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

The question whether a solution to a stochastic differential equation (hereafter SDE) on ${\mathbb{R}}^d$ 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 $x\in {\mathbb{R}}^d$ as defined, for instance, in [2] (Chapter 5); see also Definition 2 below.

To explain our motivation for this work, fix symmetric matrix $C={(c_{ij})}_{1\le i,j\le d}$ of bounded measurable functions $c_{ij}$, such that, for some $\lambda \ge 1$, and vector $\mathbf{H}=(h_1,\dots ,h_d)$ of locally bounded measurable functions. Let 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 $d\ge 3$, 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 $\alpha$-Hausdorff measure zero with sufficiently small $\alpha$; 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 ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^d$ into bounded disjoint measurable sets (choose, for instance, $\sqrt{\frac{1}{\psi }}={\sum }_{i=1}^{\infty }{\alpha }_i{1}_{A_i}$, with ${\mathbb{R}}^d={\dot{\cup }}_{i=1}^{\infty }{A}_i$, ${({\alpha }_i)}_{i\in \mathbb{N}}\subset (0,\infty )$ in Equation (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 $\mathcal{L}$ 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 $d\ge 3$) 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 $d\ge 2$, and $A={(a_{ij})}_{1\le i,j\le d}$ be a symmetric matrix of functions $a_{ij}\in H_{loc}^{1,2d+2}({\mathbb{R}}^d)\cap C({\mathbb{R}}^d)$, such that, for every open ball $B\subset {\mathbb{R}}^d$, there exist constants ${\lambda }_B,{\Lambda }_B>0$ with

Let $\psi \in L_{loc}^q({\mathbb{R}}^d)$, with $q>2d+2$, $\psi >0$ a.e., such that $\frac{1}{\psi }\in L_{loc}^{\infty }({\mathbb{R}}^d)$. Here, we assumed that expression $\frac{1}{\psi }$ stood for an arbitrary but fixed Borel measurable function satisfying $\psi ·\frac{1}{\psi }=1$ a.e., and $\frac{1}{\psi }(x)\in [0,\infty )$ for any $x\in {\mathbb{R}}^d$. Let $\mathbf{G}=(g_1,\dots ,g_d)\in L_{loc}^{\infty }({\mathbb{R}}^d,{\mathbb{R}}^d)$ be a vector of Borel measurable functions. Let ${({\sigma }_{ij})}_{1\le i\le d,1\le j\le m}$, $m\in \mathbb{N}$ arbitrary but fixed, and be any matrix consisting of continuous functions, such that $A=\sigma {\sigma }^T$. Suppose there exists a constant $M>0$, such that for a.e. $x\in {\mathbb{R}}^d$. 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 $(\mathsf{\Omega },\mathcal{F},{({\mathcal{F}}_t)}_{t\ge 0},X_t=(X_t^1,\dots ,X_t^d),W=(W^1,\dots ,W^m),{\mathbb{P}}_x)$, $x\in {\mathbb{R}}^d$, such that

Here, the solution and integrals involving the solution in Equation (4) may a priori depend on Borel versions chosen for $\sqrt{\frac{1}{\psi }}$ and $\mathbf{G}$. but Condition (5) is exactly the condition that makes these objects independent of the chosen Borel versions (cf. Lemma 2). $\sqrt{\frac{1}{\psi }}$ may, of course, be fully discontinuous, but if it takes all its values in $(0,\infty )$; then, Equation (5) is automatically satisfied. However, since $\psi \in L_{loc}^q({\mathbb{R}}^d)$, it must be a.e. finite, so that zeros Z of $\sqrt{\frac{1}{\psi }}$ 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 L on $C_0^{\infty }({\mathbb{R}}^d)$,

In Theorem 4, we found a measure $\mu :=\rho \psi dx$ with some nice regularity of $\rho$, which is an infinitesimally invariant measure for $(L,C_0^{\infty }({\mathbb{R}}^d))$, i.e.,

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 ${(T_t)}_{t\ge 0}$ on each $L^s({\mathbb{R}}^d,\mu )$, $s\ge 1$ of which the generator extended $(L,C_0^{\infty }({\mathbb{R}}^d))$, 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 ${(T_t)}_{t\ge 0}$ and its resolvent (see Section 4.3). Then, crucially using the existence of a Hunt process for a.e. starting point related to ${(T_t)}_{t\ge 0}$ 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 $\frac{1}{\psi }$ 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 $\sqrt{\frac{1}{\psi }}$, 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 $g(x,t):=P_{T-t}f(x)$, $f\in C_0^{\infty }({\mathbb{R}}^d)$. 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).

The main parts of this article are Section 4 and Section 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 $d\ge 2$. Additionally, for an open-set U in ${\mathbb{R}}^d$ and a measure $\mu$ on ${\mathbb{R}}^d$, let $L^q(U,{\mathbb{R}}^d,\mu ):=\{\mathbf{F}=(f_1,\dots ,f_d):U\to {\mathbb{R}}^d\mid f_i\in L^q(U,\mu ),1\le i\le d\}$, equipped with the norm, ${\parallel \mathbf{F}\parallel }_{L^q(U,{\mathbb{R}}^d,\mu )}:={\parallel \parallel \mathbf{F}\parallel \parallel }_{L^q(U,\mu )},\mathbf{F}\in L^q(U,{\mathbb{R}}^d,\mu )$. If $\mu =dx$, we write $L^q(U)$, $L^q(U,{\mathbb{R}}^d)$ for $L^q(U,dx)$, $L^q(U,{\mathbb{R}}^d,dx)$ respectively, and even ${\parallel \mathbf{F}\parallel }_{L^q(U)}$ for ${\parallel \mathbf{F}\parallel }_{L^q(U,{\mathbb{R}}^d)}$. Denote by $C^k(\overline{U})$, $k\in \mathbb{N}\cup \left\{0\right\}$, 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 $\overline{U}$ (as defined, for instance, in [13]). In particular, $C(\overline{U}):=C^0(\overline{U})$ is the space of continuous functions on $\overline{U}$ with supnorm ${\parallel ·\parallel }_{C(\overline{U})}$ and $C^{\infty }(\overline{U}):={\bigcap }_{k\in \mathbb{N}}{C}^k(\overline{U})$. If I is an open interval in $\mathbb{R}$ and $p,q\in [1,\infty ]$, we denoted by $L^{p,q}(U\times I)$ the space of all Borel measurable functions f on $U\times I$ for which and let $\mathrm{supp}(f):=\mathrm{supp}(|f|dxdt)$. For a locally integrable function g on $U\times I$ and $i\in \{1,\dots ,d\}$, we denoted by ${\partial }_{i}g$ the i-th weak spatial derivative on $U\times I$, by $\nabla g:=({\partial }_{1}g,\dots ,{\partial }_{d}g)$ the weak spatial gradient of g, by ${\nabla }^{2}g:={({\partial }_{ij}g)}_{1\le i,j\le d}$ the weak spatial Hessian matrix, and by ${\partial }_{t}g$ the weak time derivative on $U\times I$, provided these existed. For $p,q\in [1,\infty ]$, let $W_{p,q}^{2,1}(U\times I)$ be the set of all locally integrable functions $g:U\times I\to \mathbb{R}$ such that ${\partial }_{t}g,{\partial }_{i}g,{\partial }_i{\partial }_{j}g\in L^{p,q}(U\times I)$ for all $1\le i,j\le d$. Let $W_p^{2,1}(U\times I):=W_{p,p}^{2,1}(U\times I)$.

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.

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).