Existence, uniqueness and the strong Markov property of solutions to Kimura diffusions with singular drift
Abstract.
Motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics, we study existence, uniqueness and the strong Markov property of weak solutions to a class of degenerate stochastic differential equations. The stochastic differential equations considered in our article admit solutions supported in the set , and they are degenerate in the sense that the diffusion matrix is not strictly elliptic, as the smallest eigenvalue converges to zero proportional to the distance to the boundary of the domain, and the drift coefficients are allowed to have powertype singularities in a neighborhood of the boundary of the domain. Under suitable regularity assumptions on the coefficients, we establish existence of weak solutions that satisfy the strong Markov property, and uniqueness in law in the class of Markov processes.
Key words and phrases:
Kimura diffusions, singular drift coefficient, degenerate diffusions, degenerate elliptic operators, the strong Markov property, anisotropic Hölder spaces2010 Mathematics Subject Classification:
Primary 60J60; secondary 35J701. Introduction
The stochastic differential equations considered in our article are a generalization of continuous limits of Markov chains that arise in population genetics as random models for the evolution of gene frequencies. The solutions to such differential equations are supported in , where , , and and are nonnegative integers such that . Under suitable regularity assumptions on the coefficients of the stochastic differential equation, we prove existence of weak solutions that satisfy the strong Markov property (Theorem 3.4), and we establish that uniqueness in law holds in the class of Markov processes (Theorem 3.8). The stochastic differential equations considered in our article take the form:
(1.1)  
where and . The important features of the coefficients of the stochastic differential equation (1.1) are that the diffusion matrix is not strictly elliptic on , in that the smallest eigenvalue converges to proportional to the distance to the boundary of the domain , the components of the drift coefficient are allowed to have powertype singularities of the form , where the positive constant is suitably chosen, and the coefficient functions are assumed to be bounded from below by a positive constant on the boundary of the domain . When the coefficients , then we only require that the coefficients are nonnegative on . While the coefficients and are assumed to be Borel measurable, the coefficients , and are assumed to belong to suitable anisotropic Hölder spaces. A precise statement of the properties of the coefficients of the stochastic differential equation (1.1) is given in Assumption 3.1, and the definition of the anisotropic Hölder spaces considered in our work is given in §2.2.
The stochastic differential equations (1.1) are an extension of continuous processes that arise as continuous limits of discrete models for gene frequencies [14, 26, 15, 19, 20, 23, 12, 17], and we call them generalized Kimura stochastic differential equations with singular drift. When the coefficients , the singular drift disappears, and we call the resulting equations standard Kimura stochastic differential equations.
1.1. Outline of the article
We begin in §2 with the analysis of the standard Kimura stochastic differential equation, (2.1). Existence of solutions (Proposition 2.2) is an immediate consequence of classical results, and for this purpose the assumptions on the coefficients are more general, as outlined in Assumption 2.1. We establish uniqueness in law of solutions to the standard Kimura stochastic differential equation in Proposition 2.8, under the more restrictive Assumption 2.4. Notice that the drift coefficients are only assumed to be nonnegative on the boundary of the domain , and that the coefficient functions , and a suitable combination of the coefficients of the diffusion matrix are assumed to belong to the anisotropic Hölder spaces introduced in §2.2. This condition arises because our method of proof is based on the existence, uniqueness and regularity of solutions in anisotropic Hölder spaces to the homogeneous initialvalue problem,
(1.2)  
where the operator is the generator of standard Kimura diffusions. Regularity of solutions to parabolic equations defined by the infinitesimal generator of standard Kimura diffusions are established in [9, 10, 21]. Our definition of the anisotropic Hölder spaces in §2.2 are an adaptation to our framework of the Hölder spaces introduced in [10, Chapter 5].
In §3, we prove our main results (Theorems 3.4 and 3.8) concerning the existence and uniqueness in law of weak solutions to the singular Kimura stochastic differential equation, (1.1). Our method of the proof consists in applying Girsanov’s Theorem [16, Theorem 3.5.1] to the weak solutions of the standard Kimura stochastic differential equation, (2.1), to change the probability distribution so that, under the new measure, the solutions solve the singular Kimura stochastic differential equation, (1.1). We justify the application of Girsanov’s Theorem by proving that Novikov’s condition [16, Corollary 3.5.13] holds, a fact that uses the Markov property of the processes we consider. Because Girsanov’s Theorem is also used in the proof of uniqueness in law of weak solutions, our uniqueness result is established in the class of Markov processes. While this result is sufficient for the applications we have in mind (see §1.2), employing ideas used to prove [24, Theorem 12.2.4], it may be possible to prove that uniqueness in the class of Markov processes implies weak uniqueness. Notice though that [24, Theorem 12.2.4] does not apply directly to our framework because our drift coefficients are not necessarily bounded (see condition (3.5)). When the drift coefficients are bounded, that is, we consider the standard Kimura stochastic differential equation (2.1), then we establish the weak uniqueness of solutions (Proposition 2.8).
To prove existence and uniqueness of weak solutions to the singular Kimura stochastic differential equation, (1.1), we assume that the drift coefficient functions are bounded from below on by a positive constant, (see condition (3.2)). This is a crucial ingredient in our verification of Novikov’s condition in Lemmas 3.5 and 3.9. Notice also that the singular coefficients are assumed to satisfy the growth assumption (3.5), where and the positive constant depends on , by identity (3.1).
1.2. Applications of the main results
The motivation to study the singular Kimura stochastic differential equation, (1.1), comes from its application to the proof of the Harnack inequality for nonnegative solutions to the parabolic equation defined by the infinitesimal generator of standard Kimura diffusions, which we establish in joint work with Charles Epstein [11]. Let be the generator of standard Kimura diffusions. Our method of the proof of the Harnack inequality for nonnegative solution to the parabolic equation consists in employing a stochastic analysis method due to K.T. Sturm [25]. This makes use of the fact that we already know that the Harnack inequality holds for nonnegative solutions to a parabolic equation , where the operator is a suitable lower order perturbation of the operator . In [7, §4], C. Epstein and R. Mazzeo show that this is indeed true, when the operator is chosen to be the infinitesimal generator of singular Kimura diffusions which solve equation (1.1), where the coefficients have the form
where is a compactly supported smooth function. Notice that the preceding form of the coefficients satisfy our growth assumption (3.5), and so, Theorems 3.4 and 3.8 give us the existence and uniqueness of strong Markov solutions to the singular Kimura stochastic differential equation with logarithmic drift. This together with the strong Markov property of solutions are one of the main ingredients in our proof of the Harnack inequality for nonnegative solution to the parabolic equations defined by the generators of standard Kimura diffusions, (2.1).
1.3. Comparison with previous research
Articles which address the questions of existence and uniqueness in law of weak solutions to degenerate stochastic differential equations similar to ours are [2, 4]. While the motivation behind the work in [2, 4] are applications to superprocesses ([2, p. 3], [4, Example 1.4]), the main application of our results is to the study of diffusions arising in population genetics ([19, 20, 23], [12, §10.1], [17, §15.2.F]), and to the study of regularity of solutions to degenerate parabolic equations (see §1.2). The main difference between the Kimura stochastic differential equations (2.1) and those considered in [2, 4] consist in the fact that we allow coordinates, , of the weak solutions whose dispersion coefficients are nonzero on the boundary of the domain , and we do not require the drift coefficients to be bounded; instead we allow singularities in the drift component of the form , for , where the exponent satisfies a suitable restriction given by inequality (3.5). In the sequel, we explain in more detail the differences between the work done in [2, 4] and our results.
In [2], the authors consider diffusions corresponding to the generator
where and . Under the assumption that the coefficients of the operator are continuous functions on and that the drift coefficients are positive on , it is proved in [2] that the martingale problem associated to the operator has a unique solution. The method of the proof consists in proving estimates for the resolvent operators, employing a method of Krylov and Safonov to establish continuity of the resolvent operators [3, §V.7], and a localizing procedure due to Stroock and Varadhan [24, Theorem 6.6.1] to reduce the existence and uniqueness of solutions to a local problem. In §2, we recover and extend the results obtained in [2], under the assumption that the coefficients of the operator belong to the anisotropic Hölder spaces introduced in §2.2, and we allow the drift coefficient to be along the boundary of . Moreover, our method of the proof appears to be simpler, as we rely on existence and uniqueness of solutions in anisotropic Hölder spaces to homogeneous initialvalue parabolic equations defined by the operator . These results were established in [9, 10, 8, 21].
In [4], the authors consider a more general class of generators which are assumed to take the form
where and . In this work, the coefficient functions are are assumed to belong to suitable weighted Hölder spaces, as opposed to the anisotropic Hölder spaces introduced in §2.2, and the drift coefficient is assumed nonnegative on the boundary of the domain . Using estimates of the semigroup associated to the operator and of the resolvent operators in weighted Hölder spaces, and the localizing procedure of Stroock and Varadhan [24, Theorem 6.6.1], the authors prove existence and uniqueness of solutions to the martingale problem associated to . Our results are both more general and more restrictive in certain ways, than the ones obtained in [4]. The smallness condition [4, Inequality (1.4)] on the crossterms , for , of the operator is less restrictive than our analogous condition (2.17) of the matrix , defined in (2.15). On the other hand, we allow nongenerate directions, , in our stochastic differential equation (1.1), and we allow singular, unbounded drift coefficients.
1.4. Notations and conventions
Let be a closed set in , and be a positive integer. We let denote the set of functions, , that are continuous on , but are not necessarily bounded. The space consists of smooth functions, , with compact support in . For a Borel measurable set , we denote by the collection of Borel measurable subsets of .
1.5. Acknowledgment
The author would like to thank Charles Epstein for suggesting this problem and for many very helpful discussions on this subject.
2. Standard Kimura diffusions
To establish existence, uniqueness and the strong Markov property of weak solutions to the Kimura stochastic differential equation with singular drift (1.1), we first prove these results for the standard Kimura diffusions,
(2.1)  
where , and . We denote by , for all , the weak solution to the standard Kimura equation (2.1). We organize this subsection into three parts. In §2.1, we prove under suitable hypotheses (Assumption 2.1) that the standard Kimura stochastic differential equation (2.1) admits weak solutions, , supported in , when the initial condition is assumed to satisfy . In §2.3, we prove under more restrictive hypotheses (Assumption 2.4), that the weak solutions to the Kimura equation (2.1) are unique in law and satisfy the strong Markov property. In §2.2, we introduce the definitions of the anisotropic Hölder spaces used in the proof of uniqueness of weak solutions.
2.1. Existence of weak solutions
Existence of solutions to the standard Kimura stochastic differential equation (2.1) can be established for a more general form of the diffusion matrix than the one implied by equations (2.1). For this reason, we consider the stochastic differential equation,
(2.2)  
where , and .
To establish existence of weak solutions to the stochastic differential equation (2.2), we need the following
Assumption 2.1 (Properties of the coefficients in (2.2)).
The coefficient functions of the stochastic differential equation (2.2) satisfy the properties:

We assume that , , and .

The coefficients , and have at most linear growth in .

We assume that
(2.3) where denotes the transpose matrix of .

The drift coefficients satisfy
(2.4)
We begin with
Proposition 2.2 (Existence of weak solutions to standard Kimura diffusions).
Proof.
The method of the proof is similar to that of [13, Proposition 3.1 and Theorem 3.3]. We divide the proof into two steps. In Step 1, we continuously extend the coefficients of the stochastic differential equation (2.2) from to , and we prove that the stochastic differential equation associated to the extended coefficients, (2.5), has a weak solution. In Step 2, we prove that any weak solution to equation (2.5) is supported , when the support of the initial condition is contained in . Combining Steps 1 and 2, we obtain the existence of weak solutions supported in , to the stochastic differential equation (2.2).
Step 1 (Extension of the coefficients).
By Assumption 2.1, we can extend the coefficients of the stochastic differential equation (2.2) by continuity from to . We consider the function defined by
Because is a closed, convex set, the point is uniquely determined for all . Moreover is a continuous function and , for all . We define the coefficient functions , and , which are continuous extensions to of the coefficient functions , and , respectively. By Assumption 2.1, the extended coefficients are continuous functions on , and have at most linear growth in the spatial variable. By [16, Theorem 5.4.22], [12, Theorem 5.3.10], it follows that the stochastic differential equation,
(2.5)  
has a weak solution, , on a filtered probability space satisfying the usual conditions, , for any initial condition, .
Step 2 (Support of weak solutions).
Let , and let be a weak solution to the stochastic differential equation (2.5), with initial condition . Our goal is to show that
(2.6) 
where denotes the probability distribution of the process , with initial condition . To prove identity (2.6), it is sufficient to show that
(2.7) 
For , let be a smooth function such that for , for , and on . We see that identity (2.7) holds, if we show that for all , we have that
(2.8) 
Applying Itô’s rule [16, Theorem 3.3.6] to the process , we obtain
(2.9) 
From condition (2.4) and the construction of the extended coefficient , it follows that the drift coefficient is nonnegative on the support of the function . Using the fact that , we obtain
From condition (2.3) and the construction of the extended matrix coefficient , it follows that on the support of . Thus we have
Using now the fact that , since we choose and on , it follows from identity (2.9) that
and, because is a nonnegative function, the preceding expression holds with equality. Since was arbitrarily chosen, the preceding identity implies (2.7), for all , and so, we conclude that (2.6) holds.
Identity (2.6) proves that, when started at points in , the weak solutions to the stochastic differential equation (2.5) remain in . Because the coefficients of the stochastic differential equations (2.1) and (2.5) agree on , we obtain that the weak solutions to (2.5) also solve equation (2.1). This completes the proof of Proposition 2.2. ∎
Remark 2.3 (Existence of weak solutions to the standard Kimura equation).
We now consider a matrix coefficient, , such that it satisfies the property that by setting
(2.10)  
the matrix verifies Assumption 2.1. Then Proposition 2.2 implies that there is a weak solution, , to the standard Kimura stochastic differential equation (2.1), for any initial condition supported in , and that the solution remains supported in at all subsequence times.
2.2. Anisotropic Hölder spaces
Before we can state the assumptions on the coefficients of the Kimura stochastic differential equation (2.1) that will guarantee the uniqueness in law of weak solutions, we first need to introduce a class of anisotropic Hölder spaces adapted to the degeneracy of the diffusion matrix. The following Hölder spaces are a slight modification of the Hölder spaces introduced by C. Epstein and R. Mazzeo in their study of the existence, uniqueness and regularity of solutions to the parabolic problem defined by Kimura operators [9, 10].
Following [10, Chapter 5], we need to first introduce a distance function, , which takes into account the degeneracy of the diffusion matrix of stochastic differential equation (2.1). We let
(2.11) 
where is a distance function in the spatial variables. Because our domain is unbounded, as opposed to the compact manifolds considered in [10], the properties of the distance function depend on whether the points and are in a neighborhood of the boundary of , or far away from the boundary of . For any set of indices, , we let
(2.12) 
where we denote . The distance function has the property that there is a positive constant, , such that for all subsets , and all and , we have that
(2.13)  
Let . Following [10, §5.2.4], we let be the Hölder space consisting of continuous functions, , such that the following norm is finite
Let be a positive integer, and be a set in . We let denote the Hölder space containing functions, , such that the derivatives belong to the space , for all and , such that . We endow the space with the norm,
We fix a set of indices, . Let be a set such that . We let denote the Hölder space of functions, , such that
and such that the functions,
We endowed the space with the norm,
We now consider the case when is an arbitrary set in . Then we let denote the Hölder space consisting of functions , satisfying the property that
We endow the Hölder space with the norm
When , we write for brevity , instead of . The elliptic Hölder spaces and are defined analogously to their parabolic counterparts, and so, we omit their definitions for brevity.
2.3. Uniqueness and the strong Markov property
Our goal is to prove uniqueness in law and the strong Markov property of weak solutions to the standard Kimura stochastic differential equation (2.1). By [16, Proposition 5.4.27], to prove uniqueness in law of weak solutions to the Kimura stochastic differential equation (2.1), it is sufficient to establish that for all , any two weak solutions, , for , satisfying the property that , for , have the same onedimensional marginal distributions. That is, for all functions and , we have that
(2.14) 
where denotes the probability distribution of the process , with initial condition , for .
Before we give the proof of the uniqueness in law of weak solutions to the Kimura stochastic differential equation in (2.1), we introduce the differential operator , which will be the infinitesimal generator of the Markovian solutions to the Kimura stochastic differential equation. We let
(2.15) 
and we define
(2.16)  
for all and all . Compare the definition of the Kimura differential operator with that of the operator defined in [21, Identity (1.1)]. Similarly to [21, Assumption 3.1], we will need the following assumptions on the coefficients of the Kimura stochastic differential equation (1.1).
Assumption 2.4 (Properties of the coefficients in (2.1)).
The coefficient functions of the stochastic differential equation (1.1) satisfy the properties: Let , and assume that

The coefficient functions satisfy the nonnegativity condition (2.4), for all .

For all such that , and all , there are functions, , such that
(2.17) where denotes the Kronecker delta symbol.

The strict ellipticity condition holds: there is a positive constant, , such that for all sets of indices, , for all , and , we have
(2.18) 
The coefficient functions are Hölder continuous: for all sets of indices, , and for all , and , we have that
(2.19)
Assumption 2.4 yield some immediate boundedness conditions on the coefficients of the Kimura stochastic differential equation (2.1), which we now explain.
Remark 2.5 (Boundedness of the coefficient functions and ).
Assumption 2.4 implies that there is a positive constant, , such that for all and all , we have that
<(2.20) 