argue for instance as in the proof of [15, Proposition 2.1] with obvious changes. We divide the proof of existence of a solution into two steps. Step (3.14) Then modifying slightly the proof of [15, Lemma 2.6], we show that there exists a unique solution (Y, M) of the BSDE on the space ( , F, P z ) (for brevity, in our notation we drop the dependence of Y, M on z). Let sgn(x) = 1 if x > 0 and sgn(x) = −1 if x ≤ 0. By the Meyer-Tanaka formula (see [34, p. 216]) and the fact that A μ is continuous, From this, Eq. 3.15 and (P3), (P5) we get (3.16) which shows that in fact (Y, M) is a solution of Eq. 3.15 withf c replaced by f andḡ c replaced by g Step 2. For n ≥ 0, we set ξ n = T n (ξ ), f n (t, Step 1, for each n ≥ 0 there exists a unique solution (Y n , M n ) of the BSDE on the space ( , F, P z ) (as in Step 1, for brevity, in our notation we drop the dependence of , A m is an increasing process. Therefore using the Meyer-Tanaka formula we obtain From the above and (P3), (P5) it follows that Hence Observe that from our assumptions on the data ϕ, f, g, μ it follows that E z n → 0 as We have Applying the Meyer-Tanaka formula we get (see the proof of Eq. 3.16) For k, N ∈ N, we set From the definition of τ k,N it follows that From this, (P2) and Eq. 3.19 one can deduce that By Doob's inequality (see, e.g., [21, Theorem 1.9.1]) and Eq. 3.19, for every ε > 0 we have Similarly, by Eqs. 3.21, 3.22 and Doob's inequality, for every ε > 0. Letting n → ∞ in Eq. 3.20 and using Eqs. 3.23-3.25 we conclude that We have By the Meyer-Tanaka formula and Eq. 3.17, so applying Fatou's lemma and Eq. 3. 19 gives Hence Y z τ k,N → Y z τ k P z -a.s., and consequently, Since τ k → ζ τ as k → ∞, letting k → ∞ in Eq. 3.29 and repeating arguments used to prove (3.29) we get We may now repeat the reasoning following [18, (3.6)] with the process V from [18] replaced by · 0 g(t, Y z t ) dA μ t (see also the reasoning following Eq. 4.26 in the present paper) to prove that the pair (Y z ,M z ), whereM z is a càdlàg version of the martingale is a solution of the BSDE on ( , F, P z ). Furthermore, by [15,Remark 3.6], there exists a pair of processes Then the argument from the beginning of the proof of [14,Theorem 5.8] shows that Y t = u(X t ), t ∈ [0, ζ τ ], which implies that M is a version of the martingale M z and that (u(X), M) is a solution of Eq. 3.30 for q.e. z ∈ E 0,T . In view of our convention made at the beginning of Section 2.2, this means that (u(X), M) is a solution of Eq. 3.4 on the space ( , F, P z ) for q.e. z ∈ E 0,T . Of course, u(X) ∈ D q (P z ). Furthermore, M is a uniformly integrable martingale under P z , because under P z it is a version of the closed martingale M z . Finally, since we know that Y z t = u(X), t ∈ [0, ζ τ ], P z -a.s., Eq. 3.13 follows immediately from Eq. 3.27. This completes the proof of part (ii) of the theorem.
Part (i) follows from (ii). Indeed, since μ ∈ R(E 0,T ) and we know that Eq. 3.27 is satisfied with Y z replaced by u(X) and M is a martingale under P z for q.e. z ∈ E 0,T , putting t = 0 in Eq. 3.4 and then taking the expectation shows thatū is a solution of Eq. 3.2. To show thatū is unique one can argue as in the proof of [14,Theorem 5.8].
where · T V denotes the total variation norm. Therefore, by [14,Theorem 3.12], u ∈ L 1 (E 0,T ; m 1 ), T k u ∈ L 2 (0, T ; V ) for k > 0 (T k u is defined by Eq. 3.14) and for every k > 0 there is C > 0 depending only on k, α, T such that Moreover, if the forms (B (t) , V ) are (non-symmetric) Dirichlet forms, then by [17,Theorem 4.5], u is a renormalized solution of Eq. 3.2 in the sense defined in [17].
Remark 3.5 In Theorem 3.2 we have assumed that the AF A μ is continuous. In the general case where μ ∈ R + (E 0,T ) and A μ is possibly discontinuous, one can prove the existence of a solution of Eq. 3.2 in the following sense: there exists u : E T → R such that f u ·m, g u ·μ ∈ R(E 0,T ) and Eq. 3.5 is satisfied with g u replaced by gû, whereû is the precise version of u (for the notion of a precise version of a parabolic potential see [32]). In the paper we decided to provide the proof of less general result, because it suffices for the purposes of Sections 4-6 in which our main results are proved, and on the other hand, the proof of the general result is more technical than the proof of Theorem 3.2. Also note that by [14,Proposition 3.4], the solution u described above is quasi-càdlàg.

Convergence of BSDEs and Elliptic PDEs
In this section, we assume that Eq. 2.9. We denote by L the operator associated via (3.1) with the form (B, V ). We also assume that μ ∈ R + (E 0,T ) does not depend on time and f, g : E × R → R, i.e. f, g also do not depend on time.
For v ∈ E → R, we set To shorten notation, in what follows we denote P 0,x by P x , E 0,x by E x and · (0,x);1 by · x,1 . Under the measure P x , and In the rest of the paper we say that some property is satisfied quasi-everywhere (q.e. for brevity) if the set of those x ∈ E for which it does not hold is exceptional with respect to the form (B, V ).
Let ν ∈ S(E). We will say that a Borel measurable F : E → R is ν-quasi-integrable (F ∈ qL 1 (E; ν) in notation) if for every T > 0, P x ( Note that in case ν = m the notion of quasi-integrability was introduced in [13, Section 2]. For a comparison of the notion of m-integrability and the notion of quasi-integrability in the analytic sense see [13,Remark 2.3].
In this section and Section 5, we assume that the data satisfy the following conditions.
Definition Let x ∈ E. We say that a pair (Y x , M x ) is a solution of the BSDE on the space ( , F, P x ) if (a) Y x is an (F t )-progressively measurable process of class D under P x , Y x t∧ζ → 0, P xa.s. as t → ∞ and M x is an (F t )-local martingale under P x such that M x 0 = 0, and Suppose that for some x ∈ E for every n > 0 there exists a solution (Y n , M n ) of the BSDE on the probability space ( , F, P x ). The solutions may depend on x but for brevity, in our notation we drop the dependence of Y n , M n on x. In what follows byỸ n ,M n we denote the processes defined as Y n t = Y n t ,M n t = M n t , t < n,Ỹ n t = 0,M n t = M n n , t ≥ n. and for every q ∈ (0, 1 From the above and the fact that the process A μ is continuous it follows that the pair (Ỹ n ,M n ) defined by Eq. 4.5 satisfies where V n t = 0 ift < n, V n t = −Y n n ift ≥ n. Let δỸ =Ỹ m −Ỹ n . By Eq. 4.10, which shows Eq. 4.7. Finally, to prove (4.8), we first observe that by the Meyer-Tanaka formula, By the above inequality and Eq. 4.9, for t < n we have On the other hand, for every t ≥ 0, Assume also for some x ∈ E for each n ∈ N there exists a solution (Y n , M n ) of Eq. 4.4 on the space ( , F, P x ). If and for every q ∈ (0, 1), lim Proof From Eqs. 4.6 and 4.13, 4.14 it follows that for every x ∈ E, Y n − Y m x,1 → 0 as n, m → ∞. Hence there exists a process Y ∈ L 1 (P x ) of class D such that Eq. 4.16 is satisfied. By Eqs. 4.7, 4.13 and 4.14, lim n,m→∞ E x sup t≥0 |Y n t − Y m t | q → 0. Since the space D q (P x ) is complete, the last convergence and Eq. 4.16 imply that Y x ∈ D q (P x ) and Eq. 4.17 is satisfied. From Eqs. 4.8, 4.16, 4.17 and Fatou's lemma it follows that for every T > 0, Since 1 {n≥ζ } Y n ζ = 0 P x -a.s. for n ∈ N, from Eq. 4.17 we conclude that Y x T ∧ζ → 0 in probability P x as T → ∞. As a consequence, since Y x is of class D, E x |Y x T ∧ζ | → 0. Therefore letting T → ∞ in the last inequality we get (4.15). Using Eq. 4.17 one can show that From the definition of δ N,R and (E2), Eq. 4.17 it follows that Hence, by Doob's inequality (see, e.g., [ From the last convergence and Doob's inequality it follows that for every ε > 0, Therefore letting N → ∞ in Eq. 4.22 and using Eq. 4.15 we show that P x -a.s., We now show that τ R ∞ P x -a.s. as R → ∞. To see this, let us suppose that P x (sup R>0 τ R ≤ M) > ε for some M, ε > 0. Then Clearly, By Eq. 4.17, taking a subsequence if necessary, we may assume that sup t≤M |Y n t − Y t | → 0 P x -a.s. Therefore the random variable Z = sup n≥0 sup t≤M |Y n t − Y t | is finite a.s., which when combined with Eq. 4.25 contradicts (4.24). This proves that τ R ∞ P x -a.s. Now, letting R → ∞ and repeating argument used to prove (4.23), we get (4.23) with T ∧ ζ ∧ τ R replaced by T ∧ ζ . Since we know that E x |Y x T ∧ζ | → 0 as T → ∞, letting T → ∞ in this equation (i.e. in Eq. 4.23 with T ∧ ζ ) and repeating once again the argument used to prove (4.23) we get Hence (4.27) where M x is a càdlàg version of the martingale Indeed, by Eq. 4.26, From the above it follows that M x t∧ζ = M x t , t ≥ 0, and moreover, that Letting T → ∞ and using the fact that Y x T ∧ζ → Y x ζ = 0 P x -a.s. we obtain (4.27). Thus the pair (Y x , M x ) is a solution of Eq. 4.2.
Then there is a càdlàg (F t )-adapted process M such that M t = M x t , t ≥ 0, P x -a.s. for q.e x ∈ E and for q.e. x ∈ E the pair (v(X), M) is a unique solution of Eq. 4.2 on the space ( , F, P x ). Moreover, v(X) ∈ D q (P x ) for q ∈ (0, 1) and M is a uniformly integrable martingale under P x for q.e. x ∈ E.
Proof We first prove part (ii). The uniqueness of a solution of Eq. 4.2 follows easily from (E3), (E5) and the fact that μ is positive. To see this it suffices to modify slightly the proof of [15,Proposition 3.1]. To prove the existence of a solution, we first note that by Theorem 3.2, for q.e. x ∈ E for every n ∈ N there exists a unique solution (Y n , M n ) of the BSDE (4.4) with ϕ ≡ 0 on the space ( , F, P x ). Since f (·, 0) · m, g(·, 0) ·μ ∈ R(E), condition (4.14) is satisfied for q.e. x ∈ E. Therefore, by Proposition 4.2, for q.e. x ∈ E there exist a solution (Y x ,M x ) of BSDE (4.2). In fact, Y x is given by Eq. 4.26 andM x is a càdlàg version of the martingale (4.28). Repeating step by step the proof of [15,Theorem 4.7] one can show that there is a pair of càdlàg processes (Y, M) not depending on x such that This shows that the pair (v(X), M) is a solution of Eq. 4.2 on the space ( , F, P x ) for q.e. x ∈ E. By Proposition 4.2, v(X) ∈ D q (P x ) for q ∈ (0, 1), and M is a uniformly integrable (F t )-martingale under P x . This completes the proof of (ii). Part (i) follows immediately from (ii), because g v · μ ∈ R(E) and Eq. 4.15 is satisfied with Y x replaced by v(X), so for q.e. x ∈ E we can integrate with respect to P x both sides of Eq. 4.2 with t = 0 and Y x replaced by v(X).

Large Time Asymptotics
In this section, as in Section 4, we assume that Eq. 2.9 is satisfied and the data f, g, μ do not depend on time. We denote by L the operator corresponding to (B, V ). We continue to write P x for P 0,x and E x for E 0,x , and as in Section 4, the abbreviation "q.e." means quasi-everywhere with respect to the capacity determined by (B, V ).
Suppose that for every T > 0 there exists a unique solution u T of Eq. 3.2 with L and the data f, g, μ satisfying the above assumptions. By Remark 3.1, by putting we define a probabilistic solution u of Eq. 1.1, i.e. solution of the problem Our goal is to prove that under suitable assumptions, u(t, x) → v(x) as t → ∞ for q.e.
x ∈ E, where v is a solution of Eq. 1.2 with λ = 0, i.e. solution of the problem whereμ is determined by Eq. 2.6. We will also estimate the rate of the convergence. The proofs of these results rely on the results of Section 4. The main idea is as follows. We have where u T is a solution of the problem In particular, putting t = T , we get u(T , x) = u T (0, x). Hence, by Eq. 3.5, because ζ τ = T ∧ ζ under the measure P x . On the other hand, by Lemma 2.2, Therefore our problem reduces to showing that the right-hand side of Eq. 5.5 converges to the right-hand side of Eq. 5.6 as T → ∞, and to estimating the difference between the two expressions by some function of T .
for q.e. x ∈ E. In fact, for q.e. x ∈ E, for all T > 0.
Proof Let Y T be the first component of the solution of Eq. 4.4 (with T = n) and Y be the first component of the solution of Eq. 4.2. Since Eq. 4.14 is satisfied for q.e. x ∈ E, applying Proposition 4.2 we conclude that for every q ∈ (0, 1), for q.e. x ∈ E. On the other hand, by Theorem 3.2 and Theorem 4.3, for q.e. x ∈ E we have where u T is a solution of Eq. 5.4 and v is a solution of Eq. 4.3. In particular, for q.e. x ∈ E, for T > 0. Therefore (5.11) implies (5.9). To show Eq. 5.10, we first observe that by Eqs. 5.11 and 5.12, (5.14) By Lemma 2.2, so by the Markov property of M (0) , Combining (5.13)-(5.16) yields (5.10) but with constant 3 replaced by (1 − q) −1/q with arbitrary q ∈ (0, 1). This proves (5.10) since (1 − q) −1/q → e as q ↓ 0.
Let λ ≥ 0 and let L λ denote the operator associated with the form (B λ , V ), i.e.
L λ = L 0 − λ, (5.17) where L 0 is the operator associated with (B 0 , V ) = (B, V ). Let (P λ t ), (R λ α ) denote the semigroup and the resolvent associated with the Hunt process corresponding to (B λ , V ). It is well known that for ψ ∈ L 1 (E; m), μ ∈ R(E) we have x ∈ E. Therefore from Theorem 5.1 we immediately get the following corollary.

Remark 5.3
The results of Sections 3-5 can be carried over to quasi-regular forms. Indeed, if the forms {B(t), t ∈ [0, T ]} are quasi-regular, then by [35,Theorem IV.2.2], there exists a special standard process M properly associated in the resolvent sense with the time dependent form defined by Eq. 2.4. One can check that all the results of Sections 3 and 4 hold true for such a process. This is because in their proofs the fact that M is a Hunt process is not used and the results of [14] on which we rely in the proofs of Section 3 hold for quasiregular forms (B (t) , V ) (see [14,Remark 4.4]). Similarly, the results of [18] on which we rely in Section 4 hold for quasi-regular form (B, V ). As a consequence, Theorem 5.1 holds true in the case of quasi regular form (B, V ) (its proof for such forms requires no changes).

Applications
In this section, we give four quite different examples of forms (B, V ) and measures μ for which Theorem 5.1 applies.

Classical local Dirichlet forms
In this subsection, we assume that E = D, where D is a nonempty connected bounded open subset of R d with d ≥ 2. We denote by m the Lebesgue measure on D. We consider the classical form (B, V ) on H = L 2 (D; m) defined as We will consider two cases: V = H 1 0 (D) and V = H 1 (D). where ϕ ∈ L 1 (D; m) is nonnegative, μ = dt ⊗μ withμ ∈ M + 0,b (D) and h : R → R is a continuous function satisfying the "sign condition", i.e. ∀s ∈ R, h(s)s ≥ 0.

Equations with Dirichlet boundary conditions
(6.4) The model example is h(s) = s, s ∈ R. In Eqs. 6.2 and 6.3 gradient of the solution appears, so they are more general than the equations studied in Sections 3-5. We shall see, however, that they are closely related to equations of the forms (5.1), (5.2). We first give definitions of probabilistic solutions of Eqs. 6.2, 6.3.

Remark 6.1
If w is a solution of Eq. 6.7, then 0 ≤ w ≤ (∞) q.e. on (0, ∞)×D. Thus, we can replaceĤ by H in Eq. 6.7. Similarly, ifw is a solution of Eq. 6.8, then 0 ≤w ≤ (∞) q.e. on D. Thus, we can replaceĤ by H in Eq. 6.8. We provide the proof for Eq. 6.7. The proof for Eq. 6.8 is similar. Let T > 0 and w(t, x) = w(T − t, x). By [16,Proposition 3.7], for q.e. z ∈ D 0,T the pair is a solution of the BSDE under the measure P z , where W is some Wiener process starting from z under P z (In different words, in the case where the form (6.1) is considered, if w is a probabilistic solution of Eq. 6.7, then the martingale M appearing in Theorem 3.2 (with the data from Eq. 6.7) has the representation M t = t 0 Z r dW r with Z as above). Since, by assumption, ϕ ≥ 0, we have • ϕ ≥ 0, so from Eq. 6.9 it follows thatw ≥ 0 q.e. on D 0,T . Since T > 0 was arbitrary, w ≥ 0 q.e. on (0, ∞) × D. Since w is quasi-continuous, it is finite q.e., so w ≤ (∞) q.e.
To prove the opposite implication, we first note that if u is a solution of Eq. 6.2, then for every T > 0, for q.e. z ∈ D 0,T the pair is a solution of the BSDẼ is a solution of Eq. 6.9. From this it follows that w is a solution of Eq. 6.7. This completes the proof of (ii). The proof of (iii) is similar to that of (ii). We apply Itô's formula and the fact that in case of the form (6.1), the martingale M appearing in Theorem 4.3 has the representation M t = t 0 Z s dW s , t ≥ 0, with Z t = ∇v(X t ) if we consider Eq. 6.3, and with Z t = ∇w(X t ) if we consider (6.8) (for the representation property for M see [13,Theorem 3.5]. We now show (i). We know that g :=Ĥ satisfies the hypotheses (E2), (E5) and (E6). Therefore, by Theorem 3.2, there exists a unique solution w of Eq. 6.7, while by Theorem 4.3, there exists a unique solutionw of Eq. 6.8. Therefore (i) follows from (ii), (iii) and Remark 6.1.
and h is a continuous function satisfying (6.4). Moreover, assume that there exist L, δ > 0 such that h(s)s ≥ δ for s ∈ R such that |s| ≥ L.
(i) In [3] it is proved that under the above assumptions there exists a weak solution v ∈ H 1 0 (D) of Eq. 6.3 such that h(v)|∇v| 2 ∈ L 1 (D; m). A quasi-continuous version of v, which we still denote by v, is a probabilistic solution of Eq. 6.3. Indeed, since for every bounded w ∈ H 1 0 we have B(v, w) = D (h(v)|∇v| 2 + β)w dx, v is a solution of problem (6.3) in the sense of duality (see [15,Section 5] for the definition). Therefore, by [15,Proposition 5.1], v is a probabilistic solution of Eq. 6.3.
(ii) By the results proved in [33], there exists a weak solutionū ∈ L 2 (0, T ; H 1 0 (D)) of problem (6.5) such that h(ū)|∇ū| 2 ∈ L 1 (D T ; m 1 ). Its quasi-continuous version is a probabilistic solution of Eq. 6.5. This follows from the fact that it is a solution of Eq. 6.5 in the sense of duality (see [14,Section 4] for the definition), and hence, by [14,Corollary 4.2], a probabilistic solution of Eq. 6.5. Proposition 6.4 Let ϕ, h satisfy the assumptions of Proposition 6.2, and let μ(dx) = β(x) m(dx) for some nonnegative β ∈ L 1 (D; m). Then Proof In the proof we adopt the notation from the proof of Proposition 6.2. We know that w = (u) is nonnegative and solves (6.2) with H replaced byĤ . We also know that the initial condition • ϕ and coefficients f = 0, g :=Ĥ of that equation satisfy the assumptions (E1)-(E6). Moreover, we shall see in the proof of Proposition 6.6 (in a more general situation where is replaced by the fractional Laplacian α/2 ) that Eq. 5.7 with ϕ replaced by • ϕ is satisfied. Hence, by Theorem 5.1, w(t, x) →w(x) as t → ∞ for q.e. x ∈ D. Therefore part (i) follows from Proposition 6.2 and the fact that −1 is continuous. To prove part (ii), we first note that for every T > 0,w ≥ 0 q.e. on D 0,T , soū ≥ 0 q.e. on D 0,T . Consequently, h(ū) ≥ 0 q.e. on D 0,T since h satisfies (6.4). Therefore from Eq. 6.6 it follows that for q.e. (s, x) ∈ D 0,T , The functionû defined asû(t, x) =ū(T − t, x), (t, x) ∈ D T , is a solution of Eq. 6.2 with h ≡ 0. By Theorem 5.1,û(t, x) →v(x) as t → ∞ for q.e. x ∈ D, wherev is a solution of Eq. 6.3 with h ≡ 0. In fact, by Eq. 5.10 (see the proof of Proposition 6.6 for details), (D;m) ), t > 0, for q.e. x ∈ D. Since D is bounded, it follows thatû(t, ·) →v in L 1 (D; m) as t → ∞. From this and the fact that 0 ≤ u(t, ·) ≤û(t, ·) we conclude that the family {u(t, ·)} is uniformly integrable, which together with (i) proves (ii).

Nonlocal Dirichlet forms
Let E = R d with d ≥ 2, m be the Lebesgue measure on E and α ∈ (0, 2). We consider the form (6.18) whereû denotes the Fourier transform of u and It is known that (B, V ) is a regular Dirichlet form on L 2 (R d ; m) (see [ . From the fact that p(t, x, y) = p(t, 0, x − y) and the scaling property p(t, 0, x) = t −d/α p(1, 0, t −1/α x) it follows that p(t, x, y) ≤ Ct −d/α , t > 0 (6.20) with C = sup x∈R d p(1, 0, x). Hence