Well-posedness for a class of degenerate It\^o-SDEs with fully discontinuous coefficients

We show uniqueness in law for a general class of stochastic differential equations in $\mathbb{R}^d$, $d\ge 2$, with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. The points of degeneracy have $d$-dimensional Lebesgue-Borel measure zero. Weak existence is obtained for more general, not necessarily locally bounded drift coefficient.


Introduction
The question whether a solution to a stochastic differential equation (hereafter SDE) on R d exists that is pathwise unique and strong occurs widely in the mathematical literature; for instance, see the introduction of [1] for a recent detailed, but possibly incomplete development. Sometimes, strong solutions that are roughly described as weak solutions for a given Brownian motion are required, for instance, in signal processing, where a noisy signal is implicitly given. Sometimes, it may be impossible to obtain a strong solution, only weak solutions are important to consider, or only the strong Markov property of the solution is needed for some reason. Then, uniqueness in law, i.e., the question whether, given an initial distribution, the distribution of any weak solution no matter on which probability space it is considered is the same, plays an important role. It might also be that pathwise uniqueness and strong solution results are just too restrictive, so that one is naturally led to consider weak solutions and their uniqueness. Here, we consider weak uniqueness of an SDE with respect to all initial conditions x ∈ R d as defined, for instance, in [2] (Chapter 5); see also Definition 2 below.
To explain our motivation for this work, fix symmetric matrix C = (c ij ) 1≤i,j≤d of bounded measurable functions c ij , such that, for some λ ≥ 1, λ −1 ξ 2 ≤ C(x)ξ, ξ ≤ λ ξ 2 , for all x, ξ ∈ R d , be the corresponding linear operator and be the corresponding Itô-SDE. If the c ij are continuous and the h i bounded, then Equation (2) is well-posed, i.e., there exists a solution and it is unique in law (see [3]). If the h i are bounded, then Equation (2) is well-posed for d = 2 (see [3] Exercise 7.3.4); however, if d ≥ 3, there exists an example of a measurable discontinuous C for which uniqueness in law does not hold [4]. Hence, even in the nondegenerate case, well-posedness for discontinuous coefficients is nontrivial, and one is naturally led to search for general subclasses in which well-posedness holds. Some of these are given when C is not far from being continuous, i.e., continuous up to a small set (e.g., a discrete set or a set of α-Hausdorff measure zero with sufficiently small α; else, see, for instance, introductions of [4,5] for references). Another special subclass is given when C is a piecewise constant on a decomposition of R d into a finite union of polyhedrons [6], and the h i are locally bounded with at most linear growth at infinity. The work in [6] is one of our sources of motivation for this article. Though we do not perfectly cover the conditions in [6], we complement them in many ways. In particular, we consider arbitrary decompositions of R d into bounded disjoint measurable sets (choose, for instance, 1 (4) below). A further example for a discontinuous C, where well-posedness holds, can be found in [7]. There, discontinuity is along the common boundary of the upper-and lower-half spaces. In [5], among others, the problem of uniqueness in law for Equation (2) is related to the Dirichlet problem for L as in Equation (1), locally on smooth domains. This method was also used in [4] using Krylov's previous work. In particular, a shorter proof of the well-posedness results of Bass and Pardoux [6] and Gao [7] is presented in [5] (Theorems 2.16 and 3.11). However, the most remarkable is the derivation of well-posedness for a special subclass of processes with degenerate discontinuous C. Though discontinuity is only along a hyperplane of codimension one, and coefficients are quite regular outside the hyperplane, it seems to be one of the first examples of a discontinuous degenerate C where well-posedness still holds ( [5] (Example 1.1)). This intriguing example was another source of our motivation. As was the case for results in [6], we could not perfectly cover [5] (Example 1.1), but we again complement it in many ways. As a main observation besides the above considerations, it seems that no general subclass has been presented so far where C is degenerate (or also nondegenerate if d ≥ 3) and fully discontinuous, but well-posedness holds nonetheless. This is another main goal of this paper, and our method strongly differs from techniques used in [5,6] and in the past literature. Our techniques involve semigroup theory, elliptic and parabolic regularity theory, the theory of generalized Dirichlet forms (i.e., the construction of a Hunt process from a sub-Markovian C 0 -semigroup of contractions on some L 1 -space with a weight), and an adaptation of an idea of Stroock and Varadhan to show uniqueness for the martingale problem using a Krylov-type estimate. Krylov-type estimates have been widely used to simultaneously obtain a weak solution and its uniqueness, in particular, pathwise uniqueness. The advantage of our method is that the weak existence of a solution and uniqueness in law are shown separately of each other using different techniques. We used local Krylov-type estimates (Theorem 9) to show uniqueness in law. Once uniqueness in law holds, we could improve the original Krylov estimate, at least for the time-homogeneous case (see Remark 4). In particular, our method typically implies weak-existence results that are more general than uniqueness results (see Theorem 8 here and in [1,8]). Now, let us describe our results. Let d ≥ 2, and A = (a ij ) 1≤i,j≤d be a symmetric matrix of functions a ij ∈ H 1,2d+2 loc (R d ) ∩ C(R d ), such that, for every open ball B ⊂ R d , there exist constants λ B , Λ B > 0 with Let ψ ∈ L q loc (R d ), with q > 2d + 2, ψ > 0 a.e., such that 1 ψ ∈ L ∞ loc (R d ). Here, we assumed that expression 1 ψ stood for an arbitrary but fixed Borel measurable function satisfying ψ · 1 ψ = 1 a.e., and 1 ψ (x) ∈ [0, ∞) for any x ∈ R d . Let G = (g 1 , . . . , g d ) ∈ L ∞ loc (R d , R d ) be a vector of Borel measurable functions. Let (σ ij ) 1≤i≤d,1≤j≤m , m ∈ N arbitrary but fixed, and be any matrix consisting of continuous functions, such that A = σσ T . Suppose there exists a constant M > 0, such that for a.e. x ∈ R d . The main result of our paper (Theorem 13) was that weak existence and uniqueness in law, i.e., well-posedness, then holds for stochastic differential equation among all weak solutions (Ω, Here, the solution and integrals involving the solution in Equation (4) may a priori depend on Borel versions chosen for 1 ψ and G. but Condition (5) is exactly the condition that makes these objects independent of the chosen Borel versions (cf. Lemma 2). 1 ψ may, of course, be fully discontinuous, but if it takes all its values in (0, ∞); then, Equation (5) is automatically satisfied. However, since ψ ∈ L q loc (R d ), it must be a.e. finite, so that zeros Z of 1 ψ have Lebesgue-Borel measure zero. Nonetheless, our main result comprehends the existence of a whole class of degenerate (on Z) diffusions with fully discontinuous coefficients for which well-posedness holds. This seems to be new in the literature. For another condition that implies Equation (5), we refer to Lemma 2. For an explicit example for well-posedness, which reminds the Engelbert/Schmidt condition for uniqueness in law in dimension one (see [9]), we refer to Example 2.
We derived weak existence of a solution to Equation (4) up to its explosion time under quite more general conditions on the coefficients, see Theorem 8. In this case, for nonexplosion, one only needs that Equation (3) holds outside an arbitrarily large open ball (see Remark 3ii). Moreover, Equation (5) is always satisfied for the weak solution that we construct (see Remark 3), and our weak solution originated from a Hunt process, not only from a strong Markov process.
The techniques that we used for weak existence are as follows. First, any solution to Equation (4) determines the same (up to a.e. uniqueness of the coefficients) second-order partial differential operator In Theorem 4, we found a measure µ := ρψ dx with some nice regularity of ρ, which is an infinitesimally invariant measure for (L, Then, using the existence of a density to the infinitesimally invariant measure, we adapted the method from Stannat [10] to our case and constructed a sub-Markovian C 0 -semigroup of contractions (T t ) t≥0 on each L s (R d , µ), s ≥ 1 of which the generator extended (L, C ∞ 0 (R d )), i.e., we found a suitable functional analytic frame (see Theorem 3 that further induced a generalized Dirichlet form; see (19)) to describe a potential infinitesimal generator of a weak solution to Equation (4). This is done in Section 4, where we also derive, with the help of the results about general regularity properties from Section 3, the regularity properties of (T t ) t≥0 and its resolvent (see Section 4.3). Then, crucially using the existence of a Hunt process for a.e. starting point related to (T t ) t≥0 in Proposition 3 (which follows similarly to [11] (Theorem 6)) this leads to a transition function of a Hunt process that not only weakly solves (4), but also has a transition function with such nice regularity that many presumably optimal classical conditions for properties of a solution to Equation (4) carry over to our situation. We mention, for instance, nonexplosion Condition (3) and moment inequalities (see Remark 2). However, irreducibility and classical ergodic properties, as in [1], could also be studied in this framework by further investigating the influence of 1 ψ on properties of the transition function. Similarly to the results of [1], the only point where Krylov-type estimates were used in our method was when it came up to uniqueness. Here, because of the possible degeneracy of 1 ψ , we needed Condition (5) to derive a Krylov-type estimate that held for any weak solution to Condition (4) (see Theorem 9 which straightforwardly followed from the original Krylov estimate [12] (2. Theorem (2), p. 52)). Again, our constructed transition function had such a nice regularity that a time-dependent drift-eliminating Itô-formula held for function g(x, t) := P T−t f (x), f ∈ C ∞ 0 (R d ). In fact, it held for any weak solution to Condition (4), so that for all these, the one-dimensional and, hence, all finite-dimensional marginals coincided (cf. Theorem 12). This latter technique goes back to an idea of Stroock/Varadhan ( [3]), and we used the treatise of this technique as presented in [2] (Chapter 5).

Article Structure and Notations
The main parts of this article are Sections 4 and 5. Section 4 contains the analytic results, and Section 5 contains the probabilistic results. Section 3 also contains auxiliary analytical results that are important on their own. Section 3 could be skipped in a first reading, so the reader may directly start with Section 4. The proofs for all statements of this article and further auxiliary statements were collected in Appendix A.
Throughout, we used the same notations as in [1,8], and d ≥ 2. Additionally, for an open-set U in R d and a measure µ , the usual space of k-times continuously differentiable functions in U, such that the partial derivatives of an order less or equal to k extend continuously to U (as defined, for instance, in [13]). In particular, C(U) := C 0 (U) is the space of continuous functions on U with supnorm · C(U) and C ∞ (U) := k∈N C k (U). If I is an open interval in R and p, q ∈ [1, ∞], we denoted by L p,q (U × I) the space of all Borel measurable functions f on U × I for which and let supp( f ) := supp(| f |dxdt). For a locally integrable function g on U × I and i ∈ {1, . . . , d}, we denoted by ∂ i g the i-th weak spatial derivative on U × I, by ∇g := (∂ 1 g, . . . , ∂ d g) the weak spatial gradient of g, by ∇ 2 g := (∂ ij g) 1≤i,j≤d the weak spatial Hessian matrix, and by ∂ t g the weak time derivative on U × I, provided these existed. For p, q ∈ [1, ∞], let W 2,1 p,q (U × I) be the set of all locally integrable functions g :

New Regularity Results
In this section, we develop some new regularity estimates (Theorems 1 and 2). Theorem 1 was used to obtain the semigroup regularity in Theorem 6, and Theorem 2 was used to obtain the resolvent regularity in Theorem 5.

Regularity Estimate for Linear Parabolic Equations with Weight in Time Derivative Term
Throughout this subsection, we assume the following condition: matrix of functions on U that is uniformly strictly elliptic and bounded, i.e., there exist constants , and there exists c 0 > 0, such that c 0 ≤ ψ on U, and finally u ∈ H 1,2 (U × (0, T)) ∩ L ∞ (U × (0, T)).
Assuming Condition (I), we considered a divergence form linear parabolic equation with a singular weight in the time derivative term as follows which is supposed to hold for all ϕ ∈ C ∞ 0 (U × (0, T)). Let (x,t) be an arbitrary but fixed point in U × (0, T), and Rx(r) be the open cube in R d of edge length r > 0 centered atx. Define Q(r) := Rx(r) × (t − r 2 ,t). Theorem 1. Suppose that Q(3r) ⊂ U × (0, T). Under the assumption (I) and (7), we have where C > 0 is a constant depending only on r, λ, M and B L p (Rx(3r)) .

Elliptic Hölder Regularity and Estimate
The following theorem is an adaptation of [14] (Théorème 7.2) using [15] (Theorem 1.7.4). It might already exist in the literature, but we could not find any reference for it, and we therefore provide a proof (in Appendix A).

Theorem 2. Let U be a bounded open ball in
then for any open ball U 1 in R d with U 1 ⊂ U, we have u ∈ C 0,γ (U 1 ) and where γ ∈ (0, 1) and C > 0 are constants which are independent of u and f .

L 1 -Generator and Its Strong Feller Semigroup
In this section, we precisely describe the potential infinitesimal generator, its semigroup and resolvent, of a weak solution to Condition (4) in a suitable functional analytic frame, originally due to Stannat (Theorem 3 and (19)). Subsequently, using the regularity results from Section 3, we derived regularity properties for the resolvent and semigroup (Theorems 5 and 6). One key tool for this method is the existence of an infinitesimally invariant measure with nice density (Theorem 4).

L 1 -Generator
In this section, we use all notations and assumptions from Section 4.1. The technique of [10] (Chapter 1) to obtain a closed extension of a densely defined diffusion operator and, subsequently, a generalized Dirichlet form carried nearly one by one over to our situation; only a small structural difference occurred. Since we considered a degenerate diffusion matrix in the definition of the underlying symmetric Dirichlet form via a function ψ that also acts on the µ-divergence free antisymmetric part of drift (see Equation (13)), we considered local convergence in space H 1,2 0 (V, µ) and imposed Assumption (14) on the antisymmetric part, while [10] (Chapter 1) dealt with local convergence in space H 1,2 0 (V, µ). As a first step, the following proposition was derived in a nearly identical manner to [10] (Proposition 1.1). We therefore omitted the proof.
By means of Proposition 1, the following Theorem 3 was also derived in a nearly identical manner to [10] (Theorem 1.5).

Theorem 3.
There exists a closed extension (L, D(L)) of Lu : satisfying the following properties: be a family of bounded open subsets of R d satisfying U n ⊂ U n+1 and R d = n≥1 U n . Then

Existence of Infinitesimally Invariant Measure and Strong Feller Properties
Here, we state some conditions that were used as our assumptions.
(A1) p > d is fixed, and A = (a ij ) 1≤i,j≤d is a symmetric matrix of functions that are locally uniformly or equivalently, From now on, we assume that Condition (A1) holds and fix A, ψ, ρ, B as in Theorem 4. Then, A, ψ, ρ, B satisfy all assumptions of Section 4.1. As in Section 4.1 µ := ρψ dx, A := 1 ψ A. By Theorem 3, there existed a closed extension (L, D(L)) of Denote by (L r , D(L r )) the corresponding closed generator with graph norm and by (G α ) α>0 the corresponding resolvent. (T t ) t>0 and (G α ) α>0 can also be uniquely defined on L ∞ (R d , µ), but are no longer strongly continuous there.
by Theorem 5, f has a locally Hölder continuous µ-version on R d and where c 3 > 0, γ ∈ (0, 1) are constants independent of f . In particular, T t f ∈ D(L r ) and T t f hence has a continuous µ-version, say P t f , with c 3 is independent of t ≥ 0 and f . The following lemma is quite important later to show the joint continuity of P · g(·) for g ∈ ∪ ν∈[ 2p p−2 ,∞] L ν (R d , µ). Due to Equation (21), it can be proven as in [1] (Lemma 4.13). : where C 1 is a constant that depends on U × [τ 1 , τ 2 ], V × (τ 3 , τ 4 ), but is independent of f .

Well-Posedness
With the help of the regularity results, Theorems 5 and 6 of Section 4, and the mere existence of a Hunt process for a.e. starting point (Proposition 3), we constructed a weak solution to Equation (4) (Theorems 7 and 8). Then, using a local Krylov-type estimate and Itô-formula (Theorems 9 and 10), uniqueness in law was derived for weak solutions to Equation (4) that spend zero time at the points of degeneracy of the dispersion matrix (Theorems 12 and 13). The method to derive uniqueness in law is an adaptation of the Stroock and Varadhan method ( [3]) via the martingale problem.

Weak Existence
The following assumption in particular is necessary to obtain a Hunt process with transition function (P t ) t≥0 (and consequently a weak solution to the corresponding SDE for every starting point). It is first used in Theorem 7 below.
Condition (A4) is not necessary to get a Hunt process (and consequently a weak solution to the corresponding SDE for merely quasi-every starting point) as in the following proposition.
with life timeζ := inf{t ≥ 0 |X t = ∆} and cemetery ∆, such that E is (strictly properly) associated withM and for strictly E -q.e. x ∈ R d ,
Analogously to [1] (Theorem 3.12), we obtained having transition function (P t ) t≥0 as the transition semigroup, such that M has continuous sample paths in the one-point compactification R d ∆ of R d with cemetery ∆ as point at infinity, i.e., for any x ∈ R d ,

Remark 2.
The analogous results to [1] (Lemma 3.14, Lemma 3.15, Proposition 3.16, Proposition 3.17, Theorem 3.19) hold in the situation of this paper. One of the main differences is that q = dp d+p > d 2 of [1] is replaced by s > d 2 of (A2). A Krylov-type estimate for M of Theorem 7 especially holds as stated in Equation (25) right below. Let g ∈ L r (R d , µ) for some r ∈ [s, ∞] be given. Then, for any ball B, there exists a constant C B,r , depending in particular on B and r, such that for all t ≥ 0, The derivation of Equation (25) is based on Theorem 5, of which the proof uses the elliptic Hölder estimate of Theorem 2. This differs from the proof of the Krylov-type estimates in [1,8] that are based on an elliptic H 1,p -estimate. Finally, one can get the analogous conservativeness and moment inequalities to [1] (Theorem 4.2, Theorem 4.4(i)) in this paper.

Example 1.
Given p > d, let A = (a ij ) 1≤i,j≤d be a symmetric matrix of functions on R d that is locally uniformly strictly elliptic and a ij ∈ H 1,p for some α > 0 and consider following conditions. Any and is nonexplosive if Equation (3) holds a.e. outside an arbitrarily large compact set.
Using Theorem 9 and Equation (25), the proof of the following lemma is straightforward. (29):

Lemma 2. Let M be a weak solution to Equation (28) . Then, either of the following conditions implies Equation
where ψ denotes the extended Borel measurable version as explained in Remark 3(i). Moreover, Equation (5) is equivalent to Equation (29).
In particular, if the weak solution that is constructed in Theorem 8 is nonexplosive, then Condition (ii) always holds for this solution and (29) implies in general that integrals of the form t 0 f ( X s , s)ds are, whenever they are well-defined, independent of the particular Borel version that is chosen for f . Theorem 10 (Local Itô-formula). Assume (A4) and let M be a weak solution to (28) such that (29) holds. Let R 0 > 0, T > 0. Let u ∈ W 2,1 2d+2 (B R 0 × (0, T)) ∩ C(B R 0 × [0, T]) be such that ∇u ∈ L 4d+4 (B R 0 × (0, T)). Let R > 0 with R < R 0 . Then P x -a.s. for any x ∈ R d , where Lu := 1 2 trace( A∇ 2 u) + G, ∇u .

Definition 2.
We say that uniqueness in law holds for Equation (28) if, for any two weak solutions, We say that the stochastic differential Equation (28) is well-posed if there exists a weak solution to it, and uniqueness in law holds.

Theorem 12. Assume Condition (A4) . Consider two arbitrarily given weak solutions to Equation
Then, P x • X −1 = P x • X −1 for all x ∈ R d . In particular, under Assumption (A4) , any weak solution to Equation (28) is a strong Markov process.

Remark 4. Once uniqueness in law holds for Equation
On the other hand, if Assumption (A4) holds with q = ∞, and 1 ψ is supposed to be locally pointwise bounded below and above by strictly positive constants, we may choose s = d 2 + ε for arbitrarily small ε > 0, and we obtain for g ∈ L s (R d ) 0 with supp(g) ⊂ V, V, B and x as above,

Example 2.
Consider the situation in Example 1 except for Conditions (a), (b), (c). Let p := 2d + 2, and assume G ∈ L ∞ loc (R d , R d ). Let α ≥ 0 be such that α(2d + 2) < d. Take q ∈ (2d + 2, d α ). Then A, G, and ψ satisfy Assumption (A4) . Therefore, Hunt process M of Theorem 8 solves weakly P x -a.s. for any x ∈ R d , Assume Equation (3). Then ζ = ∞ and by Theorem 13, M is the unique (in law) solution to Equation (31) that satisfies P x (Λ(Z M )) = 1, for all x ∈ R d . If we choose the following Borel measurable version of x α/2 , namely, where γ is an arbitrary but fixed strictly positive real number, then P is automatically satisfied by Lemma 2(i) for any weak solution M to Thus, under Equation (3)

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Proofs and Auxiliary Statements
In this section, we collect all proofs of statements given in this article, and the statement of several auxiliary Lemmas.
The following lemma is a slight modification of [17] (Lemma 6) and involves a weight function ψ.
Exactly in the same way, but with u replaced by −u, we obtain Equation (8) where C > 0 only depends on q, U. In particular, applying the Sobolev inequality, we get where C > 0 only depends on q, U.
Proof of Lemma A3. By [19] (Theorem 9.15 and Lemma 9.17), there exists where C 1 > 0 is a constant only depending on q, U. Let F := ∇u.
Proof of Theorem 2. Without loss of generality, we may assume that d where C 1 > 0 is a constant only depending on q and U 2 . Then, Equation (9) implies .

Proof of Theorem 4. By [8] (Theorem 3.6), there exists
The equivalence of Equation (18) and (6) follows where L 0 is as in Equation (12) and by elementary calculation R d L 0 f dµ = 0 for any f ∈ C ∞ 0 (R d ).
Proof of Theorem 5. Let f ∈ C ∞ 0 (R d ) and α > 0. Then, by Theorem 3, ρ is locally bounded below and above on R d and ρψB ∈ L p loc (R d , R d ), αρψ ∈ L q loc (R d ). Let B and B be open balls in R d satisfying B ⊂ B . Since 1 ψ ∈ L ∞ (B ), G α f ∈ H 1,2 (B ). Thus, by Theorem 2, there exist a Hölder continuous µ-version R α f of G α f on R d and constants γ ∈ (0, 1), c 1 > 0 that are independent of f , such that where (inf B ρψ) 1/s . Using the Hölder inequality and the contraction property, Assumption (A12) extends to f ∈ ∪ r∈[s,∞) L r (R d , µ). In order to extend Assumption (A12) to f ∈ L ∞ (R d , µ), let f n : The following well-known fact is stated without proof.
Thus, using the sub-Markovian property and Lebesgue's Theorem in Assumption (A15), (P · f n (·)) n≥1 is a Cauchy sequence in C(U × [τ 1 , τ 2 ]). Hence, we can again define For each t > 0, P t f n converges uniformly to P t f in U; hence, in view of Assumption (A17), T t f has continuous µ-version P t f and P · f ∈ C(U × [τ 1 , τ 2 ]). Therefore, Assumption (A16) extends to all f ∈ L ∞ (R d , µ). Since U and [τ 1 , τ 2 ] were arbitrary, it holds for any Proof of Proposition 3. The first shows the quasiregularity of the generalized Dirichlet form (E , D(L 2 )) associated with (L 2 , D(L 2 )), and the existence of a µ-tight special standard process associated with (E , D(L 2 )). This can be done exactly as in [10] (Theorem 3.5). One only has to take care that space Y, as defined in the proof of [10] (Theorem 3.5), is replaced because of a seemingly uncorrected version of the paper by to guarantee the convergence at the end of the proof. In particular, D(L) b is an algebra that can be proven in a similar way to [10] (Remark 1.7iii). Then, the assertion follows exactly as in [11] (Theorem 6), using for the proof instead G there the space Y defined above and defining E k ≡ R d , k ≥ 1.

Proof of Theorem 10. Take
Then, it holds that For sufficiently large n ∈ N, let ζ n be a standard mollifier on R d+1 and u n := u * ζ n . Then it holds u n ∈ C ∞ (B R × [0, T]), such that lim n→∞ u n − u W 2,1 2d+2 (B R ×(0,T)) = 0 and lim n→∞ ∇u n − ∇u L 4d+4 (B R ×(0,T)) = 0 . By Itô's formula, for x ∈ R d , it holds for any n ≥ 1 By Sobolev embedding, there exists a constant C > 0, independent of u n and u, such that Thus, lim n→∞ u n (x, 0) = u(x, 0) and where C > 0 is a constant that is independent of u and u n . Using Jensen's inequality, Itô isometry, and Theorem 9, we obtain Letting n → ∞ in Assumption (A18), the assertion holds.
Proof of Theorem 12. Let x ∈ R d be arbitrary. Let Q x = P x • X −1 and Q x = P x • X −1 respectively. Then Q x , Q x are two solutions of the time-homogeneous martingale problem with initial condition x and coefficients ( σ, G) as defined in [2] (Chapter 5, 4.15 Definition). Let f ∈ C ∞ 0 (R d ). For T > 0, define g(x, t) := u f (x, T − t), (x, t) ∈ R d × [0, T], where u f is defined as in Theorem 11. Then by Theorem 11, and it holds ∂g ∂t + Lg = 0 a.e. in R d × (0, T), g(x, T) = f (x) for all x ∈ R d .
Applying Theorem 9 to M, for x ∈ R d , R > 0, it holds that ∇g(X s , s) σ(X s )dW s , P x -a.s.
Therefore E x g(X T∧D R , T ∧ D R ) = g(x, 0).
Analogously for M, we obtain E x [ f ( X T )] = g(x, 0). Thus, Therefore, Q x and Q x have the same one-dimensional marginal distributions, and we can conclude as in [2] (Chapter 5, proof of 4.27 Proposition) that Q x = Q x .