Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients

Article

Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients

Haesung Lee¹and Gerald Trutnau^2,*

1 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany;

fthslt@math.uni-bielefeld.de

2 Department of Mathematical Sciences and Research Institute, Mathematics of Seoul National University, 1 Gwanak-Ro, Gwanak-Gu, Seoul 08826, Korea

* Correspondence: trutnau@snu.ac.kr

Received: 14 February 2020; Accepted: 12 March 2020; Published: 5 April 2020 Abstract:We show uniqueness in law for a general class of stochastic differential equations inR^d,d≥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. Points of degeneracy have ad-dimensional Lebesgue–Borel measure zero. Weak existence is obtained for a more general, but not necessarily locally bounded drift coefficient.

Keywords: degenerate stochastic differential equation; uniqueness in law; martingale problem;

weak existence; strong Feller semigroup.

MSC:primary: 60H20, 47D07, 35K10; secondary: 60J60, 60J35, 31C25, 35B65

1. Introduction

The question whether a solution to a stochastic differential equation (hereafter SDE) onR^dexists 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 conditionsx∈R^das defined, for instance, in [2] (Chapter 5); see also Definition2below.

To explain our motivation for this work, fix symmetric matrixC= (c_ij)_1≤i,j≤dof bounded measurable functionsc_ij, such that, for someλ≥1,

λ⁻¹kξk²≤ hC(x)ξ,ξi ≤λkξk², for allx,ξ∈R^d,

Symmetry2020,12, 570; doi:10.3390/sym12040570 www.mdpi.com/journal/symmetry

and vectorH= (h₁, ...,h_d)of locally bounded measurable functions. Let Lf =

∑

c_ij 2∂ijf +

∑

hi∂if (1)

be the corresponding linear operator and Xt=x+

Z _t 0

√C(Xs)dWs+ Z _t

0 H(Xs)ds, t≥0, x∈R^{d, (2)}

be the corresponding Itô-SDE. If thec_ijare continuous and theh_ibounded, then Equation (2) is well-posed, i.e., there exists a solution and it is unique in law (see [3]). If thehiare bounded, then Equation (2) is well-posed ford=2 (see [3] Exercise 7.3.4); however, ifd≥3, there exists an example of a measurable discontinuousC 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 whenC is not far from being continuous, i.e., continuous up to a small set (e.g., a discrete set or a set ofα-Hausdorff measure zero with sufficiently small α; else, see, for instance, introductions of [4,5] for references). Another special subclass is given whenCis a piecewise constant on a decomposition ofR^dinto a finite union of polyhedrons [6], and thehiare 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^dinto bounded disjoint measurable sets (choose, for instance,q

1

ψ = _∑^∞_i=1α_i1A_i, withR^d =∪˙^∞_i=1Ai, (α_i)_i∈N⊂(0,∞)in Equation (4) below). A further example for a discontinuousC, 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 forLas 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 discontinuousC. 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 degenerateCwhere 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 whereCis degenerate (or also nondegenerate ifd≥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-MarkovianC0-semigroup of contractions on someL¹-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 (Theorem9) 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 Remark4). In particular, our method typically implies weak-existence results that are more general than uniqueness results (see Theorem8here and in [1,8]).

Now, let us describe our results. Letd≥2, andA= (a_ij)_1≤i,j≤dbe a symmetric matrix of functions a_ij ∈H^1,2d+2_loc (R^d)∩C(R^d), such that, for every open ballB⊂R^d, there exist constantsλ_B,ΛB>0 with

λ_Bkξk²≤ hA(x)ξ,ξi ≤_ΛBkξk², for allξ∈R^{d, x} ∈B.

Letψ ∈ L^q_loc(R^d)_{, with}q > 2d+_2,ψ > 0 a.e., such that _ψ¹ ∈ L^∞_loc(R^d). Here, we assumed that expression _ψ¹ stood for an arbitrary but fixed Borel measurable function satisfying ψ· ¹_ψ = 1 a.e., and _ψ¹(x)∈[0,∞)for anyx ∈R^d. LetG= (g1, . . . ,gd)∈ L^∞_loc(R^d,R^d)be a vector of Borel measurable functions. Let(σ_ij)1≤i≤d,1≤j≤m,m∈ Narbitrary but fixed, and be any matrix consisting of continuous functions, such thatA=σσ^T. Suppose there exists a constantM>0, such that

−

(_ψ¹A)(x)x,x kxk²+1 +¹

2trace (¹

ψA)(x)+G(x),x

≤M

kxk²+1 ln(kxk²+1) +1

(3) for a.e.x∈R^d. The main result of our paper (Theorem13) was that weak existence and uniqueness in law, i.e., well-posedness, then holds for stochastic differential equation

Xt=x+ Z _t

0

s1 ψ·σ

(Xs)dWs+ Z _t

0 G(Xs)ds, t≥0, x∈R^{d. (4)}

among all weak solutions(_Ω,F,(F_t)_t≥0,Xt= (X_t¹, . . . ,X^d_t),W = (W¹, . . . ,W^m),Px),x∈R^d, such that Z _∞

0 1^q₁

ψ=0 (Xs)ds=0 Px-a.s. ∀x∈R^d. (5) Here, the solution and integrals involving the solution in Equation (4) may a priori depend on Borel versions chosen forq

1

ψ andG. but Condition (5) is exactly the condition that makes these objects independent of the chosen Borel versions (cf. Lemma2).q

1

ψ may, of course, be fully discontinuous, but if it takes all its values in(0,∞); then, Equation (5) is automatically satisfied. However, sinceψ∈L^q_loc(R^d), it must be a.e. finite, so that zerosZofq

1

ψ have Lebesgue–Borel measure zero. Nonetheless, our main result comprehends the existence of a whole class of degenerate (onZ) 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 Lemma2. 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 Example2.

We derived weak existence of a solution to Equation (4) up to its explosion time under quite more general conditions on the coefficients, see Theorem8. In this case, for nonexplosion, one only needs that Equation (3) holds outside an arbitrarily large open ball (see Remark3ii). Moreover, Equation (5) is always satisfied for the weak solution that we construct (see Remark3), 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 LonC₀^∞(R^d),

L f =

∑

1 ψa_ij

2 ∂_ijf+

∑

g_i∂_if, f ∈C^∞₀(R^d).

In Theorem 4, we found a measure µ := ρψdx with some nice regularity of ρ, which is an infinitesimally invariant measure for(L,C₀^∞(R^d)), i.e.,

Z

R^d

L f dµ= Z

R^d

∑

1 ψaij

2 ∂ijf +

∑

gi∂if

dµ=0, ∀f ∈C₀^∞(R^d). (6)

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-MarkovianC₀-semigroup of contractions (Tt)t≥0on eachL^s(R^d,µ),s≥1 of which the generator extended(L,C₀^∞(R^d)), i.e., we found a suitable functional analytic frame (see Theorem3that 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 Section4, where we also derive, with the help of the results about general regularity properties from Section3, the regularity properties of(Tt)t≥0and its resolvent (see Section4.3). Then, crucially using the existence of a Hunt process for a.e. starting point related to(Tt)t≥0in Proposition3(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 Remark2). However, irreducibility and classical ergodic properties, as in [1], could also be studied in this framework by further investigating the influence of _ψ¹ 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 ofq

1

ψ, we needed Condition (5) to derive a Krylov-type estimate that held for any weak solution to Condition (4) (see Theorem9which 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 functiong(x,t):=PT−tf(x), f ∈C₀^∞(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. Theorem12). 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).

2. Article Structure and Notations

The main parts of this article are Sections4and5. Section4contains the analytic results, and Section5 contains the probabilistic results. Section3also contains auxiliary analytical results that are important on their own. Section3could be skipped in a first reading, so the reader may directly start with Section4.

The proofs for all statements of this article and further auxiliary statements were collected in AppendixA.

Throughout, we used the same notations as in [1,8], andd ≥ 2. Additionally, for an open-setU inR^dand a measureµonR^d, letL^q(U,R^d,µ):={F= (f1, . . . ,f_d):U→R^d | fi∈ L^q(U,µ), 1≤i≤d}, equipped with the norm,kFk_Lq(U,R^d,µ) := kkFkk_Lq(U,µ),F ∈ L^q(U,R^d,µ). Ifµ = dx, we write L^q(U), L^q(U,R^d)forL^q(U,dx),L^q(U,R^d,dx)respectively, and evenkFk_Lq(U)forkFk_Lq(U,R^d). Denote byC^k(U), k∈N∪ {0}, the usual space ofk-times continuously differentiable functions inU, such that the partial derivatives of an order less or equal tokextend continuously toU(as defined, for instance, in [13]).

In particular,C(U) := C⁰(U)is the space of continuous functions onUwith supnorm k · k_C(U) and C^∞(U):=^T_k∈NC^k(U). IfIis an open interval inR^andp,q∈[1,∞], we denoted byL^p,q(U×I)the space of all Borel measurable functions f onU×Ifor which

kfk_Lp,q(U×I):=kkf(·,·)k_Lp(U)k_Lq(I)<_∞,

and let supp(f) _:= _supp(|f|dxdt). For a locally integrable function g on U×I and i ∈ {1, . . . ,d}, we denoted by∂_igthei-th weak spatial derivative onU×I, by∇g:= (∂₁g, . . . ,∂_dg)the weak spatial gradient ofg, by∇²g:= (∂_ijg)_1≤i,j≤dthe weak spatial Hessian matrix, and by∂tgthe weak time derivative onU×I, provided these existed. Forp,q ∈ [1,∞], letWp,q^2,1(U×I)be the set of all locally integrable functionsg:U×I →Rsuch that∂_tg,∂_ig,∂_i∂_jg ∈ L^p,q(U×I)for all 1≤i,j ≤d. LetW^2,1p (U×I):= Wp,p^2,1(U×I).

3. New Regularity Results

In this section, we develop some new regularity estimates (Theorems1and2). Theorem1was used to obtain the semigroup regularity in Theorem6, and Theorem2was used to obtain the resolvent regularity in Theorem5.

3.1. Regularity Estimate for Linear Parabolic Equations with Weight in Time Derivative Term Throughout this subsection, we assume the following condition:

(I) U×(0,T)is a bounded open set inR^d×R^,T >0, A= (a_ij)_1≤i,j≤dis a (possibly nonsymmetric) matrix of functions onUthat is uniformly strictly elliptic and bounded, i.e., there exist constants λ>0,M>0, such that, for allξ= (ξ₁, . . . ,ξ_d)∈R^d,x∈U, it holds

∑

a_ij(x)ξ_iξ_j ≥λkξk², max

1≤i,j≤d|a_ij(x)| ≤M,

B∈L^p(U,R^d)withp>d, ψ∈L^q(U),q∈[2∨ ^p₂,p), and there existsc0>0, such thatc0≤ψonU, 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

Z Z

U×(0,T)(u∂tϕ)ψdxdt= Z Z

U×(0,T)

A∇u,∇ϕ

+hB,∇uiϕdxdt, (7)

which is supposed to hold for allϕ∈C^∞₀(U×(0,T)).

Let(x, ¯¯ t)be an arbitrary but fixed point inU×(_0,T)_{, and}Rx¯(r)be the open cube inR^dof edge lengthr>0 centered at ¯x. DefineQ(r):=Rx¯(r)×(t^¯−r², ¯t).

Theorem 1. Suppose that Q(3r)⊂U×(0,T). Under the assumption(I)and(7), we have kuk_L∞(Q(r))≤Ckuk

L

2p

p−2 ,2(Q(2r))

, (8)

where C>₀is a constant depending only on r,λ, M andkBk_Lp(Rx¯(3r)). 3.2. 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 AppendixA).

Theorem 2. Let U be a bounded open ball inR^{d. Let A} = a_ij

1≤i,j≤dbe as in(I). AssumeB ∈ L^p(U,R^d), c∈L^q(U), f ∈ L^qe(U)for some p>d, q,eq> ^d₂. If u∈ H^1,2(U)satisfies

Z

U

hA∇u,∇ϕi+ (hB,∇ui+cu)ϕdx= Z

Ufϕdx, for allϕ∈C₀^∞(U), (9) then for any open ball U₁inR^{dwith U}1⊂U, we have u∈C^0,γ(U₁)and

kuk_C0,γ(U₁)≤C

kuk_L1(U)+kfk_Leq(U)

, whereγ∈(0, 1)and C>0are constants which are independent of u and f . 4.L¹-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 (Theorem3and (19)). Subsequently, using the regularity results from Section3, we derived regularity properties for the resolvent and semigroup (Theorems5and6). One key tool for this method is the existence of an infinitesimally invariant measure with nice density (Theorem4).

4.1. Framework

Letρ∈H^1,2_loc(R^d)∩L^∞_loc(R^d),ψ∈L¹_loc(R^d)be a.e. strictly positive functions satisfying¹_ρ,_ψ¹ ∈L^∞_loc(R^d). Here, we assumed that expressions¹_ρ, _ψ¹, denoted any Borel measurable functions satisfyingρ·¹

ρ =1 andψ· _ψ¹ = 1 a.e., respectively (later, especially in Section5it is important which measurable Borel version_ψ¹ we choose, but for the moment it does not matter). Setµ:=ρψdx. IfUis any open subset of R^d; then, bilinear formR

Uh∇u,∇vidx, u,v∈ C₀^∞(U)is closable inL²(U,µ)by [16] (Subsection II.2a)).

DefineHb₀^1,2(U,µ)as the closure ofC^∞₀(U)inL²(U,µ)with respect to norm R

Uk∇uk²dx+R

Uu²dµ1/2

. Thusu∈ Hb₀^1,2(U,µ), if and only if there exists(un)_n≥1⊂C₀^∞(U)such that

n→lim∞un =u inL²(U,µ)_{, lim}

n,m→∞ Z

Uk∇(un−um)k²dx=_{0; (10)}

moreover,Hb₀^1,2(U,µ)is a Hilbert space with inner product hu,vi

Hb₀^1,2(U,µ)= lim

n→∞ Z

U

h∇un,∇vnidx+ Z

Uuv dµ, where(un)_n≥1,(vn)_n≥1⊂C₀^∞(U)are arbitrary sequences that satisfy Equation (10).

Ifu ∈ Hb^1,2₀ (V,µ)for some bounded open subsetVofR^d, thenu ∈ H₀^1,2(V)∩L²(V,µ)and there exists(un)n≥1⊂C^∞₀(V), such that

n→lim∞un =u inH^1,2₀ (V) and inL²(V,µ). Consider a symmetric matrix of functionsA= (aij)_1≤i,j≤dsatisfying

a_ij=a_ji∈ H^1,2_loc(R^d), 1≤i,j≤d,

and assume Ais locally uniformly strictly elliptic, i.e., for every open ball B, there exist constants λ_B,ΛB>0, such that

λ_Bkξk²≤ hA(x)ξ,ξi ≤ΛBkξk², for allξ∈R^d, x ∈B. (11) DefineAb:= _ψ¹A. By [16] (Subsection II.2b)), the symmetric bilinear form

E⁰(f,g):= ¹ 2 Z

R^d

hAb∇f,∇gidµ, f,g∈C₀^∞(R^d),

is closable inL²(R^d,µ), and its closure(E⁰,D(E⁰))is a symmetric Dirichlet form inL²(R^d,µ)(see [16]

((II. 2.18))). Denote the corresponding generator of (E⁰,D(E⁰)) by (L⁰,D(L⁰)). Let f ∈ C₀^∞(R^d). Using integration by parts, for anyg∈C₀^∞(R^d),

E⁰(f,g) = − Z

R^d

1

2trace(Ab∇²f) +h ¹

2ψ∇A+ ^A∇ρ 2ρψ

| {z }

=:β^ρ,A,ψ

,∇fig dµ.

Thus, f ∈ D(L⁰). This impliesC₀^∞(R^d)⊂D(L⁰)and L⁰f = ¹

2trace(Ab∇²f) +hβ^ρ,A,ψ,∇fi ∈L²(R^d,µ). (12) Let (T_t⁰)_t>0 be the sub-Markovian C₀-semigroup of contractions on L²(R^d,µ) associated with (L⁰,D(L⁰)). It is well-known that T_t⁰|_L1(R^d,µ)∩L^∞(R^d,µ)can be uniquely extended to a sub-Markovian C0-semigroup of contractions(T_t⁰)_t>0onL¹(R^d,µ).

Now, letB∈L²_loc(R^d,R^d,µ)be weakly divergence-free with respect toµ, i.e., Z

R^d

hB,∇uidµ=0, for allu∈C^∞₀(R^d). (13) Assume

ρψB∈L²_loc(R^d,R^d). (14) By routine arguments, Equation (13) extends to allu∈Hb^1,2₀ (R^d,µ)_0,b, and

Z

R^d

hB,∇uivdµ=− Z

R^d

hB,∇viudµ, for allu,v∈Hb₀^1,2(R^d,µ)_0,b. DefineLu:=L⁰u+hB,∇ui, u∈D(L⁰)_0,b. Then,(L,D(L⁰)_0,b)is an extension of

1

2trace(Ab∇²u) +hβ^ρ,A,ψ+B,∇ui, u∈C^∞₀(R^d). For any bounded open subsetVofR^d,

E^0,V(f,g):= ¹ 2 Z

VhAb∇f,∇gidµ, f,g∈C₀^∞(V).

is also closable on L²(V,µ) by [16] (Subsection II.2b)). Denote by (E^0,V,D(E^0,V)) the closure of (E^0,V,C₀^∞(V))inL²(V,µ). Using (11) and 0<infVρ≤sup_Vρ<∞, it is clear thatD(E^0,V) =Hb₀^1,2(V,µ)

since the normsk · k_D(E0,V)andk · k

Hb₀^1,2(V,µ)are equivalent. Denote by(L^0,V,D(L^0,V))the generator of (E^0,V,D(E^0,V)).

4.2. L¹-Generator

In this section, we use all notations and assumptions from Section4.1.

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 functionψthat also acts on theµ-divergence free antisymmetric part of drift (see Equation (13)), we considered local convergence in spaceHb₀^1,2(V,µ)and imposed Assumption (14) on the antisymmetric part, while [10] (Chapter 1) dealt with local convergence in spaceH₀^1,2(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.

Proposition 1. Let V be a bounded open subset ofR^d. (i) Operator(L^V,D(L^0,V)_b)on L¹(V,µ)defined by

L^Vu:=L^0,Vu+hB,∇ui, u∈D(L^0,V)_b

is dissipative, and hence closable on L¹(V,µ). Closure(L^V,D(L^V))generates a sub-Markovian C₀-semigroup of contractions(T^V_t )_t>0on L¹(V,µ).

(ii) D(L^V)_b⊂ Hb₀^1,2(V,µ)and E^0,V(u,v)−

Z

VhB,∇uivdµ= Z

VL^Vu·vdµ, for all u∈D(L^V)_b, v∈ Hb₀^1,2(V,µ)_b. (15) Now, letV be a bounded open subset ofR^d. Denote by(G^V_α)_α>0the resolvent associated with (L^V,D(L^V))onL¹(V,µ). Then,(G^V_α)_α>0could be extended toL¹(R^d,µ)by

G^V_αf := (

G^V_α(f1V) on V

0 on R^d\V, f ∈L¹(R^d,µ), (16)

Letg∈ L¹(R^d,µ)_b. ThenG^V_α(g1V)∈ D(L^V)_b ⊂Hb₀^1,2(V,µ), henceG^V_αg∈Hb₀^1,2(V,µ).

Ifu ∈ D(E^0,V), then by definition it holdsu ∈ D(E⁰)andE^0,V(u,u) = E⁰(u,u). Therefore, we obtained

E⁰(G^V_αⁿg,G^V_αⁿg) =E^0,Vn G^V_αⁿ(g1_Vn)_,G^V_αⁿ(g1_Vn)_{. (17)} By means of Proposition1, the following Theorem3was also derived in a nearly identical manner to [10] (Theorem 1.5).

Theorem 3. There exists a closed extension(L,D(L))of Lu := L⁰u+hB,∇ui, u ∈ D(L⁰)_0,bon L¹(R^d,µ) satisfying the following properties:

(a) (L,D(L))generates a sub-Markovian C0-semigroup of contractions(Tt)_t>0on L¹(R^d,µ).

(b) Let(Un)n≥1be a family of bounded open subsets ofR^dsatisfying Un ⊂Un+1andR^d=^S_n≥1Un. Then limn→∞G^U_αⁿf = (α−L)⁻¹f in L¹(R^d,µ), for all f ∈ L¹(R^d,µ)andα>0.

(c) D(L)_b ⊂D(E⁰)and for all u∈D(L)_b,v∈Hb₀^1,2(R^d,µ)_0,bit holds E⁰(u,u)≤ −

Z

R^d

Lu·udµ, and E⁰(u,v)− Z

R^d

hB,∇uivdµ=− Z

R^d

Lu·vdµ.

4.3. Existence of Infinitesimally Invariant Measure and Strong Feller Properties Here, we state some conditions that were used as our assumptions.

(A1)p > dis fixed, andA = (a_ij)_1≤i,j≤dis a symmetric matrix of functions that are locally uniformly strictly elliptic on R^d, such that aij ∈ H_loc^1,p(R^d)∩C(R^d) for all 1 ≤ i,j ≤ d. ψ ∈ L¹_loc(R^d)is a positive function, such that _ψ¹ ∈L^∞_loc(R^d)andGis a Borel measurable vector field onR^dsatisfying ψG∈L_loc^p (R^d,R^d).

(A2)ψ∈ L^q_loc(R^d)withq∈(^d₂,∞]. Fixs∈(^d₂,∞)such that ¹_q +¹_s < ²_d. (A3)q∈[^p₂∨2,∞].

Theorem 4. Under Assumption(A1), there existsρ ∈ H_loc^1,p(R^d)∩C(R^d)satisfyingρ(x)>0for all x∈R^d

such that _Z

R^d

hG−β^ρ,A,ψ,∇ϕiρψdx=0, for allϕ∈C₀^∞(R^d), (18) or equivalently,(6)holds. Moreover,ρψB∈L^p_loc(R^d,R^d), whereB:=_G−β^ρ,A,ψ.

From now on, we assume that Condition(A1)holds and fixA,ψ,ρ,Bas in Theorem4. Then,A,ψ, ρ,Bsatisfy all assumptions of Section4.1. As in Section4.1µ:=ρψdx,Ab:= _ψ¹A.

By Theorem3, there existed a closed extension(L,D(L))of L f =L⁰f +hB,∇fi, f ∈D(L⁰)_0,b,

on L¹(R^d,µ) that generates a sub-Markovian C₀-semigroup of contractions (Tt)_t>0 on L¹(R^d,µ). Restricting (Tt)_t>0 to L¹(R^d,µ)_b, it is well-known by Riesz–Thorin interpolation that(Tt)_t>0 could be extended to a sub-MarkovianC0-semigroup of contractions (Tt)_t>0 on eachL^r(R^d,µ),r ∈ [1,∞). Denote by(Lr,D(Lr))the corresponding closed generator with graph norm

kfk_D(Lr):=kfk_Lr(R^d,µ)+kLrfk_Lr(R^d,µ),

and by (G_α)_α>0 the corresponding resolvent. (Tt)_t>0 and (G_α)_α>0 can also be uniquely defined on L^∞(R^d,µ), but are no longer strongly continuous there.

For f ∈C₀^∞(R^d), we have

L f =L⁰f+hB,∇fi= ¹

2trace(Ab∇²f) +hG,∇fi. Define

L^∗f : = L⁰f− hB,∇fi= ¹

2trace(Ab∇²f) +hG^∗,∇fi, with

G^∗:= (g₁^∗, . . . ,g^∗_d) =2β^ρ,A,ψ−_G=β^ρ,A,ψ−_B∈ L²_loc(R^d,R^d,µ).

Denote by(L^∗_r,D(L^∗_r))operators corresponding toL^∗for the cogenerator onL^r(R^d,µ)_,r∈[_1,∞)_, (T_t^∗)_t>0 for the cosemigroup, (G_α^∗)_α>0 for the coresolvent. As in ([10], Section 3), we obtained a corresponding bilinear form with domainD(L2)×L²(R^d,µ)∪L²(R^d,µ)×D(L^∗₂)by

E(f,g):= ( −R

R^d L₂f·g dµ for f ∈ D(L₂)_, g∈ L²(R^d,µ)_,

−R

R^d f·L^∗₂g dµ for f ∈ L²(R^d,µ), g∈D(L^∗₂). (19) E is called thegeneralized Dirichlet form associated with(L2,D(L2)).

Theorem 5. Assume Conditions (A1) and (A2), and let f ∈ ∪_r∈[s,∞]L^r(R^d,µ). Then, G_αf has a locally Hölder continuous µ-version R_αf onR^d. Furthermore for any open balls B, B⁰ satisfying B ⊂ B⁰, we have the following estimate:

kR_αfk_C0,γ(B)≤c2

kfk_Ls(B⁰,µ)+kG_αfk_L1(B⁰,µ)

, (20)

where c2>0,γ∈(0, 1)are constants that are independent of f .

Let f ∈ D(Lr)for somer ∈ [s,∞). Then f = G₁(1−Lr)f; hence, by Theorem5, f has a locally Hölder continuousµ-version onR^dand

kfk_C0,γ(B) ≤ c3kfk_D(Lr),

wherec3>0,γ∈(0, 1)are constants independent of f. In particular,Ttf ∈D(Lr)andTtf hence has a continuousµ-version, sayPtf, with

kPtfk_C0,γ(B)≤c₃kPtfk_D(Lr). (21) c3 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).

Lemma 1. Assume Conditions(A1),(A2). For any f ∈^S_r∈[s,∞)D(Lr), map (x,t)7→Ptf(x)

is continuous onR^d×[0,∞).

Theorem 6. Assume Conditions(A1),(A2), and(A3), and let f ∈^S

ν∈[_p²₋^p₂,∞]L^ν(R^d,µ), t>0. Then, Ttf has a continuousµ-version Ptf onR^d, and P·f(·)is continuous onR^d×(0,∞). For any bounded open set U, V inR^{dwith U}⊂ V and0< τ3 <τ₁ <τ2< τ₄, i.e.,[τ₁,τ2]⊂ (τ3,τ₄), we have the following estimate for all

f ∈ ∪

ν∈[_p^2p₋₂,∞]L^ν(R^d,µ):

kP·f(·)k_C(U×[τ

1,τ2])≤C1kP·f(·)k

L

2p

p−2 ,2(V×(τ₃,τ₄))

, (22)

where C1is a constant that depends on U×[τ₁,τ₂],V×(τ₃,τ₄), but is independent of f .

By Theorems5and6, exactly as in [1] (Remark 3.7), we obtained resolvent kernels and resolvent kernel densitiesR_α(x,dy),r_α(x,y), corresponding to resolvent(R_α)_α>0, as well as transition kernels and transition-kernel densitiesPt(x,dy),pt(x,y), corresponding to transition function(Pt)t≥0.

Proposition 2. Assume Conditions(A1),(A2), and(A3), and let t,α>0. Then, it holds that (i) Gαg has a locally Hölder continuousµ-version

Rαg= Z

R^d

g(y)Rα(·,dy) = Z

R^d

g(y)rα(·,y)µ(dy), ∀g∈ ^[

r∈[s,∞]

L^r(R^d,µ). (23)

In particular, Equation (23) extends by linearity to all g ∈ L^s(R^d,µ) +L^∞(R^d,µ), i.e., (Rα)_α>0 is L^[s,∞](R^d,µ)-strong Feller.

(ii) Ttf has a continuousµ-version Ptf =

Z

R^d

f(y)Pt(·,dy) = Z

R^d

f(y)pt(·,y)µ(dy), ∀f ∈ ^[

ν∈[_p^2p₋₂,∞]

L^ν(R^d,µ). (24)

In particular,Equation (24) extends by linearity to all f ∈ L

2p

p−2(R^d,µ) +L^∞(R^d,µ), i.e., (Pt)_t>0 is L^[

2p p−2,∞]

(R^d,µ)-strong Feller.

Finally, for anyα>0,x∈R^{d, g}∈ L^s(R^d,µ) +L^∞(R^d,µ) R_αg(x) =

Z _∞

0 e^−αtPtg(x)dt.

5. Well-Posedness

With the help of the regularity results, Theorems5and6of Section4, and the mere existence of a Hunt process for a.e. starting point (Proposition3), we constructed a weak solution to Equation (4) (Theorems7and8). Then, using a local Krylov-type estimate and Itô-formula (Theorems 9and10), uniqueness in law was derived for weak solutions to Equation (4) that spend zero time at the points of degeneracy of the dispersion matrix (Theorems12and13). The method to derive uniqueness in law is an adaptation of the Stroock and Varadhan method ([3]) via the martingale problem.

5.1. Weak Existence

The following assumption in particular is necessary to obtain a Hunt process with transition function (Pt)_t≥0(and consequently a weak solution to the corresponding SDE for every starting point). It is first used in Theorem7below.

(A4) G∈L^s_loc(R^d,R^d,µ), wheresis as in(A2).

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.

Proposition 3. Assume Conditions(A1),(A2), and(A3). Then, there exists a Hunt process M˜ = (Ω, ˜^˜ F,(F^˜)t≥0,(X^˜t)t≥0,(P^˜x)_x∈

R^d∪{∆})

with life timeζ˜:=inf{t≥0|X^˜t=∆}and cemetery∆, such thatEis (strictly properly) associated withM˜ and for strictlyE-q.e. x∈R^d,

P˜x

ω∈_Ω^˜ |X^˜·(ω)∈C [0,∞),R^d_∆^{, ˜X}t(ω) =_∆,∀t≥ζ(ω) =1.

Remark 1. (i) Assume Conditions(A1),(A2),(A3), andG ∈ L

sq q−1

loc (R^d,R^d). Then, for any bounded open subset V ofR^d, it holds that

Z

VkGk^sdµ≤ kGk^s

L

sq q−1(V)

kρψk_Lq(V); hence, Condition(A4)is satisfied.

(ii) Two simple examples where Conditions(A1),(A2),(A3), and(A4)are satisfied are given as follows: for the first example, let A,ψsatisfy the assumptions of(A1),ψ∈ L_loc^p (R^d), s= _2p−d^dp +ε, andG∈L^∞_loc(R^d,R^d); for the second, let A,ψsatisfy the assumptions of(A1),ψ∈L^2p_loc(R^d), s= _4p−d^2pd +εandG∈L^2p_loc(R^d,R^d). In both cases,ε>₀can be chosen to be arbitrarily small.

Analogously to [1] (Theorem 3.12), we obtained

Theorem 7. Under Assumptions(A1),(A2),(A3),(A4), there exists a Hunt process M= (_Ω,F,(F_t)_t≥0,(Xt)_t≥0,(Px)_x∈

R^d∪{∆})

with state spaceR^dand life time

ζ=inf{t≥0|Xt=∆}=inf{t≥0|Xt∈/R^d},

having transition function(Pt)t≥0as the transition semigroup, such thatMhas continuous sample paths in the one-point compactificationR^d_∆^ofR^dwith cemetery∆as point at infinity, i.e., for any x∈R^d,

Px

ω∈_Ω|X·(ω)∈C [0,∞),R^d_∆^{, X}t(ω) =_∆,∀t≥ζ(ω) =1.

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 = _d+p^dp > ^d₂ of [1] is replaced by s> ^d₂ of(A2). A Krylov-type estimate forM^{of Theorem7}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,

sup

x∈B

Ex

Z _t

0 |g|(Xs)ds

<e^tCB,rkgk_Lr(R^d,µ). (25)

The derivation of Equation(25)is based on Theorem5, of which the proof uses the elliptic Hölder estimate of Theorem2. 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.

The following theorem can be proved exactly as in [1] (Theorem 3.19).

Theorem 8. Assume Conditions (A1), (A2), (A3), and (A4) are satisfied. Consider Hunt process M from Theorem 7with co-ordinates Xt = (X¹_t, . . . ,X_t^d). Let(σ_ij)1≤i≤d,1≤j≤m, m ∈ N arbitrary but fixed, be any locally uniformly strictly elliptic matrix consisting of continuous functions for all1≤i≤d,1≤ j≤m, such that A=σσ^T, i.e.,

a_ij(x) =

∑

σ_ik(x)σ_jk(x)_, ∀x∈R^d, 1≤i,j≤d.

Set

bσ= s

1

ψ·σ, i.e.,bσ_ij= s

1

ψ·σ_ij, 1≤i≤d, 1≤j≤m.

(Recall that expression _ψ¹ denotes an arbitrary Borel measurable function satisfyingψ·_ψ¹ =1a.e.).

Then, on a standard extension of(_Ω,F,(Ft)_t≥0,Px), x∈R^d, which we denote for notational convenience again by (Ω,F,(F_t)t≥0,Px), x ∈ R^d, there exists a standard m-dimensional Brownian motion W = (W¹, . . . ,W^m)starting from zero, such thatPx-a.s. for any x= (x1, . . . ,x_d)∈R^{d, i}=1, . . . ,d

Xⁱ_t=xi+

∑

Z _t

0 bσ_ij(Xs)dWs^j+ Z _t

0 gi(Xs)ds, 0≤t<ζ, (26)

in short

Xt=x+ Z t

0 bσ(Xs)dWs+ Z t

0 G(Xs)ds, 0≤t<ζ.

If Equation(3)holds a.e. outside an arbitrarily large compact set, thenPx(ζ = _∞) = 1for all x ∈ R^d (cf. [1] (Theorem 4.2)).

Example 1. Given p>d, let A= (aij)_1≤i,j≤dbe a symmetric matrix of functions onR^dthat is locally uniformly strictly elliptic and aij ∈ H_loc^1,p(R^d)∩C(R^d)for all1≤i,j ≤d. Given m ∈N, letσ= (σ_ij)1≤i≤d,1≤j≤mbe a matrix of functions satisfyingσ_ij ∈C(R^d)for all1≤i≤d, 1≤j≤m, such that A=σσ^T. Letφ∈L^∞_loc(R^d) be such that for any open ball B, there exist strictly positive constants cB, CBsuch that

c_B≤φ(x)≤C_B for every x∈ B.

Let _ψ¹(x):= ^kxk_φ(x)^α, x∈R^{d, for someα}>0and consider following conditions.

(a) αp<d, G∈ L^∞(B_ε(0),R^d)∩L^p_loc(R^d\B_ε(0),R^d)for someε>0, (b) 2αp<d, G∈L^2p(Bε(0),R^d)∩L_loc^p (R^d\Bε(0),R^d)for someε>0,

(c) α·(^p₂∨2) < d, G ≡ 0on Bε(0)andG ∈ L^s_loc(R^d\Bε(0),R^d)for someε > 0, where s > d so that (^p₂∨2)⁻¹+¹_s < ²_d.

Any of Conditions (a), (b), or (c) imply Assumptions(A1),(A2),(A3), and(A4). Indeed, for an arbitrary ε>0take q= p, s= _2p−d^pd +εin the case of Condition (a), q=2p, s= _4p−d^2pd +εin the case of Condition (b), and q= ^p₂∨2, s>d defined by Condition (c) in the case of Condition (c). Assuming Condition (a), (b), or (c), Hunt processMas in Theorem8solves weakly Px-a.s. for any x∈R^d,

Xt=x+ Z t

0 kXsk^α/2· √^σ

φ(Xs)dWs+ Z t

0 G(Xs)ds, 0≤t<ζ (27)

and is nonexplosive if Equation(3)holds a.e. outside an arbitrarily large compact set.

5.2. Uniqueness in Law Consider

(A4)⁰:(A1)holds withp = 2d+2,(A2)holds with someq ∈(2d+2,∞],s∈ (^d₂,∞)is fixed, such that

1

q+¹_s < ²_d, andG∈L^∞_loc(R^d,R^d).

Definition 1. Suppose Assumptions(A1),(A2),(A3), and(A4)hold (for instance, if(A4)⁰holds). Let expression

1

ψ denote an arbitrary but fixed Borel measurable function satisfyingψ· ¹_ψ =1a.e. and _ψ¹(x)∈[0,∞)for any x ∈R^{d. Let}

Me = (Ω,e Fe,(Fet)_t≥0,(Xet)_t≥0,(Wet)_t≥0,(Pex)_x∈

R^d) be such that for any x= (x1, . . . ,x_d)∈R^d

(i) (Ω,e Fe,(Fe_t)_t≥0,Pex)is a filtered probability space, satisfying the usual conditions, (ii) (Xet= (Xe¹_t, . . . ,Xe_t^d))t≥0is an(Fet)t≥0-adapted continuousR^d-valued stochastic process,

(iii) (Wet= (We_t¹, . . . ,We_t^m))_t≥0is a standard m-dimensional((Fe_t)_t≥0,ePx)-Brownian motion starting from zero, (iv) for the (real-valued) Borel measurable functionsbσ_ij,gi,_ψ¹,bσ_ij =^q_ψ¹σ_ij, withσis as in Theorem8, it holds

Pex

^Z t

0 bσ_ij²(Xes) +|gi(Xes)|ds<∞

=1, 1≤i≤d, 1≤j≤m, t∈[0,∞), and for any1≤i≤d,

Xeⁱ_t=_xi+

∑

Z _t

0 bσ_ij(Xes)_dWes^j+ Z _t

0 gi(Xes)_{ds, 0}≤t<_∞, Pex-a.s., in short

Xet=x+ Z _t

0 bσ(Xes)dWes+ Z _t

0 G(Xes)ds, 0≤t<∞, ePx-a.s. (28) Then,Me is called a weak solution to Equation(28). In this case,(t,ωe)7→_bσ(Xet(ω_e))and(t,ωe)7→G(Xet(ω_e)) are progressively measurable with respect to(Fe_t)_t≥0, and

De_R:=inf{t≥0|Xet∈R^d\B_R} %_∞ Pex-a.s. for any x∈R^d.

Remark 3. (i) In Definition1, the (real-valued) Borel measurable functionsbσ_ij,gi,_ψ¹ are fixed. In particular, the solution and the integrals involving the solution in Equation(28)may depend on the versions that we choose.

When we fix the Borel measurable version _ψ¹ with_ψ¹(x)∈[0,∞)for all x∈R^d, as in Definition1, we always consider corresponding extended Borel measurable functionψdefined by

ψ(x):= ₁¹

ψ(x)^{, if} 1

ψ(x)∈(0,∞), ψ(x):=_∞, if 1

ψ(x) =0.pt

Thus, the choice of the special version forψdepends on the previously chosen Borel measurable version _ψ¹. (ii) IfM^{of Theorem8}is nonexplosive (has infinite lifetime for any starting point), then it is a weak solution

to Equation(28). Thus, a weak solution to Equation(28)exists just under Assumptions(A1),(A2),(A3), and(A4), and a suitable growth condition (cf. Remark2) on the coefficients. For this special weak solution, we know that integrals involving the solution do not depend on the chosen Borel versions. This follows similarly to [1] (Lemma 3.14(i)).

Theorem 9(Local Krylov-type estimate). Assume(A4)⁰, and letMe be a weak solution to Equation(28). Let

Z^Me(ω_e):={t≥0| s1

ψ(Xet(ω_e)) =0}