Higher-Order Relations for Derivatives of Nonlinear Diffusion Semigroups

Ukrainian Mathematical Journal, Vol. 53, No. 1, 2001

HIGHER-ORDER RELATIONS FOR DERIVATIVES OF NONLINEAR DIFFUSION SEMIGROUPS

A. Val. Antonyuk and A. Vik. Antonyuk UDC 519.217.4, 517.955.4, 517.956.4, 517.958:536.2

We show that a special choice of the Cameron – Martin direction in the characterization of the Wienermeasure via the formula of integration by parts leads to a set of natural representations for derivatives ofnonlinear diffusion semigroups. In particular, we obtain a final solution of the non-Lipschitz singular-ities in the Malliavin calculus.

1. Introduction

Exact representations of derivatives of semigroups generated by differential operators of the second order play akey role in understanding smooth properties of solutions of associated parabolic equations [1 – 6].

In particular, the representation

∂∂

( ) = ( ) −

∫xP f x f

Du dW D

Dt t

t

u t

t

s s ut

u t

E ξ ξξ

ξξ

01

00

1

0 , (1)

which is standard in the investigation of the regular properties of diffusion semigroups within the framework of the

Malliavin calculus, allows us to conclude that the diffusion semigroup Pt f ( x ) = E f (ξt x0( )) increases the smooth-

ness of the original function, i.e., it follows from the inclusion f ∈ Cb that, for every t > 0, we have Pt f ∈ C1 if

the expression … on the right-hand side of (1) is sufficiently regular. Here, Du tξ0 is the stochastic derivative in

a Wiener space, ξt1 = ∂

∂x tξ0 is the first variation of the process, and

ξ ξ ξt

t

s s

t

sx x B x dW F x ds0

0

0

0

0( ) = + ( ) − ( )∫ ∫( ) ( ) , (2)

where B ∈ C∞ ( R

d, L ( Rd

) ) , F ∈ C∞ ( R

d, Rd ) , and B is uniformly nondegenerate.

Representation (1) follows directly from the main formula of integration by parts on a Wiener space [5 – 9]:

E Du G = E G 0

∞

∫ ⟨ ut , dWt ⟩ R

d , u ∈ J. (3)

Here, E is the mathematical expectation with respect to the Wiener measure P on the metric space Ω =

C0 ( [ 0, ∞ ), Rd ) , J is the set of continuous integrable processes ut (ω) adapted with respect to the canonical filtra-

tion Ft and such that E0

1Tt

pu dt∫ + < ∞ for any T , p > 0, and D u G is the strong stochastic derivative.

A Wiener functional G = G ( ω ), ω ∈ Ω, G ∈ p≥1I L

p ( Ω, P ) is called strongly stochastically differentiable [5,

6] if, for any u ∈ J, there exists a strong stochastic derivative Du G ∈ p≥1I Lp

( Ω, P ) such that

Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 53, No. 1,pp. 117–122, January, 2001. Original article submitted February 2, 1999.

134 0041–5995/00/5301–0134 $25.00 © 2001 Plenum Publishing Corporation

HIGHER-ORDER RELATIONS FOR DERIVATIVES OF NONLINEAR DIFFUSION SEMIGROUPS 135

∀p ≥ 1 ∃ ( ) − ( ) − ( )→

limε

εω ωε

ω0

EG G

D Gu

p

= 0,

where ωε denotes a perturbed trajectory, i.e., ωε = ωt + ε 0

t

∫ us ds t ≥ 0 , in the direction of infinitesimal incre-

ment along the process u. Correspondingly, a Wiener functional G ∈ p≥1I Lp

( Ω, P ) is stochastically differen-

tiable if, for G, there exist strong stochastic derivatives in the directions of bounded processes u ∈ J and the

functional Du G (which is linear in u ) possesses the following property: It follows from the convergence of

bounded processes un ∈ J to an integrable process u ∈ J ( ∀p, T > 0 E0

1Tn

pu u dt∫ − + → 0, m , n → ∞) that

the strong stochastic derivatives are fundamental ( ∀p, T > 0 E | Dun G – Dum

G | 1

+

p → 0, n, m → ∞) . The limit

limn→∞

Dun G = Du G is unique; in what follows, it is called the stochastic derivative of a functional G . Note that

formula (3) can easily be extended to the class of Ft -measurable stochastically differentiable Wiener functionals for

any s > 0.

One can directly verify that the equations for the first variation ξt1 and inverse stochastic derivative 1 / Du tξ0

have coercive and dissipative coefficients in the dominant parts only in the case where the Lipschitz conditions are

imposed on the diffusion and translation B, F . For this reason, the earlier investigations of the property of in-crease in smoothness were concentrated at diffusions with globally Lipschitz coefficients, which naturally arise, e.g.,

on compact C∞ Riemannian and spin manifolds [3 – 8, 10]. In fact, for essentially nonlinear diffusions, it is impos-

sible to obtain even estimates for moments, and the singularities of the process 1 / Du tξ0 lead to the violation of re-

lation (1) [11].

2. Main Assumptions and Auxiliary Statements

Below, we present formulas for derivatives of heat semigroups of any order, which enable us to consider essen-tially nonlinear diffusions and do not contain singular terms such as the inverse Malliavin determinant or inversestochastic derivative.

We assume that the coefficients B and F in Eq. (2) satisfy the following conditions:

A. Coercivity and dissipativity. ∀ M ∃ KM , K1 , K2

⟨ x – y, F ( x ) – F ( y ) ⟩ R

d – M B x B y HS( ) − ( ) 2 ≥ KM || x – y || 2, (4)

⟨ x, F ( x ) ⟩ R

d – M B x HS( ) 2 ≥ – K1 || x || 2 – K2 .

B. Non-Lipschitz parameters. The mapping F is quasimonotone and ∃ k F , k B ≥ – 1 with 2k B ≤ k F

such that ∀ n ∈ N ∃ Cn ∀ i = 1, … , n ∀ x, y ∈ Rd

|| F( i

)

( x ) – F( i

)

( y ) || i ≤ Cn || x – y || ( 1 + || x || + || y || ) k

F,

|| B( i

)

( x ) – B( i

)

( y ) || i ≤ Cn || x – y || ( 1 + || x || + || y || ) k

B.

In the above relations, || ⋅ || HS denotes the Hilbert – Schmidt norm of a matrix in Rd and || ⋅ || i is understood

as the norm of the i th Frechét derivative of the corresponding mapping.

136 A. VAL. ANTONYUK AND A. VIK. ANTONYUK

Lemma 1. For u ∈ J, the stochastic derivative Du tξ0 satisfies the equation

D B u ds B x D dW F x D dsu t

t

s s

t

s u s s

t

s u sξ ξ ξ ξ ξ ξ0

0

0

0

0 0

0

0 0= ( ) + ′ ( ) [ ] − ′ ( ) [ ]∫ ∫ ∫( ) ( ) , (5)

where G′ ( x ) [ v ] is the Gâteaux derivative of the mapping G at the point x in the direction v.

Proof. First, we note that relation (5) follows from (2) and the general properties of the stochastic derivativeDu [3 – 7, 10]:

Du f ( ξ1, … , ξn ) =

i

n

=∑

1

∂i f ( ξ1, … , ξn ) Du ξi, D f ds D f dsu

t

s

t

u s0 0∫ ∫= , (6)

D g dW D g dW u g dsu

t

s s

t

u s s

t

s s0 0 0∫ ∫ ∫= + .

I. Construction of a perturbed process ξεt x u( ), . To prove Lemma 1, for bounded u ∈ J we consider a proc-

ess ξ ωεt x u( ), , = ξ ω εt sx u ds0

0, . +( )∫ , ω ∈ Ω, that is a solution of the equation

ξ ε ξ ξ ξε ε ε εt

t

t s

t

t s

t

tx u x B x u u ds B x u dW F x u ds( ) = + ( ) + ( ) − ( )∫ ∫ ∫( ) ( ) ( ), , , ,0 0 0

. (7)

Since ε || B ( ⋅ ) u || ≤ 12

12

2 2 2ε u B HS+ (⋅) , we conclude that, for bounded u ∈ J, the conditions of coer-

civity and dissipativity are satisfied for Eq. (5). Hence, according to [12 – 14], Eq. (7) has a unique strong solutioncontinuous with respect to the initial condition x.

II. Construction of the stochastic derivative D xu tξ0( ) . By analogy, substituting x = y + α h in the coercivity

condition (4), multiplying by 1 / α2, and passing to the limit as α → 0, we establish that ∀ M ∃ KM ∀ x, h ∈ Rd

⟨ h, F′ ( x ) h ⟩ – M B x h HS′( )[ ] 2 ≥ KM || h || 2, (8)

i.e., we establish the coercivity and dissipativity of the linear inhomogeneous equation (5). The solvability of Eq. (5)follows from the criteria presented in [12 – 14]. Moreover, by using the Itô formula and estimates for the moments

of the process ξt x0( ) , we obtain the following inequality, which is analogous (in a certain sense) to relation (5.7) in[11]:

E EkD x Ke x u dtu tp pMt p

t

tpBξ0 1

0

21( ) ≤ +( ) ( + ) ∫ , (9)

where the non-Lipschitz parameter k B appears in the course of estimation of the inhomogeneous term in (5). We

have also used the coercivity and dissipativity of Eq. (5) and the property D xu t tξ00( ) = = Du x = 0.


III. Existence of strong stochastic derivatives of the process ξt x0( ) . The existence of strong stochastic deriv-

atives of the process ξt x0( ) follows from the convergence

∀p, T > 0 ∃ ( ) − ( ) − ( )→ ∈[ ]

lim sup,

,ε

εξ ξε

ξ0 0

00E

t T

t tu t

px u x

D x = 0

for bounded processes u ∈ J. This statement can be proved by using the Itô formula and assumption B concerningthe non-Lipschitz behavior of the coefficients of diffusion and translation. At the first step, we estimate the expres-sion

∆ε ( t ) = ξ ξ

εξ

εt t

u tx u x

D x( ) − ( ) − ( ), 0

0 .

We get

ht = E || ∆ε ( t ) || p ≤ K h ds C x ep

t

sp

pp

pM u tF s t

s

0

2 11 0∫ + +( ) ( + )( )

∈[ ]ε ksup

, . (11)

Here, one should use the equality ∆ε ( 0 ) = 0 and the formula

G ( x ) – G ( y ) = G′ ( y ) [ x – y ] + 0

1

∫ G′ ( y + l ( x – y ) ) – G′ ( y ) [ x – y ] d

for the separation of the coercive part B′ ( ξ0 ) [ ∆ε ], F′ ( ξ0

) [ ∆ε ] in the differential d ∆ε ( t ) and the parameters of

nonlinearity k F and k B , which appear in the term with 0

1

∫ … d.

At the second step, by using (11) and the Doob inequality (inequality (3.7 ′ ) in [15, Chap. 7, Sec. 3]), one shouldestablish the convergence of (10), i.e., one should perform the commutation of E and sup

,t T∈[ ]0.

IV. Finally, the stochastic differentiability of the process ξt x0( ) follows from the strong stochastic differenti-

ability (see steps I – III), the linearity of D xu u t1 2

0− ( )ξ = D x

u t10ξ ( ) – D x

u t20ξ ( ), and inequality (9). Applying in-

equality (9) again, one can pass to the limit in Eq. (5) and establish it for an integrable process u ∈ J. Lemma 1 isproved.

We denote by ξt x1( ) = ∂ ( )

∂

=

it

ji j

dx

x

ξ0

1, the matrix of the first variation of the process ξt x0( ) with respect to the

initial condition x. This matrix satisfies the equation

∀ v ∈ Rd

ξ ξ ξ ξ ξt

t

s s s

t

s sx B x x dW F x x ds1

0

0 1

0

0 1( ) = + ′ ( ) ( ) − ′ ( ) ( )∫ ∫( )[ ] ( )[ ]v v v v , (12)

which, according to [12 – 14], has a unique solution under condition (8).


In the statement below, we calculate the stochastic derivative in certain directions in terms of the first variation.Denote

Ψt v = B x xt t− ( )( ) ( )1 0 1ξ ξ v ∈ J, v ∈ Rd. (13)

Theorem 2. The stochastic derivative in the direction Ψt v is equal to

D x t xt tΨv vξ ξ0 1( ) = ( ) . (14)

Proof. By using the method of variation of constants, we can guess a representation for Du tξ0 in terms of the

first variation ξt1. Writing the differential for [ ]−ξ ξt u tD1 1 0 , we establish that a solution of (5) admits the formal

representation

∀ u ∈ J D x x B x u dsu t t

t

s s sξ ξ ξ ξ0 1

0

1 1 0( ) = ( ) [ ] ( )∫ − ( ) (15)

if, e.g., Eq. (2) has Lipschitz coefficients and [ ]−ξt1 1 is well defined. Substituting (13), we obtain equality (14).

A correct proof of representation (14) can be realized by the comparison of differentials. According to Lem-

ma 1, the process D xtΨvξ0( ) satisfies the equation

D x B ds B D dW F D dst

t

s s

t

s s s

t

s sΨ Ψ ΨΨv v vvξ ξ ξ ξ ξ ξ0

0

0

0

0 0

0

0 0( ) = ( ) + ′ − ′∫ ∫ ∫( )[ ] ( )[ ] . (16)

Substituting (13) and the equality D xtΨvξ0( ) = t xtξ1( )v in (16) and using representation (12) of the stochastic dif-

ferential for ξt x1( ) , we establish that Eq. (16) is identically true. The statement of Theorem 2 follows from the uni-queness of solutions.

3. Main Result

In the next theorem, we obtain a formula for the higher-order derivatives of the diffusion semigroup Pt f ( x ) =

E f ( ξt x0( ) ) by using the Malliavin-calculus approach. Note that, by virtue of (14), we can reduce the singular frac-

tion in (1) and eliminate the non-Lipschitz singularity of the inverse stochastic derivative, which usually arises in theMalliavin calculus [3 – 11].

We introduce the operator ϒv that acts according to the rule

ϒv K ( t, x, ω ) = t < v, ∂x > K + K0

1

∫ < Ψs v, dWs > – DΨv K

on smooth functionals defined on the Cartesian product of Rd and the Wiener space. In what follows, we denote by

∂xn f =

∂ ( )∂ … ∂

… =

n

j j j j

df x

x xn n1 1 1, ,

the n th derivative of a function f ∈ C dpol∞ ( )R , where C d

pol∞ ( )R is the class of con-

tinuously differentiable functions with derivatives of at most polynomial growth.


Theorem 3. The n th-order derivative ∂n Pt f ( x ) of the semigroup Pt admits the following representation:

∀ t > 0 ∀ f ∈ C dpol∞ ( )R vi ∈ Rd

⟨ v1 ⊗ … ⊗ vn , ∂n Pt f ( x ) ⟩ = 1 0

tf xn tE ( )( )ξ ϒvn

… ϒv1 1. (17)

Remark. For n = 1, the statement of Theorem 3 gives the Elworthy formula

∀ t > 0 ⟨ v, ∂ Pt f ( x ) ⟩ R

d = 1 0

0t

f xt

t

E ( )( ) ∫ξ ⟨ Ψs v, dWs ⟩ R

d ,

which was first obtained in [16] and then used in [17 – 19]. In comparison with the approach of inverse parabolicequations applied in [16 – 19], we develop a non-Lipschitz Malliavin-calculus approach and obtain the Elworthy for-mula as a special case of representations of arbitrary order.

Proof. We prove Theorem 3 by induction on n ≥ 0. The initial representation is realized by the classical for-

mula ( Pt f ) ( x ) = E f xt( )( )ξ0 .

Consider the following simple modification of (3): For an Ft-measurable differentiable functional G, we have

EGf t

t

( ) ∫ξ0

0

⟨ us, dWs ⟩ = ED Gfu t[ ]( )ξ0 = E ( ) + ⟨∂ ( ) ⟩f D G G f Dt u t u tξ ξ ξ0 0 0,

or

E EG f D f G u dW D Gt u t t

t

s s u⟨ ⟩∂ ( ) = ( ) ⟨ ⟩ −

∫ξ ξ ξ0 0 0

0

, , . (18)

By using relation (14) and the chain rule (6), we obtain

⟨ ∂ ⟩ ( ) = ∂ ( ) ( )+ +( ) ⟨ ( ) ⟩v vn x t t t nf x f x x10 0 1

1, ,ξ ξ ξ =

1 10 0 01 1t

f x D xt

D f xt t tn n⟨ ( ) ⟩ ( )∂ ( ) ( ) = ( )

+ +ξ ξ ξ, Ψ Ψv v . (19)

By using the formula of integration by parts (18), the induction assumption (17), and property (19) of the sto-chastic derivative, we get

⟨ v1 ⊗ … ⊗ vn , ∂n +

1

Pt f ( x ) ⟩ = ⟨ vn + 1 , ∂x ⟩ 1 0

tf xn tE ( )( )ξ ϒvn

… ϒv1 1

=

11

0

tfn n x tE[ ]⟨ ∂ ⟩ ( )+v , ξ ϒvn

… ϒv1 1 + 1 0

tfn tE ( )ξ ⟨ vn + 1 , ∂x ⟩ϒvn

… ϒv1 1

=

11

01t

D fn tn+ [ ]+

( )E Ψv ξ ϒvn … ϒv1

1 + 1 0

tfn tE ( )ξ ⟨ vn + 1 , ∂x ⟩ϒvn

… ϒv1 1

=

1 110

10

1 1 1tf t dW Dn t n x

t

s n s n n+ + +( ) ⟨ ∂ ⟩ + ⟨ ⟩ −

ϒ … ϒ

∫ +E ξ v v v v v, ,Ψ Ψ ,

i.e., we obtain the statement of the induction step.


The stochastic derivatives D D

xx

n

m

m tΨ Ψv v1

1… ∂∂

( )ξ ω, , which appear in the representation of the kernel

ϒ …ϒv vn 1

1, are solutions of coercive systems whose coefficients have at most polynomial growth by virtue of the

conditions imposed on B, F . The application of the methods presented in [11, 20] shows that the processes

ϒ …ϒv vn 1

1 are well defined.

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Higher-Order Relations for Derivatives of Nonlinear Diffusion Semigroups