Large time behaviour of solutions to parabolic equations with Dirichlet operators and nonlinear dependence on measure data

We study large time behaviour of solutions of the Cauchy problem for equations of the form $\partial_tu-L u+\lambda u=f(x,u)+g(x,u)\cdot\mu$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form ${\mathcal{E}}$ and $\mu$ is a nonnegative bounded smooth measure with respect to the capacity determined by ${\mathcal{E}}$. We show that under the monotonicity and some integrability assumptions on $f,g$ as well as some assumptions on the form ${\mathcal{E}}$, $u(t,x)\rightarrow v(x)$ as $t\rightarrow\infty$ 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 locally compact separable metric space, m be 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 behaviour 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 (1.1), ϕ : E → R, f, g : E × R → R are measurable functions, µ is a smooth measure with respect to the capacity associated with (B, V ) and λ ≥ 0. 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 [10,14,17,21,24]. 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 ma.e. finite. It is known that if the form (B, V ) is transitive or λ > 0 then R + (E) contains the class M + 0,b (E) of positive bounded smooth measures on E. In general, the inclusion M + 0,b (E) ⊂ R + (E) is strict. Elliptic equations with unbounded measures of class R + (E) are considered for instance in the monograph [22]; 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, |u(t, x) − v(x)| ≤ 3P t |ϕ|(x) + 3P t (R 0 (|f (·, 0)| + |g(·, 0)| ·μ))(x), t > 0. (1.4) The quantities on the right hand-side of (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, they generalize the results obtained in the paper [11] in which g ≡ 1 and L is a uniformly elliptic divergence form operator. Note, however, that in [11] systems of equations are treated. We also strengthen slightly the results of [19] concerning asymptotic behaviour of 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 behaviour, which are not covered by our approach, are to be found in [26,27,28]. In [27,28] equations involving Leray-Lions type operators and smooth measure data are considered while [26] deals with linear equations with general, possibly singular, bounded measure µ. Note that the methods used in [26,27,28] do not provide estimates between the parabolic solution and the corresponding stationary solution.
In order to prove (1.3), (1.4) we develop the probabilistic approach initiated in [11]. 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 behaviour for solutions of (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 they proofs rely on our earlier results proved in [14,17] 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 (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 (nonsymmetric) Dirichlet forms, the probabilistic solutions coincide with the renormalized solutions defined in [16] (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 [8,29] and also [15]).

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 = dt ⊗ m . For T > 0 we write E T = [0, T ]×E, E 0,T = (0, T ]×E. By B b (E) we denote the set of all real bounded Borel measurable functions on E and by B + b (E) we denote the subset of B b (E) consisting of all nonnegative functions. The sets B b (E 1 ), B + b (E 1 ) are defined analogously.

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 [24,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 where B (t) λ (ϕ, ψ) = B (t) (ϕ, ψ) + λ(ϕ, ψ) for λ ≥ 0, and 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, byB (t) we denote 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. By · V we denote the norm in V , i.e. ϕ 2 V = B 1 (ϕ, ϕ), ϕ ∈ V . By V ′ we denote 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 By E we denote 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 [33,Example I.4.9(iii)]). Given a time dependent form (2.4) we define quasi notions with respect to E (exceptional sets, nests, quasi-continuity as in [24,Section 6.2]. Note that by [24, Theorem 6.2.11] each element u of W has a quasi-continuous m 1 -version. We will denote it bỹ u. Quasi-notions with respect to (B, V ) are defined as in [24,Section 2.2].
By S(E) we denote the set of all smooth measures on E with respect to the form (B, V ) (see, e.g., [24,Section 4.1] for the definition). By S(E 1 ) we denote the set of all smooth measures on E 1 with respect to E (see [13]) and by S(E 0,T ) the set of all smooth measures on E 1 with support in E 0,T . By M b (E 0,T ) we denote 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.
We will say that a Borel measure By the above inequality and (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 [24,Eq. (6.2.21)] and (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 ∂ 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 ϕ on E is extended to E 1 by setting ϕ(t, x) = ϕ(x), (t, x) ∈ E 1 , and every function f on E 1 (resp. E 0,T ) is extended to E 1 ∪ {∂} by setting f (∂) = 0 (resp. f (z) = 0 for z ∈ E 1 ∪ {∂} \ E 0,T ). Let E be the form defined by (2.4). By [24, Theorem 6.3.1] there exists a Hunt process M = (Ω, (F t ) t≥0 , (X t ) t≥0 , (P z ) z∈E 1 ∪{∂} ) with state space E 1 , life time ζ and cemetery state ∂ associated with E in the resolvent sense, i.e. for every α > 0 and is an E-quasi-continuous m 1 -version of the resolvent associated with the form E. By [24,Theorem 6 is a decomposition of X into the process on R and on E then τ is the uniform motion to the right, i.e. τ (t) = τ (0) + t, τ (0) = s, P z -a.s. for z = (s, x) ∈ E 1 . Moreover, one can check that for every s ∈ R the process M (s) = (Ω, (F s+t ) t≥0 , (X s+t ) t≥0 , (P s,x ) x∈E∪{∂} ) is a Hunt process with life time ξ s = inf{t ≥ 0 : X s+t ∈ ∂} associated with the form (B (s) , 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 [13,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 . On the contrary, if A is a positive natural AF of M then modifying the proof of [10, Lemma 5.1.7] (we replace quasi-notions and facts used in the proof in [10] by corresponding quasi-notions and facts from [24,; for the case of (non-symmetric) Dirichlet form see also [23,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 [13,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 [13,Proposition 3.8], if (B, V ) is a (non-symmetric) Dirichlet form or, more generally, a semi-Dirichlet form satisfying the duality condition (see [13] for the definition), then M 0,b (E 0,T ) ⊂ R(E 0,T ). The inclusion may be strict (see [ We set (2.9) Lemma 2.2. Assume (2.9).
) 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. Then A 0,μ t = t 0 e rÃ r , whereÃ t = lim n→∞Ã n t andÃ n t = t 0 e −r f n (X r ) dr for some f n ∈ L 1 (E; m) (see the proof of [10, Theorem 5.1.1] or [24, Theorem 4.1.10]). From this and the first part we deduce that (i) is satisfied for everyμ ∈ S 0 (E). By [24,Lemma 4.1.14] there exists a nest {F n } such that 1 Fn ·μ ∈ S 0 (E) for each n ∈ N. Since we already know that (i) holds forμ replaced by 1 Fn ·μ, applying the monotone convergence theorem we conclude that it holds forμ replaced by 1 ∞ n=1 Fn ·μ, and hence forμ, because the set E \ ∞ n=1 F n is exceptional.
One can check that A is a CAF of M. Let ν denote its Revuz measure. Then for every 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 (B (t) , V ), i.e.
(see [21,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: Definition. Let z ∈ E T . We say that a pair (Y z , M z ) is a solution of the BSDE on the space (Ω, F, P z ) if (a) Y z is an (F t )-progressively measurable process of class D under P z , M z is an (F t )-martingale under P z such that M z 0 = 0, 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 [9, 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 (F t )-stopping time τ , which shows that {Y τ } is uniformly integrable (see [32,Theorem I.11]). To simplify notation, in what follows we write x, u(t, x)), g u (t, x) = g(t, x, u(t, x)).
Definition. (a) We say that u : E 0,T → R is a solution of problem (3.2) if f u · m ∈ R(E 0,T ), g u · µ ∈ R(E 0,T ) and for q.e. z ∈ E 0,T , is a solution of the Cauchy problem with terminal condition of the form Remark 3.1. If equation (3.6) has the uniqueness property (i.e. has a unique solution v T for every T > 0) then for every a > 0, To see this, let us write x)). With this notation, It is known (see [13, p. 1213]) that there exists a generalized nest {F n } on E 0,T +a such that Φ n,T +a : denote the solution of the linear equation By [13,Theorem 3.7], v n T +a is a weak solution of (3.10). Therefore making a simple change of variables shows that v n T +a,a is a weak solution of the linear equation Using the probabilistic representation of the solution of (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 (3.12) one can show that v n T +a,a converges pointwise as n → ∞ to the solution of (3.8), that is to v T . This and (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 will 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 [13,Section 5]) (see also [12,Section 2]).
Our basic assumptions on the data are the following.
In what follows by D q (P z ), q > 0, we denote the space of all (F t )-progressively measurable càdlàg processes Y such that E z sup t≥0 |Y t | q < ∞. (ii) Let Proof. By using the standard change of variables (see, e.g., the beginning of the proof of [3, 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 [14, Proposition 2.1] with obvious changes. The proof of the existence of a solution we divide into 2 steps.
Step 1. Let ξ = ϕ(X ζτ ), f (t, y) = f (X t , y), g(t, y) = g(X t , y) and let A be a continuous increasing (F t )-adapted process. Assume that (3.14) Then modifying slightly the proof of [14, Lemma 2.6] we show that there exists a unique solution (Y, M ) of the BSDE on the space (Ω, F, P z ) (for brevity, in notation we drop the dependence of Y, M on z). Let sgn(x) = 1 if x > 1 and sgn(x) = −1 if x ≤ 0. By the Meyer-Tanaka formula (see [32, p. 216]) and the fact that A µ is continuous, From this, (3.15) and (P3), (P5) we get 16) which shows that in fact (Y, M ) is a solution of (3.15) withf c replaced by f andḡ c replaced by g. Step on the space (Ω, F, P z ) (as in Step 1, for brevity, in notation we drop the dependence of Since µ ∈ R + (E 0,T ), 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 Ψ n → 0 as n → ∞ for q.e. z ∈ E 0,T . By (3.18), Applying the Meyer-Tanaka formula we get (see the proof of (3.16)) From the definition of τ k,N it follows that From this, (P2) and (3.19) one can deduce that By Doob's inequality (see, e.g., [20, Theorem 1.9.1]) and (3.19), for every ε > 0 we have Similarly, by (3.21), (3.22) and Doob's inequality, for every ε > 0. Letting n → ∞ in (3.20) and using (3.23)-(3.25) we conclude that But the Meyer-Tanaka formula and (3.17), Hence, so applying Fatou's lemma and (3.19) gives s., and consequently, lim since Y z is of class D. Letting N → ∞ in (3.26) and using (3.27), (3.28) and Doob's inequality we obtain Since τ k → ζ τ as k → ∞, letting k → ∞ in (3.29) and repeating arguments used to prove (3.29) we get We may now repeat the reasoning following Eq. (3.6) in [17] with the process V from [17] replaced by · 0 g(t, Y z t ) dA µ t (see also the reasoning following (4.26) in the present paper) to prove that the pair (Y z ,M z ), whereM z is a càdlàg version of the martingale Then the argument from the beginning of the proof of [13,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 (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 (3.4) on the space (Ω, 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 = u(X), t ∈ [0, ζ τ ], P z -a.s., (3.13) follows immediately from (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 (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 (3.4) and then taking the expectation shows thatū is a solution of (3.2). To show thatū is unique we may argue as in the proof of [13, Theorem 5.8].
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 (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 (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 [30]). 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 [13, Proposition 3.4] the solution u described above is quasi-càdlàg.

Convergence of BSDEs and Elliptic PDEs
In this section we assume that (2.9). By L we denote 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. To shorten notation, in what follows we denote P 0,x by P x and E 0,x by E x . Under the measure P x , and A µ t = A 0,μ t , t ≥ 0, whereμ is determined by (2.6).
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 : Note that in case ν = m the notion of quasi-integrability was introduced in [12, Section 2]. For a comparison of the notion of m-integrability and the notion of quasiintegrability in the analytic sense see [12,Remark 2.3]).
In this section and Section 5 we will 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 (Ω, 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 x -a.s. as t → ∞ and M x is an (F t )-local martingale under P x such that M 0 = 0, Definition. We say that v : 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 (Ω, F, P x ). The solutions may depend on x but for brevity in notation we drop the dependence of Y n , M n on x. In what follows byỸ n ,M n we denote 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). Moreover, for every t ≥ 0, Proof. By (4.4), From the above and the fact that the process A µ is continuous it follows that the pair (Ỹ n ,M n ) defined (4.5) satisfies where V n t = 0 if t < n, V n t = −Y n n if t ≥ n. Let δỸ =Ỹ m −Ỹ n . By (4.10), By the Meyer-Tanaka formula (see [32, p. 216]), for t < m we have where sgn(x) = 1 if x > 0 and sgn(x) = −1 if x ≤ 0. Therefore, for t < m, From this it follows that for t ∈ [0, m], Therefore, for t ∈ [0, m] we have This implies (4.6). By [3, Lemma 6.1], which shows (4.7). Finally, to prove (4.8), let us first observe that by the Meyer-Tanaka formula, By the above inequality and (4.9), for t < n we have On the other hand, for every t ≥ 0, and similarly, which when combined with (4.12) proves (4.8). Assume for some x ∈ E for each n ∈ N there exists a solution (Y n , M n ) of (4.4) on the space (Ω, F, P x ). If then there exists a solution (Y x , M x ) of (4.2) on (Ω, F, P x ). Moreover, Y x ∈ D q (P x ) for q ∈ (0, 1), M x is a uniformly integrable (F t )-martingale under P x and and for every q ∈ (0, 1), lim Proof. From (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 (4.16) is satisfied. By (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 (4.16) imply that Y x ∈ D q (P x ) and (4.17) is satisfied. From (4.8), (4.16), (4.17) and Fatou's lemma it follows that for every T > 0, Since Y n ζ = 0, P x -a.s. for n ∈ N, from (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 (4.17) one can show that ζ 0 |g(X t , Y n t ) − g(X t , Y x t )| dA µ t → 0 in probability P x (see the proof of [11, Eq. (6.16)]). Set F R (t, x) = |f (t, x, −R)| ∨ |f (t, x, R)|, G R (t, x) = |g(t, x, −R)| ∨ |g(t, x, R)| and for N, R > 0 and n ∈ N define the stoping times By (4.4), for T < n we have Since Y n t = Y n t∧ζ and T ∧ζ∧δ N,R t∧ζ∧δ N,R dM n s = T t dM n s∧ζ∧δ N,R and the martingale M n stopped at ζ ∧ δ N,R is still a martingale (see [32,Theorem I.18]), it follows that By Doob's inequality (see, e.g., [20, Theorem 1.9.1]) and (4.16) for every ε > 0 we have From the definition of δ N,R and (E2), (4.17) it follows that Hence, by Doob's inequality (see, e.g., [20, Theorem 1.9.1]), for every ε > 0. Letting n → ∞ in (4.18) and using (4.17) and (4.19)-(4.21) we conclude that P x -a.s.
From the last convergence and Doob's inequality it follows that for every ε > 0, lim Therefore letting N → ∞ in (4.22) and using (4.15) we show that P x -a.s., We now show that τ R ր ∞, 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 (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 (4.25) contradicts (4.24). This proves that τ R ր ∞, 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 (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 (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 (4.2).  (ii) Let 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 (4.2) on the space (Ω, 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 (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 [14,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 (Ω, 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 (4.26) andM x is a càdlàg version of the martingale (4.28). Repeating step by step the proof of [14,Theorem 4.7] one can show that there is a pair of càdlàg processes (Y, M ) not depending x ∈ E, and secondly, that in This shows that the pair (v(X), M ) is a solution of (4.2) on the space (Ω, 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 (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 (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 (2.9) is satisfied and the data f, g, µ do not depend on time. By L we denote 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 (3.2) with L and the data f, g, µ satisfying the above assumptions. By Remark 3.1, putting we may define a probabilistic solution u of (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 (1.2), i.e. solution of the problem 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 (3.5), because ζ τ = T ∧ ζ under the measure P x . On the other hand, Therefore our problem reduces to showing that the right-hand side of (5.5) converges to the right-hand side of (5.6) as T → ∞ and to estimating the difference between the two expressions by some function of T .
where A 0,ν is the continuous AF of M (0) associated with ν in the Revuz sense. Note that if (B, V ) is transient then R α ν is defined for α = 0. Before stating our main result, let us note that with the convention made at the beginning of Section 2.2, E x 1 {ζ>t} ψ(X t ) = P t ψ(x) for ψ ∈ L 1 (E; m), t ≥ 0. Therefore (4.13) is equivalent to lim Clearly, assumption (4.14) is equivalent to By remarks given in Section 2.2, if f (·, 0) · m ∈ R(E) and g(·, 0) ·μ ∈ R(E) then (5.8) is satisfied for q.e. x ∈ E.
Theorem 5.1. Assume that the assumptions of Theorem 4.3 hold, and moreover, (5.7) is satisfied. Let u be a solution of (5.1) and v be a solution of (5.2). Then 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 (4.4) (with T = n) and Y be the first component of the solution of (4.2). Since (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.
Let λ ≥ 0 and let L λ denote the operator associated with the form (B λ , V ), i.e.
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 for q.e. x ∈ E. Therefore from Theorem 5.1 we immediately get the following corollary.
Corollary 5.2. Let the assumptions of Theorem 5.1 hold. Let u, v be solutions of (5.1) and (5.2), respectively, with L = L λ defined by (5.17). Then for q.e. x ∈ E,  4). One can check 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 [13] on which we rely in the proofs of Section 3 hold for quasi-regular forms (B (t) , V ) (see [13,Remark 4.4]). Similarly, the results of [17] 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. where ϕ ∈ L 1 (D; m), µ ∈ M + 0,b (D) and h : R → R is a continuous function satisfying the "sign condition", i.e.

Classical local Dirichlet forms
∀s ∈ R, h(s)s ≥ 0 (6.4) In the above equations the gradient of the solution appears, so they are more general that 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 (6.2), (6.3).
Proof. We first prove (ii). Let w be a solution of (6.7) and letw(t, x) = w(T − t, x). By [15,Proposition 3.7], for q.e. z ∈ E 0,T the pair under the measure P z , where W is some Wiener process starting from z under P z (In different words, in case of the form (6.1), if w is a probabilistic solution of (6.7) then the martingale M of Theorem 3.2 (with the data from (6.7)) has the representation M t = t 0 Z r dW r with Z as above). Letū = Φ −1 (w). Since Φ −1 is of class C 2 , applying Itô's formula we get a.e. with respect to the Lebesgue measure. Hence, Sinceū(X ζτ ) = ϕ(X ζτ ) and Taking the expectation with respect to P x we see thatū = Φ −1 (w) is a probabilistic solution of (6.5). Hence u = Φ −1 (w) is a probabilistic solution of (6.2).
To prove the opposite implication, let us first note that if u is a solution of (6.2) then for q.e. z ∈ E 0,T the pair is a solution of the BSDẼ under the measure P z . In case h(ū)|∇ū| 2 ∈ L 1 (D 0,T ; m 1 ) this follows directly from [15,Proposition 3.7], while in case h(ū)|∇ū| 2 · m ∈ R(D 0,T ) follows from [15,Proposition 3.7] by simple approximation. Putw = Φ(ū). Applying Itô's formula we show that the pair is a solution of (6.9). Therefore taking t = 0 in (6.9) and then the expectation with respect to P x shows that w is a solution of (6.5).
The proof of assertion (iii) is similar to (ii). We apply Itô's formula and the fact that in case of the form (6.1), the martingale M of 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 equation (6.3), and with Z t = ∇w(X t ) if we consider (6.8) (for the representation property for M see [12,Theorem 3.5]. Finally, assertion (i) follows from (ii), (iii) and Theorems 3.4 and 4.3.
(i) In [2] it is proved that under the above assumptions there exists a weak solution v ∈ H 1 0 (D) of (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 (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 [14,Section 5] for the definition). Therefore, by [14,Proposition 5.1], v is a probabilistic solution of (6.3). (ii) By results proved in [31] 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 (6.5). This follows from the fact that it is a solution of (6.5) in the sense of duality (see [13,Section 4] for the definition), and hence, by [13,Corollary 4.2], a probabilistic solution of (6.5). Proposition 6.3. Let ϕ, h satisfy assumptions of Proposition 6.1 and let µ(dx) = β(x) m(dx) for some nonnegative β ∈ L 1 (D; m).
Proof. In the proof we adopt the notation of Proposition 6.1. Since H is bounded nonincreasing and Φ is bounded, the initial condition Φ•ϕ and coefficients f = 0, g = H satisfy the assumptions (E1)-(E6). Moreover, we shall see in the proof of Proposition 6.5 (in a more general situation where ∆ is replaced by the fractional Laplacian (∆) α/2 ) that (5.7) 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 6.1 and the fact that Φ −1 is continuous. To prove part (ii), let us first observe that from the formulā w(z) = E z ϕ(X ζτ ) + ζτ 0 H(w(X t )) dA µ t and the fact that H ≥ 0 it follows immediately that if ϕ ≥ 0 thenw ≥ 0. Since Φ(s) ≥ 0 for s ≥ 0, we conclude from this and Proposition 6.1 thatū ≥ 0. Consequently, h(ū) ≥ 0 since h satisfies (6.4). Therefore from (6.6) it follows that for q.e. z ∈ E 0,T , The functionû defined asû(t, x) =ū(T − t, x), (t, x) ∈ D T , is a solution of (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 (6.3) with h ≡ 0. In fact, by (5.10) (see the proof of Proposition 6.5 for details), for q.e. x ∈ D. Henceû(t, ·) →v in L 1 (D; m) as t → ∞. From this and the fact that 0 ≤ u(t, ·) ≤û(t, ·) it follows that the family {u(t, ·)} is uniformly integrable, which together with (i) proves (ii).
Remark 6.7. (i) An analogue of Proposition 6.5 holds true for D as before and form (6.16) replaced by any regular transient symmetric Dirichlet form (B, V ) on L 2 (R d ; dx) whose semigroup possesses a kernel p satisfying uniform estimate of the form (6.18) with α/d replaced by κ, i.e.
p(t, x, y) ≤ Ct −κ , t > 0 (6.23) for some C, κ > 0. Indeed, an inspection of the proof of Proposition 6.5 shows that for such a form estimate (6.17) holds with α/d replaced by κ. A characterization of symmetric Dirichlet forms satisfying (6.23) in terms of Dirichlet form inequalities of Nash's type is given in [4]. For a concrete example of a class of forms satisfying (6.23) and containing the form (

Local semi-Dirichlet forms
Let D ⊂ R d , m, H be as in Section 6.1 and let a : D → R d ⊗ R d , b : D → R d be measurable functions such that for every x ∈ D, for some λ ≥ 1. Set V = H 1 0 (D) and Of course, the operator L determined by (B, V ) has the form It is well known that the transition density p of the process associated with (B, V ) has the property that p(t, x, y) ≤ Ct −d/2 , t > 0, for some C > 0 (i.e. (6.18) with α = 2 is satisfied). Therefore there is an analogue of Proposition 6.5 for equations involving the operator L defined by (6.24).