Potential Analysis 14: 123–148, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

123

On Dirichlet Processes Associated with Second Order Divergence Form Operators† ANDRZEJ ROZKOSZ Faculty of Mathematics and Informatics, Nicholas Copernicus University, ul. Chopina 12/18, 87–100 Toru´n, Poland (e-mail: [email protected]) (Received: 8 September 1998; accepted: 7 September 1999) Abstract. Let {(X, P x ); x ∈ Rd } be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous ϕ in the Sobolev space Wq1 (Rd ) the process ϕ(X) admits under P x a decomposition into a martingale additive functional (AF) M ϕ and a continuous AF Aϕ of zero quadratic variation for almost every starting point x if q = 2, for quasi every x if q > 2 and for every x ∈ Rd if ϕ is continuous, d = 1 and q ≥ 2 or d > 1 and q > d. Our decomposition enables us to show that in the case of symmetric operator the energy of Aϕ equals zero if q = 2 and that the decomposition of ϕ(X) into the martingale AF M ϕ and the AF of zero energy Aϕ is strict if ϕ ∈ W21 (Rd ) ∩ Wq1 (Rd ) for some q > d. Moreover, our decomposition provides a probabilistic representation of Aϕ . Mathematics Subject Classifications (2000): 60J60, 60J45, 60J57. Key words: divergence form operator, diffusion, Dirichlet process, additive functional.

1. Introduction and Notation For x ∈ Rd let P x be a measure on the σ - field of Borel subsets of C([0, ∞); Rd ) whose one-dimensional distributions are determined by Z P x (X0 = x) = 1, P x (Xt ∈ 0) = p(t, x, y) dy, t > 0. 0

Here X is a canonical process on C([0, ∞); Rd ), p( · , · , · ) is the weak fundamental solution of the second order linear differential operator La,b = 12 Dj (a ij (x)Di ) + bi (x)Di , where Di = ∂/∂xi and we employ the convention of summation over repeated indices. We assume that for i, j = 1, . . . , d the coefficients a ij , bi are real measurable functions on Rd verifying a ij (x) = a j i (x),

λ|ξ |2 ≤ a ij (x)ξi ξj ≤ 3|ξ |2 ,

|bi (x)| ≤ 3,

x, ξ ∈ Rd (1.1)

for some constants 0 < λ ≤ 3 < ∞ (for construction of P x see e.g. [21]). † Research supported by Komitet Bada´n Naukowych under grant PB 483/P03/97/12.

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For t ∈ [0, T ] set X¯ t = XT −t , F t = σ (Xs , s ∈ [0, t]), F¯t = σ (X¯ s , s ∈ [0, t]). In [19] (see, however, Remark 2.6 in [20]) and in [20] it was proved that for each x ∈ Rd the components X i are ({Ft }, P x ) - Dirichlet processes on [0, T ] in the sense of Föllmer [3] admitting the Lyons-Zheng [14] decomposition Xti = X0i + Mtx,i + Ax,i t ,

t ∈ [0, T ] P x - a.s.,

where x,i x,i x,i x,i 1 Ax,i t = 2 (−Mt + NT −t − NT − Vt ),

t ∈ [0, T ].

Here M x,i (resp. N x,i ) is an ({Ft }, P x ) (resp. ({F¯t }, P x )) - square-integrable martingale on [0, T ] with quadratic variation Z t Z t x,i ii x,i hM it = a (Xs ) ds, (resp. hN it = a ii (X¯ s ) ds), t ∈ [0, T ] 0

and

Z Vti

0

t

=

{−2bi (Xs ) + a ij (Xs ) 0

Dj p (s, x, Xs )} ds, p

t ∈ [0, T ]

is an {Ft } - adapted process of integrable variation on [0, T ]. Moreover, for any continuous ϕ in the Sobolev space Wq1 (Rd ) with q > 2 ∨ d, ϕ(X) is a Dirichlet process as well and ϕ(X) has the decomposition x,ϕ

ϕ(Xt ) = ϕ(X0 ) + Mt where x,ϕ

Mt

Z

x,ϕ

+ At ,

t

=

Di ϕ(Xs ) dMsx,i , 0

and

Z x,ϕ Nt

t

= 0

x,ϕ

At

Di ϕ(X¯ s ) dNsx,i ,

t ∈ [0, T ]

x,ϕ

= 12 (−Mt Z

x,ϕ Vt

=

P x -a.s.,

x,ϕ

(1.2)

x,ϕ

+ NT −t − NT

x,ϕ

− Vt

)

(1.3)

t

Di ϕ(Xs ) dVsx,i

(1.4)

0

for t ∈ [0, T ]. In the present paper we show that the processes M x,ϕ , N x,ϕ , V x,ϕ , Ax,ϕ in the above decomposition of ϕ(X) under P x can be choosen independently of x and that similar decompositions hold under weaker regularity assumptions imposed on ϕ. In particular, we strenghen results of Lyons and Zheng [14] on a decomposition of ϕ(X) under P x for fixed x ∈ Rd . In case b = 0 we show also how to pass from a decomposition of ϕ(X) into a square-integrable martingale and a process of zero quadratic variation to Fukushima’s [6, 7] decomposition into a Martingale additive functional (MAF) of finite energy and an additive functional (AF) of zero energy.

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125

More precisely, consider the measurable space (, F∞ ), where  = µ µ C([0, ∞); Rd ). Let F∞ be the P µ - completion of F∞ and for t ≥ 0 let Ft (resp. µ µ F¯t ) be the σ - field generated by Ft (resp. F¯t ) and the P µ - null sets of F∞ . Set \ \ µ \ µ G∞ = F¯t , t ≥ 0, F∞µ , Gt = Ft , G¯ t = µ

µ

µ

the intersections being taken over all probability measures on a Borel σ - field B(Rd ). In Section 3 we show that in the case of a continuous ϕ ∈ Wq1 (Rd ), where q > d if d > 1 and q ≥ 2 if d = 1 there exist AF in the strict sense M ϕ of (X, P x ) and {Gt } - adapted processes Y ϕ , V ϕ such that P x (Mtϕ = Mtx,ϕ , Ntϕ = Ntx,ϕ , Vtϕ = Vtx,ϕ , t ∈ [0, T ]) = 1 for every x ∈ Rd , where Ntϕ = YTϕ−t − YTϕ , t ∈ [0, T ]. Moreover, for every x ∈ Rd , M ϕ is a ({Gt }, P x ) - martingale, N ϕ is a ({G¯ t }, P x ) - martingale and ϕ(X) is a ({Gt }, P x ) - Dirichlet process on [0, T ] admitting the decomposition ϕ

ϕ

ϕ(Xt ) = ϕ(X0 ) + Mt + At ,

t ∈ [0, T ] P x - a.s.,

(1.5)

where Aϕt = 12 (−Mtϕ + NTϕ−t − NTϕ − Vtϕ ),

t ∈ [0, T ].

(1.6)

For a quasi continuous ϕ ∈ Wq1 (Rd ) with d > 1 and q > 2 we prove existence of the decompositions (1.2)–(1.4) and (1.5)–(1.6) for quasi every (q.e.) x ∈ Rd , that is except for a set of zero capacity, while in the case d > 1 and q = 2 for almost every (a.e.) x ∈ Rd . In fact, in the latter case decompositions similar to (1.2)–(1.4) and (1.5)–(1.6) hold for q.e. x ∈ Rd . Unfortunately we know only that Ax,ϕ , Aϕ are zero quadratic variation processes for a.e. x ∈ Rd . Consequently, we know only that ϕ(X) is a Dirichlet process under P x for a.e. x ∈ Rd . In Section 4 we consider symmetric diffusions corresponding to La,0 . Starting from (1.5)–(1.6) and using some stochastic calculus with respect to Dirichlet processes we show that the energy of Aϕ equals zero for any quasi continuous ϕ ∈ W21 (Rd ). Thus we arrive at Fukushima’s [7] decomposition of ϕ(X) into a MAF of finite energy and an AF of zero energy. Moreover, from results of Section 3 it follows immediately that if d = 1 or ϕ ∈ W21 (Rd ) ∩ Wq1 (Rd ) with q > d if d > 1 then the decomposition (1.5) is strict. This follows also from a general Fukushima’s [7] result on a strict decomposition of functionals of symmetric Markov processes, because if either d = 1 and ϕ ∈ W21 (Rd ) or d > 1 and ϕ ∈ W21 (Rd ) ∩ Wq1 (Rd ) with q > d then the energy measure dµ = (a ij Di ϕDj ϕ)(x) dx of ϕ is smooth in the strict sense (see Remark 4.3). Let us stress, however, that our proof gives the representation (1.6), which refines (up to localization) known results concerning the Lyons-Zheng decomposition of ϕ(X) (see [14] and [7,R 11, 12, 13, 22] for a corresponding decomposition under the measure P λ ( · ) = P x ( · ) dx).

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In Section 2 we give basic definitions concerning additive functionals and Dirichlet processes and we prove some auxillary results. In Appendix we recall Aronson’s [1] estimates and prove an estimate on the transition density p, which will be needed in estimations of a variation of V x,ϕ . We adopt the following notation. C(Rd ) is the set of all continuous functions on Rd . C0∞ (Rd ) is a subset of C(Rd ) of smooth functions having compact support. Lp (Rd ), p ≥ 1 is the space Lp (Rd ; dx) with the usual norm k - kp and Wp1 (Rd ) is the Banach space consisting of all elements ϕ of Lp (Rd ) having generalized derivatives Di ϕ from Lp (Rd ). (E1 , W21 (Rd )) is a symmetric form on L2 (Rd ) defined by E1 (ϕ, ψ) = (−La,0ϕ, ψ)+ (ϕ, ψ), where ( · , · ) is the usual inner product in L2 (Rd ). For ϕ ∈ Wp1 (Rd ) we set ∇ϕ = (D1 ϕ, . . . , Dd ϕ). Given a process Y on [0, T ] we set Y¯t = YT −t , Y˜t = YT −t − YT , t ∈ [0, T ]. L[Y |P x ] is the law of Y under P x . “⇒" denotes convergence in law. R For a given measure µ on B(Rd ) we set P µ ( · ) = Rd P x ( · ) µ(dx). E x , E µ denote expectations with respect to P x and P µ , respectively. The abbreviation “q.e." or “quasi everywhere" means “except for a set of E1 capacity zero, and “a.e." means “except for a set of the Lebesgue measure zero". From Lemma 5.5.2 in [7] it follows that any ϕ ∈ Wq1 (Rd ) with q ≥ 2 has a quasi continuous modification ϕ˜ (i.e. ϕ = ϕ˜ a.e.), whereas by the Sobolev imbedding theorem (see e.g. Theorem II.2.1 in [10]), if q > d then ϕ has a continuous modification. We will therefore always assume that ϕ denotes the quasi continuous (continuous) modification of a given function from Wq1 (Rd ) with q ≥ 2 (q > d). The letter C will denote a positive constant which may vary from one expression to another one, however depends only on λ, 3, d, T and possibly some parameters α, q. By Dj p(s, x, y) we will denote the distributional derivative of p( · , x, · ) evaluated at (s, y). 2. Basic Definitions and Auxiliary Results Let (X, Qx ) be a symmetric diffusion corresponding to La,0 . By Theorem 4.2.1 in [7] a set E ⊂ Rd is of E1 - capacity zero iff it is exceptional, and therefore, by Theorems 4.1.2 and 4.2.4, iff it is polar for (X, Qx ), that is there is a nearly Borel set F ⊃ E such that Qx (σF < ∞) for every x ∈ Rd , where σF = inf{t > 0 : Xt ∈ F }. Let us recall also that according to Theorem 2.2.3 in [7], E1 - capacity of a Borel set E ⊂ Rd equals zero iff ν(E) = 0 for every ν ∈ S00 . Here S00 = {ν ∈ S0 : ν(Rd ) < ∞, kU1 νk∞ < ∞}, where S0 is a set of positive Radon measures on Rd of finite E1 - energy integrals, U1 is a 1 - potential of ν and k - k∞ denotes the norm in L∞ (Rd ) (see §2.2 in [7]). The following simple lemma shows that P x is equivalent to Qx for every x ∈ d R . Consequently, a set E is of E1 - capacity zero iff it is polar for (X, P x ), that is if P x (σF < ∞) for x ∈ Rd .

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

127

LEMMA 2.1. Let (X, P x ) be a diffusion starting from x ∈ Rd corresponding to La,b and let (X, Qx ) corresponds to La,0 . Then P x is equivalent to Qx on FT . In fact dP x ˆ T ≡ exp(Zˆ T − 1 hZi ˆ T ), = E(Z) 2 dQx FT dQx = E(Z)T ≡ exp(ZT − 12 hZiT ), dP x FT where Zˆ t =

Z

t

a −1 b(Xs ) dMˆ sx ,

Z Zt =

0

t

a −1 b(Xs ) dMsx ,

t ∈ [0, T ]

0

and Mˆ x (resp. M x ) is the martingale part in the decomposition of X under Qx (resp. P x ). Proof. For each n ∈ N choose smooth an , bn satisfying (1.1) so that ij an → a ij , bni → bi a.e. for i, j = 1, . . . , d. Set θn = (θni , . . . , θnd ), where θni = ij 1 D a , i = 1, . . . , d. Let Pnx , Qxn denote solutions to the Martingale problems for 2 j n (an , θn + bn ) and (an , θn ), respectively, starting from x. By the Cameron-MartinGirsanov formula, Pnx  Qxn and the density process equals {E(Zˆ n )t , t ∈ [0, T ]}, where Z t Z t n −1 x,n x,n Zˆ t = an bn (Xs ) dMˆ s , Mˆ t = Xt − x − θn (Xs ) ds, t ∈ [0, T ]. 0

0

It is well known that ⇒P , ⇒ Q (see [21]). Moreover, by (2.34) in [20], L[(X, Mˆ x,n )|Qxn ] ⇒ L[(X, Mˆ x )|Qx ] in C([0, T ]; R2d ), so applying Lemma 1.1 in [19] we conclude that L[(X, Z n )|Qxn ] ⇒ L[(X, Z)|Qx ] in C([0, T ]; Rd+1 ) and hence that Pnx

x

Qxn

x

L[(X, E(Z n ))|Qxn ] ⇒ L[(X, E(Z))|Qx ] in C([0, T ]; Rd+1 ), by Corollary VI.6.6 in [8] and the continuous mapping theorem. Similarly, Qxn  Pnx , the density process equals {E(Z n )t , t ∈ [0, T ]}, where Z t Z t n −1 x,n x,n Zt = an bn (Xs ) dMs , Mt = Xt − x − (θn + bn )(Xs ) ds, t ∈ [0, T ], 0

0

and x ˆ L[(X, E(Zˆ n ))|Pnx ] ⇒ L[(X, E(Z))|P ]

in C([0, T ]; Rd+1 ). Finally, observe that the expectation of E(Z)T under Qx equals ˆ T under P x , so the lemma follows from Corol1 and so is the expectation of E(Z) lary V.1.12 and Theorem X.3.3 in [8]. 2

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ANDRZEJ ROZKOSZ

In the next sections we will prove several decompositions of ϕ(X) into a martingale and a process of zero quadratic variation and into a MAF and an AF of zero energy. To avoid confusion we recall basic definitions. Let 5m denote a partition of [0, T ] of the form 0 = t0 < t1 < . . . < tk(m) = T and let k5m k = max1≤k≤k(m) |tk − tk−1 |. Let {Ht } = {Ft } or {Ht } = {Gt }. We will say that ϕ(X) is an ({Ht }, P x ) - Dirichlet process on [0, T ] if it admits the decomposition ϕ(Xt ) = ϕ(X0 ) + Mt + At ,

t ∈ [0, T ] P x - a.s.,

where M is an ({Ht }, P x ) - square-integrable martingale such that M0 = 0 and A is an {Ht } - adapted process of zero quadratic variation, that is A0 = 0 and X Qm (A) ≡ |Atk+1 − Atk |2 → 0 in P x as m → ∞ T tk ∈5m

for each sequence {5m} of partitions of [0, T ] such that k5m k → 0. We will call a {Gt } - adapted process A a continuous AF (CAF) of (X, P x ) if there is a set 0 ∈ G∞ and an exceptional set E ⊂ Rd such that P x (0 ) = 1 for x ∈ E c , θs 0 ⊂ 0 for s ≥ 0, where θs :  → , (θs ω)t = ωs+t , and for each ω ∈ 0 A0 (ω) = 0, A· (ω) is continuous and As+t (ω) = As (ω) + At (θs ω) for s, t ≥ 0. The set 0 in the above definition will be called a definig set and E an exceptional set of A. In case E = ∅ we will say that A is a CAF in the strict sense. If A is a CAF (CAF in the strict sense) and its quadratic variation under P x equals zero for q.e. x ∈ Rd (every x ∈ Rd ) we call A a CAF of zero quadratic variation (CAF of zero quadratic variation in the strict sense). For any CAF A of (X, P x ) we define its energy e(A) by 1 e(A) = lim E λ A2t t &0 t whenever the limit exists, where λ is the Lebesgue measure. If e(A) = 0 we call A a CAF of zero energy. We call M a MAF of (X, P x ) with exceptional set E if M is a CAF of (X, P x ) with exceptional set E and for each x ∈ E c , E x Mt2 < ∞ and E x Mt = 0 for all t ≥ 0. If E = ∅ we will say that M is a MAF in the strict sense. Let us recall that if M is a MAF of (X, P x ) then for any 0 ≤ s ≤ t and x ∈ E c E x (Mt |Gs ) = E x (Ms + Mt −s ◦ θs |Gs ) = Ms + E Xs Mt −s = Ms , so M is a ({Gt }, P x ) - square-integrable martingale for each x ∈ E c . The following lemma will prove extremely useful in the next section. LEMMA 2.2. Let ϕ ∈ C0∞ (Rd ). Then there exist a MAF in the strict sense M ϕ of (X, P x ) and a continuous {G¯ t } - adapted process N ϕ on [0, T ] such that N˜ ϕ is {Gt } - adapted and for every x ∈ Rd (i) N ϕ is a ({G¯ t }, P x ) - square-integrable Martingale

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS ϕ

x,ϕ

ϕ

129

x,ϕ

(ii) P x (Mt = Mt , Nt = Nt , t ∈ [0, T ]) = 1. Proof. Let {Rα , α > 0} be a resolvent of (X, P x ) and let ϕk = αk Rαk ϕ, where {αk } is choosen so that k∇(ϕk − ϕk+1 )k2 ≤ 3−k for k ∈ N. Set Mtk = ϕk (Xt ) − ϕk (X0 )− Z

t

−

k(ϕk − ϕ)(Xs ) ds, 0

Mtk,δ = Mtk∨δ − Mδk ,

t ≥ 0.

First we are going to show that for each δ ∈ (0, T ) Z t ∨δ x k,δ P (Mt = Di ϕk (Xs ) dMsx,i , t ∈ [0, T ]) = 1

(2.1)

δ

for every x ∈ Rd . For this purpose, for fixed k ∈ N let us choose {ψn } ⊂ C0∞ (Rd ) such that ψn → ϕk in W21 (Rd ) and uniformly in compact sets in Rd as n → ∞. Set Z t ∨δ Z t ∨δ x,n,δ x,i x,n,δ Mt = Di ψn (Xs ) dMs , Nt = Di ψn (X¯ s ) dNsx,i , δ

and

Z

Vtx,n,δ

=

t ∨δ

δ

(−2bi + a ij

δ

Dj p )(s, Xs )Di ψn (Xs ) dMsx,i p

for t ∈ [0, T ]. By Theorem 2.2 and Remark 2.5 in [20], Y n,δ ≡ ψn (X·∨δ ) − ψn (Xδ ) is an ({Ft }, P x ) - Dirichlet process on [0, T ] admitting the decomposition Ytn,δ = Mtx,n,δ + Ax,n,δ t = Mtx,n,δ + 12 (−Mtx,n,δ + N˜ tx,n,δ − Vtx,n,δ ),

t ∈ [0, T ].

By the continuous mapping theorem, sup |Ytn,δ − ϕk (Xt ∨δ ) + ϕk (Xδ )| → 0

0≤t ≤T

in P x

(2.2)

as n → ∞. On the other hand {Y n,δ }n∈N satisfies the so-called conditon UTD introduced in [2], that is { sup |Ax,n,δ |}n∈N t 0≤t ≤T

is bounded in P x

(2.3)

and x,n,δ ∀ε>0 lim sup P x (Qm ) > ε) = 0 T (A m→∞ n≥1

(2.4)

130

ANDRZEJ ROZKOSZ

(see Corollary 1 in [2] or [20]). Indeed, (2.3) follows from the fact that Z T E x hM n,x,δ iT = E x hN n,x,δ iT = E x a ij Di ψn Dj ψn (Xs ) ds 0

≤ Cδ −d/2 k∇ψn k22 , by (5.1), and that E x Var(V x,n,δ )T ≤ Cδ −d/4k∇ψn k22 ,

(2.5)

because kDj p( · , x , · )kL2 ((δ,T )×Rd ) ≤ Cδ −d/4 by Theorem 10 in [1]. As for (2.4), observe that for fixed ε > 0 x,n,δ E x Qm − N˜ x,n,δ − M x,N,δ + N˜ x,N,δ ) T (M

x,n,δ ˜ x,n,δ − N˜ x,N,δ ) ≤ 2E x Qm − M x,N,δ ) + 2E x Qm T (M T (N Z T = 4E x a ij Di (ψn − ψN )Dj (ψn − ψN )(Xs ) ds

≤ Cδ

δ −d/2

k∇(ψn − ψN )k22

for all n, m, N ∈ N. At the same time, since {ψn } ⊂ C0∞ (Rd ), x,j,δ lim P x (Qm − N˜ x,j,δ ) > ε) = 0 T (M

m→∞

for n ∈ N. Therefore x,n,δ ∀ε>0 lim sup P x (Qm − N˜ x,n,δ ) > ε) = 0, T (M m→∞ n≥1

which implies (2.4) in view of (2.5) and Lemma 1.4 in [20]. Thus {Y n,δ } satisfies UTD under P x for each x ∈ Rd . Therefore, by (2.2) and Theorem 2 in [2], sup |Mtx,n,δ − Mtk,δ | → 0 in P x

0≤t ≤T

as n → ∞. We have also Z ·∨δ  x x,i lim E Di (ψn − ϕk )(Xs ) dMs = 0. n→∞

δ

T

Accordingly, (2.1) holds for every x ∈ Rd . By the above, Doob’s inequality and (5.1), Z ·∨δ  x k,δ k+1,δ −k k x x,i P ( sup |Mt − Mt |>2 ) ≤ 4 E Di (ϕk − ϕk+1 )(Xs ) dMs 0≤t ≤T

≤ 4 Cδ k

δ −d/2

T

k∇(ϕk − ϕk+1 )k22

for x ∈ Rd . Hence, by the Borel–Cantelli lemma, P x (0 c ) = 1 for each x ∈ Rd , where 0 = {ω ∈  : {Mtk,δ (ω)} converges uniformly in t on finite intervals}. Set

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

131

Mtδ (ω) = limk→∞ Mtk,δ (ω), t ≥ 0 for ω ∈ 0. Then M δ is a CAF of (X, P x ) with defining set 0. Moreover, since Z ·∨δ  lim E x Di (ϕk − ϕ)(Xs ) dMsx,i = 0, k→∞

δ

T

it follows that x,ϕ P x (Mtδ = Mtx,ϕ ∨δ − Mδ , t ∈ [0, T ]) = 1

(2.6)

for every x ∈ Rd . Now, set δk = 8−k , k ∈ N. Since Z δk δk+1 δk x −k k P ( sup |Mt − Mt | > 2 ) ≤ 4 a ij Di ϕDj ϕ(Xs ) ds, 0≤t ≤T

δk+1

we see that P x (c0 ) = 1 for x ∈ Rd , where 0 = {ω ∈  : {Mtδk (ω)} converges uniformly in t on finite intervals}. Therefore, if we set Mtϕ (ω) = limk→∞ Mtδk (ω), T∞ t ≥ 0 for ω ∈ 0 ∩ 00 , where 00 = k=1 0k and 0k is a defining set for M δk , then M ϕ is a CAF of (X, P x ) with defining set 0 ∩ 00 (see the proof of Theorem 5.2.1 in [7]). Furthermore, by (2.6), P x (Mtϕ = Mtx,ϕ , t ∈ [0, T ]) = 1 for every x ∈ Rd . We now turn to the proof of existence of N ϕ . Since p ∈ W20,1,loc ((δ, T ) × R2d ) for every δ ∈ (0, T ) and the distributional derivative of p( · , · , · ) at (s, x, y) equals Dj p(s, x, y) for a.e. (s, x, y) ∈ (δ, T )×R2d , for each x ∈ Rd we can choose a version of Dj p( · , x, · ) such that (s, x, y) 7 → Dj p(s, x, y) is measurable. Then [0, t] 3 (s, ω) 7 → {−2bi (Xs (ω)) +a ij (Xs (ω))Dj p/p(s, X0 (ω), Xs (ω))}Di ϕ(Xs (ω)) is B([0, t]) ⊗ Gt - measurable for t ∈ (0, T ]. Accordingly, by Fubini’s theorem, V ϕ defined to be V X0 ,ϕ is {Gt } - adapted. Of course V ϕ is P x - indistinguishable from V x,ϕ , so ϕ

ϕ

ϕ

ϕ

Nt = (2Xt − Mt + Vt )˜,

t ∈ [0, T ]

is P x - indistinguishable from N x,ϕ . To prove that N ϕ is {G¯ t } - adapted, we set ˜ m = {0 = s0 < s1 < . . . < sk(m) }, where sj = T − tk(m)−j for j = 0, . . . , k(m) 5 ¯ is an ({F¯t }, P x ) - Dirichlet process and recall that by Corollary 2.3 in [20], ϕ(X) ˜ along {5m } admitting the decomposition ϕ(X¯ t ) − ϕ(X¯ 0 ) = Ntx,ϕ + Btx,ϕ ,

t ∈ [0, T ].

Hence, for every ε, R > 0, X x,ϕ P x ( sup | E x (ϕ(X¯ sj ) − ϕ(X¯ sj−1 )|F¯sj−1 ) − Bt |2 ≥ ε) 0≤t ≤T

0
= P x ( sup | 0≤t ≤T

X 0
E x (1j |F¯sj−1 ) − Bt |2 ≥ ε) x,ϕ

(2.7)

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ANDRZEJ ROZKOSZ

       X ε ≤ P x  sup | (E x (1j |F¯sj−1 ) − 1j )|2 ≥ ∩ max |1j | ≤ 2R  0≤t ≤T 0
+P (τR ≤ T ) + P x

x

max

sup

j ≥1 sj−1 ≤t
x,ϕ |Bt

−

Bsx,ϕ |2 j−1

! ε ≥ , 4

x,ϕ

where τR = inf{t ≥ 0 : |Bt | > R} and 1j = Bsx,ϕ − Bsx,ϕ . Furthermore, j j−1 ¯ ¯ ¯ if we consider the filtration {Ft } defined by Ft = Fsj for t ∈ [sj , sj +1 ), j = P 0, . . . , k(m) − 1, then t 7 → | 0 0. Since B is a continuous process of zero ˜ m }, it follows from the above and (2.7) that for each quadratic variation along {5 t ∈ [0, T ], X Ztm ≡ ϕ(X¯ t ) − ϕ(X¯ 0 ) − E x (ϕ(X¯ sj ) − ϕ(X¯ sj−1 )|F¯sj−1 ) → Ntx,ϕ ˜ m , 0
in P x for every x ∈ Rd , hence that Ztm → Ntϕ in P x for every x ∈ Rd , and ϕ finally that Ztm → Nt in P µ for every probability measure µ on B(Rd ). At the same time, by the definition, for each t ∈ [0, T ] the variables Ztm , m ∈ N are F¯t measurable, and so N ϕ is {G¯ t } - adapted, as desired. 2 The next lemma will not be needed until Section 4. LEMMA 2.3. Let ϕ ∈ C0∞ (Rd ). Define M x,ϕ , N x,ϕ , V x,ϕ , Ax,ϕ by (1.3)–(1.4). Then for every x ∈ Rd there exist the integrals Z t X x,ϕ x,ϕ ϕ(Xs ) dAx,ϕ = lim ϕ(Xtk )(Atk+1 − Atk ), t ∈ (0, T ], s m→∞

0

Z 0

t

tk ∈5m ,tk
X

Msx,ϕ dAx,ϕ = lim s

m→∞

tk ∈5m ,tk
x,ϕ Mtx,ϕ (Ax,ϕ tk+1 − Atk ), k

t ∈ (0, T ]

as limits in P x . Actually, Z t Z t Z T 1 x,ϕ x,ϕ ϕ(Xs ) dAs = − ϕ(Xs ) dMs + ϕ(X¯ s ) dNsx,ϕ + hϕ(X)it 2 0 0 T −t  Z t + ϕ(Xs ) dVsx,ϕ 0

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

and

Z

Z

t

Msx,ϕ

dAx,ϕ s

=

x,ϕ x,ϕ Mt At

t

−

0

133

x,ϕ Ax,ϕ s dMs . 0

for t ∈ (0, T ]. Finally, Z · Z x,ϕ ϕ(Xs ) dMs ,

T

T −·

0

ϕ(X¯ s ) dNsx,ϕ ,

Z

·

x,ϕ Ax,ϕ s dMs

0

are Martingales on [0, T ] under P . Proof. Follows easily from the representation (1.2)–(1.4) and well known properties of stochastic integrals with respect to semimartingales (see the proof of Theorem 3.1 in [19] and [14]). x

3. Main Theorems The following theorem generalizes the decomposition of ϕ(X) obtained in [20] (see also [19] and Remark 2.6 in [20]) to a broader class of functions and strengthens it slightly in the case d = 1. THEOREM 3.1. Let ϕ ∈ Wq1 (Rd ), q ≥ 2. Then (i) For q.e. x ∈ Rd the integrals M x,ϕ , N x,ϕ in (1.3)–(1.4) are well defined, M x,ϕ is an ({Ft }, P x ) - square-integrable martingale and N x,ϕ is an ({F¯t }, P x ) square-integrable martingale. (ii) For a.e. x ∈ Rd the integral V x,ϕ in (1.4) is well defined and V x,ϕ is a process of P x - integrable variation on [0, T ]. (iii) For a.e. x ∈ Rd ϕ(X) is an ({Ft }, P x ) - Dirichlet process on [0, T ] admitting the decomposition (1.2)–(1.4) In fact, if q > 2 then (ii), (iii) hold with “a.e" replaced by “q.e." while if d = 1 and q ≥ 2 or d > 1 and q > d then (i)–(iii) hold for every x ∈ Rd . For q > 2 ∨ d the theorem was proved in [20], so we consider only other cases. First assume that d = 1 and q = 2. By (4.1), Z TZ Z T Z |ϕ 0 (y)|2 p(s, x, y) ds dy ≤ M s −1/2 ds |ϕ 0 (y)|2 dy. 0

R

0

for every x ∈ R. Therefore M and

x,ϕ

,N

x,ϕ

are square-integrable martingales on [0, T ]

E x hM x,ϕ iT = E x hN x,ϕ iT ≤ Ckϕ 0 k22 . Furthermore, by Schwarz’s inequality, Z TZ 0

R

|aϕ 0 p 0 |(s, x, y) ds dy

2

R

(3.1)

134

ANDRZEJ ROZKOSZ

≤3

Z TZ s 1/4 R

0

a(p 0 )2 (s, x, y) ds dy p

Z TZ 0

R

s −1/4 (ϕ 0 )2 p(s, x, y) ds dy

(p 0 denotes the distributional derivative of p with respect to the space variable). Hence, by (5.1) and Lemma 5.2, E x (Var V x,ϕ)T ≤ Ckϕ 0 k2 ,

(3.2)

and (i), (ii) are proved. Now take a sequence {ϕk } ⊂ C0∞ (R) such that ϕk → ϕ in W21 (R) and uniformly on compacts in R, and define M x,ϕk , N x,ϕk , V x,ϕk by (1.3)–(1.4) with ϕ replaced by ϕk . By the definition, M x,ϕk − M x,ϕ = M x,ϕk −ϕ , N x,ϕk − N x,ϕ = N x,ϕk −ϕ , V x,ϕk − V x,ϕ = V x,ϕk −ϕ . Hence, by (3.1), lim E x hM x,ϕk − M x,ϕ iT = lim E x hN x,ϕk − N x,ϕ iT = 0,

k→∞

k→∞

(3.3)

while by (3.2), x,ϕk

lim E x sup |Vt

k→∞

0≤t ≤T

x,ϕ

− Vt

| = 0.

At the same time, by Remark 2.5 in [20], for each k ∈ N k ϕk (Xt ) = ϕk (X0 ) + Mtx,ϕk + Ax,ϕ , t

t ∈ [0, T ]

P x - a.s.

(3.4)

for every x ∈ Rd , where k Ax,ϕ = 12 (−Mtx,ϕk + N˜ tx,ϕk − Vtx,ϕk ), t

t ∈ [0, T ],

(3.5)

so applying the continuous mapping theorem shows that ϕ(X) has the representation (1.2) with Ax,ϕ given by (1.3). The proof is completed by showing that Ax,ϕ is a process of zero quadratic variation on [0, T ]. Since we know that V x,ϕ is a process of finite variation on [0, T ] for every x ∈ Rd , we are reduced to proving that hM x,ϕ − N˜ x,ϕ iT exists and equals zero P x - a.s. for x ∈ Rd . We have x,ϕ lim sup P x (Qm − N˜ x,ϕ ) ≥ ε) T (M m→∞

n   ε ε x,ϕ x,ϕk ≤ lim sup P x Qm − M x,ϕk ) ≥ − N˜ x,ϕk ) ≥ + P x Qm T (M T (M 9 9 m→∞  ε o x,ϕk x,ϕ ˜ ˜ +P x Qm ( N − N ) ≥ T 9 ≤ P x (hM x,ϕ − M x,ϕk iT ≥ ε) + P x (hN x,ϕk − N x,ϕ iT ≥ ε)

(3.6)

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

135

for ε > 0, because for each k ∈ N, x,ϕ Qm − M x,ϕk ) → hM x,ϕ − M x,ϕk iT , T (M

˜ x,ϕk − N˜ x,ϕ ) → hN x,ϕk − N x,ϕ iT Qm T (N and x,ϕk Qm − N˜ x,ϕk ) → hM x,ϕk − N˜ x,ϕk iT = 0 T (M x,ϕ in P x for x ∈ Rd . Combining (3.3) with (3.6) we see that Qm − N˜ x,ϕ ) → 0 T (M x d in P for every x ∈ R , and the proof in the case d = 1 is complete. Now assume that d > 1 and q R> 2. Let W x be a standard d - dimensional · Wiener measure starting at x. Since 0 |∇ϕ(Xs )|q ds is a PCAF of (X, W x ) with Revuz measure |∇ϕ(y)|q dy, it follows from (5.1) and Lemma 5.1.9 in [17] that Z T ν E |∇ϕ(Xs )|q ds ≤ CkU1 νk∞ k∇ϕkqq < ∞ (3.7) 0

for every ν ∈ S00 . Hence, by Theorems 2.2.3 and 4.2.1 in [7], Z T Ex |∇ϕ(Xs )|q ds < ∞

(3.8)

0

for q.e. x ∈ Rd . In particular, by Hölder’s inequality, Z T Z T x 2 x,i x E |Di ϕ(Xs )| dhM is = E |Di ϕ(X¯ s )|2 dhN x,i is < ∞ 0

(3.9)

0

for q.e. x ∈ Rd . Clearly, for each x such that (3.9) is satisfied M x,ϕ , N x,ϕ are square-integrable martingales on [0, T ]. Next, putting α = (q − 2)/(4q) and applying Hölder’s inequality and Lemma 5.2 we obtain Z T Dj p Ex |a ij Di ϕ|(s, x, Xs ) ds p 0 Z T Dj p −α x =E |s α a ij s Di ϕ|(s, x, Xs ) ds p 0 Z ≤C

T

Z s Rd

0

Z

T

×

s

−1/2

2α

a ij Di pDj p (s, x, y) ds dy 2p

2α  Z Ex ds

0

T

1/q |∇ϕ(Xs )| ds q

0

 Z ≤ C Ex 0

1/q

T

|∇ϕ(Xs )|q ds

1/2

.

136

ANDRZEJ ROZKOSZ

Also, Z

Z

T

T

|bi Di ϕ(Xs )| ds ≤ C(E x

Ex 0

|∇ϕ(Xs )|q ds)1/q ,

0

so for x for which (3.8) is satisfied V is well defined and E x (VarV x,ϕ )T < ∞. This proves (i) and (ii). To prove (iii) let us take a sequence {ϕk } ⊂ C0∞ (Rd ) such that ϕk → ϕ in Wq1 (Rd ), k∇(ϕk − ϕk+1 )kq ≤ 3−k for k ∈ N, and define M x,ϕk , N x,ϕk , V x,ϕk by (1.3)–(1.4) with ϕ replaced by ϕk . We first show that x,ϕ

sup |Mtx,ϕk − Mtx,ϕ | + sup |Ntx,ϕk − Ntx,ϕ | → 0

0≤t ≤T

0≤t ≤T

in P x

(3.10)

for q.e. x ∈ Rd . To this end, let us denote by M ϕk , N ϕk the processes of Lemma 2.2, which for each x ∈ Rd are P x - indistinguishable from M x,ϕk and N x,ϕk , respectively. Since   ϕk+1 ϕk ν −k P sup |Mt ≤ 4k E ν hM ϕk+1 − M ϕk iT − Mt | > 2 0≤t ≤T

Z =4

k

Rd

E x hM x,ϕk+1 −ϕk iT ν(dx)

2 ≤ 4k C(ν(Rd ))(q−2)/q kU1 νk2/q ∞ k∇(ϕk+1 − ϕk )kq

and

 P

ν

sup

0≤t ≤T

ϕ |Nt k+1

−

−k

ϕ Nt k |



>2

2 ≤ 4k C(ν(Rd ))(q−2)/q kU1 νk2/q ∞ k∇(ϕk+1 − ϕk )kq

for every ν ∈ S00 , arguing as in the proof of Theorem 5.2.1 in [7] gives existence of a MAF M ϕ of (X, P x ) and a continuous {G¯ t } - adapted process N ϕ such that N˜ ϕ is {Gt } - adapted, N ϕ is a ({G¯ t }, P x ) - square-integrable Martingale on [0, T ] for q.e. x ∈ Rd and ϕ

ϕ

ϕ

sup |Mt k − Mt | → 0,

ϕ

sup |Nt k − Nt | → 0 P x - a.s.

0≤t ≤T

0≤t ≤T

(3.11)

for q.e. x ∈ Rd . In fact, ϕ

x,ϕ

x,ϕ

P x (Mt = Mt , N ϕ = Nt

, ∈ [0, T ]) = 1

(3.12)

for q.e. x ∈ Rd . To see this, let us put Lx = M ϕ − M x,ϕ . Then Lx is an ({Gt }, P x ) Martingale for q.e. x ∈ Rd . By Schwarz’s inequality, for q.e. x ∈ Rd ,   ε ε P x (hLx iT > ε) ≤ P x hM ϕ − M ϕk iT > + P x hM x,ϕk − M x,ϕ iT > 2 2

137

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

for ε > 0, k ∈ N. From (3.4) and Corollary VI.6.6 in [8] we see that hM ϕ − M ϕk iT → 0 in P x for q.e. x ∈ Rd . Since Z k ≡ hM x,ϕk − M x,ϕ iT does not depend on x and for every ν ∈ S00 , P ν (|Z k+1 − Z k | > 2−k ) Z ≤ 2 3E k

T

ν

|∇(ϕk+1 − ϕk )| · |∇(ϕk+1 + ϕk )|(Xs ) ds

0

≤ 2 C(ν(R )) k

d

(q−2)/q

 Z Eν

T

|∇(ϕk+1 − ϕk )|

q/2

|∇(ϕk+1 + ϕk )|

q/2

2/q (Xs ) ds

0

≤ 2k 3−k C(ν(Rd ))(q−2)/q kU1 νk2/q ∞ k∇(ϕk+1 + ϕk )kq for all k ∈ N, it follows from the Borel-Cantelli lemma and the inequalities Z T ν k ν E Z ≤ dE |Di (ϕk − ϕ)(Xs )|2 dhM x,i is 0 2 ≤ C(ν(Rd ))(q−2)/q kU1 νk2/q ∞ k∇(ϕk − ϕ)kq

that P ν ({ω ∈  : {Z k (ω)} converges to zero}c ) = 0 for ν ∈ S00 . In particular, by Theorem 2.2.3 in [7], Z k → 0 in P x for q.e. x ∈ Rd . Accordingly, P x (hLx iT = 0) = 1 for q.e. x ∈ Rd , and consequently P x (Mtϕ = Mtx,ϕ , t ∈ [0, T ]) = 1. Since similar considerations apply to N ϕ and N x,ϕ , we obtain (3.5), and hence (3.3), by (3.4). We next show that sup |Vtx,ϕk − Vtx,ϕ | → 0

0≤t ≤T

in P x

(3.13)

RT for q.e. x ∈ Rd . For this purpose set R k = ( 0 |∇(ϕk −ϕ)(Xs )|q ds)1/q and observe that by Hölder’s inequality and Lemma 5.2, for every ε, K > 0   x,ϕk x,ϕ x P sup |Vt − Vt | > ε 0≤t ≤T

Z ≤P

T

x

ε |2b Di (ϕk − ϕ)(Xs )| ds > 2



i

0

Z

T

Dj p ε +P |a Di (ϕk − ϕ)|(s, x, Xs ) ds > p 2 0     ε ε ≤ P x R k > (ν(Rd )(1−q)/q + P x R k > C CK Z T  a ij Di pDj p 2 +P x . s 2α (s, x, X ) ds > K s 2p 2 0 x

ij



138

ANDRZEJ ROZKOSZ

By Lemma 5.2 the third summand on the right-hand side of the last inequality is less or equal to C/K 2 . Furthermore, for every ν ∈ S00 we have P ν (|R k+1 − R k | > 2−k ) ≤ 2 (ν(R )) k

d

(q−1)/q

 Z Eν

T

1/q |∇(ϕk+1 − ϕk )(Xs )| ds q

0

≤ 2k C(ν(Rd ))(q−1)/q kU1 νk1/q ∞ k∇(ϕk+1 − ϕk )kq and E ν R k ≤ C(ν(Rd ))(q−1)/q kU1 νk1/q ∞ k∇(ϕk − ϕ)kq for all k ∈ N. Therefore, by the Borel-Cantelli lemma, P ν ({ω ∈  : {R k (ω)} converges to zero}c ) = 0 for ν ∈ S00 , so applying once again Theorem 2.2.3 in [7] we see that R k → 0 in P x for q.e. x ∈ Rd . From the above (3.6) easily follows. Combining (3.10) and (3.13) with (3.4)–(3.5) we conclude that {ϕk (X)} converges in C([0, T ]; R) in P x for q.e. x ∈ Rd . Moreover, by Theorem 4.2.2 in [7], ϕ(X) is a continuous process on [0, T ] under P x for q.e. x ∈ Rd , while by the dominated convergence theorem, the finite dimensional distributions of ϕk (X) under P x converge weakly to those of ϕ(X). Therefore in fact (ϕk (X), M x,ϕk , N x,ϕk , V x,ϕk ) → (ϕ(X), M x,ϕ , N x,ϕ , V x,ϕ )

in P x

(3.14)

in C([0, T ]; R4 ) for q.e. x ∈ Rd . Hence, by the continuous mapping theorem, ϕ(X) admits the decomposition (1.2)–(1.4). To prove that Ax,ϕ is a process of zero quadratic variation on [0, T ] for q.e. x ∈ Rd we proceed as in the case d = 1. Since we know already that E x (VarV x,ϕ )T < ∞ for q.e. x ∈ Rd , it suffices to show that hM x,ϕ − N˜ x,ϕ iT exists and equals zero P x - a.s. for q.e. x ∈ Rd . To this end, observe that (3.14) holds for q.e. x ∈ Rd and that from (3.3) and Corollary VI.6.6 in [8] it follows that hM x,ϕ − M x,ϕk iT + hN x,ϕk − N x,ϕ iT → 0 in P x x,ϕ for q.e. x ∈ Rd . Thus Qm − N˜ x,ϕ ) → 0 in P x for q.e. x ∈ Rd , as required. T (M Finally, assume that d > 1 and q = 2. Since (3.0) holds for q = 2 as well, (3.1) is satisfied for q.e. x ∈ Rd . In fact, all the estimates concerning M x,ϕ , N x,ϕ made above hold for q = 2, as is easy to check. Therefore, M x,ϕ , N x,ϕ are squareintegrable martingales on [0, T ] such that the quadratic variation of M x,ϕ − N˜ x,ϕ along {5m } exists and equals zero for q.e. x ∈ Rd . Moreover, by (5.1), Lemma 5.2 and Schwarz’s inequality, for every finite absolutely continuous measure ν on B(Rd ) with bounded density, Z t Dj p 1/2 Eν |(2bi + a ij k∇ϕk2 , )Di ϕ|(s, x, Xs ) ds ≤ C(ν(Rd )1/2 kf k∞ p 0

139

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

where f is the density of ν. This forces E x (VarV x,ϕ )T < ∞ for a.e. x ∈ Rd , and (i), (ii) are proved. The proof of (iii) runs as before. 2 REMARK 3.2. Let d > 1. At the end of the proof of Theorem 3.1 we pointed out that if ϕ ∈ W21 (Rd ) then M x,ϕ − N˜ x,ϕ is a zero quadratic variation process on [0, T ] for q.e. x ∈ Rd . Moreover, we have x,ϕ

ϕ(Xt ) = ϕ(X0 ) + 12 (Mt

x,ϕ x,ϕ + N˜ t − Vt ),

t ∈ [0, T ] P x - a.s.

(3.15)

for q.e. x ∈ Rd , where Vtx,ϕ = lim Vtx,ϕ,δ , δ&0

Z x,ϕ,δ Vt

t ∨δ

= δ



 Dj p −2b (Xs ) + a (Xs ) (s, x, Xs ) ds p i

ij

(3.16)

with convergence in P x . Unfortunately, for ϕ ∈ W21 (Rd ) we do not know whether V x,ϕ has finite variation on [0, T ] for q.e. x ∈ Rd . Therefore we do not know whether Ax,ϕ defined by x,ϕ

At

x,ϕ

= 12 (−Mt

+ N˜ t

x,ϕ

x,ϕ

− Vt

t ∈ [0, T ]

),

(3.17)

is a zero quadratic variation process, and, in consequence, that ϕ(X) is a Dirichlet process under P x for q.e. x ∈ Rd . To prove (3.15), take the sequence {ϕk } considered at the end of the proof of Theorem 3.1. Then for arbitrary but fixed δ ∈ (0, T ) and all t ∈ [δ, T ] x,ϕk

ϕk (Xt ) = ϕk (Xδ ) + 12 (Mt

x,ϕk

− Mδ

x,ϕ x,ϕ x,ϕ x,ϕ + N˜ t k − N˜ δ k − Vt k + Vδ k )

P x - a.s. for x ∈ Rd . By Theorem 10 in [1], p( · , x, · ) ∈ W20,1 ((δ, T ) × Rd ), so x,ϕ x,ϕ x,ϕ,δ V x,ϕ,δ is well defined and E x supδ≤t ≤T |Vt k − Vδ k − Vt | → 0 as k → ∞ for x ∈ Rd . As in the proof of (3.9) one can show now that (ϕk (X), M x,ϕk , N x,ϕk , V x,ϕk − Vδx,ϕk ) → (ϕ(X), M x,ϕ , N x,ϕ , V x,ϕ,δ )

in P x

in C([δ, T ]; R4 ) for q.e. x ∈ Rd . Therefore, by the continuous mapping theorem, x,ϕ ˜ x,ϕ − Vtx,ϕ,δ ), ϕ(Xt ∨δ ) = ϕ(Xδ ) + 12 (Mtx,ϕ + N˜ tx,ϕ ∨δ − Mδ ∨δ − Nδ

P x - a.s. for q.e. x ∈ Rd . Letting δ & 0 yields (3.10).

t ∈ [0, T ] 2

The next theorem strengthens Lemma 2.2. THEOREM 3.3. Let ϕ ∈ Wq1 (Rd ) for some q ≥ 2. Then there exist a MAF M ϕ of (X, P x ) and a continuous {G¯ t } - adapted process N ϕ on [0, T ] such that N˜ ϕ is {Gt } - adapted and (i), (ii) of Lemma 2.2 hold for q.e. x ∈ Rd . Moreover, if d = 1

140

ANDRZEJ ROZKOSZ

or d > 1 and q > d then M ϕ , N ϕ can be choosen so that M ϕ is a MAF in the strict sense and (i), (ii) of Lemma 2.2 hold for every x ∈ Rd . Proof. The first part follows immediately from the proof of Theorem 3.1, so it remains to consider the case d = 1, q ≥ 2 and d > 1, q > d. In both cases, by (5.1), M x,ϕ defined by (1.3) is an ({Ft }, P x ) - square-integrable martingale and N x,ϕ defined by (1.4) is an ({F¯t }, P x ) - square-integrable martingale on [0, T ] for every x ∈ Rd . Let us choose {ϕk } ⊂ C0∞ (Rd ) such that ϕk → ϕ in Wq1 (Rd ), k∇(ϕk − ϕk+1 )kq ≤ 3−k for k ∈ N. Define M x,ϕk , N x,ϕk by (1.3)–(1.4) with ϕ replaced by ϕk . Since E x hM x,ϕ − M x,ϕk iT = E x hN x,ϕ − N x,ϕk iT Z T x = E a ij Di (ϕ − ϕk )Dj (ϕ − ϕk )(Xs ) ds 0

≤ Ck∇(ϕ − ϕk )k2q , (3.10) holds for every x ∈ Rd . Let us denote by M ϕk , N ϕk the processes of Lemma 2.2 such that ϕ

x,ϕk

P x (Mt k = Mt

ϕ

x,ϕk

, Nt k = Nt

, t ∈ [0, T ]) = 1

By Doob’s inequality and (5.1), P x ( sup |Mt k+1 − Mt k | > 2−k ) ≤ 4k E x hM ϕk+1 − M ϕk iT ϕ

ϕ

0≤t ≤T

= 4k E x hM x,ϕk+1 −ϕk iT ≤ 4k Ck∇(ϕk+1 − ϕk )k2q and likewise P x ( sup |Nt k+1 − Ntϕk | > 2−k ) ≤ 4k Ck∇(ϕk+1 − ϕk )k2q . ϕ

0≤t ≤T

T ϕk Let 0 = ∞ k=1 0k , where 0k is a defining set for M . From the estimates above ϕ and the Borel-Cantelli lemma it follows that if we set 1 = {ω ∈  : {Mt k (ω)} ϕ ϕk converges uniformly in t on finite intervals} and Mt (ω) = limk→∞ Mt (ω), t ≥ 0, for ω ∈ 0 ∩ 1 , then M ϕ is a MAF in the strict sense of (X, P x ) with defining ϕ set 0 ∩ 1 . Moreover, if we set 2 = {ω ∈  : {Nt k (ω)} converges uniformly ϕ ϕk in t on finite intervals} and Nt (ω) = limk→∞ Nt (ω), t ≥ 0, for ω ∈ 2 then N ϕ a ({G¯ t }, P x ) - square-integrable martingale on [0, T ] for each x ∈ Rd and N˜ ϕ is {Gt } - adapted which completes the proof. 2 We are ready to prove that the processes appearing in the decomposition (1.2)– (1.4) can be choosen independently of the starting point x.

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

141

THEOREM 3.4. Let ϕ ∈ Wq1 (Rd ), q ≥ 2. Then there exist continuous processes M ϕ , N ϕ , V ϕ on [0, T ] such that (i) M ϕ is a MAF of (X, P x ), N ϕ is a ({G¯ t }, P x ) - square-integrable martingale for q.e. x ∈ Rd , N˜ ϕ is {Gt } - adapted and Aϕ = 12 (−M ϕ + N˜ ϕ − V ϕ ) is a CAF of (X, P x ). (ii) For a.e. x ∈ Rd ϕ(X) is a ({Gt }, P x ) - Dirichlet process on [0, T ] admitting the decomposition ϕ(Xt ) = ϕ(X0 ) + Mtϕ + Aϕt , t ∈ [0, T ]. (iii) P x (Mtϕ = Mtx,ϕ , Ntϕ = Ntx,ϕ , t ∈ [0, T ]) = 1 for q.e. x ∈ Rd where M x,ϕ , N x,ϕ are defined by (1.3)–(1.4). (iv) P x (Vtϕ = Vtx,ϕ , t ∈ [0, T ]) = 1 for a.e. x ∈ Rd if V x,ϕ is defined by (1.4) and for q.e. x ∈ Rd if V x,ϕ is defined by (3.16). In fact, if q > 2 then (ii)–(iv) hold with “a.e” replaced by “q.e.” Moreover, if d = 1 or d > 1 and q > d then M ϕ , N ϕ , V ϕ can be choosen so that M ϕ is a MAF in the strict sense, N ϕ is a ({G¯ t }, P x ) - square-integrable martingale for every x ∈ Rd , Aϕ is a CAF in the strict sense and (ii)–(iv) hold for every x ∈ Rd . Proof. Consider the processes M ϕ , N ϕ of Theorem 3.3 and set Vtϕ = −2ϕ(Xt ) + 2ϕ(X0 ) + Mtϕ − N˜ tϕ ,

t ∈ [0, T ].

(3.18)

Then the theorem follows immediately from Theorem 3.1, Remark 3.2 and properties of M ϕ , N ϕ . COROLLARY 3.5. Let ϕ ∈ Wq1 (Rd ), q ≥ 2. Suppose ϕ ϕ ϕ(Xt ) = ϕ(X0 ) + Mˆ t + Aˆ t ,

t ∈ [0, T ] P x - a.s.

(3.19)

for q.e. x ∈ Rd , where Mˆ ϕ is a MAF of (X, P x ) and Aˆ ϕ is a CAF of (X, P x ) of zero quadratic variation. If q > 2 then P x (Mˆ tϕ = Mtx,ϕ , Aˆ ϕt = Ax,ϕ t , t ∈ [0, T ]) = 1

(3.20)

for q.e. x ∈ Rd , where M x,ϕ , Ax,ϕ are defined by (1.3)–(1.4). Moreover, if d = 1 and q ≥ 2 or d > 1 and q > d, Mˆ ϕ is a MAF in the strict sense, Aˆ ϕ is a CAF of zero quadratic variation in the strict sense and (3.16) holds for every x ∈ Rd then (3.17) holds for every x ∈ Rd , too. Proof. Let M ϕ , Aϕ be processes defined in Theorem 3.4. Then P x (Mtϕ = Mtx,ϕ , Aϕt = Ax,ϕ t , t ∈ [0, T ]) = 1

(3.21)

for q.e. x ∈ Rd if q > 2 and for every x ∈ Rd if d = 1 and q ≥ 2 or d > 1 and q > d. Therefore the corollary is a consequence of uniqueness of the decomposition of a Dirichlet process into a Martingale and a process of zero quadratic variation. 2 REMARK 3.6. If d > 1, ϕ ∈ W21 (Rd ) and Ax,ϕ is defined by (3.12) then (3.18) still holds for q.e. x ∈ Rd , however from Theorem 3.4 it follows only that (3.17) is satisfied for a.e. x ∈ Rd . 2

142

ANDRZEJ ROZKOSZ

4. Symmetric Case In this section we will show that for diffusions corresponding to the operator La,0 the CAF Aϕ of Theorem 3.4 is of zero energy. As a result we obtain Fukushima’s [5, 6, 7] decomposition of ϕ(X) (see also [9, 22] for the non-symmetric case and [16, 17], where time-inhomogeneous diffusions are considered) and at the same time we characterize its zero energy part. THEOREM 4.1. Assume b = 0 and let ϕ ∈ W21 (Rd ). Then there exist continuous processes M ϕ , Aϕ on [0, T ] such that (i) M ϕ is a MAF of (X, P x ) of finite energy and Aϕ is a CAF of (X, P x ) of zero energy. ϕ ϕ (ii) ϕ(Xt ) = ϕ(X0 ) + Mt + At , t ∈ [0, T ] P x - a.s. for every x ∈ Rd . Moreover, if d = 1 or d > 1 and ϕ ∈ W21 (Rd ) ∩ Wq1 (Rd ) with q > d then Aϕ , M ϕ can be choosen so that M ϕ is a MAF of finite energy in the strict sense and Aϕ is a CAF of zero energy. Proof. Let M ϕ , N ϕ , V ϕ be processes considered in the proof of Theorem 3.4 and let Aϕt = ϕ(Xt ) − ϕ(X0 ) − Mtϕ = 12 (−Mtϕ + N˜ tϕ − Vtϕ ),

t ∈ [0, T ].

Since 1 1 e(M ϕ ) = lim E λ (Mtϕ )2 = lim E λ hM ϕ it = ka ij Di ϕDj ϕk1 < ∞, t &0 t t &0 t

(4.1)

to complete the proof it sufficies to show that e(Aϕ ) = 0 if ϕ ∈ W21 (Rd ). To this end, we first assume additionally that ϕ ∈ C0∞ (Rd ). Then from (iii)-(iv) of R Theorem 3.4 and Lemma 2.3 it follows that for every x ∈ Rd there exists Aϕs dAϕs as limit in P x of Riemann sums and Z t (Aϕt )2 = 2 (Aϕt − Aϕs ) dAϕs 0

Z

t

=2

Z (ϕ(Xt ) −

ϕ(Xs )) dAϕs

0

t

−2

(Mtϕ − Msϕ ) dAϕs

0

= ϕ(Xt )(−Mt + N˜ t ) − ϕ

Z

ϕ

t

(ϕ(Xt ) − ϕ(Xs )) dVsϕ 0

Z

Z

t

+

ϕ(Xs ) dMsϕ

+

0

T T −t

ϕ(X¯ s ) dNsϕ + hϕ(X)it − 2

for all t ∈ (0, T ]. Observe also that E x ϕ(Xt )N˜ tϕ = E x E x (ϕ(Xt )(NTϕ−t − NTϕ )|F¯T −t ) = 0,

Z

t

Aϕs dMsϕ 0

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

143

since ϕ(Xt ) is F¯T −t - measurable, and that E x {−ϕ(Xt )Mtϕ + hϕ(X)it } = E x {−ϕ(X0)Mtϕ − (Mtϕ )2 − Aϕt Mtϕ + hM ϕ it } = −E x Aϕt Mtϕ . Therefore E

x

Z

(Aϕt )2

= −E

x

Aϕt Mtϕ

−E

t

x

(ϕ(Xt ) − ϕ(Xs )) dVsϕ

0

for every x ∈ Rd . Let {P t } be the semigroup of operators on L2 (Rd ) corresponding to La,0 . By the Markov property, Z t Ex (ϕ(Xt ) − ϕ(Xs )) dVsϕ 0 Z t a ij Dj p = Ex { (s, x, Xs )Di ϕ(Xs )E Xs (ϕ(Xt −s ) − ϕ(X0 ))} ds p 0 Z tZ = a ij Dj p(s, x, y)Di ϕ(y)(P t −s ϕ(y) − ϕ(y)) ds dy. Rd

0

Hence

Z

E since

t

(ϕ(Xt ) − ϕ(Xs )) dVsϕ = 0,

λ 0

R Rd

Dj p(s, x, y) dx = 1. From what has already been proved it follows that ϕ

ϕ

ϕ

E λ ((At )2 + At Mt ) = 0.

(4.2)

On the other hand,

Z 1 1 lim E λ (ϕ(Xt ) − ϕ(X0 ))2 = lim ϕ(x)(ϕ(x) − P t ϕ(x)) dx t &0 t t &0 t Rd = ka ij Di ϕDj ϕk1 ,

which when combined with (4.1) gives 1 1 ϕ ϕ ϕ ϕ lim E λ {(At )2 + 2At Mt } = lim E λ {(ϕ(Xt ) − ϕ(X0 ))2 − (Mt )2 } = 0. t &0 t t &0 t From this and (4.2) we have e(Aϕ ) = 0. To prove the general case, take a sequence {ϕk } ⊂ C0∞ (Rd ) such that ϕk → ϕ in W21 (Rd ). Since Aϕ = ϕ(X) − ϕk (X) − M ϕ + M ϕk + Aϕk , it follows that e(Aϕ ) ≤ 3e(ϕ(X) − ϕk (X)) + 3e(M ϕ−ϕk ) = 6ka ij Di (ϕ − ϕk )Dj (ϕ − ϕk )k1

144

ANDRZEJ ROZKOSZ

for each k ∈ N, hence that e(Aϕ ) = 0, as required.

2

From the preceding results and uniqueness of Fukushima’s decomposition we obtain immediately the following refinement of the Lyons-Zheng [14] decomposition of ϕ(X) (see also remarks at the end of [14] and [7, 11, 12, 13, 22] for a corresponding decomposition under P λ ). COROLLARY 4.2. Let ϕ ∈ W21 (Rd ) ∩ Wq1 (Rd ), q ≥ 2. Suppose ϕ(Xt ) = ϕ(X0 ) + Mˆ tϕ + Aˆ ϕt ,

t ∈ [0, T ] P x - a.s.

(4.3)

for q.e. x ∈ Rd , where Mˆ ϕ is a MAF of (X, P x ) of finite energy and Aˆ ϕ is a CAF of (X, P x ) of zero energy. Then P x (Mˆ tϕ = Mtx,ϕ , Aˆ ϕt = Ax,ϕ t , t ∈ [0, T ]) = 1

(4.4)

for q.e. x ∈ Rd , where M x,ϕ is defined by (1.3), Ax,ϕ is defined by (3.012) if q = 2 and by (1.3) if q > 2. Moreover, if d = 1 or d > 1 and q > d, Mˆ ϕ is a MAF of finite energy in the strict sense and (4.4) holds for every x ∈ Rd then (4.4) holds for every x ∈ Rd , too. Proof. Follows from (3.21) and uniqueness of the decomposition of an AF into a MAF of finite energy (or MAF in the strict sense of finite energy) and a CAF of zero energy (see Section 5.2 in [7]). 2 REMARK 4.3. In case ϕ ∈ W21 (Rd ), Theorem 4.1 is nothing but well known Fukushima’s decomposition of ϕ(X) (see Example 4.5.2 and Theorem 5.2.2 in [7]). Assume now that d = 1 or d > 1 and ϕ ∈ W21 (Rd ) ∩ Wq1 (Rd ) with q > d. Let µ denote the energy measure of ϕ, that is dµ = (a ij Di ϕDj ϕ)(x)dx. Then, by (5.1), for every x ∈ Rd Z ∞ x R1 µ(x) = E e−t a(Xt )(ϕ 0 (Xt ))2 dt 0 Z ∞ Z −1/2 −t ≤ M3 t e dt |ϕ 0 (y)|2 dy ≤ if d = 1 and

Z

0 0 2 Ckϕ k2

∞

R1 µ(x) = E x

R

e−t (a ij Di ϕDj ϕ)(Xt )dt

0

Z ≤ 0

∞ Z

Rd

|e−t /2 (a ij Di ϕDj ϕ)(y)|q/2 dy

!(q−d)/(2q)

4/(q−d) dt

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

Z ×

∞

Z Rd

0

Z ≤ Ck∇ϕk2q

∞

|e

−t /2

2(q−2)/(q+d) q/(q−2)

p(t, x, y)|

dy

145

!(q+d)/(2q) dt

e−t /2 t −2d/(q+d) dt

0

≤ Ck∇ϕk2q if d > 1. Accordingly, in both cases µ is a smooth measure in the strict sense and Gk = {x ∈ Rd : |x| < k}, k ∈ N is an associated exhaustive sequence of open sets. Moreover, as above we check that in both cases Z t x E (a ij Di ϕDj ϕ)(Xs ) ds < ∞ 0

for every x ∈ Rd and t ≥ 0. Therefore the second part of Theorem 4.1 follows from Theorem 1 in [6] providing necessary and sufficient conditions for the strict decomposition of additive functionals of symmetric Markov processes into martingale and a zero energy parts. Our proof is different and applies only to processes associated with La,0 . It gives, however, a probabilistic representation of Aϕ . Let us note also that the CAF Aϕ appearing in Theorem 4.1 is not of bounded variation unless ϕ is a potential of a signed smooth measure in the strict sense (see [5] and Theorems 5.2.5, 5.4.2 and 5.5.5 in [7] for details). 5. Appendix Throughout the section, p denotes the fundamental solution of La,b with coefficients satisfying (1.1). THEOREM 5.1. There exists an M > 0 depending only on λ, 3, d and T such that M −1 t −d/2 exp(−M|x − y|2 /t) ≤ p(t, x, y) ≤ Mt −d/2 exp(−M −1 |x − y|2 /t) (5.1) for all (t, x, y) ∈ (0, T ] × Rd × Rd . Proof. See [1, 21]. LEMMA 5.2. For every α > 0 there exists C > 0 depending only on λ, 3, d, T and α such that Z TZ Di pDj p s α a ij (y) (s, x, y) ds dy ≤ C d 2p 0 R for all x ∈ Rd .

146

ANDRZEJ ROZKOSZ

Proof. First we assume additionally that a, b have bounded derivatives of all orders. Following [15], for fixed x ∈ Rd we define the entropy by Z Qx (s) = − p(s, y) ln p(s, y) dy, s ∈ (0, T ], Rd

where for simplicity of notation we write p(s, y) instead of p(s, x, y). Since ∂p (s, y) = { 12 Di (a ij Dj p) − Di (bp)}(s, y), ∂s for (s, y) ∈ (0, T ] × Rd we have Z dQx ∂p (1 + ln p) (s, y) dy (s) = − d ds ∂s ZR = − (1 + ln p){ 12 Di (a ij Dj p) − Di (bp)}(s, y) dy Rd Z a ij Di pDj p = ( − bi Di p)(s, y) dy. d 2p R Therefore, Z TZ Rd

0

Z =

a ij Di pDj p (s, y) ds dy 2p Z TZ α dQx s s α bi Di p(s, y) ds dy (s) ds + ds 0 Rd

sα T

0

Z ≤

T

sα 0

dQx (s) ds + C, ds

by Theorem 5 in [1]. Furthermore, integrating by parts and using (5.1) we obtain Z T dQx sα (s) ds − T α Qx (T ) ds 0 Z T = −α s α−1 Qx (s) ds 0

  |x − y|2 s −d/2 exp − Ms Rd 0   d ln s |x − y|2 × ln M − − dy 2 Ms Z T ≤C s α−1 (1 + | ln s|) ds, Z

≤ αM

0

Z

T

s α−1 ds

ON DIRICHLET PROCESSES ASSOCIATED WITH DIVERGENCE FORM OPERATORS

147

which proves the lemma in case of smooth coefficients. In the general case, for ij n = 1, 2, . . . choose smooth an , bn satisfying (1.1) so that an → a ij , bni → bi a.e. for i, j = 1, . . . , d and denote by pn ( · , · , · ) the fundamental solution of the operator Lan ,bn . Then for arbitrary x ∈ Rd and δ ∈ (0, T ), un = pn ( · , x, · ) is the classical solution to the Cauchy problem   ∂un = 1 D (a ij D u ) − D (b u ) for (s, y) ∈ (δ, T ] × Rd j n i n n 2 i n ∂s  limun (s, y) = pn (δ, x, y) for y ∈ Rd . s&δ

It is known that pn ( · , x, · ) → p( · , x, · ) uniformly on compacts in (0, T ]×Rd (see [1, 21]). Hence pn (δ, x, · ) → p(δ, x, · ) pointwise in Rd , and consequently, by (5.1) and the dominated convergence theorem, in L2 (Rd ). Therefore analysis similar to that in the proof of Theorem III.4.5 in [10] shows that Di un → Di u in ij L1 ((δ, T ) × Rd ) for i = 1, . . . , d. From the above it follows that s α pn−1 an Di pn Dj pn → s α p −1 a ij Di pDj p in Lebesgue measure on (δ, T )×Rd for every δ ∈ (0, T ), so applying Fatou’s lemma gives Z TZ a ij Di pDj p sα (s, y) ds dy p 0 Rd Z TZ ij α an Di pn Dj pn ≤ lim lim inf s (s, y) ds dy, δ&0 n→∞ pn Rd δ and the proof is complete.

2

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