Non-symmetric distorted Brownian motion: strong solutions, strong Feller property and non-explosion results

Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first show weak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strict positivity of the unique continuous version of the density to its invariant measure. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically has an unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of \cite{KR} that the constructed weak solution is indeed strong and weakly as well as pathwise unique up to its explosion time. As a consequence of our approach, we can use the theory of Dirichlet forms to prove further properties of the solutions. More precisely, we obtain new non-explosion criteria for them. We finally present concrete existence and non-explosion results for non-symmetric distorted Brownian motion related to a class of Muckenhoupt weights and corresponding divergence free perturbations.


Introduction
In this paper we are concerned with the non-symmetric Dirichlet form given by (the closure of) on L 2 (R d , m), m := ρ dx, and the corresponding stochastic differential equation (SDE) where x ∈ R d , ζ is the lifetime (=explosion time). Our conditions on ρ and B are formulated as Hypotheses (H1)-(H3) in Section 2 below. It is well-known that starting with (1.1) by Dirichlet form theory one can construct a weak solution to (1.2) for quasi-every starting point x ∈ R d , and usually there is no analytic characterization (in terms of ρ and B) of the set of "allowed" starting points. In case B ≡ 0, it was however shown in [1] (see also [3], [8], for extensions of this result to other situations), that (1.2) has a weak solution for every x ∈ {ρ > 0} in the sense of the martingale problem, whereρ is the continuous version of ρ (which exists as a consequence of (H1)) and that for such starting points the process X t stays in {ρ > 0} before its lifetime ζ. The identification of (1.2) with B ≡ 0 for any x ∈ {ρ > 0} in the sense of a weak solution of an SDE related to the form in (1.1) has been worked out as a part of a general framework in [16,Section 4].
The first aim of this paper is to generalize these results to B 0, i.e. to the non-symmetric case (see Remark 2.2). The proof follows ideas from [1], but requires some modifications. For example, one observation is that the elliptic regularity results in weighted spaces from [1] extend to the non-symmetric case. The corresponding result is formulated as Theorem 3.6 in Section 3 below.
It is well-known by [12,Theorem 2.1] (see also [9], [21]) that for every x ∈ {ρ > 0} there exists a strong solution (i.e. adapted to the filtration generated by (W t ) t≥0 )) to (1.2), which is pathwise and weak unique. Hence this solution coincides with our weak solution (which is hence a strong solution) from Theorem 3.6. Thus we have identified the Dirichlet form associated to the Markov processes, given by the laws P x , x ∈ {ρ > 0}, of these strong solutions, to be the closure of (1.1). As a consequence, we can apply the theory of Dirichlet forms to obtain further properties of the solutions to (1.2) for every starting point in {ρ > 0}.
In this paper, as our second aim, we concentrate on proving non-explosion results for (1.2) using Dirichlet form theory, which means (cf. Remark 2.13) that the process started in x ∈ {ρ > 0} will neither go to infinity nor hit any point in {ρ = 0} in finite time. Non-explosion criteria from Dirichlet form theory are of analytic nature and different from the usual ones known from the theory of SDE (e.g. the one proved in [12], see Remark 4.2 (ii) below), but very useful in applications.
Finally, we present a number of concrete applications where the density ρ = dm dx is in certain Muckenhoupt classes. Our main result here is Theorem 5.5.
The organization of this paper is as follows. After, this introduction in Section 2 we recall some important elliptic regularity results for the Kolmogorov operator corresponding to (1.2), i.e. the generator of the Dirichlet form (1.1), under the assumption (H1) on ρ and (H2) on B. Subsequently, we present their analytic consequences associated to the closure of (1.1). In Section 3 we construct the weak solutions of (1.2) for every x ∈ {ρ > 0}. In Section 4 we show that by [12, Theorem 2.1] these solutions are strong, pathwise and weak unique. Section 5 is devoted to the mentioned applications.

Elliptic regularity and construction of a diffusion process
As usual dx denotes Lebesgue measure on R d and the Sobolev space H 1,q (E, dx), q ≥ 1 is defined to be the set of all functions f ∈ L q (E, dx) such that ∂ j f ∈ L q (E, dx), j = 1, . . . , d, and Here C ∞ 0 (E) denotes the set of all in-finitely differentiable functions with compact support in E. We also denote the set of continuous functions on E, the set of continuous bounded functions on E, the set of compactly supported continuous functions in E by C(E), C b (E), C 0 (E), respectively. C ∞ (E) denotes the space of continuous functions on E which vanish at infinity. We equip R d with the Euclidean norm · with corresponding inner product ·, · and write B r (x) : We shall assume (H1)-(H3) below throughout up to including section 3: By (H1) the symmetric positive definite bilinear form is closable in L 2 (R d , m) and its closure (E 0 , D(E 0 )) is a symmetric, strongly local, regular Dirichlet form. We further assume where p is the same as in (H1) and and (H3) where c 0 is some constant (independent of f and g) and E 0 α (·, ·) := E 0 (·, ·) + α(·, ·) L 2 (R d ,m) , α > 0.
Next, we consider the non-symmetric bilinear form -semigroup (resp. cosemigroup) and resolvent (resp. coresolvent) associated to (E, D(E)) and (L, D(L)) (resp. (L, D(L))) be the corresponding generator (resp. cogenerator) (see [13,Diagram 3,p. 39]). Using properties (H2) and [13, I 4.7] (cf. also [13, II 2. d)]), it is straightforward to see that (T t ) t>0 as well as (T t ) t>0 are submarkovian. Here an operator S is called submarkovian if 0 ≤ f ≤ 1 implies 0 ≤ S f ≤ 1. It is then further easy to see that (T t ) t>0 (resp. (G λ ) λ>0 ) restricted to L r (R d , m) ∩ L ∞ (R d , m) can be extended to strongly continuous contraction semigroups (resp. strongly continuous contraction resolvents) on all L r (R d , m), r ∈ [1, ∞) (see [13, I. 1] for the definition of a strongly continuous contraction semigroup (resp. resolvent)). We denote the corresponding operator families again by (T t ) t>0 and (G λ ) λ>0 and let (L r , D(L r )) be the corresponding generator on L r (R d , m). Since by (H1), (H2), and assumes that if has an infinitesimally (not necessarily probability) invariant measure m, i.e. m is a nonnegative Radon measure m on R d , such that b ∈ L p loc (R d , m) and Because then it follows by Proposition 2.1 that m = ρdx and that ρ satisfies (H1). Defining it satisfies (H2). So, we have the above decomposition in a natural way.
From now on, we shall always consider the continuous dx-version of ρ and denote it also by ρ.
and for any open ball , the assertion for general g ∈ L r (R d , m) follows by continuity and (2.4).

Remark 2.5. By [13, I. Corollary 2.21], it holds that
. By Stein interpolation (cf. e.g. [2, Lecture 10, Theorem 10.8]) (T t ) t>0 is also analytic on L r (R d , m) for all r ∈ (2, ∞). We would like to thank Hendrik Vogt for pointing this out to us as well as a misprint in the mentioned Theorem 10.8. There θ τ should be defined as τ · θ and not as (1 − τ) · θ.
. Then the above statements still hold with (2.5) replaced by Remark 2.7. By (2.5) and Sobolev imbedding, for r ∈ [p, ∞), R > 0 the set From now on, we shall keep the notation is a (temporally homogeneous) submarkovian transition function (cf. [6, 1.2]) and an m-version of T t f for any f ∈ ∪ r≥p L r (E, m). Moreover, letting P 0 := id, it holds We further consider (H4) (E, D(E)) is conservative.
In particular (T t ) t>0 (and also (T t ) t>0 ) can be defined as semigroups on and by [ Thus (2.11) is equivalent to (H4).
Following [1, Proposition 3.8], we obtain: Proposition 2.9. If (H4) holds (additionally to (H1)-(H3)), then: Therefore, we may assume that Cap(R d \ F k ) < ∞ for any k ≥ 1. Hence where K is the sector constant. Therefore, For a Borel set B ⊂ R d , we define The last and previous imply that is an open cover of K n for every n ≥ 1. (2.12) and (2.13) now easily imply the assertion.

Theorem 2.12. There exists a Hunt process
with state space E, having the transition function (P t ) t≥0 as transition semigroup. In particular M satisfies the absolute continuity condition, because for any t > 0 and f ∈ B b (E) with compact support (i.e. | f |dm has compact support). Thus the absolute continuity condition is satisfied.

Theorem 3.3.
Let u ∈ C ∞ 0 (R d ) and Proof. By Lemma 3.1 and the Markov property Then since u ∈ D(L p ) (cf. (2.3)), it follows by Lemma 3.1 that (M t ) t≥0 and all integrands below are integrable w.r.t. P x . Using Lemma 3.2 we get for s ∈ [0, t) Taking conditional expectation, it follows P x -a.s.
Using Lemma 3.1(iii) this simplifies to Note that the first term of the right hand side satisfies and the second term satisfies s. and the assertion follows.

Lemma 3.4. Let (B k ) k≥1 be an increasing sequence of relatively compact open sets in E with
Note that by Lemma 2.11 for m-a.e. x ∈ E P x Λ) = 1.
Then for x ∈ E and s > 0 Define where S is a countable dense set in (0, ∞). Fix ω ∈ Ω x . By the continuity of X t (ω) there is s ′ ∈ S such that X t (ω) ∈ B¯k, t ∈ [0, s ′ ], for somek ∈ N. This implies for k ≥k and since ζ(ω) ≥ s ′ , we get Putting all together and noting that θ s ′ (ω) ∈ Λ, we obtain Hence Ω x ⊂ Λ. Since P x (Ω x ) = 1, the assertion follows.

Remark 3.5.
For an alternative proof of Lemma 3.4, which does not require the absolute continuity condition, we refer to Lemma 6.1 in Section 6.
Theorem 3.6. Under (H1)-(H3) after enlarging the stochastic basis (Ω, F , (F t ) t≥0 , P x ) appropriately for every x ∈ E, the process M satisfies Proof. Let is a continuous (F t ) t≥0 -martingale under P x . By Theorem 3.3 M u t ∈ L 2 (Ω, F , P x ) and its quadratic variation satisfies M u t = t 0 ∇u 2 (X s ) ds. Suppose ζ < ∞. Then it is standard that there is an enlargement (Ω,F ,P x ) (since ∇u is degenerate) of the underlying probability space (Ω, F , P x ) and a Brownian motion (W t ) t≥0 on (Ω,F ,P x ) such that The identification of X up to ζ is now obtained by using Lemma 3.4 with an appropriate localizing sequence as in Lemma 2.11 for which the coordinate projections on E coincide locally with C ∞ 0 (E)-functions and noting that W t = t 0 1 E (X s )dW s on {t < ζ}. If ζ = ∞, then using the same localization, we obtain that M u i t = t 0 1 E (X s ) ds = t for t < ∞, where u i is the i-th coordinate projection. Thus M u i is a Brownian motion by Lévy's characterization and we do not need an enlargement of stochastic basis. The localization of the drift part is trivial.

Pathwise uniqueness and strong solutions
We first recall that by [12, Theorem 2.1] under the conditions (H1), (H2) ((H3) is not needed), for every stochastic basis and given Brownian motion (W t ) t≥0 there exists a strong solution to (3.1) which is pathwise unique among all solutions satisfying In addition, one has pathwise uniqueness and weak uniqueness in this class.
In the situation of Theorem 3.6 it follows, however immediately from Lemma 3.4 that (4.1) holds for the solution there. Hence we obtain the following:

Applications to Muckenhoupt A β -weights
In this section we present a class of examples of ρ and B satisfying our assumptions (H3) and (H4). Throughout, we assume (H1) and (H2) to hold.
where c r is some constant, N > 2 and where c B,K is some constant, i.e. (H3) holds.
Proof. For r 0 > 0 such that K ⊂ B r 0 (0) The last inequality follows from assumption (i) and · ∞,K c denotes the L ∞ (R d , m)-norm on K c .

Lemma 5.2. Let ρ be a Muckenhoupt
where C x,r is some constant and N ≥ βd + log 2 A, A is the A β constant of ρ.
where u x,r = 1 m(Br(x)) Br(x) u dm and c is some constant. Consequently, [15, Theorem 2.1], the doubling property, and the scaled Poincaré inequality imply the Sobolev inequality, i.e. for where c x,r is some constant and N ≥ βd + log 2 A. Then using a cutoff function like for instance g r (y) := 1 r (2r − x − y ) + , we see that for x ∈ R d , r > 0 where C x,r is some constant and N > 2 as well as N ≥ βd + log 2 A.
where c B,K is some constant, i.e. (H3) holds.

Appendix
We present here an alternative proof of Lemma 3.4, which does not require the absolute continuity condition.