Probab. Theory Relat. Fields (2007) 139:415–449 DOI 10.1007/s00440-006-0047-9
The submartingale problem for a class of degenerate elliptic operators Richard F. Bass · Alexander Lavrentiev
Received: 15 December 2005 / Revised: 8 November 2006 / Published online: 12 December 2006 © Springer-Verlag 2006
Abstract We consider the degenerate elliptic operator acting on Cb2 functions on [0, ∞)d : Lf (x) =
d i=1
ai (x)xαi i
∂ 2f ∂f (x) + bi (x) (x), 2 ∂xi ∂xi d
i=1
where the ai are continuous functions that are bounded above and below by positive constants, the bi are bounded and measurable, and the αi ∈ (0, 1). We impose Neumann boundary conditions on the boundary of [0, ∞)d . There will not be uniqueness for the submartingale problem corresponding to L. If we consider, however, only those solutions to the submartingale problem for which the process spends 0 time on the boundary, then existence and uniqueness for the submartingale problem for L holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations dXti =
2ai (Xt )(Xti )αi /2 dWti + bi (Xt ) dt + dLX t , i
Xti ≥ 0,
i i where Wti are independent Brownian motions and LX t is a local time at 0 for X .
Research partially supported by NSF grant DMS-0244737. R. F. Bass (B) Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA e-mail:
[email protected] A. Lavrentiev Department of Mathematics, University of Wisconsin, Fox Valley Menasha, WI 54952, USA e-mail:
[email protected]
416
R. F. Bass, A. Lavrentiev
Keywords Martingale problem · Stochastic differential equations · Degenerate elliptic operators · Speed measure · Perturbation · Bessel process · Littlewood-Paley Mathematics Subject Classification (2000)
Primary 60H10 · Secondary 60H30
1 Introduction We consider the degenerate elliptic operator acting on Cb2 functions on [0, ∞)d defined by
Lf (x) =
d
ai (x)xαi i
i=1
∂ 2f ∂f (x) + bi (x) (x). 2 ∂xi ∂xi d
(1.1)
i=1
We assume here that the bi are bounded and measurable, the ai are continuous and bounded above and below by positive constants, and each αi ∈ (0, 1). We impose zero Neumann boundary conditions on ∂(Rd+ ), where we write R+ = [0, ∞). In this paper we investigate whether there is at most one process corresponding to the operator L. We formulate this question in terms of a submartingale problem. Let = C([0, ∞); Rd+ ), the continuous functions from [0, ∞) to Rd+ . Define the canonical process X by Xt (ω) = ω(t) and let Ft be the filtration generated by X. Let x ∈ Rd+ . We say that a probability measure P on is a solution to the submartingale problem for L started at x if P(X0 = x) = 1 and whenever f ∈ Cb2 (Rd+ ) such that for each i we have in addition ∂f /∂xi ≥ 0 on {x = (x1 , . . . , xd ) ∈ Rd+ : xi = 0}, t then f (Xt ) − f (X0 ) − 0 Lf (Xs )ds is a submartingale with respect to P. If Yt is a one dimensional process on [0, ∞), by local time at 0 of Y we mean Y any continuous nondecreasing process LY t such that L increases only when Y is at 0. Closely related to the operator L is the system of equations dXti =
2ai (Xt )(Xti )αi /2 dWti + bi (Xt )dt + dLX t , i
i
i = 1, . . . , d,
(1.2)
where X0i = xi , Xti ≥ 0 for all t, LX is a local time at 0 for X i , and the W i are independent one dimensional Brownian motions started at 0. We say a weak solution to (1.2) exists if there is a probability P such that (1.2) holds and the W i are independent Brownian motions under P. Weak uniqueness holds if given any two solutions (Xj , Wj , Pj ), j = 1, 2, the joint law of (X1 , W1 ) under P1 is equal to the joint law of (X2 , W2 ) under P2 . For weak uniqueness to hold for (1.2) it is only necessary that the law of X1 under P1 equals the law of X2 under P2 ; see Remark 2.2. We have assumed that each αi is in the interval (0, 1), so in fact uniqueness for the submartingale problem for L does not hold. This can be seen even in
The submartingale problem for a class of degenerate elliptic operators
417
one dimension: if one looks at the one-dimensional diffusion on natural scale with speed measure m, where m(dx) = x−α dx for x positive, m is 0 on (−∞, 0), and m has an atom of mass λ at 0, one can let λ be an arbitrary non-negative number and obtain different processes. If, however, one restricts attention to those solutions to the submartingale problem L for which the process spends zero time at the boundary, then uniqueness of the submartingale problem does hold. Our main theorem is the following. Let = ∂(Rd+ ). Theorem 1.1 Let x ∈ Rd+ (a) There exists one and only one solution to the submartingale problem for L started at x that spends zero time in , i.e., ∞ 1 (Xs )ds = 0,
P − a.s.
0
(b) A weak solution to (1.2) exists that spends zero time in . Weak uniqueness holds if we restrict attention to those weak solutions that spend zero time in . Our paper continues the study of degenerate diffusions in the positive orthant begun in [1] and [5]. Those papers concerned the operator L where all the αi were equal to 1. In some sense, if all the αi are equal to 1, we are in the critical case, in that then the exact values of the drift coefficients bi make a large difference to the behavior of the resulting process. When αi > 1 and the drift coefficients are zero, then either the process never attains the boundary, or if it starts on the boundary, never leaves, so the problem then becomes a lower dimensional one. This paper deals with the case αi < 1. Although the values of the drift coefficients play less of a role, the results of this paper are not a subset of those in [1]. In fact, they could not be, because here we need the additional assumption that the process spends zero time on the boundary in order to have uniqueness, while no such assumption is needed in [1]. If αi < 1/2, one can check that a Girsanov transformation allows one to assume that the corresponding bi can be taken to zero; see Remark 7.1. We wanted to allow the full range of αi and drift coefficients, so we did not restrict the values of the αi to (0, 1/2). As is often the case, uniqueness for a martingale or submartingale problem is often related to the existence of a solution to a PDE problem. That is also the case here, but we do not pursue this connection. We believe our techniques could also be applied to diffusions on Rd whose coefficients decay near the ith axis like |xi |αi . Consider the diffusion on Rd which corresponds to the operator d i=1
ai (x)|xi |αi
∂ 2f ∂f (x) + bi (x) (x). 2 ∂xi ∂xi d
i=1
418
R. F. Bass, A. Lavrentiev
The drift coefficients play much less of a role than in [1]. The drift can be handled either by a Girsanov transformation when all the αi < 1/2 (see Remark 7.1) or by the perturbation approach of Sect. 5 in the more general case when all the αi < 1. The arguments of Sect. 6 should go through with little if any change. The papers [14], [10], and [6] also consider diffusions with reflection; the latter two consider pathwise uniqueness. Although some smoothness of the domain is needed in these papers, the key difference is that degeneracies of the type given in (1.1) and (1.2) are not allowed. Our methods differ substantially from those in [1]. That paper used L2 estimates only, and required a use of the Krylov-Safonov Harnack inequality. The proof of Lemma 5.1 of [1] does not extend to the present situation, and we were forced to use a different approach. In this paper we proceed by proving an analogue of Krylov’s inequality (see [8] and Theorem 3.2) and then using Littlewood-Paley theory to obtain Lp estimates. The use of Littlewood-Paley theory is of independent interest, and could potentially have applications to many other types of martingale problems. (The paper [5] uses Cα estimates and is quite different, both in results and in methods.) We give a brief overview of the proof. Existence is done by taking smooth approximations to (1.1), showing the laws of the corresponding processes are tight, and looking at a convergent subsequence. Both in order to show that the weak limit of this subsequence spends zero time on the boundary of Rd+ and that the limit corresponds to (1.1), we need an inequality of Krylov type that gives a bound on the amount of time spent in small sets. To prove uniqueness, we suppose we have two solutions P1 and P2 to the martingale problem and define resolvent functionals S1λ and S2λ by (7.6). We let be the norm of S1λ − S2λ as a functional on Lp0 (µ) for a certain positive real p0 and a certain measure µ. The Krylov inequality that we prove guarantees that < ∞. Some stochastic calculus shows that ≤ BRλ p0 for certain operators B and Rλ defined by (7.3) and (6.1). If we knew that BRλ p0 < 12 , we would conclude that = 0 and therefore S1λ = S2λ ; from here on the proof follows a familiar path. The principal work in this paper is to establish this bound on the norm of BRλ . The first order terms are handled by calculations with transition densities of the process corresponding to L0 defined in (7.2) together with a variant of Young’s inequality (Proposition 5.2). The second order terms are where Littlewood-Paley theory comes in. Littlewood-Paley theory refers to a large body of literature where Lp estimates on operators are obtained by looking at square functions and quadratic variation. Our application is a mixture of analytic and probabilistic techniques. After a section on preliminaries, we prove our inequality of Krylov type in Sect. 3. Section 4 concerns existence of solutions. Sections 5 and 6 give the required estimates for the first and second order terms, respectively. The proof
The submartingale problem for a class of degenerate elliptic operators
419
of Theorem 1.1 is completed in Sect. 7. Some of the calculations needed in Sect. 5 are deferred to Sect. 8, which is an appendix. 2 Preliminaries We often write fi and fii for the first and second partial derivatives of f with respect to xi . The Lebesgue measure of a Borel set B will be written |B|. We use R+ = [0, ∞) and for our state space we will use Rd+ . We use Cb2 (Rd+ ) to refer to the collection of functions which are C2 on Rd+ and such that the function and its first and second partial derivatives are bounded on Rd+ ; for points on ∂(Rd+ ) we use instead the appropriate one-sided partial derivatives. We will later introduce a measure µ (see (3.14)) andf p will denote the Lp norm with respect to µ. When we want the Lp norm with respect to Lebesgue measure we will write f Lp (dx) . We use the letter c with or without subscripts to denote finite positive constants whose values are unimportant and may change from place to place. Let i = {x = (x1 , . . . , xd ) ∈ Rd+ : xi = 0},
= ∪di=1 i .
(2.1)
For any process Z and any Borel set C we let TC (Z) = TC = inf{t : Zt ∈ C},
τC (Z) = τC = inf{t : Zt ∈ / C}.
(2.2)
When C is a single point {y}, we write instead Ty = Ty (Z) and τy = τy (Z). The connection between Theorem 1.1(a) and (b) is the following. Proposition 2.1 There exists a solution to the submartingale problem for L started at x if and only if there exists a weak solution to (1.2). The solution to the submartingale problem will be unique if and only if there is weak uniqueness to (1.2). These assertions continue to hold if we restrict attention to probability measures P such that Xt spends zero time in , P-a.s. Proof The proof of this proposition is very similar to the nondegenerate diffusion case (see [3], Theorem V.1.1) and we give only a brief sketch. If P is a weak solution to (1.2), then an application of Ito’s formula shows that P will be a solution to the submartingale problem. If P is a solution to the submartingale problem, M > 0, and we take f (x) to be a Cb2 function which equals xi on [0, M]d , then by the definition of submartingale problem, t f (Xt ) −
t Lf (Xs )ds =
0
Xti
−
bi (Xs )ds 0
is a submartingale, and so by the Doob-Meyer decomposition can be written as a martingale Mti plus an increasing process Lit . Similarly to the nondegenerate
420
R. F. Bass, A. Lavrentiev
t diffusion case, one can show that Xti − 0 bi (Xs ) ds is a martingale away from t the boundary with quadratic variation 0 ai (Xs )(Xsi )αi ds up until the time of first exiting [0, M]d . We then let M → ∞. This implies that Lit increases only when Xti is at 0, and hence is a local time at 0 for X i . We then proceed as in the proof of the nondegenerate case. Remark 2.2 To show that the laws of (X, W, L) under P1 and P2 are the same, it is enough to show equality of the laws of X under P1 and P2 . To see this, observe that we can recover L and W from X by the following formulas:
i LX t
t 1i (Xs ) dXsi ,
= 0
t Wti =
(2ai (Xs ))−1/2 (Xsi )−αi /2 (dXsi − bi (Xs ) ds − dLX s ). i
0
Hence the joint law of (X, W, L) is determined by the law of X. In this argument, we use the fact that X spends zero time on the boundary to verify that W is a Brownian motion. Any weak solution to (1.2) satisfies a uniform tightness estimate. By this we mean the following. Proposition 2.3 If M > 0, δ > 0, η > 0, and t0 > 0, there exists ε > 0 such that if / [0, M]d }, then (X, W, P) is any weak solution to (1.2) and SM = inf{t : Xt ∈
P
sup
s,t≤SM ∧t0 ,|t−s|<ε
|Xt − Xs | > δ
< η.
Proof It suffices to consider each component of X separately. Fix i. Let Aε =
sup
s≤t≤s+ε≤SM ∧(t0 +ε)
|(Xti
i i − LX t ) − (Xs
i − LX s )|
> δ/4 .
(2.3)
By standard estimates for stochastic integals (see, e.g., [3], Proposition I.8.1) we can find ε such that if X is any solution to (1.2), then P(Aε ) < η. We claim that if ω ∈ Acε , then sups≤t≤s+ε≤t0 +ε |Xt −Xs | ≤ δ. Proving this claim will complete the proof. Suppose ω ∈ Acε and suppose there exist s ≤ t ≤ t0 + ε i Xi Xi with t − s < ε such that |Xti − Xsi | > δ. Then LX t − Ls ≥ 3δ/4. Since L increases only when X i is at 0, there exist s < t such that s ≤ s < t ≤ t, i Xi Xsi = Xti = 0, and LX t − Ls > δ/2. But then
The submartingale problem for a class of degenerate elliptic operators i
i
421 i
i
i X X X 0 = Xti − Xsi = (Xti − LX t ) − (Xs − Ls ) + (Lt − Ls ) δ δ ≥ − , 2 4
a contradiction; the claim is proved.
When it comes to proving uniqueness of the submartingale problem, it suffices to consider only solutions defined on the canonical probability space C([0, ∞); Rd+ ). If S is a stopping time, we let QS be a regular conditional probability P( · ◦ θS | FS ), where θS is the usual shift operator of Markov process theory. We denote the corresponding expectation by EQS . Just as in the nondegenerate case, it is easy to see that with probability one QS will be a solution to the submartingale problem started at XS ; cf. [3], Proposition VI.2.1. 3 Occupation time estimates We start with an estimate on how long the solution to a one dimensional SDE can spend near 0. We proceed by proving that the process X is always at least as large as Ut defined by (3.5). Since Ut is a one-dimensional strong Markov process, we can calculate occupation times and number of crossings, and we then use the comparison with X to derive the corresponding facts about X. Theorem 3.1 Suppose x0 ∈ [0, ∞), Wt is a Brownian motion, at and bt are adapted processes, c−1 1 ≤ at ≤ c1 a.s. for each t, and |bt | ≤ c1 a.s. for each t. Suppose either (a) X solves dXt =
√
α/2
at Xt
dWt + bt dt + dLX t ,
Xt ≥ 0, X0 = x0 ,
(3.1)
and X spends zero time at 0 or (b) for some ε > 0 dXt =
√ at (Xt + ε)α/2 dWt + bt dt + dLX t ,
Xt ≥ 0, X0 = x0 ,
(3.2)
where LX t is a continuous nondecreasing process that increases only when Xt is at 0. Let K > 0. There exists c2 , c3 , c4 depending only on K and c1 such that for each γ ≤ K the probability of more than m upcrossings of [γ /2, γ ] by X before time TK (X) is bounded by (3.3) c2 (1 − c3 γ )m and if η ≤ K, then
T K (X)
1[0,η] (Xs ) ds ≤ c4 η1−α .
E
(3.4)
0
Both (3.3) and (3.4) are used in the proof of Theorem 3.2, and (3.4) is also used in Proposition 4.1.
422
R. F. Bass, A. Lavrentiev
Proof By first performing a nondegenerate time change, we may without loss of generality suppose that at ≡ 1. In case (b), Girsanov’s theorem for general continuous semimartingales ([2], Theorem I.6.4) and the fact that the diffusion coefficient is bounded below away from 0 tell us that the solution to (3.2) will spend zero time at 0. So we can consider both cases at once if we let ε ≥ 0 and specify that Xt spends 0 time at 0. We next define Ut . Let Ut be the solution to dUt = (Ut + ε)α/2 dWt − c1 dt + dLU t ,
(3.5)
α/2 ∧ n, where Ut ≥ 0 for all t and LU t is a local time at 0 for U. If an (x) = (x + ε) the strong existence and pathwise uniqueness of the solution to n
dUtn = an (Utn ) dWt − c1 dt + dLU t , n
where Utn ≥ 0 for all t and LU is a local time at 0 for U n is guaranteed for all times by Theorem 1.3 of [4]. The strong existence and pathwise uniqueness of the solution to (3.5) up to time TK (U) follows easily. Let Px be the law of the process U started at the point x; then the family {Px } forms a continuous strong Markov process corresponding to the operator 12 (x + ε)α f (x) − c1 f (x) with Neumann boundary conditions at 0 (i.e., reflecting at 0). We want to show that if U0 ≤ X0 , then Ut ≤ Xt for all t ≤ T0 (X). We show this by applying a stochastic comparison theorem. If U0 ≤ X0 , by Theorem VI.1.1 of [7] we see that Ut ≤ Xt for all t ≤ T0 (X)∧T0 (U). Let V1 = inf{t : Ut ≥ 12 Xt } and i : Ut ≥ 1 Xt }. Recall i = inf{t > Vi : Ut = 0} and Vi+1 = inf{t > V for i ≥ 1, let V 2 that QVi is the regular conditional probability for P(· ◦ θVi | FVi ). Under QVi the process X satisfies the hypotheses of Theorem 3.1 and U0 ◦ θVi = 12 X0 ◦ θVi ≤ X0 ◦θVi . Applying [7], Theorem VI.1.1, to Xt ◦θVi and Ut ◦θVi under the probability measure QVi , we conclude that Ut ≤ Xt for t ≤ (T0 (X) ◦ θVi ) ∧ (T0 (U) ◦ θVi ). Therefore i ] | FVi ) = 1, P(Ut ≤ Xt for t ∈ [Vi , V
P − a.s.
i ], then Ut ≤ Xt , P-almost surely. By the defiSo if t < T0 (X) and t ∈ [Vi , V i , Vi+1 ], then again Ut ≤ Xt . If nition of Vi and Vi , if t < T0 (X) and t ∈ [V Vi → ∞, we conclude Ut ≤ Xt for all t ≤ T0 (X). Suppose, on the other hand, that Vi ↑ V∞ < ∞. Since UV i = 0, by the continuity of the paths of U we see that UV∞ = 0. By the continuity of the paths of X, XV∞ = lim XVi ≤ 2 lim UVi = 2UV∞ = 0, i→∞
i→∞
so T0 (X) ≤ V∞ . Therefore in this case too, we again have Ut ≤ Xt for all t ≤ T0 (X). Recall that the scale function for U on (0, ∞) is any function s(x) satisfying for x > 0 the differential equation
The submartingale problem for a class of degenerate elliptic operators α 1 2 (x + ε) s (x) − c1 s (x)
423
= 0.
We compute the scale function s(x) for the process Ut and find that it is determined by log s (x) =
x
2c1 dy + c5 . (y + ε)α
0
We can take c5 = 0 and s(x) = 1−α e2c1 ε /(1−α)
x 0
e2c1 (y+ε)
1−α /(1−α)
dy. Note that s(x)/x →
since α < 1. It follows that s(Ut ) corresponds to the operator Aε (x)(x + ε)α f (x)
where Aε (x) is a function of x satisfying c6 ≤ Aε (x) ≤ c7 ,
|x| ≤ K,
and 0 < c6 < c7 < ∞ do not depend on ε; furthermore the speed measure for the process has no atom at 0. Moreover, we see that s(η)/η is bounded by c8 for η small. R R We now prove (3.3). For any process R on R, let SR 0 = 0, Si = inf{t > Si−1 : R R Rt = γ /2}, Si = {t > Si : Rt = γ }, i ≥ 1. Since s(U) is on natural scale, s(γ ) (S1 Pγ (SU 1 < TK (U)) = P
s(U)
< Ts(K) (s(U))) = 1 −
s(γ ) s(K)
≤ 1 − c9 γ .
(3.6)
If X0 = γ , then γ U P(SX 1 < TK (X)) ≤ P (S1 < TK (U)) ≤ 1 − c9 γ .
Under Q SX the process Xt satisfies the hypotheses of Theorem 3.1, and so by i what has just been proved X X P( SX SX (S1 < TK (X)); Si ≤ TK (X)] i+1 ≤ TK (X)) ≤ E[Q i
≤ (1 − c9 γ )P( SX i ≤ TK (X)). By induction, i P( SX i ≤ TK (X)) ≤ c10 (1 − c9 γ ) ,
which implies (3.3).
424
R. F. Bass, A. Lavrentiev
We turn to the proof of (3.4). First of all, very similarly to the proof of (3.3), the probability of more than m upcrossings of [η/4, η/2] before time T6η (X) is bounded by c11 cm 12 for some c12 < 1 independent of m and η. The main difference here is that s(η/2)/s(6η) is bounded below independently of η provided η is small; cf. (3.6). Next we obtain a bound on the amount of time X spends in [η/2, η] before Bi−1 : Xt = η/2}, and Bi = inf{t > Bi : hitting 6η. Let B0 = 0, Bi = inf{t > Xt = η/4}. Observe T6η (X)
E
1[η/2,η] (Xr ) dr = E
∞
Bi ∧T6η (X)
1[η/2,η] (Xr ) dr.
(3.7)
i=0 B ∧T (X) i 6η
0
Suppose X0 = η/2 and U is the solution to (3.5) with U0 = η/2. If r ≤ B1 ∧ T6η (X), then Xr ≥ Ur , and so T6η (X) ≤ T6η (U). Therefore B1 ∧T6η (X)
1[η/2,η] (Xr ) dr ≤ E[T6η (U)].
E
(3.8)
0
The expected amount of time until U hits 6η is the same as the expected amount of time until s(U) hits s(6η), and by symmetry this is the same as the expected amount of time until U exits [−s(6η), s(6η)], where U is the process on R on natural scale corresponding to the operator Aε (|x|)(|x| + ε)α f (x), with speed measure not having an atom at 0. Using [3,Corollary 2.4, Eq. (2.1) and Theorem 3.2], we have that B1 ∧T6η (X)
1[η/2,η] (Xr ) dr ≤ c13 η2−α .
E
(3.9)
0
Under QBi , Xt ◦ θBi is a solution to (1.2) and therefore we conclude Bi ∧T6η (X)
E Bi ∧T6η (X)
⎡
⎢ 1[η/2,η] (Xr ) dr = E ⎣EQBi
B1 ∧T6η (X)
⎤
⎥ 1[η/2,η] (Xr ) dr; Bi ≤ T6η (X)⎦
0
≤ c13 P(Bi ≤ T6η (X)).
(3.10)
For Bi to be less than or equal to T6η (X), the process must have at least i − 1 upcrossings of [η/4, η/2] by time T6η (X), and the probability of this is bounded
The submartingale problem for a class of degenerate elliptic operators
425
i−1 by c11 c12 . Combining this, (3.7), (3.9), and (3.10), we have T6η (X)
1[η/2,η] (Xr ) dr ≤ c14 η2−α .
E
(3.11)
0
Another upcrossing argument gives us a bound on the amount of time X spends in [η/2, η] before time TK (X). Let C0 = 0, Ci = inf{t > Ci−1 : Xt = 3η}, and Ci = inf{t > Ci : Xt = 6η}. Similarly to the above and using (3.11), we have T K (X)
E
1[η/2,η] (Xr ) dr ≤
∞
Ci ∧T K (X)
E
i=0
0
≤
∞
1[η/2,η] (Xr ) dr
Ci ∧TK (X)
c14 η2−α P(Ci ≤ TK (X)).
(3.12)
i=0
In order for Ci to be less than or equal to TK (X), we must have at least i − 1 upcrossings of [3η, 6η] by time TK (X), and by (3.3) the probability of this is bounded by c15 (1 − c16 η)i−1 . Substituting in (3.12), we conclude T K (X)
1[η/2,η] (Xr ) dr ≤ c14 η2−α
E
∞
c15 (1 − c16 η)i−1 ≤ c17 η1−α .
(3.13)
i=0
0
To conclude the proof of (3.4), we apply (3.13) with η replaced by η2−k , k = 0, 1, . . . , and sum over k. Since X spends zero time at 0, we obtain (3.4). Define µ on [0, ∞)d by µ(dx) =
d
i x−α i dx.
(3.14)
i=1
This measure will be used later on; it is characterized by the fact that the oper ator di=1 xαi i fii is self-adjoint with respect to µ. As mentioned in Sect. 2, f p denotes the Lp norm of f with respect to µ while f Lp (dx) is the Lp norm with respect to Lebesgue measure. Now we turn to our inequality of Krylov type. Theorem 3.1 shows that the process does not spend too much time near the boundary of Rd+ . So we can restrict our attention to functions whose support are a positive distance from the boundary. We do a transformation of the state space so that we have a uniformly elliptic operator with a large drift, and use Krylov’s inequality ([8]).
426
R. F. Bass, A. Lavrentiev
Theorem 3.2 Let M > 1 be fixed and λ > 0. There exist p0 and c1 depending on α1 , . . . , αd , M, λ, and d such that if f is supported in [0, M]d , then for any solution X to (1.2) ∞ E e−λs f (Xs ) ds ≤ c1 f p0
(3.15)
0
∞ E e−λs f (Xs ) ds ≤ c1 f p0 L (dx) .
and
(3.16)
0
Proof Let us set D = [0, 2M]d and E = [0, M]d . Let A ⊂ E and let ε = |A|. Let δ > 0 be a small positive real to be chosen later and let F = [εδ , 2M]d . Let A = A ∩ F. Our first goal is to show that there exists K1 and γ depending only on d, α1 , . . . , αd , M, and λ, but not ε, such that τF E
1A (Xs ) ds ≤ c2 ε−δK1 +γ .
(3.17)
0
Define the map : Rd+ → Rd+ by α1 −1 αd −1 1−(α1 /2) 1−(αd /2) . (x1 ) ,..., 1 − (xd ) (x1 , . . . , xd ) = 1 − 2 2 Let Yt = (Xt ). We use Ito’s formula to see that for t ≤ τF (X) dYti =
√
2ai (Xt ) dWti +
αi i −αi /(2−αi ) αi −1 αi dt. (Yt ) 1− bi (Xt ) − 1 − 2 2 4Yti
Since X is in the set F, there is no issue of the degeneracy of Xti at 0 causing problems. Notice that for X ∈ F the drift coefficient of Y i is bounded by c3 ε−K2 δ . To obtain (3.17) it suffices to bound τ(F)
E
1(A ) (Ys ) ds. 0
Note |(A )| ≤ c4
d i=1
(εδ )−αi /2 |A | ≤ c5 ε−δK3 +1
(3.18)
The submartingale problem for a class of degenerate elliptic operators
427
for some K3 > 0. Let Y be the process whose coefficients agree with those of Y up until the time that the process Y leaves (F) and is a d-dimensional Brownian motion thereafter. Let G be a ball of radius c6 M that contains F. We need a bound on the expected time to leave G. By (3.4), E[τ(F) (Y)] = E[τF (X)] is bounded by a constant not depending on ε or δ. Since Y is a Brownian motion for t > τ(F) , it is then clear that E[τG (Y)] is also bounded by a constant independent of ε and δ. We use Krylov [8] to obtain the bound τ G (Y)
1(A ) (Y s ) ds ≤ c7 (1 + c8 ε−K2 δ E[τG (Y)])|(A )|1/d .
E 0
This inequality follows from a passage to the limit (as t → ∞) in equation (4) of [8]. We have τ(F) (Y)
τ F (X)
E
1A (Xs ) ds ≤ E 0
1(A ) (Ys ) ds 0 τG(Y)
≤E
1(A ) (Y s ) ds.
(3.19)
0
We deduce τ F (X)
1A (Xs ) ds ≤ c9 (1 + ε−K2 δ )ε(−δK3 +1)/d .
E 0
This proves (3.17) with K1 = K2 + K3 /d and γ = 1/d. Next we will show that if A ⊂ E, then there exist K4 and K5 depending only on d, αi , . . . , αd , λ, and M, such that τD E
1A (Xs ) ds ≤ c10 (ε−K4 δ+γ + εδK5 ).
(3.20)
0
Write A = A1 ∪ A2 , where A1 = A ∩ (∪di=1 {0 ≤ xi ≤ εδ }) and A2 = A \ A1 . Note A2 ⊂ F. By Theorem 3.1, we know τD E 0
τD d 1A1 (Xs ) ds ≤ E 1[0,εδ ] (Xsi ) ds ≤ c11 εδK5 . i=1
0
(3.21)
428
R. F. Bass, A. Lavrentiev
So we need to bound τD 1A2 (Xs ) ds.
E 0
We need to insure that X does not visit F too many times before exiting D. Let / [εδ /2, 2M]d }, T0 = 0, Si = inf{t > Ti−1 : Xt ∈ F}, and Ti = inf{t > Si : Xt ∈ i ≥ 1. Recall that QSi is used for a regular conditional probability. Then using (3.17) τD E
⎤ ⎡ Ti ∞ ⎥ ⎢ 1A2 (Xs ) ds = E ⎣ 1A2 (Xs ) ds; Si < τD ⎦ i=1
0
Si
⎤ T1 ∞ ⎥ ⎢ = E ⎣EQSi 1A2 (Xs ) ds; Si < τD ⎦ ⎡
i=1
≤ c12 ε−K1 δ+γ
0 ∞
P(Si < τD ).
(3.22)
i=1
Now in order for Si to be less than τD , we must have for some j ≤ d at least (i − 1)/d upcrossings of X j from εδ /2 to εδ before hitting the level 2M. We know by Theorem 3.1 that the probability of this happening is less than c13 (1 − c14 εδ )(i−1)/d . Therefore ∞
P(Si < τD ) ≤ c15 ε−K6 δ .
i=1
Combining with (3.21), we have τD E
1A (Xs ) ds ≤ c16 (εδK5 + ε−K1 δ+γ −K6 δ ),
(3.23)
0
which yields (3.20) with K4 = K1 + K6 . The next step is to show that there exists K7 such that if A ⊂ E, then ∞ E
e−λs 1A (Xs ) ds ≤ c17 (ε−K7 δ+γ + εδK5 ).
(3.24)
0
Our approach is to show that if X is not in D, then X will not reach E too quickly. / D}, i ≥ 1. Let V0 = 0, Ui = inf{t > Vi−1 : Xt ∈ E}, Vi = inf{t > Ui : Xt ∈
The submartingale problem for a class of degenerate elliptic operators
429
Suppose X starts at x ∈ / [0, 2M)d . Then there exist c18 and c19 not depending / [0, 2M)d , then at least on x such that P(U1 > c18 ) > c19 . This is because if x ∈ i0 one coordinate, say, X0 , of X0 is greater than or equal to 2M, and in the range (M, 3M), X i0 is a diffusion whose diffusion coefficients are bounded above and below and whose drift coefficient is bounded above; therefore there exist c18 and c19 such that P
sup s≤c18
|Xsi0
i − X00 |
> M/2 ≤ 1 − c19 .
We conclude Ee−λU1 ≤ P(U1 ≤ c18 ) + e−λc18 P(U1 > c18 ) = (1 − e−λc18 )P(U1 ≤ c18 ) + e−λc18 ≤ (1 − e−λc18 )(1 − c19 ) + e−λc18 := ρ. Note that ρ < 1. We then have Ee−λUi = E e−λVi−1 E e−λ(Ui −Vi−1 ) | FVi−1 ≤ E e−λUi−1 EQV e−λU1 ≤ ρEe−λUi−1 . i−1
So by induction Ee−λUi ≤ ρ i−1 . Then using (3.20) ∞ E
−λs
e
Vi ∞ 1A (Xs ) ds = E e−λs 1A (Xs ) ds
0
i=1
Ui
⎡ ⎤ V1 ∞ ⎢ ⎥ E ⎣e−λUi EQUi e−λs 1A (Xs ) ds⎦ = i=1
0
≤ c20 (ε−K4 δ+γ + εK5 δ )
∞ i=1
≤ c21 (ε which proves (3.24).
−K4 δ+γ
+ε
K5 δ
),
Ee−λUi
430
R. F. Bass, A. Lavrentiev
We now choose δ = γ /(2K7 ). Substituting this value of δ in (3.24), we obtain ∞ E
e−λs 1A (Xs ) ds ≤ c22 εγ ,
(3.25)
0
where γ = (γ /2)(1 + K5 /K7 ). Some real analysis finishes the proof. Let p0 = 2/γ . Now suppose f ∈ p L 0 (Rd+ ) with support in [0, M]d . By multiplying by a constant, it suffices to consider the case where f Lp0 (dx) = 1. Without loss of generality we may also suppose f ≥ 0. Let An = {x : f (x) ≥ 2n }. Then p
|An | ≤
f L0p0 (dx) (2n )p0
= 2−np0 .
We then have ∞ E
e−λs 1An (Xs ) ds ≤ c22 2−np0 γ .
0
Thus ∞ E
−λs
e
∞
1 n+1 f (Xs ) ds ≤ + 2 E λ n=0
0
≤
1 + λ
∞
∞
e−λs 1An (Xs ) ds
0
2n+1 c22 2−np0 γ ≤ c23 < ∞.
n=0
Since f Lp0 (dx) = 1, the proof of (3.15) is complete. If A ⊂ [0, M]d , then |A| = [0,M]d
d
xαi i 1A (x) µ(dx) ≤ Md µ(dx).
i=1
From this and (3.25) we see ∞ E
e−λs 1A (Xs ) ds ≤ c22 |A|γ ≤ c24 µ(A)γ .
0
Proceeding exactly as in the previous paragraph, we obtain (3.16).
Remark 3.3 Once uniqueness is established, this theorem shows that the resolvent of X maps Lp0 functions with support in [0, M]d into L∞ .
The submartingale problem for a class of degenerate elliptic operators
431
4 Existence In this section we prove existence of a solution to the submartingale problem. for the operator L defined in (1.1) with Neumann boundary conditions on . There are two complications that are not present in the usual case: we need to show that our solution spends zero time on the set defined in (2.1); and unless the αi are small, we cannot use the Girsanov transformation to reduce to the case of zero drift. For ε > 0 let Lε be the operator defined by Lε f (x) =
d
ai (x)(xi + ε)αi fii (x) +
i=1
d
bi (x)fi (x),
i=1
again with Neumann boundary conditions on = ∂(Rd+ ). The diffusion coefficients are uniformly positive definite and continuous and are of at most linear growth, so there exists a unique solution to the submartingale problem for Lε started from x0 for every x0 ; let us denote it Pε . (We reflect the coefficients over the coordinate axes, construct the solution to the corresponding martingale problem on Rd , and then look at the law of (|X 1 |, . . . , |X d |).) We remark that the statement and proof of Proposition 2.3 apply equally well to solutions to (3.2). It is now standard (cf. Sect. VI.1 of [3]) that the Pε are a tight sequence of probability measures on C([0, ∞); Rd+ ) and there must exist a subsequence εj such that Pεj converges weakly. Denote the limit measure by P and the corresponding expectation by E. It is obvious that P(X0 = x0 ) = 1. Proposition 4.1 Under P the process spends zero time in , i.e., ∞ 1 (Xs )ds = 0,
P − a.s.
0
Proof Under Pε , the ith component of Xt will satisfy an SDE of the form dXti =
2ai (Xt )(Xti + ε)αi /2 dWti + bi (Xt ) dt + dLX t , i
i
where LX is a local time at 0. Applying Theorem 3.1 the amount of time Xti spends in [0, η] before exceeding K under Pε is bounded by c1 η1−αi , where c1 may depend on K but not ε. Taking a limit as ε → 0 τK(X i )
1[0,η] (Xsi )ds ≤ c1 η1−αi .
EP 0
432
R. F. Bass, A. Lavrentiev
Letting η → 0, we have τK(X i )
EP
1i (Xs ) ds = 0. 0
Since K is arbitrary, and using this argument for each i = 1, . . . , d, we obtain ∞ EP
1 (Xs )ds = 0,
(4.1)
0
and hence the amount of time spent in is 0 almost surely.
Proposition 4.2 P is a solution to the submartingale problem for L started at x0 . Proof The proof of this proposition is very similar to the proof of Theorem VI.1.3 in [3] and we give only a brief sketch, leaving the details to the reader. Note that Theorem 3.2 is needed because the drift coefficients are not assumed to be continuous. If f ∈ Cb2 is such that fi ≥ 0 on i , then Mtε
t = f (Xt ) − f (X0 ) −
Lε f (Xr ) dr
(4.2)
0
is a submartingale. The goalis to show that one can take a limit as ε → 0. It suffices to show that if Y = ni=1 gi (Xri ), where the gi are bounded continuous functions with compact support and r1 ≤ r2 ≤ · · · ≤ rn ≤ s ≤ t, then E[Mt Y] ≥ E[Ms Y], t where Mt = f (Xt ) − f (X0 ) − 0 Lf (Xr ) dr. We know Eε [Mtε Y] ≥ Eε [Msε Y], where Eε is the law of the process corresponding to Lε . The key step is to show ⎡ Eε ⎣Y
t
⎤
⎡
Lf (Xr ) dr⎦ → E ⎣Y
0
t
⎤ Lf (Xr ) dr⎦ ;
(4.3)
0
because the bi are not continuous, then Lf is not continuous. We approximate Lf by a bounded continuous function G, that is, so that LF − GLp0 (dx) is small. Then by the definition of weak convergence, ⎡ Eε ⎣Y
t 0
⎤
⎡
G(Xr ) dr⎦ → E ⎣Y
t 0
⎤ G(Xr ) dr⎦ ,
(4.4)
The submartingale problem for a class of degenerate elliptic operators
433
provided we take the limit along the appropriate subsequence of ε’s. Theorem 3.2 shows that the left and right hand sides of (4.4) are close to the left and right hand sides of (4.3), resp. 5 First order estimates We first consider the continuous strong Markov process Zt on [0, ∞) associated with the operator AZ f (x) = xα f (x). Here α ∈ (0, 1) and we impose Neumann (i.e., reflecting) boundary conditions at 0. More precisely, we have a process on natural scale whose speed measure has no atom at 0 and does not charge (−∞, 0]. Let b=1−
α , 2
and note that b ∈ ( 12 , 1). If we set 1 Yt = √ Ztb , b 2 a straightforward calculation shows that Y is a continuous strong Markov process on [0, ∞) associated to the operator AY f (x) = 12 f (x) +
b−1 f (x) 2bx
with reflection at 0, i.e., a Bessel process of order δ = b−1 b + 1 with reflection at 0. By [13] the transition densities of Y (with respect to Lebesgue measure) are given by pY (t, x, y) =
y ν y x
t
e−(x
2 +y2 )/2t
Iν (xy/t),
1 and Iν is the standard modified Bessel function. where ν = 2δ − 1 = − 2b A change of variables then gives
pZ (t, x, y) =
c1 2b− 3 −c2 y2b /2t 1 −c2 x2b /2t 2e x2 e Iν (c2 xb yb /t) y t
(5.1)
and we have the scaling relationship pZ (t, x, y) = t−1/2b pZ (1, xt−1/2b , yt−1/2b ).
(5.2)
434
R. F. Bass, A. Lavrentiev
We will need the following lemma, the proof of which is given in the appendix. The proof consists of lengthy calculation. Lemma 5.1 There exists a constant c1 such that ∞ ∂ 1 sup pZ (t, x, y) dy ≤ c1 t− 2b ; ∂x x
(5.3)
∞ ∂ α pZ (t, x, y) x−α dx ≤ c1 t− 2b1 . sup y ∂x y
(5.4)
0
0
Let Pt be the semigroup for Zt , i.e., Pt f (x) = Ex f (Zt ). Let µ(dx) = x−α dx and we consider the space L2 (R+ , µ). We use the above estimates to prove the following proposition, which is a variant of the classical Young inequality (cf. [2], Theorem IV.5.1). Proposition 5.2 Suppose p ∈ (1, ∞). There exists a constant c1 depending only on p such that (Pt f ) p ≤ c1 t−1/2b f p ,
f ∈ L2 (R+ , µ).
∂ p(t, x, y)|. Let q be the conjugate expoProof Fix t > 0 and write K(x, y) for | ∂x nent to p. Then by Lemma 5.1 we have
(Pt f ) p ≤ p
R+
= R+
≤ R+
⎡ ⎢ ⎣
⎤p ⎥ K(x, y)|f (y)| dy⎦ x−α dx
R+
⎡ ⎢ ⎣
⎤p ⎥ K(x, y)1/q K(x, y)1/p |f (y)| dy⎦ x−α dx
R+
⎡ ⎢ ⎣
⎤p/q ⎡ ⎥ K(x, y) dy⎦
R+
≤ c2 t−(1/2b)(p/q)
⎢ ⎣
⎤ ⎥ K(x, y)|f (y)|p dy⎦ x−α dx
R+
K(x, y)x−α dx |f (y)|p dy
R+ R+
≤ c3 t−(1/2b)((p/q)+1)
|f (y)|p y−α dy.
R+
Now take pth roots of both sides.
The submartingale problem for a class of degenerate elliptic operators
435
Now let us turn to the d-dimensional case. The analogue of Proposition 5.2 follows readily from that proposition. We suppose Zti is the process on R corresponding to the operator (5.5) Ai f (x) = xαi f (x), with the speed measure having no atom at 0 and not charging (−∞, 0) and αi ∈ (0, 1). Then Zti ≥ 0 for all t and is reflecting at 0. Set bi = 1 − α2i . We let pi (t, x, y) denote the transition densities of Zti , Pti the corresponding semigroups, let Zt = (Zt1 , . . . , Ztd ), and let p(t, x, y) = di=1 pi (t, xi , yi ) be the transition densities for Z when x = (x1 , . . . , xd ), y = (y1 , . . . , yd ). Let Pt now denote the semigroup for Z. Let µi be the measure on R+ whose Radon-Nikodym derivi and let µ be the ative with respect to d-dimensional Lebesgue measure is x−α i d measure on R+ given by d µ(dx) = µi (dxi ). (5.6) i=1
We have the analogue of Proposition 5.2. Proposition 5.3 There exists a constant c1 such that for each i ∂(Pt f ) −1/2bi f p , ∂x ≤ c1 t i p
f ∈ Lp (Rd+ , µ).
Proof We will prove this in the case i = 1, the case for other i’s being exactly similar. Let x = (x2 , . . . , xd ). Let f ∈ L2 (Rd+ , µ) and set F(x1 ; x) =
···
d
pj (t, xj , yj )f (x1 , y2 , . . . , yd ) dy2 · · · dyd .
j=2
Then ∂ ∂ 1 ∂x Pt f (x) = ∂x Pt F(x1 ; x) , 1
1
and so by Proposition 5.2 we have ∂Pt f (x) p µ1 (dx1 ) ≤ c2 t−p/2b1 |F(x1 ; x)|p µ1 (dx1 ). ∂x 1 If we integrate both sides with respect to µ(dx2 · · · dxd ) = have our result provided we show
d
j=2 µj (dxj ),
we will
|F(x1 ; x)|p µ(dx) ≤
|f |p µ(dx).
(5.7)
436
R. F. Bass, A. Lavrentiev
To prove (5.7) let Pt be the semigroup corresponding to (Zt2 , . . . , Ztd ). It is easy to check that dj=2 Aj is self-adjoint with respect to the measure µ. Therefore, using Jensen’s inequality, g : Rd−1 + → R,
Pt gLp (µ) ≤ gLp (µ) , or
p
|g(x)|p µ(dx).
|Pt g(x)| µ(dx) ≤
(5.8)
We hold x1 fixed and apply this to g(x; x1 ) = f (x1 , . . . , xd ). Note Pt g(·; x1 ) = F(x1 ; x). So applying (5.8) to this g, we have |F(x1 ; x)|p µ(dx) ≤ |f (x1 , . . . , xd )|p µ(dx). (5.7) follows by integrating both sides of this equation with respect to µ1 (dx1 ). Our main result of this section is the following. Let ∞ Rλ f =
e−λt Pt f dt.
0
Theorem 5.4 There exists c1 such that for each i 1
∂(Rλ f )/∂xi p ≤ c1 λ 2bi
−1
f p .
Proof Since −1/(2bi ) > −1, the result follows from Proposition 5.3, dominated convergence, and Minkowski’s inequality for integrals: ∞ ∞ ∂ −λt −λt ∂Pt f e P f dt = e dt t ∂x ∂xi i 0
0
p
∞ ≤
p
e−λt c2 t−1/(2bi ) dtf p .
0
Remark 5.5 Only very minor changes are needed to get the same conclusion as in Theorem 5.4 if we instead set Rλ to be the resolvent for the operator d i=1 ai Ai , where the ai are strictly positive finite constants. We need one more estimate.
The submartingale problem for a class of degenerate elliptic operators
437
Lemma 5.6 If g is in C2 with compact support contained in (0, ∞)d , then for each t > 0 and λ > 0 we have that Pt g and Pt Rλ g are C2 on Rd+ and for each i we have (Pt Rλ g)i = 0 on i . Proof We have a formula for the derivative of the transition density in the one-dimensional case in (8.6) below in the appendix. If we differentiate once more and use the fact that the transition densities for the process factor as a product of transition densities of one-dimensional processes, then tedious calculations show that Pt g is C2 with normal derivative 0 on the boundary. (This is somewhat easier than in the proof of Lemma 5.1 since we can use the fact that g has compact support.) Moreover one can show that the second derivatives of ∞ Pt g grow with t at most polynomially. Since Pt Rλ g = 0 e−λs Ps+t g ds by the semigroup property, the lemma follows. 6 Second order estimates Let Ai be defined by (5.5), and let Pti be the semigroup corresponding to the process Zti associated with Ai that spends zero time at 0. We let Pt be the semigroup corresponding to the process Zt = (Zt1 , . . . , Ztd ), where the Zti are independent. The independence implies that if f (z) = di=1 f (i) (zi ) and z = (z1 , . . . , zd ), then Pt f (z) = di=1 Pti f (i) (zi ). Let ∞ Rλ f (z) =
e−λt Pt f (z) dt,
λ ≥ 0,
(6.1)
0
be the resolvent for Z. We let Ut be the Poisson semigroup defined in terms of Pt : Ut =
∞ 0
t 2 √ e−t /4s s−3/2 Ps ds; 2 π
(6.2)
see [11], p. 127; Ut is also known as the Cauchy semigroup. The semigroup Pti is self-adjoint on (R+ , µi ), where µi (dz) = |zi |−αi dzi . We let µ(dz) = di=1 µi (dzi ) be the product measure on Rd . We use spectral theory to prove the following. Lemma 6.1 Let f , h be C2 on Rd+ with compact support in (0, ∞)d . We have the identity ∞
(Ai R0 f (z))h(z) µ(dz) =
t(Ai Ut/2 f (z))(Ut/2 h(z)) dt µ(dz). 0
(6.3)
438
R. F. Bass, A. Lavrentiev
Proof Using the spectral theorem, there exists (see [12], Theorem 13.30) a spectral representation ∞ Pti
=
e−λi t dEλi i ,
i = 1, . . . , d.
0
Write s(λ) =
d
i=1 λi
if λ = (λ1 , . . . , λd ). If f (z) = ∞
Pt f =
∞ ···
0
∞
and since
0
d
i=1 f
(i) (z ), i
then
e−ts(λ) dEλ1 1 (f (1) ) · · · dEλdd (f (d) ),
0
e−ts(λ) dt = 1/s(λ), then ∞ R0 f =
∞ ···
0
1 dE1 (f (1) ) · · · dEλdd (f (d) ). s(λ) λ1
0
We have by (6.2) and ([11], p. 127, Eq. (5)) that ∞ Ut f =
∞ ···
0
e−t
√
s(λ)
dEλ1 1 (f (1) ) · · · dEλdd (f (d) ).
0
Note also ∞ λi dEλi i .
Ai = 0
Therefore, if h(z) = ∞
∞ ···
0
d
(i) i=1 h (zi ),
the left hand side of (6.3) is
λi d(Eλ1 1 (f (1) ), Eλ1 1 (h(1) )) · · · d(Eλdd (f (d) ), Eλdd (h(d) )). s(λ)
0
We use here ( ·, · ) for the inner product in L2 (µ).
(6.4)
The submartingale problem for a class of degenerate elliptic operators
439
Similarly, the right hand side of (6.3) is ⎡ ∞ ∞ ∞ √ √ ⎣ · · · t λi e−(t/2) s(λ) e−(t/2) s(λ) d(Eλ1 (f (1) ), Eλ1 (h(1) )) 1 1 0
0
0
⎤
· · · d(Eλdd (f (d) ), Eλdd (h(d) ))⎦ dt ⎡ ∞ ∞ ∞ √ ⎣ · · · t λi e−t s(λ) d(Eλ1 (f (1) ), Eλ1 (h(1) )) = 1 1 0
0
0
⎤
· · · d(Eλdd (f (d) ), Eλdd (h(d) ))⎦ dt.
Since ∞
te−Kt dt =
1 , K2
0
this is equal to (6.4). Linear combinations of functions of the form di=1 f (i) (zi ) are dense in L2 (µ), and an approximation argument completes the proof. We use the notation uf (z, t) = Ut f (z) and similarly with f replaced by h. For t > 0 and f ∈ C2 (Rd+ ) with support disjoint from the boundary, we have that ∂uf /∂zi exists by Lemma 5.6. The main theorem of this section is the following. Theorem 6.2 Let 1 < p < ∞. There exists a constant c1 depending only on p such that Ai R0 f p ≤ c1 f p . Proof Let f and h be C2 with compact support contained in (0, ∞)d . Using Lemma 6.1, integration by parts, and a change of variables, we have (Ai R0 f (z))h(z) µ(dz) ∞ = t(Ai Ut/2 f (z))(Ut/2 h(z)) dt µ(dz) 0
440
R. F. Bass, A. Lavrentiev
∞ ∂U ∂U f h t/2 t/2 αi = t|zi | (z) (z) dt µ(dz) ∂z ∂z i i 0 ∞ ∂Ut h αi ∂Ut f = 4 t|zi | (z) (z) dt µ(dz) ∂zi ∂zi 0
∞ ≤4 0
∂Ut f t|zi |αi (z) ∂zi
∂Ut h dt µ(dz). (z) ∂z i
(6.5)
Let Wt be a Brownian motion independent of Z and define the measure ξs (dz dt) on Rd+ × [0, ∞) by ξs (dz dt) = µ(dz)δs (dt), where δs is point mass at s. Let τ = inf{t : Wt = 0}. An application of Ito’s formula shows that uf (Zt∧τ , Wt∧τ ) is a martingale, and ⎛ ⎞ 2 2 t∧τ d ∂uf ∂uf j ⎝ (Zs , Ws ) ⎠ ds. |Zs |αj (Zs , Ws ) + uf (Z, W)t = ∂z ∂t 0
(6.6)
j
j=1
We claim that if F ≥ 0
ξs
τ
E
∞ F(Zr , Wr ) dr = (t ∧ s)F(z, t) dt µ(dz).
0
(6.7)
0
To see this, it suffices to prove it for F of the form F(z, t) = F1 (z)F2 (t) and then use linearity and an approximation procedure. Recall that the Green function for Brownian motion on [0, ∞) killed on hitting 0 is G(s, t) = s ∧ t; this is easily derived from equation (2.1) of [3] by taking a limit. By the independence of Z and W, the product structure of ξs , and the fact that µ is an invariant measure for Z,
ξs
τ
E
τ s
F1 (Zr )F2 (Wr ) dr =
F1 (z) µ(dz) E
0
F2 (Wr ) dr 0
=
∞ F1 (z) µ(dz) (t ∧ s)F2 (t) dt, 0
The submartingale problem for a class of degenerate elliptic operators
441
and (6.7) follows. Hence ∞ ∂Ut f (t ∧ s)|zi |αi (z) 4 ∂zi 0
ξs
τ
= 4E
0
∂Ut h dt µ(dz) (z) ∂z i
∂uf |Zri |αi (Zr , Wr ) ∂zi
∂uh (Z , W ) r r dr. ∂z i
(6.8)
Using Cauchy-Schwarz, Hölder’s inequality and (6.6), the right hand side of (6.8) is bounded by ⎡⎛ ⎞1/2 ⎛ τ ⎞1/2 ⎤ 2 2 τ ∂uf ∂uh ⎢ ⎥ 4 Eξs ⎣⎝ |Zri |αi (Zr , Wr ) dr⎠ ⎝ |Zri |αi (Zr , Wr ) dr⎠ ⎦ ∂zi ∂zi 0
≤ ≤
0
ξs
1/2 4E [uf (Z, W)1/2 τ uh (Z, W)τ ] 1/p 1/q ξs q/2 4 Eξs uf (Z, W)p/2 E u (Z, W) , h τ τ
(6.9)
where q is the conjugate exponent to p. Let νn,s be the restriction of ξs to [0, n]d × [0, ∞), so that νn,s is a finite measure. By the Burkholder-Davis-Gundy inequalities and Doob’s inequality, Eνn,s uf (Z, W)p/2 ≤ c2 Eνn,s |uf (Zτ , Wτ )|p + c2 Eνn,s |uf (Z0 , W0 )|p . τ Letting n → ∞, we have Eξs uf (Z, W)p/2 ≤ c2 Eξs |uf (Zτ , Wτ )|p + c2 Eξs |uf (Z0 , W0 )|p . τ Now Eξs |uf (Zτ , Wτ )|p =
|uf (z, 0)|p µ(dz) =
|f (z)|p µ(dz),
and since µ is an invariant measure for Pt , and hence for Ut , an application of Jensen’s inequality yields ξs
p
E |uf (Z0 , W0 )| =
p
|uf (z, s)| µ(dz) ≤
|f (z)|p µ(dz).
We have a similar inequality for uh and q. Therefore for each s 1/p 1/q ξs q/2 E u (Z, W) ≤ c3 f p hq . 4 Eξs uf (Z, W)p/2 h τ τ
442
R. F. Bass, A. Lavrentiev
Let s → ∞; by (6.5) and Fatou’s lemma (Ai R0 f (z))h(z) µ(dz) ≤ c3 f p hq . Taking the supremum over h with hq ≤ 1, the duality of Lp and Lq implies that Ai R0 f p ≤ c3 f p if f is C2 with support disjoint from the boundary. An approximation argument allows us to extend this inequality to all f ∈ Lp . Corollary 6.3 Let λ > 0. Let 1 < p < ∞. There exists a constant c1 depending only on p such that Ai Rλ f p ≤ c1 f p . Proof If f ∈ Lp , then f − λRλ f is also in Lp with f − λRλ f p ≤ 2f p . Our result now follows from Theorem 6.2 because Rλ f = R0 (f − λRλ f ). Remark 6.4 As with the first order estimates, only minor changes are needed if Rλ is the resolvent for di=1 ai Ai , where the ai are finite positive constants. 7 Uniqueness We now can complete the proof of Theorem 1.1. The existence part of Theorem 1.1 was done in Section 4. It remains to prove uniqueness. Proof of uniqueness in Theorem 1.1. Fix x0 ∈ Rd+ and let ε > 0 be specified later. Let M > 0. As in the nondegenerate case, to prove uniqueness it suffices to localize, that is, to consider only the case where d
|ai (y) − ai (x0 )| ≤ ε,
y ∈ Rd+ ,
(7.1)
i=1
and in addition ai (y) = ai (x0 ) and bi (y) = 0 if y ∈ / [0, M]d , i = 1, . . . , d; see [3], Sect. VI.3 and note that the continuity of the aij is used here. Let p0 be the positive real given by Theorem 3.2. Set L0 f (x) =
d i=1
ai (x0 )xαi i
∂ 2f (x) ∂x2i
(7.2)
The submartingale problem for a class of degenerate elliptic operators
443
and let B = L − L0 .
(7.3)
[0, M]d .
Note Bf (y) = 0 if y ∈ / Let Rλ and Pt be the resolvent and semigroup, respectively, for the operator L0 . Taking into account Remarks 5.5 and 6.4, by Theorem 5.4 and Corollary 6.3 we have ⎛ ⎞ d 1 −1 λ 2bi bi ∞ ⎠ f p0 . (7.4) BRλ f p0 ≤ c1 ⎝dε + i=1
Let us now choose ε small enough and λ large enough so that by (7.4) we have BRλ f p0 ≤ 12 f p0 .
(7.5)
Let P1 and P2 be any two solutions to the submartingale problem for L started at x0 , where we continue to assume (7.1) holds. We also assume that under each Pi the process spends zero time on . Define ∞ Siλ h
= Ei
e−λt h(Xt ) dt,
i = 1, 2.
(7.6)
0
Let RK = inf{t : for each i, then
d
Xi i=1 Lt
≥ K}. By Ito’s formula, if f ∈ Cb2 and fi = 0 on i t∧R K
Ei f (Xt∧RK ) − f (x0 ) = Ei
Lf (Xs ) ds,
i = 1, 2.
0
We let K → ∞, so that RK → ∞. we then multiply both sides by λe−λt and integrate over t from 0 to ∞ to obtain λSiλ f − f (x0 ) = Siλ Lf = Siλ L0 f + Siλ Bf .
(7.7)
Now let g be Cb2 with compact support contained in (0, ∞)d and let f = Pt Rλ g. By Lemma 5.6 we can apply (7.7) to f . Note L0 f = L0 Rλ Pt g = λRλ Pt g − Pt g. Therefore (7.7) becomes Siλ Pt g = Rλ Pt g(x0 ) + Siλ BRλ Pt g. Let = sup{|S1λ h − S2λ h| : h supported in [0, M]d , hp0 ≤ 1}.
(7.8)
444
R. F. Bass, A. Lavrentiev
By Theorem 3.2 we know < ∞. By (7.8) and (7.5) and recalling that BRλ g is supported in [0, M]d , |S1λ Pt g − S2λ Pt g| ≤ BRλ Pt gp0 ≤ 12 Pt gp0 ≤ 12 gp0 . The last inequality follows by Jensen’s inequality and the fact that Pt is selfadjoint with respect to µ. Since the support of g is disjoint from , we can let t → 0 and obtain |S1λ g − S2λ g| ≤ 12 gp0 . We now take the supremum over all such g that in addition satisfy gp0 ≤ 1. Since neither S1λ nor S2λ charge , we then have ≤ 12 . Since < ∞, we conclude = 0. From this point on we follow the proof of the nondegenerate case; see [3], Theorem VI.3.2. Very briefly, the argument can be summarized as follows. Since S1λ = S2λ , by the uniqueness of the Laplace transform, E1 h(Xt ) = E2 h(Xt )for almost every t. If h is continuous and bounded, the continuity implies this equality for all t. Therefore the one-dimensional distributions of X under E1 and E2 are equal. An argument using regular conditional probabilities is then used to show that the finite dimensional distributions of X under E1 and E2 are the same. Finally, one shows that the localization assumptions can be dispensed with. Remark 7.1 We show how Girsanov’s theorem allows us to dispense with the drift when all the αi are small enough. Suppose each αi < 1/2 and under P, X is a solution to i dXti = 2ai (Xt )(Xti )αi /2 dWti + dLX t , i
where LX is a local time at 0 for X i , the W i are independent one-dimensional Brownian motions, and Xti ≥ 0 for all t. If we define dQ by dQ = exp Mt − 12 Mt , dP Ft where Mt = −
t d 0
i=1
√
bi (Xt ) dWsi , 2ai (Xt )(Xsi )αi /2
The submartingale problem for a class of degenerate elliptic operators
445
then τ[0,M]d (X)
d (Xsi )−αi ds
E[Mτ[0,M]d ] ≤ c1 E
i=1
0
≤ c2
d
i τ[0,M] (X )
(Xsi )−αi ds.
E
i=1
0
Using (3.4), this can be seen to be finite if αi < 1/2 for each i. It is readily checked using Girsanov’s theorem for continuous semimartingales (see [2], Theorem I.6.4) that under Q, Xt solves (1.2); cf. the proof of Theorem VI.3.1 in [3]. This simplifies the argument in the case where all the αi < 1/2, both for existence and for uniqueness. Appendix We prove Lemma 5.1. We will use the well known facts (see [9], pp. 150–152): p Ip (x), x p Ip (x) = Ip−1 (x) − Ip (x), x 1 xp , x → 0, Ip (x) ∼ p 2 (p + 1) 1 ex Ip (x) ∼ √ √ , x → ∞. 2π x Ip (x) = Ip+1 (x) +
In what follows we will take p = ν or ν + 1 and ν = −1/(2b). If we let F(x) = Iν+1 (x) − Iν (x), then from (8.1) and (8.2) we have F (x) = −F(x) −
ν+1 ν Iν+1 (x) − Iν (x). x x
Using (8.4) |F (x) + F(x)| ≤ c1
ex , x3/2
or |(ex F(x)) | ≤ c1
e2x . x3/2
(8.1) (8.2) (8.3) (8.4)
446
R. F. Bass, A. Lavrentiev
Therefore x x
|e F(x)| ≤ |eF(1)| + c1
e2y dy. y3/2
1
By l’Hôpital’s rule, the integral is bounded by c2 e2x /x3/2 , and so we deduce |Iν+1 (x) − Iν (x)| ≤ c3
ex x3/2
(8.5)
for x ≥ 1. Proof of Lemma 5.1. We start with (5.3). By scaling it suffices to do the case t = 1. Differentiating (5.1) we have 1 3 ∂pX (1, x, y) 2b 2b = cxb− 2 y2b− 2 e−K(x +y )/2 [−xb Iν (Kxb yb ) + yb Iν+1 (Kxb yb )], ∂x (8.6)
where K is some fixed positive constant. We write ∞ 1/x ∞ ∂pX (1, x, y) dy = + := S1 + S2 . ∂x 0
0
1/x
Using the bounds on Iν ,
2b−1 −Kx2b /2
S1 ≤ cx
e
∞ 2b [y2b−2 + y4b−2 ]e−Ky /2 dy. 0
Since 2b−2 > −1, the integral term is finite. Since 2b−1 > 0, the factor in front of the integral is bounded independently of x, so S1 is bounded independently of x. Since | − xb Iν (Kxb yb ) + yb Iν+1 (Kxb yb )| = |(yb − xb )Iν (Kxb yb ) + yb (Iν+1 (Kxb yb ) − Iν (Kxb yb ))| b b b b b b ≤ c|yb − xb |eKx y x− 2 y− 2 + cyb (eKx y x−b y−b )
The submartingale problem for a class of degenerate elliptic operators
447
for y ≥ 1/x, to bound S2 we need to bound ∞ |yb − xb |x
b−1 2
y
3b−3 2
e−K(y
b −xb )2 /2
dy
1/x
∞ +
1
3
x− 2 y2b− 2 e−K(y
b −xb )2 /2
dy
1/x
= S3 + S4 . For S3 we make the substitution z = yb − xb and then ∞ S3 = c
|z|x
b−1 2
(xb + z)
b−1 2b
c−Kz
2 /2
dz.
x−b −xb
Since (b − 1)/(2b) < 0 and xb + z ≥ x−b , this is less than ∞ |z|x
c
b−1 2
(x−b )
b−1 2b
e−Kz
2 /2
dz
−∞
which is bounded independently of x. For S4 we use the same substitution. Since 2b − 1 > 0, we have (xb + z)
2b−1 2b
≤ c(x
2b−1 2
+z
2b−1 2b
).
Hence ∞ S4 ≤
1
x− 2 (xb + z)
x−b −xb ∞
≤c
2b−1 2b
1
(xb−1 + x− 2 z
e−Kz
2b−1 2b
2 /2
dz
)e−Kz
2 /2
dz.
x−b −xb
For each p ≥ 1 and q > 0 there exists c(p, q) such that ∞ 2 (1 + z)q e−Kz /2 dz ≤ c(p, q)a−p , a
a > 1.
(8.7)
448
R. F. Bass, A. Lavrentiev
From this we see that S4 is bounded independently of x for x ≤ 1. On the other hand, for x ≥ 1, ∞ S4 ≤ c
(1 + |z|
2b−1 2b
)e−Kz
2 /2
dz ≤ c.
−∞
We now turn to the proof of (5.4). Again by scaling we may assume t = 1. 1/y Looking at 0 and using the bounds on Iν , 1/y ∂pX (1, x, y) −α x dx y ∂x α
0
α+2b−2
α+4b−2
+y
≤ c[(y
−Ky2b /2
)e
∞ ]
x2b−1−α e−Kx
2b /2
dx.
0
Since α + 2b − 2 = 0, α + 4b − 2 > 0, and 2b − 1 − α = 1 − 2α > −1, the integral is finite and the ∞expression in brackets is bounded in y. To look at 1/y , we rewrite the integral as in S2 and see that we have to bound
3
3
cy 2 b− 2 +α
∞
b
1
x 2 − 2 −α |yb − xb |e−K(y
b −xb )2 /2
dx
1/y 2b− 32 +α
∞
+ cy
1
x− 2 −α e−K(y
b −xb )2 /2
dx
1/y
= S5 + S6 . Letting z = xb − yb as in S3 , 3
∞
3
S5 ≤ cy 2 b− 2 +α
(yb + z)−
b−1+2α 2b
ze−z
2 /2
dz
y−b −yb
∞ ≤ cyb−1
ze−z
2 /2
dz.
y−b −yb
When y ≤ 1 this is bounded using (8.7). When y ≥ 1 this is bounded because b − 1 < 0.
The submartingale problem for a class of degenerate elliptic operators
449
For S6 we have ∞
2b− 32 +α
S6 ≤ cy
(yb + z)−
2b−1+2α 2b
e−Kz
2 /2
dz
y−b −yb
∞ b−1
≤ cy
e−Kz
2 /2
dz.
y−b −yb
This is seen to be bounded in y for y ≤ 1. This is bounded in y for y > 1 because b − 1 is negative. References 1. Athreya, S.R., Barlow, M.T., Bass, R.F., Perkins, E.A.: Degenerate stochastic differential equations and super-Markov chains. Probab. Theory Relat. Fields 123, 484–520 (2002) 2. Bass, R.F.: Probabilistic Techniques in Analysis. Springer, Berlin Heidelberg New York (1995) 3. Bass, R.F.: Diffusions and Elliptic Operators. Springer, Berlin Heidelberg New York (1997) 4. Bass, R.F., Burdzy, K., Chen, Z.-Q.: Pathwise uniqueness for a degenerate stochastic differential equation, preprint 5. Bass, R.F., Perkins, E.A.: Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains. Trans. Am. Math. Soc. 355, 373–405 (2003) 6. Dupuis, P., Ishii, H.: SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21, 554–580 (1993) 7. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. Elsevier North-Holland, Amsterdam (1989) 8. Krylov, N.V.: certain estimate from the theory of stochastic integrals. Theor. Probab. Appl. 16, 438–448 (1971) 9. Lebedev, N.N.: Special Functions and their Applications. Dover, New York (1972) 10. Lions, P.-L., Sznitman, A.-S.: Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37, 511–537 (1984) 11. Meyer, P.A.: Démonstration probabiliste de certaines inégalités de Littlewood-Paley. I. Les inégalités classiques. Séminaire de Probabilités, X, pp. 125–141. Springer, Berlin Heidelberg New york (1976) 12. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973) 13. Revuz, D., Yor, M.: Continuous martingales and Brownian motion, 3rd edn. Springer, Berlin Heidelberg New york (1999) 14. Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24, 147–225 (1971)
