ON DIFFERENTIABILITY OF STOCHASTIC FLOW FOR A

ON DIFFERENTIABILITY OF STOCHASTIC FLOW FOR A MULTIDIMENSIONAL SDE WITH DISCONTINUOUS DRIFT

arXiv:1306.4816v2 [math.PR] 9 Apr 2014

OLGA V. ARYASOVA AND ANDREY YU. PILIPENKO Abstract. We consider a d-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data.

Introduction Consider an SDE of the form ( dϕt (x) = a(ϕt (x))dt + dwt , (1) ϕ0 (x) = x, where x ∈ Rd , d ≥ 1, (wt )t≥0 is a d-dimensional Wiener process, a = (a1 , . . . , ad ) is a bounded measurable mapping from Rd to Rd . According to [23] there exists a unique strong solution to equation (1). It is well known that if a is continuously differentiable and its derivative is bounded, then equation (1) generates a flow of diffeomorphisms. It turns out that this condition can be essentially reduced [12], and a flow of diffeomorphisms exists in the case of possible unbounded H¨ older continuous drift vector a. Recently the case of discontinuous drift was studied in [10, 11, 19, 20] and the weak differentiability of the solution to (1) was proved under rather weak assumptions on the drift. The authors of [10] consider a drift vector belonging to Lq (0, T ; Lp(Rd )) for some p, q ∈ Rd such that p ≥ 2, q > 2,

d 2 + < 1. p q

They establish the existence of the Gˆ ateaux derivative in L2 (Ω × [0, T ]; Rd). In [20] it is proved that for a bounded measurable drift vector a the solution belongs to the space L2 (Ω; W 1,p (U )) for each t ∈ Rd , p > 1, and any open and bounded U ∈ Rd . The Malliavin calculus is used in [19, 20]. The aim of our paper is to find a natural representation of the derivative ∇x ϕt (x) if a is discontinuous. We suppose that for 1 ≤ i ≤ d, ai is a function of bounded variation ∂ai is a signed measure on on Rd , i.e. for each 1 ≤ j ≤ d, a generalized derivative µij = ∂x j Rd . Let µij,+ , µij,− be measures from Hahn-Jordan decomposition µij = µij,+ − µij,− . Denote |µij | = µij,+ + µij,− . Assume that for all 1 ≤ i, j ≤ d, |µij | satisfies the following condition (see Condition (A) in Section 1)    Z Z t ky − xk2 1 ds |µij |(dy) = 0. exp − lim sup d/2 t↓0 x∈Rd Rd 2s 0 (2πs) The condition we impose on the drift is more restrictive then that of [10, 20], but it allows us to obtain an explicit representation for the derivative in terms of intrinsic parameters of the initial equation (see Theorem 3). Our methods are different from the ones used 2000 Mathematics Subject Classification. 60J65, 60H10. Key words and phrases. Stochastic flow; Continuous additive functional; Differentiability with respect to initial data.

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OLGA V. ARYASOVA AND ANDREY YU. PILIPENKO

in the papers cited above. We show that the derivative Yt (x) in x is a solution of the following integral equation Z t Yt (x) = E + dAs (ϕ(x))Ys (x), 0

where At (ϕ(x)) is a continuous additive functional of the process (ϕt (x))t≥0 which is equal Rt to 0 ∇a(ϕs (x))ds if a is differentiable, E is a d-dimensional identity matrix. This representation is a natural generalization of the expression for the derivative in the smooth case. In the one-dimensional case (see [3, 4]) the derivative was represented via the local time of the process. It is well known that the solution of (1) does not have a local time at a point in multidimensional situation. We use continuous additive functionals for the representation of the derivative. This method can be considered as a generalization of the local time approach to the multidimensional case. Our method is likely to can be used in the case of non-constant diffusion. The paper is organized as follows. In Section 1 we collect some definitions and statements concerning continuous additive functionals. The main result of the paper is formulated in Section 2 (see Theorem 3). For the proof we approximate equation (1) by equations with smooth coefficients. The definitions and properties of approximating equations are given in Sections 3, 4. We prove Theorem 3 is Section 5. 1. Preliminaries: W-functionals

In this section we collect some facts about continuous additive functionals which will be used in the sequel. Further information can be found in [8], Ch. 6–8; [13], Ch. II, §6; see also very detailed exposition on continuous additive functionals in [6]. Let (Ω, F , P ) be a probability space with filtration {Ft : t ≥ 0}. Let (ξt )t≥0 be a continuous Rd -valued homogeneous Markov process with infinite lifetime adapted to the filtration Ft . Denote Nt = σ{ξ(s) : 0 ≤ s ≤ t}. Definition 1. A random function At , t ≥ 0, adapted to the filtration {Nt } is called a continuous additive functional of the process (ξt )t≥0 if it is • non-negative; • continuous in t; • homogeneous additive, i.e. for all t ≥ 0, s > 0, x ∈ Rd , (2)

At+s = As + θs At Px − almost surely,

where θ is a shift operator. If additionally for each t ≥ 0, sup Ex At < ∞, x∈Rd

then At , t ≥ 0, is called a W-functional. Remark 1. It follows from Definition 1 that a W-functional is non-decreasing in t, and for all x ∈ Rd Px {A0 = 0} = 1. Definition 2. The function ft (x) = Ex At is called a characteristic of W -functional At . Proposition 1 (See [8], Theorem 6.3). A W-functional is defined by its characteristic uniquely up to equivalence. The following theorem states the connection between convergence of functionals and convergence of their characteristics.

ON DIFFERENTIABILITY OF STOCHASTIC FLOW

3

Theorem 1 (See [8], Theorem 6.4). Let An,t , n ≥ 1, be W-functionals of the process (ξt )t≥0 and fn,t (x) = Ex An,t be their characteristics. Suppose that for each t > 0, a function ft (x) satisfies the condition (3)

lim sup sup |fn,u (x) − fu (x)| = 0.

n→∞ 0≤u≤t x∈Rd

Then ft (x) is the characteristic of a W-functional At . Moreover, At = l.i.m. An,t , n→∞

where l.i.m. denotes the convergence in mean square (for any initial distribution ξ0 ). Proposition 2 (See [8], Lemma 6.1′ ). If for any t ≥ 0 a sequence of non-negative additive functionals {An,t : n ≥ 1} of the Markov process (ξt )t≥0 converge in probability to a continuous functional At , then the convergence in probability is uniform, i.e. ∀ T > 0 sup |An,t − At | → 0, n → ∞, in probability. t∈[0,T ]

Let h be a non-negative continuous bounded function on Rd , let the process (ξt )t≥0 has a transition probability density pt (x, y). Then Z t At := h(ξs )ds 0

is a W -functional of the process (ξt )t≥0 and its characteristic is equal to  Z Z t Z ft (x) = ps (x, y)ds h(y)dy = kt (x, y)h(y)dy, Rd

Rd

0

where kt (x, y) =

Z

t

ps (x, y)ds.

0

R Let a measure ν be such that Rd kt (x, y)ν(dy) is a function continuous in (t, x). If we can choose a sequence of non-negative bounded continuous functions {hn : n ≥ 1} such that for each T > 0, Z Z lim sup sup kt (x, y)ν(dy) = 0, kt (x, y)hn (y)dy − n→∞ t∈[0,T ] x∈Rd

Rd

Rd

then by Theorem 1 there exists R a W-functional corresponding to the measure ν with its characteristic being equal to Rd kt (x, y)ν(dy). Formally we will denote this functional Rt by 0 dν dy (ξs )ds. A sufficient condition for the existence of a W-functional corresponding to a given measure is stated in the following theorem. Theorem 2 (See [8], Theorem 6.6). Let the condition Z (4) lim sup ft (x) = lim sup kt (x, y)ν(dy) = 0 t↓0 x∈Rd

t↓0 x∈Rd

Rd

hold. Then ft (x) is the characteristic of a W-functional Aνt . Moreover, Z t fh (ξu ) Aνt = l.i.m. du, h→0 0 h Rt u) and the sequence of characteristics of functionals 0 fh (ξ du converge to ft (x) in sense h of relation (3).

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OLGA V. ARYASOVA AND ANDREY YU. PILIPENKO

Let us return to SDE (1). Let (ϕt )t≥0 be a solution of equation (1) with bounded measurable a. The transition probability density pϕ t (y, z) of the process (ϕt )t≥0 satisfies the Gaussian estimates (see [2])     K1 K2 ky − zk2 ky − zk2 ϕ (5) ≤ p (y, z) ≤ exp −k exp −k 1 2 t t t td/2 td/2 valid in every domain of the form t ∈ [0, T ], y ∈ Rd , z ∈ Rd , where T > 0, K1 , k1 , K2 , k2 are positive constants that depend only on d, T, and kak∞ . Denote by ktw (x, y) the kernel kt (x, y) built on the transition density of the Wiener process, i.e.   Z t 1 ky − xk2 w ds. (6) kt (x, y) = exp − d/2 2s 0 (2πs) It is easily to see ([8], Ch. 8, §1) that for all x ∈ Rd , y ∈ Rd , x 6= y, ktw (x, y) = e kt (kx − yk), where Z ∞ 1 2−d e r sd/2−2 e−s ds, r > 0. (7) kt (r) = (2π)d/2 r 2 /2t

Therefore, the kernel ktw (x, y) has a singularity if x = y (for d > 1) and the integral Z ft (x) = ktw (x, y)ν(dy) Rd

is well defined not for all measures. It follows from (5) that a measure ν satisfies condition (4) if and only if it satisfies (4) for ktw (x, y). Therefore (4) is equivalent to the following condition. Condition (A). Z (8) lim sup ktw (x, y)ν(dy) = t↓0 x∈Rd

Rd

lim sup

t↓0 x∈Rd

Z

Rd

Z

0

t

   ky − xk2 1 ds ν(dy) = 0. exp − 2s (2πs)d/2

Remark 2. A measure ν satisfies Condition (A) if and only if Z ν(dy) < ∞, when d = 1; sup x∈R |x−y|≤1 Z 1 lim sup ln ν(dy) = 0, when d = 2; ε↓0 x∈R2 |x−y|≤ε |x − y| Z lim sup |x − y|2−d ν(dy) = 0, when d ≥ 3. ε↓0 x∈Rd

|x−y|≤ε

A proof is a slight modification of that for the case of ν(dx) = f (x)dx given in [1], Theorem 4.5 (see also [22], Exercise 1 on p. 12). Here f is a non-negative Borel measurable function. We use representation (7) in the proof. If a measure satisfies Condition (A) then it is called a measure of Kato’s class (see [16]). Example 1. Let d = 1. For each y ∈ Rd , the measure ν = δy satisfies Condition (A) and corresponds to the W-functional Z t 1 1[y−ε,y+ε] (ws ) ds, Lt (y) = lim ε↓0 2ε 0 which is called a local time of Wiener process at the point y. Assume that ν is a measure satisfying Condition (A). This means now that supx∈R ν([x, x + 1]) < ∞. Then (see [21],

ON DIFFERENTIABILITY OF STOCHASTIC FLOW

5

Ch. X, §2) the corresponding W-functional can be represented in the form Z Lt (y)ν(dy). Aνt = R

Remark 3. If d ≥ 2, then δy does not satisfy Condition (A). This agrees with the wellknown fact that the local time for a multidimensional Wiener process does not exist. Example 2. If ν(dy) = f (y)dy, where f is a non-negative bounded function, then ν Rt satisfies Condition (A) and Aνt = 0 f (ξs )ds.

Example 3. Let S ⊂ Rd be a compact (d − 1)-dimensional C 1 -manifold. Denote by σS the surface measure. Then for any non-negative bounded function f , the measure ν(dy) = f (y)σS (dy) satisfies Condition (A). Example 4. Let d ≥ 2. Assume that a measure ν is such that ∃k, γ > 0 ∀x ∈ Rd ∀ρ ∈ (0, 1] : Then (c.f. [5], §2) d

∃c = c(d, γ) ∀x ∈ R ∀ρ ∈ (0, 1] :

ν(B(x, ρ))≤ kρd−2+γ .

Z

|x − y|2−d ν(dy) ≤ ckργ .

B(x,ρ)

This inequality together with Remark 2 yields that ν satisfies Condition (A). In particular, the Hausdorff measure on the Sierpinski carpet in R2 satisfies (A) (see [5], Example 2.2). We will need the following modification of Khas’minskii’s Lemma (see [14] or [22], Ch.1 Lemma 2.1). For the convenience of reader we give a proof of this variant of Lemma. Lemma 1. Let the W-function ft satisfies condition (4). Let At be the corresponding W-functional. Then for all p > 0, t ≥ 0, there exists a constant C depending on p, t, and kft k∞ such that for all x ∈ Rd , sup Ex exp {pAt } ≤ C.

(9)

x∈Rd

To prove the Lemma we make use of the following proposition. Proposition 3 ([13], Ch. II, §6, Lemma 3). For all m ≥ 1, t > 0,  m (10) sup Ex (At )m ≤ m! sup ft (x) . x∈Rd

x∈Rd

Proof of Lemma 1. Fix p > 0. Condition (4) implies that there exists t0 > 0 such that for all t ≤ t0 , 1 sup |ft0 (x)| < . p d x∈R Then, by Proposition 3, for all t ∈ [0, t0 ], ∞ ∞ X X (pAt )m Ex exp{pAt } = Ex ≤ (pkft k∞ )m < C. m! k=1

k=1

Taking into account (2), we see that (9) is valid for all t ≥ 0. Lemma 1 is proved.



By the definition of W-functional, Aνt is measurable w.r.t. σ-algebra Nt = σ{ϕs : 0 ≤ s ≤ t}. That is, there exists a measurable function on C([0, t], Rd ) (denote it by d Aνt (·) = Aν,ϕ t (·)) such that for any x ∈ R , At = Aν.ϕ t (ϕ) Px − almost surely. The exceptional set may depend on x. Here the measure Px is a distribution of the process (ϕt (x))t≥0 . To emphasize that we consider the functional w.r.t. Px we will write Aνt (ϕ(x)).

6

OLGA V. ARYASOVA AND ANDREY YU. PILIPENKO

If for the measure ν Condition (A) holds, then the functionals of the processes (ϕt )t≥0 and (wt )t≥0 are well defined. Denote the corresponding measurable mappings by Aν,ϕ t ν,w and Aν,w and the corresponding additive functionals by Aν,ϕ t t (ϕ) and At (w). By the Girsanov theorem, for each x ∈ Rd , the distributions of the processes (ϕt (x))t≥0 and (x + wt )t≥0 are equivalent. The question naturally arises whether the mappings Aν,ϕ t and Aν,w are the same. The answer is positive and it is formulated in the next Lemma. t Lemma 2. Let ν satisfy Condition (A). Then for any x ∈ Rd , ν,ϕ Aν,w t (ϕ) = At (ϕ) Px − almost surely.

Proof. For x ∈ Rd , denote by (wt (x))t≥0 the process (x + wt )t≥0 . According to Theorem 2, Z t w fh (ws (x)) Aν,w (w(x)) = l.i.m. ds. t h↓0 h 0 Then by the Girsanov theorem, Z t w fh (ϕs (x)) ds, (11) Aν,w (ϕ(x)) = P−lim t h h↓0 0 where P−lim means the limit in probability. R t fhw (ϕs (x)) ds converge uniformly to h R It ϕremains to show that the characteristics of 0 k (x, y)ν(dy) (see Theorem 1). This proof is routine and technical, so we postpone Rd t it to Appendix.  2. The main result Let a be a bounded measurable function of bounded variation. Denote by ∇a the ∂ai . Further on we suppose that for all 1 ≤ i, j ≤ d, the measure matrix ∂x i j 1≤i,j≤d ∂a |µij | = ∂x satisfies Condition (A). j ij,±

,w By Theorem 2, there exist W -functionals Aµt (we will denote the corresponding ij,± mappings by At (·)) with their characteristics defined according to the formula Z ij,± ktw (x, y)µij,± (dy). ft (x) =

Denote

Aij t

=

Aij,+ t

−

Aij,− , t

At =

Rd ij (At )1≤i,j≤d .

Remark 4. Recall that the mappings Aij,+ , Aij,− are continuous and monotonous in t. t t ij So the function t → At is a continuous function of bounded variation on [0, T ] almost surely. The main result on differentiability of a flow generated by equation (1) with respect to the initial conditions is given in the following theorem. Theorem 3. Let a : Rd → Rd be such that for all 1 ≤ i ≤ d, ai is a function of ∂ai bounded variation on Rd . Put µij = ∂x , 1 ≤ i, j ≤ d. Assume that the measures j ij |µ |, 1 ≤ i, j ≤ d, satisfy Condition (A). Let Yt (x), t ≥ 0, be a solution to the integral equation Z t (12) Yt (x) = E + dAs (ϕ(x))Ys (x), 0

where E is a d × d-identity matrix, the integral in the right-hand side of (12) is the Lebesgue-Stieltjes integral with respect to the continuous function of bounded variation t → At (ϕ(x)). Then Yt (x) is a derivative of ϕt (x) in Lp -sense: for all p > 0, x ∈ Rd , h ∈ Rd , t ≥ 0,

p

ϕt (x + εh) − ϕt (x)

− Yt (x)h (13) E

→ 0, ε → 0. ε

ON DIFFERENTIABILITY OF STOCHASTIC FLOW

7

Moreover,

 1 P ∀t ≥ 0 : ϕt (·) ∈ Wp,loc (Rd , Rd ), ∇ϕt (x) = Yt (x) for λ-a.a. x = 1,

where λ is the Lebesgue measure on Rd .

Remark 5. The differentiability was proved in [10, 20]. We give a representation for the derivative. Note that the Sobolev derivative is defined up to the Lebesgue null set. Remark 6. Consider a non-homogeneous SDE ( dϕt (x) = a(t, ϕt (x))dt + dwt , ϕ0 (x) = x.

Similar to reasoning of Section 1 a theory of non-homogeneous additive functionals of non-homogeneous Markov processes can be constructed. All the formulations and proofs can be literally rewritten with natural necessary modifications. Unfortunately, there are no corresponding references, therefore we did not carry out the corresponding reasonings. Consider examples of functions a for which |µij |, 1 ≤ i, j ≤ d, satisfy Condition (A). Example 5. Let for all 1 ≤ i ≤ d, ai be a Lipschitz function. By Rademacher’s theorem ∂ai exist almost surely w.r.t. the Lebesgue measure. [9] the Frech´et derivatives µij = ∂x j It is easy to verify that they are bounded and the Frech´et derivative coincides with the derivative considered in the generalized sense. Then |µij | satisfies Condition (A). Let now h ∈ C 1 (Rd , Rd ), D be a bounded domain in Rd with C 1 boundary ∂D. Put a(x) = h(x)1x∈D . It follows from Example 3 that for all 1 ≤ i, j ≤ d, |µij | also satisfies Condition (A) because (cf. [24]) µij (dx) =

∂ai (x)1x∈D dx + h(x) cos(nj (x))σ∂D (dx), ∂xj

where n(x) = (n1 (x), . . . , nd (x)) is the outward unit normal vector at the point x ∈ ∂D. Condition (A) is also satisfied by the measure generated by a being a linear combination of the form m X (14) h0 (x) + hk (x)1x∈Dk , k=1

d

d

1

d

where h0 ∈ Lip(R , R ), hk ∈ C (R , Rd ), 1 ≤ k ≤ d, Dk is a bounded domain in Rd with C 1 boundary. Further examples of a can be obtained as the limits of sequences of the functions of form (14). In one-dimensional case all the functions of bounded variation generate measures satisfying Condition (A) (see Example 1). See also Example 4 showing that if |µij | are “Hausdorff-type” measures with a parameter greater than (d − 1), then a satisfies assumptions of the Theorem. The idea of the proof of Theorem 3 is to approximate the solution of equation (1) by solutions of SDEs with smooth coefficients.The definition and properties of approximating equations are given in Sections 3, 4. The proof of the Theorem itself is represented in Section 5. 3. Approximation by SDEs with smooth coefficients R For n ≥ 1, let gn ∈ C0∞ (Rd ) be a non-negative function such that Rd gn (z)dz = 1, and gn (x) = 0, |x| ≥ 1/n. Put Z gn (x − y)a(y)dy, x ∈ Rd , n ≥ 1, (15) an (x) = (gn ∗ a)(x) = Rd

8

OLGA V. ARYASOVA AND ANDREY YU. PILIPENKO

where the function a satisfies the assumptions of Theorem 3. Note that sup kan k∞ ≤ kak∞ ,

(16)

n

d

and an → a, n → ∞, in L1,loc (R ). Passing to subsequences we may assume without loss of generality that an (x) → a(x), n → ∞, for almost all x w.r.t. the Lebesgue measure. Consider an SDE ( dϕn,t (x) = an (ϕn,t (x))dt + dwt , (17) ϕn,0 (x) = x, x ∈ Rd .  i ∂a . Denote by Yn,t (x) the matrix of derivatives of ϕn,t (x) in x, Put ∇an = ∂xnj

ij i.e., Yn,t (x) =

1≤i,j≤d ∂ϕin,t (x) . Then ∂xj

Yn,t (x) satisfies the equation Z t ∇an (ϕn,s (x))Yn,s (x)ds, Yn,t (x) = E +

(18)

0

where E is a d-dimensional identity matrix.

Lemma 3. For each p ≥ 1, 1) for all t ≥ 0 and any compact set U ∈ Rd , sup x∈U, n≥1

(E(kϕnt (x)kp + kϕt (x)kp )) < ∞;

2) for all x ∈ Rd , T ≥ 0,   n p E sup kϕt (x) − ϕt (x)k → 0 as n → ∞, 0≤t≤T

where k · k is a norm in the space Rd . Proof. Statement 1) follows from the uniform boundedness of the coefficients and the finiteness of the moments of a Wiener process; 2) is proved in [18], Theorem 3.4.  For 1 ≤ i, j ≤ d, put µij n = function (see [24], Ch. 2, §7),

∂ain ∂xj .

By the properties of convolution of generalized

∇an = ∇a ∗ gn . For each n ≥ 1, 1 ≤ i, j ≤ d, the measure µij n satisfies Condition (A) (see Example 2). ij,+ Let µij − µij,− be the Hahn-Jordan decomposition of the measure µij n = µn n n . Then, d according to Theorem 2, there exist W-functionals Aij,± n,t (w) of a Wiener process on R ij,± which correspond to the measures µn and have characteristics of the form Z ij,± ktw (x, y)µij,± (19) fn,t (x) = n (dy), 1 ≤ i, j ≤ d. Rd

ij,+ Since the measures µij,± have continuous densities, the functional Aij n n,t (w) = An,t (w) − Aij,− n,t (w) is given by the formula Z t i ∂an (wu )du. (20) Aij (w) = n,t 0 ∂xj

Lemma 4. For each T > 0, 1 ≤ i, j ≤ d, ij sup Aij n,t (w) − At (w) → 0, n → ∞, in probability. 0≤t≤T

The following simple proposition used for the proof of Lemma 4 is easily checked.

Proposition 4. Let ν, νn be W-measures, and the representation νn = gn ∗ ν, hold true. Then the relation fn,t = gn ∗ft is fulfilled for characteristics corresponding W-functionals from the Wiener process.

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Proof of Lemma 4. To prove the convergence of functionals in mean square it is sufficient to show that for each T > 0, 1 ≤ i, j ≤ d, (21)

ij sup sup |fn,t (x) − ftij (x)| = 0

lim

n→∞ 0≤t≤T x∈Rd

(see Theorem 1). Then the uniform convergence in probability follows from Proposition 2. For each 0 < δ < t, Z
ij ij = I + II, (x) − ftij (x) = sup sup fn,t ktw (x, y) µij (dy) − µ (dy) n x∈Rd

x∈Rd

where (22)

Z   Z δ 2
ky − xk 1 ij ds , I = sup µij exp − n (dy) − µ (dy) d/2 2s x∈Rd Rd 0 (2πs) Z   Z
t 1 ky − xk2 ij II = sup µij (dy) − µ (dy) ds . exp − n d/2 2s d d (2πs) x∈R

We have

Rd

I ≤ sup x∈Rd

Z

Rd

δ

R

  ky − xk2 1 ds+ exp − d/2 2s 0 (2πs)   Z Z δ ky − xk2 1 sup exp − |µij |(dy) ds = I1 + I2 . d/2 2s x∈Rd Rd 0 (2πs)

|µij n |(dy)

Z

δ

Because of Condition (A), for each ε > 0, we can choose δ so small that I2 is less then ε/4. To obtain the same estimate for I1 , note that by the associative, distributive and commutative properties of convolution (see [24], Ch. II, §7), I1 = sup (|µn | ∗ kδ )(x) = ((|µ| ∗ gn ) ∗ kδ ) (x) = sup (|µ| ∗ (gn ∗ kδ )) (x) = x∈Rd

x∈Rd

sup (|µ| ∗ (kδ ∗ gn )) (x) = sup ((|µ| ∗ kδ ) ∗ gn ) (x) ≤ sup (|µ| ∗ kδ ) (x) = I2 < ε/4. x∈Rd

x∈Rd

x∈Rd

We get I < ε/2. Consider II. The function Z Z ij ij qδ,t (x) := µ (dy) Rd

δ

t

  kx − yk2 1 ds exp − 2s (2πs)d/2

is equicontinuous in x for t ∈ [δ, T ]. We have

ij ij sup II = sup sup |(qδ,t ∗ gn )(x) − qδ,t (x)| → 0, n → ∞.

δ 1. Thus all the assertions of Proposition 5 are fulfilled and we have ij sup Aij n,t (ϕn (x)) − At (ϕ(x)) → 0, n → ∞, 0≤t≤T

in probability. The Lemma is proved.



4. Convergence of the derivatives of solutions Recall that Yt (x), Yn,t (x), t ≥ 0, x ∈ Rd , are the solutions of equations (12), (18), respectively. In this section we show the convergence of the sequence {Yn,t (x) : n ≥ 1} in probability uniformly in t. This together with Lemma 3 allow us to prove Theorem 3.

ON DIFFERENTIABILITY OF STOCHASTIC FLOW

11

Lemma 6. 1) For all p > 0, sup E sup kYn,t (x)kp < ∞,

n≥1

0≤t≤T

d

2) For all T ≥ 0, x ∈ R , p > 0, E sup kYn,t (x) − Yt (x)kp → 0, n → ∞, 0≤t≤T

where kY k = max |Y ij |. 1≤i,j≤d

For the proof we need the following two propositions. The first one is a variant of the Gronwall-Bellman inequality and can be obtained by a standard argument. Proposition 6. Let x(t) be a continuous function on [0, +∞), C(t) be a non-negative continuous function on [0, +∞), K(t) be a non-negative, non-decreasing function, and K(0) = 0. If for all 0 ≤ t ≤ T , Z t x(t) ≤ C(t) + x(s)dK(s) , 0

then

x(T ) ≤



 sup C(t) exp{K(T )}.

0≤t≤T

Proposition 7. For all p > 0, 1 ≤ i, j ≤ d, there exists a constant C such that  n o n o ij,± (23) sup sup E exp pAij,± (ϕ(x)) < C. n,t (ϕn (x)) + exp pAt x∈Rd n≥1

Proof. The statement of the Proposition follows from Lemma 1 and inequalities (5), which allow us to obtain the estimates uniform in n ≥ 1.  Proof of Lemma 6. For all t > 0, define the variation of Aij · on [0, t] by ij,+ Var Aij (ϕ(x)) + Aij,− (ϕ(x)), t (ϕ(x)) := At t

and put

Var At (ϕ(x)) := Σ1≤i,j≤d Var Aij t (ϕ(x)). The variation of An,t (ϕn (x)) are defined similarly. The proof of 1). We have

Z t

Z t

kYn,t (x)k ≤ 1 + kYn,s (x)k d (Var An,s (ϕn (x))) . (dAn,s (ϕn (x))) Yn,s (x) ≤ 1 +

0

0

Making use of the Gronwall-Bellman lemma we get

kYn,t (x)k ≤ exp {Var An,t (ϕn (x))} ≤ exp {Var An,T (ϕn (x))} .

(24)

The statement 1) follows now from estimate (24) and Proposition 7. The proof of 2). We have

Z t



kYn,t (x) − Yt (x)k ≤ (dAn,s (ϕn (x)) − dAs (ϕ(x))) Ys (x)

+ 0

Z t



≤ dA (ϕ (x)) (Y (x) − Y (x)) n,s n n,s s

0

Z t

Z t

kYn,s (x) − Ys (x)k d (Var An,s (ϕ(x))) . (dAn,s (ϕn (x)) − dAs (ϕ(x))) Ys (x)

+ 0

0

By Proposition 6, (25)

Z

kYn,t (x)−Yt (x)k ≤ sup

0≤u≤t

0

u

(dAn,s (ϕn (x)) − dAs (ϕ(x))) Ys (x)

exp {Var An,t (ϕn (x))} .

12

OLGA V. ARYASOVA AND ANDREY YU. PILIPENKO

To estimate the right-hand side of (25) we make use of the following Proposition. Proposition 8. Let {gn : n ≥ 1} be a sequence of continuous monotonic functions on [0, T ], and f ∈ C([0, T ]). Suppose that for each t ∈ [0, T ], gn (t) → g(t), as n → ∞. Then Z t Z t sup f (s)dgn (s) − f (s)dg(s) → 0, n → ∞. t∈[0,T ]

0

0

We get

(26)

Z u



sup (dAn,s (ϕn (x)) − dAs (ϕ(x))) Ys (x)

exp {Var An,t (ϕn (x))} ≤ 0≤u≤t 0

Z u


+ +

exp {Var An,t (ϕn (x))} + sup dA (ϕ (x)) − dA (ϕ(x)) Y (x) n s s n,s

0≤u≤t 0

Z u


− −

exp {Var An,t (ϕn (x))} . sup dA (ϕ (x)) − dA (ϕ(x)) Y (x) n s n,s s

0≤u≤t

0

Consider the first summand in the right-hand side of (26). Put gn (s) = A+ n,s (ϕn (x)), + g(s) = As (ϕ(x)), and f (s) = Ys (x). Then Lemma 5, Proposition 7, and Proposition 8 provide that

Z u


+ +

sup dAn,s (ϕn (x)) − dAs (ϕ(x)) Ys (x)

exp {Var An,t (ϕn (x))} → 0 as n → ∞, 0≤u≤t 0

in probability. Similarly it is proved that the second summand on the right-hand side of (26) tends to 0 as n → ∞. This and statement 1) entail statement 2) of the Lemma.  5. The proof of Theorem 3 Proof. Define approximating equations by (17), where an , n ≥ 1, are determined by (15). From Lemma 3 and the dominated convergence theorem we get the relation Z E sup |ϕin,t (x) − ϕit (x)|p dx → 0, n → ∞, t∈[0,T ]

U

valid for any bounded domain U ⊂ Rd , T > 0, p ≥ 1, and 1 ≤ i ≤ d. So for each 1 ≤ i ≤ d, there exists a subsequence {nik : k ≥ 1} such that Z sup |ϕini ,t (x) − ϕit (x)|p dx → 0 a.s. as k → ∞. t∈[0,T ]

U

k

Without loss of generality we can suppose that Z (27) sup |ϕin,t (x) − ϕit (x)|p dx → 0 a.s. as n → ∞. t∈[0,T ]

U

Arguing similarly and taking into account Lemma 6 we arrive at the relation Z ij (28) sup |Yn,t (x) − Ytij (x)|p dx → 0, n → ∞, almost surely, t∈[0,T ]

U

that is fulfilled for all 1 ≤ i, j ≤ d, p ≥ 0. Since the Sobolev space is a Banach space, relations (27), (28) mean that Yt (x) is the matrix of derivatives of the solution to (1). Let us verify (13). We have for all x, h ∈ Rd , α ∈ R, Z α Yn,t (x + uh)du. ϕn,t (x + αh) = ϕn,t (x) + 0

It follows from Lemmas 3 and 6 that (29)

ϕt (x + αh) = ϕt (x) +

Z

0

α

Yt (x + uh)du.

ON DIFFERENTIABILITY OF STOCHASTIC FLOW

13

Similarly to the proof of Lemma 5 and 6 it can be verified that ∀y0 ∈ Rd : At (ϕ(y)) → At (ϕ(y0 )), y → y0 , ∀y0 ∈ Rd : Yt (y) → Yt (y0 ), y → y0 , in probability, and hence in all Lp . So (29) implies (13). The Theorem is proved.



6. Appendix: The proof of Lemma 2 R t f w (ϕ (x)) Note that Bth (ϕ(x)) = 0 h hs ds is a W-functional. Let us estimate its characteristic.  Z Z Z t Z Z t w 1 h fh (ϕs (x)) ϕ ds = du ds pw (z, y)p (x, z)dz ν(dy). EBth (ϕ(x)) = E u s h h 0 Rd 0 Rd 0 From estimates (5) we obtain (see also the proof of Theorem 6.6. in [8])

EBth (ϕ(x)) ≤      Z Z t Z Z 1 h kky − zk2 kkz − xk2 K K exp − exp − dz ν(dy) = du ds d/2 h 0 u s sd/2 Rd Rd u 0    Z h Z Z t kky − xk2 1 e1 exp − ds ν(dy) = K du d/2 h 0 u+s Rd 0 (2π(u + s))    Z Z t+u Z kky − xk2 1 h 1 e exp − ds ν(dy) = K du h 0 s (2πs)d/2 Rd u  !  Z h Z Z (t+u)/2k 2 1 1 ky − xk b ds ν(dy) = du K exp − h 0 2s (2πs)d/2 Rd u/2k Z h  w w b1 (x) du. (x) − fu/2k K f(t+u)/2k h 0

e = K 2 π 2 (2/k)d/2 , K b = 2K 2 k 1−d π d . Taking into account (2), we get where K w w w w w k∞ . (x) ≤ kft/2k ft/2k (x) = Tu/2k (x) − fu/2k f(t+u)/2k

By Proposition 3, (30)

2 
2 w sup Ex Bth (ϕ) ≤ 2 kft/2k k∞ .

x∈Rd

Therefore, the second moment of Bth (ϕ) is bounded uniformly in h. This implies the uniform integrability and, consequently the convergence in L1 holds in (11). Then the characteristic of the functional Aν,w t (ϕ(x)) is equal to Z t w fh (ϕs (x)) ds. fet (x) = lim E h↓0 h 0 If we show that Z (31) fet (x) = ktϕ (x, y)ν(dy), Rd

then the statement of the Lemma follows from Proposition 1. We have Z t w Z fh (ϕs (x)) ϕ E kt (x, y)ν(dy) ≤ ds − h 0 Rd  Z δ Z Z δ w fh (ϕs (x)) pϕ (x, y)ν(dy) ds+ ds + E s h Rd 0 0 Z t w  Z t Z fh (ϕs (x)) ϕ E ds − ps (x, y)ν(dy) ds = I + II + III. h δ δ Rd

14

OLGA V. ARYASOVA AND ANDREY YU. PILIPENKO

Consider I. Arguing as in the proof of (30) we arrive at the inequality w k∞ . I ≤ kfδ/2k

Making use of (5) and changing the variables we get   Z δ/2k Z ky − xk2 1 ν(dy) II ≤ 2Kπ d/2 (k)1−d/2 exp − ds d/2 2s 0 Rd (2πs)

≤ 2Kπ d/2 (k)1−d/2 kfδ/2k k∞ .

For each ε > 0, Condition (A) allows us to choose δ so small that (32)

I < ε/3, II < ε/3.

Further,

Z ! Z Z t Z 1 h w ϕ ϕ pu (z, y)du dz . III = (ps (x, z) − ps (x, y)) ν(dy) ds δ h 0 Rd Rd  R h The measure h1 0 pw u (z, y)du dz converges weakly to δ-measure at the point y. The

d function pϕ s (x, y) is equicontinuous in y for s ∈ [δ, t], x ∈ R . So ! Z Z 1 h w ϕ ϕ p (z, y)du dz → 0, h ↓ 0, (ps (x, z) − ps (x, y)) h 0 u Rd

uniformly in x and s. Besides, from (5) Z ! Z 1 h w ϕ ϕ pu (z, y)du dz ≤ (ps (x, z) − ps (x, y)) Rd h 0      Z kkx − yk2 kkx − zk2 K + exp − exp − d/2 s s Rd s

1 h

Z

h 0

!

pw u (z, y)du dz ≤

2K . sd/2

By the dominated convergence theorem, (33)

III → 0 as h ↓ 0.

Now equality (31) follows from (32) and (33). The Lemma is proved. Acknowledgements The authors are grateful to the anonymous referee for his thorough reading and valuable comments which help to improve essentially the exposition. We also thank Alexei Kulik for helpful discussions. References [1] M. Aizenman and B. Simon. Brownian motion and harnack inequality for schrdinger operators. Communications on Pure and Applied Mathematics, 35(2):209–273, 1982. [2] D. G. Aronson. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc., 73:890–896, 1967. [3] O. V. Aryasova and A. Yu. Pilipenko. On properties of a flow generated by an SDE with discontinuous drift. Electron. J. Probab., 17:no. 106, 1–20, 2012. [4] S. Attanasio. Stochastic flows of diffeomorphisms for one-dimensional SDE with discontinuous drift. Electron. Commun. Probab., 15:no. 20, 213–226, 2010. [5] R. F. Bass and Z.-Q. Chen. Brownian motion with singular drift. The Annals of Probability, 31(2):791–817, 04 2003. [6] R. M. Blumenthal and R. K. Getoor. Markov Processes and Potential Theory. Reprint of the 1968 ed. Mineola, NY: Dover Publications. vi, 313 p., 2007. [7] V. I. Bogachev. Measure Theory, volume 2. Springer, Berlin, 2007. [8] E. B. Dynkin. Markov Processes. Fizmatlit, Moscow, 1963. [Translated from the Russian to the English by J. Fabius, V. Greenberg, A. Maitra, and G. Majone. Academic Press, New York; Springer, Berlin, 1965. vol. 1, xii + 365 pp.; vol. 2, viii + 274 pp.].

ON DIFFERENTIABILITY OF STOCHASTIC FLOW

15

[9] Federer H. Geometric Measure Theory, volume 153 of Die Grundlehren der mathematischen Wissenschaften. New York, Springer-Verlag New York Inc. edition, 1969. [10] E. Fedrizzi and F. Flandoli. H¨ older flow and differentiability for SDEs with nonregular drift. Stochastic Analysis and Applications, 31(4):708–736, 2013. [11] E. Fedrizzi and F. Flandoli. Noise prevents singularities in linear transport equations. Journal of Functional Analysis, 264(6):1329 – 1354, 2013. [12] F. Flandoli, M. Gubinelli, and E. Priola. Flow of diffeomorphisms for SDEs with unbounded H¨ older continuous drift. Bulletin des Sciences Mathematiques, 134(4):405 – 422, 2010. [13] I. I. Gikhman and A. V. Skorokhod. The Theory of Stochastic Processes. II. Nauka, Moscow, 1973. [Translated from the Russian by S. Kotz. Corrected printing of the first edition. Berlin: Springer, 2004. viii, 441 p.]. [14] R. Z. Khasminskii. On positive solutions of the equation Au + vu = 0. Theory of Probability and Its Applications, 4(3):309–318, 1959. [15] A. M. Kulik and A. Yu. Pilipenko. Nonlinear transformations of smooth measures on infinite-dimensional spaces. Ukrainian Mathematical Journal, 52:1403–1431, 2000. 10.1023/A:1010380119199. [16] K. Kuwae and M. Takahashi. Kato class measures of symmetric Markov processes under heat kernel estimates. Journal of Functional Analysis, 250(1):86 – 113, 2007. [17] R. S. Liptser and A. N. Shiryayev. Statistics of Random Processes. Nauka, Moscow, 1974. [Translated from the Russian to the English by A. B. Aries. Springer-Verlag, New York, 1977]. [18] D. Luo. Absolute continuity under flows generated by SDE with measurable drift coefficients. Stochastic Processes and their Applications, 121(10):2393 – 2415, 2011. [19] T. Meyer-Brandis and F. Proske. Construction of strong solutions of SDE’s via Malliavin calculus. Journal of Functional Analysis, 258(11):3922 – 3953, 2010. [20] S.E.A. Mohammed, T. Nilssen, and F. Proske. Sobolev differentiable stochastic flows of SDE’s with measurable drift and applications. 2012. arXiv/1204.3867. [21] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin, 1999. [22] A.-S. Sznitman. Brownian motion, obstacles, and random media. Springer monographs in mathematics. Springer, 1 edition, 1998. [23] A. Y. Veretennikov. On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sborn, 39(3):387–403, 1981. [24] V. S. Vladimirov. The Equation of Mathematical Phisics. Nauka, Moscow, 1967. [Translated from the Russian to the English by A. Littlewood. Marcel Dekker, INC., New York, 1971.]. Institute of Geophysics, National Academy of Sciences of Ukraine, Palladin pr. 32, 03680, Kiev-142, Ukraine E-mail address: [email protected] Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska str. 3, 01601, Kiev, Ukraine; National Technical University of Ukraine ”KPI”, Kiev, Ukraine E-mail address: [email protected]