Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
We study the dependence with respect to the initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may be not everywhere twice differentiable, we show that under some monotonicity conditions on coefficients and...
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nasplib_isofts_kiev_ua-123456789-1647592025-02-09T14:47:21Z Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds Неперервність за початковими умовами та підхід теорії абсолютно неперервних функцій до регулярносі першого порядку для нелінійних дифузiй на некомпактних рiманових багатовидах Antoniouk, A.Val. Antoniouk, A.Vict. Статті We study the dependence with respect to the initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may be not everywhere twice differentiable, we show that under some monotonicity conditions on coefficients and curvature of manifold there are estimates exponential in time on the continuity of diffusion process with respect to the initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to the initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on tangent space, nor uses embeddings of manifold to linear spaces of higher dimensions. Досліджено залежність за початковими умовами для розв'язків дифузійних рівнянь з глобально неліпшицевими коефіцієнтами на некомпактних багатовидах. Хоча функція метричної відстані може бути не скрізь двічі диференційовною, показано, що за певних умов монотонності на коефіцієнти та кривину багатовиду існують експоненціальні за часом оцінки на неперервність дифузійного процесу за початковими умовами. У поєднанні з методами теорії абсолютно неперервних функцій ці оцінки приводять до першого порядку регулярності розв'язків за початковими умовами. Запропонований підхід не використовує техніку моментів часу виходу процесу з локальних координатних околів, а також експоненціальних відображень з дотичного простору або вкладення багатовиду до лінійного простору більшої розмірності. 2008 Article Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds / A.Val. Antoniouk, A.Vict. Antoniouk // Український математичний журнал. — 2008. — Т. 60, № 10. — С. 1299–1316. — Бібліогр.: 14 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164759 519.217.4, 517.955.4, 517.956.4, 517.958:536.2 en Український математичний журнал application/pdf Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Antoniouk, A.Val. Antoniouk, A.Vict. Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds Український математичний журнал |
| description |
We study the dependence with respect to the initial data for solutions of diffusion equations with globally
non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may be not everywhere
twice differentiable, we show that under some monotonicity conditions on coefficients and curvature of
manifold there are estimates exponential in time on the continuity of diffusion process with respect to the
initial data.
These estimates are combined with methods of the theory of absolutely continuous functions to
achieve the first-order regularity of solutions with respect to the initial data. The suggested approach
neither appeals to the local stopping time arguments, nor applies the exponential mappings on tangent
space, nor uses embeddings of manifold to linear spaces of higher dimensions. |
| format |
Article |
| author |
Antoniouk, A.Val. Antoniouk, A.Vict. |
| author_facet |
Antoniouk, A.Val. Antoniouk, A.Vict. |
| author_sort |
Antoniouk, A.Val. |
| title |
Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds |
| title_short |
Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds |
| title_full |
Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds |
| title_fullStr |
Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds |
| title_full_unstemmed |
Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds |
| title_sort |
continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| topic_facet |
Статті |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/164759 |
| citation_txt |
Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds / A.Val. Antoniouk, A.Vict. Antoniouk // Український математичний журнал. — 2008. — Т. 60, № 10. — С. 1299–1316. — Бібліогр.: 14 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT antonioukaval continuitywithrespecttoinitialdataandabsolutecontinuityapproachtothefirstorderregularityofnonlineardiffusionsonnoncompactmanifolds AT antonioukavict continuitywithrespecttoinitialdataandabsolutecontinuityapproachtothefirstorderregularityofnonlineardiffusionsonnoncompactmanifolds AT antonioukaval neperervnístʹzapočatkovimiumovamitapídhídteorííabsolûtnoneperervnihfunkcíjdoregulârnosíperšogoporâdkudlânelíníjnihdifuzijnanekompaktnihrimanovihbagatovidah AT antonioukavict neperervnístʹzapočatkovimiumovamitapídhídteorííabsolûtnoneperervnihfunkcíjdoregulârnosíperšogoporâdkudlânelíníjnihdifuzijnanekompaktnihrimanovihbagatovidah |
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2025-11-27T00:46:05Z |
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| fulltext |
UDC 519.217.4, 517.955.4, 517.956.4, 517.958:536.2
A. Val. Antoniouk, A. Vict. Antoniouk (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
CONTINUITY WITH RESPECT TO THE INITIAL DATA
AND ABSOLUTE-CONTINUITY APPROACH
TO THE FIRST-ORDER REGULARITY
OF NONLINEAR DIFFUSIONS
ON NONCOMPACT MANIFOLDS
НЕПЕРЕРВНICТЬ ЗА ПОЧАТКОВИМИ УМОВАМИ
ТА ПIДХIД ТЕОРIЇ АБСОЛЮТНО НЕПЕРЕРВНИХ
ФУНКЦIЙ ДО РЕГУЛЯРНОСТI ПЕРШОГО ПОРЯДКУ
ДЛЯ НЕЛIНIЙНИХ ДИФУЗIЙ НА НЕКОМПАКТНИХ
РIМАНОВИХ БАГАТОВИДАХ
We study the dependence with respect to the initial data for solutions of diffusion equations with globally
non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may be not everywhere
twice differentiable, we show that under some monotonicity conditions on coefficients and curvature of
manifold there are estimates exponential in time on the continuity of diffusion process with respect to the
initial data.
These estimates are combined with methods of the theory of absolutely continuous functions to
achieve the first-order regularity of solutions with respect to the initial data. The suggested approach
neither appeals to the local stopping time arguments, nor applies the exponential mappings on tangent
space, nor uses embeddings of manifold to linear spaces of higher dimensions.
Дослiджено залежнiсть за початковими умовами для розв’язкiв дифузiйних рiвнянь з глобально не-
лiпшицевими коефiцiєнтами на некомпактних багатовидах. Хоча функцiя метричної вiдстанi може
бути не скрiзь двiчi диференцiйовною, показано, що за певних умов монотонностi на коефiцiєн-
ти та кривину багатовиду iснують експоненцiальнi за часом оцiнки на неперервнiсть дифузiйного
процесу за початковими умовами.
У поєднаннi з методами теорiї абсолютно неперервних функцiй цi оцiнки приводять до першого
порядку регулярностi розв’язкiв за початковими умовами. Запропонований пiдхiд не використовує
технiку моментiв часу виходу процесу з локальних координатних околiв, а також експоненцiаль-
них вiдображень з дотичного простору або вкладення багатовиду до лiнiйного простору бiльшої
розмiрностi.
1. Introduction. In this article we study the continuous dependence and the first-order
regularity with respect to the initial data for Ito – Stratonovich diffusion
δyx
t = A0(yx
t )dt+
d∑
α=1
Aα(yx
t )δWα
t , yx
0 = x, (1.1)
on noncompact oriented smooth complete connected Riemannian manifold M without
boundary. Here A0, Aα, α = 1, . . . , d, are globally defined C∞-smooth vector fields on
M and δWα denote Stratonovich differentials of one dimensional independent Wiener
processes Wα
t , α = 1, . . . , d.
c© A. VAL. ANTONIOUK, A. VICT. ANTONIOUK, 2008
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 10 1299
1300 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
Under the solution of (1.1) it is understood a continuous adapted process yx
t such that
for any C∞-function f with a compact support on manifold M the following stochastic
integral equation is satisfied
f(yx
t ) = f(x) +
t∫
0
(A0f)(yx
s )ds+
d∑
α=1
t∫
0
(Aαf)(yx
s )δWα
s . (1.2)
Traditionally the continuity of diffusion process yx
t with respect to the initial data,
i.e., estimate
∃Kp : Eρp(yx
t , y
z
t ) ≤ eKptρp(x, z), (1.3)
is obtained using the geodesic deviations formulas and Jacobi fields approach. The natu-
rally arising conditions are related with the global Lipschitz assumptions on coefficients
of diffusion equation and semiboundedness of curvature of manifold, e.g. [1 – 5].
It is known that under the same global Lipschitz assumptions and semiboundedness
of curvature the process yx
t is first-order regular with respect to the initial data x and
there is derivative
∂yx
t
∂x
, e.g. [1] (Ch. 4, § 3), [2] (Ch. VIII). The restriction on curva-
ture is related with the use of uniform exponential charts of manifold M with further
estimation of local difference expressions for derivatives
yx+εh
t − yx
t
ε
− ∂yx
t
∂x
[h]. (1.4)
Later in [6] (Ch. 4, § 8), by application of local stopping time techniques, it was demon-
strated that the mapping M 3 x → yx
t ∈ M represents a diffeomorphism till the first
explosion time. However, it is still not clear, what global assumptions on coefficients and
curvature, besides global Lipschitzness and semiboundedness of curvature, may lead to
non-explosion and first-order regularity of nonlinear diffusion processes on noncompact
manifolds.
In [7] it was found a way to avoid the application of geodesic deviation’s techniques.
In [8] these results were used to prove the non-explosion of nonlinear diffusion yx
t , i.e.,
the existence and uniqueness of solutions of (1.1) for all t ≥ 0. The proposed conditions
on the coefficients of diffusion equation and curvature represent a manifold analogue
of coercitivity and dissipativity conditions, known previously for nonlinear monotone
equations on linear spaces [9, 10].
In this article we prove that under the same conditions process yx
t continuously
depends on the initial data x. Moreover, using the methods of absolute continuous
functions theory, we demonstrate that the existence of first-order derivative
∂yx
t
∂x
with
respect to the initial data is a direct consequence of the continuity estimates (1.3).
In Section 2 we formulate the main result of the article. Sections 3 and 4 are devoted
to the proof of continuous dependence of diffusion process yx
t with respect to the initial
data x. In comparison to the Euclidean space with C∞-smooth square of metric distance
ρ2(x, y) = ‖x− y‖2, for the general manifold the square of metric distance ρ2(x, y),
defined as a shortest geodesic distance
ρ2(x, z) = inf
1∫
0
|γ̇(`)|2d`, γ(0) = x, γ(1) = z
for γ̇(`) =
∂
∂`
γ(`), (1.5)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 10
CONTINUITY WITH RESPECT TO THE INITIAL DATA AND ABSOLUTE-CONTINUITY ... 1301
may be not twice differentiable everywhere, i.e., the following second-order operator:
Lf(x, z) =
{
AI
0(x) +AII
0 (z) +
1
2
d∑
α=1
(
AI
α(x) +AII
α (z)
)2
}
f(x, z)
with AI(x), AII(z) acting correspondingly on the first x and second z variables of
function f(x, z), may be undefined.
Therefore, the direct application of Ito formula
Eρp(yx
t , y
z
t ) = ρp(x, z) +
t∫
0
E(Lρp)(yx
s , y
z
s )ds (1.6)
in order to obtain (1.3) from upper bounds
Lρp(x, z) ≤ Kpρ
p(x, z) (1.7)
becomes impossible.
In Section 3 we develop results of [7, 8] and replace the strong estimates (1.7) by
the weak estimates on operators of structure (1.6), acting on the metric function on the
product of manifolds M ×M . The main difference from [8] is that we have to work
with two point functions ρ(x, z) instead of estimation of ρ(o, x) for some fixed o ∈M .
In Section 4, following the arguments of [11], we apply weak estimates on operators
L to show that for coercitive and dissipative diffusion there is a constant K such that
process
ρ2(yx
t , y
z
t )−K
t∫
0
ρ2(yx
s , y
z
s )ds
represents a supermartingale. This, in fact, replaces the Ito formula approach (1.6) and
leads to (1.3).
In Section 5 we demonstrate that the solution y(1)
t (x) of the first-order variational
equation represents the first-order derivative y(1)
t (x) =
∂yx
t
∂x
with respect to the initial
data.
First we construct special coordinate systems xi = ρ(oi, x) in small local vicinities
of manifold. The use of these particular coordinates and special transfer of relation (1.2)
from manifold M to IRdim M permit to obtain the first-order regularity from continuity
estimates (1.3).
Namely, estimate (1.3) implies that for Lipschitz continuous path [a, b] 3 u →
→ h(u) ∈M on manifold the mapping
[a, b] 3 u→ y
h(u)
t (1.8)
is Lipschitz continuous, i.e., for a.e. u ∈ [a, b] there is derivative θt(u) =
dy
h(u)
t
du
with
|θt(u)| ∈ L∞
(
[a, b]× [0, T ], Lp(Ω)
)
for all T > 0, p ≥ 1. By definition of solution (1.2)
we have for f ∈ C∞0 (M)
b∫
a
〈
∇f(yh(u)
t ), θt(u)
〉
du = f(yh(b)
t )− f(yh(a)
t ) =
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 10
1302 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
= f(h(b))− f(h(a)) +
t∫
0
[
(Aαf)(yh(b)
s )− (Aαf)(yh(a)
s )
]
δWα
s +
+
t∫
0
[
(A0f)(yh(b)
s )− (A0f)(yh(a)
s )
]
ds =
=
b∫
a
〈
∇f(h(u)), h′(u)
〉
du+
t∫
0
b∫
a
〈
∇(Aαf)(yh(u)
s ), θs(u)
〉
du
δWα
s +
+
t∫
0
b∫
a
〈
∇(A0f)(yh(u)
s ), θs(u)
〉
du
ds. (1.9)
Removing the integral
∫ b
a
. . . ds we obtain equation on the first-order variation for pro-
cess θt(u). Therefore the derivative with respect to the initial data θt(u) must represent
solution to the first-order variational equation θt(u) = y
(1)
t (h(u)). In Section 5 formal
reasoning (1.8), (1.9) is made rigorous.
Finally remark that the use of arbitrary paths h ∈ Lip
(
[a, b],M
)
permits to avoid the
separate conditions on the curvature of manifold, related with the existence of uniform
exponential charts, like in [1, 2]. Moreover, the application of absolute continuous func-
tions theory has also given a possibility to avoid the use of pure stochastic techniques
of local stopping times, necessary for the estimation of difference expressions (1.4) in
local coordinate systems, like in [4, 6]. Actually, we demonstrate that the first-order
regularity is a direct consequence of continuity estimates (1.3).
2. Main result. Let us suppose that the coefficients of equation (1.1) and curvature
tensor of manifold fulfill the following assumptions:
coercitivity: ∃ o ∈M such that ∀C ∈ R+ ∃KC ∈ R1 such that ∀x ∈M
〈
Ã0(x),∇xρ2(o, x)
〉
+ C
d∑
α=1
‖Aα(x)‖2 ≤ KC(1 + ρ2(o, x)); (2.1)
dissipativity: ∀C, C ′ ∈ R+ ∃KC ∈ R1 such that ∀x ∈M, ∀h ∈ TxM
〈
∇Ã0(x)[h], h
〉
+ C
d∑
α=1
∥∥∇Aα(x)[h]
∥∥2−
− C ′
d∑
α=1
〈
Rx(Aα(x), h)Aα(x), h
〉
≤ KC‖h‖2, (2.2)
where Ã0 = A0 +
1
2
∑d
α=1
∇Aα
Aα and notation ∇H[h] means covariant derivative
in direction h (
∇H(x)[h]
)i = ∇jH
i(x) · hj . (2.3)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 10
CONTINUITY WITH RESPECT TO THE INITIAL DATA AND ABSOLUTE-CONTINUITY ... 1303
and
[
R(A, h)A
]m = R m
p `qA
pA`hq for (1,3) — curvature tensor with components
R m
p `q =
∂Γ m
p `
∂xq
−
∂Γ m
p q
∂x`
+ Γ j
p `Γ
m
j q − Γ j
p qΓ
m
j `; (2.4)
nonlinear behaviour of coefficients and curvature: for any n there are constants
k0, kα, kR such that for all j = 1, . . . , n and ∀x ∈M :
‖(∇)jÃ0(x)‖ ≤ (1 + ρ(x, z))k0 ,
‖(∇)jAα(x)‖ ≤ (1 + ρ(x, z))kα , (2.5)
‖(∇)jR(x)‖ ≤ (1 + ρ(x, z))kR .
Denote by Lip
(
[a, b],M
)
the space of Lipschitz paths on [a, b] with values in M . It
is formed from continuous paths h ∈ C
(
[a, b],M
)
such that ∃Kh ∀ c, d ∈ [a, b] there
is estimate on metric distance ρ(h(c), h(d)) ≤ Kh|c − d|. In particular, by theory of
absolute continuous functions this means that ‖h′‖ ∈ L∞
(
[a, b], TM
)
and Lipschitzness
constant Kh = sup
z∈[a,b]
‖h′(z)‖Th(z)M .
Theorem 2.1. Under conditions (2.1), (2.2), and (2.5) the solution yx
t of diffusion
equation (1.1) is differentiable with respect to the initial data.
Its derivative y(1)
t (x) =
∂yx
t
∂x
represents a unique solution to the first-order varia-
tional equation, written in local coordinates
δ[y(1)
t (x)]mk = −Γ m
p q(y
x
t )
[
(y(1)
t (x)
]p
k
δ(yx
t )q +∇pA
m
α (ξx
t )
[
y
(1)
t (x)
]p
k
δWα
t +
+∇pA
m
0 (ξx
t )
[
y
(1)
t (x)
]p
k
dt (2.6)
with initial data y(1)
0 (x) = I given by identity matrix.
Moreover, for any path h ∈ Lip([a, b],M) and f ∈ C∞0 (M) a.e. integral relation is
true:
f(yh(b)
t )− f(yh(a)
t ) =
b∫
a
〈
∇f(yh(z)
t ), y(1)
t (h(z))
[
h′(z)
]〉
T
y
h(z)
t
dz. (2.7)
Under solution of (2.6) it is understood a continuous adapted integrable process
R+ ×M 3 (t, x) −→
−→ y
(1)
t (x) ∈ L∞
(
[0, T ], Lp(Ω, Tyx
t
M ⊗ T ∗xM)
)
, for all T > 0, p > 1,
such that for any f ∈ C∞0 (M), h ∈ TxM〈
∇f(yx
t ), y(1)
t (x)[h]
〉
Tyx
t
M
=
〈
∇f(x), y(1)
0 (x)[h]
〉
TxM
+
+
t∫
0
〈
∇(Aαf)(yx
s ), y(1)
t (x)[h]
〉
δWα
s +
t∫
0
〈
∇(A0f)(yx
s ), y(1)
t (x)[h]
〉
ds. (2.8)
The proof is conducted in further sections.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 10
1304 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
3. Weak estimates on diffusion generators. Consider open set U⊂M with com-
pact closure U and function ζU such that
√
ζU ∈ C∞0 (M, [0, 1]) and ζU (z) = 1 for
z ∈ U, 0 ≤ ζU < 1 otherwise. Consider differential operator on M ×M
LUf(x, z) = ζU (x)ζU (z)Lf(x, z).
Being a second-order differential operator with localized coefficients, LU corresponds
to the localized Stratonovich diffusion yU
t (x, z) on M ×M
δyU
t (x, z) =
d∑
α=1
√
ζU (yI,U
t )
√
ζU (yII,U
t )
{
AI
α(yI,U
t ) +AII
α (yII,U
t )
}
δWα +
+ ζU (yII,U
t )
{
ζU (yI,U
t AI
0(y
I,U
t ) −
− 1
2
√
ζU (yI,U
t )
d∑
α=1
(Aα
√
ζU )(yI,U
t )AI
α(yI,U
t )
}
dt +
+ζU (yI,U
t )
{
ζU (yII,U
t AII
0 (yII,U
t ) −
− 1
2
√
ζU (yII,U
t )
d∑
α=1
(Aα
√
ζU )(yII,U
t )AII
α (yII,U
t )
}
dt, (3.1)
with initial data yU
0 (x, z) = (x, z), where yI,U
t and yII,U
t are first and second com-
ponents of process yU
t = (yI,U
t , yII,U
t ) on the product M ×M. Remark that, due to
property ζU
∣∣
U
= 1, for initial data x, z ∈ U process yU
t (x, z) coincides with process
(yx
t , y
z
t ) till the first exit time t ≤ τ(ω) = inf
{
t : yx
t (ω) 6∈ U or yz
t (ω) 6∈ U
}
.
Equation (3.1) has globally Lipschitz coefficients with all bounded derivatives, there-
fore it has unique solution that C∞-regularly depends on the initial data x [4 – 6, 11].
Its diffusion semigroup (PU
t f)(x) = Ef(yU
t (x, z)) preserves the space C∞0,+(M ×M)
of non-negative C∞-functions with compact support.
Main result of this section lies in the weak uniform with respect to U estimates on
generators LU .
Theorem 3.1. Under conditions (2.1), (2.2), and (2.5) there is K such that ∀ ζU ∈
C∞0 (M, [0, 1]) with ζU
∣∣
U
= 1 and ∀ϕ ∈ C∞0,+(M ×M)∫
M×M
([LU ]∗ϕ)ρ2(x, z)dσ(x)dσ(z) ≤ K
∫
M×M
ϕ(x, z)ρ2(x, z)dσ(x)dσ(z). (3.2)
Proof. As [LU ]∗ = [ζU (x)ζU (z)L]∗ = L∗ζU (x)ζU (z), estimate (3.2) will follow
from the weak estimate on operator L: ∃K ∀ψ ∈ C∞0,+(M ×M)∫
M×M
(L∗ψ(x, z))ρ2(x, z)dσ(x)dσ(z) ≤ K
∫
M×M
ψ(x, z)ρ2(x, z)dσ(x)dσ(z) (3.3)
if one substitutes first ψ(x, z) = ζU (x)ζU (z)ϕ(x, z) and then applies 0 ≤ ζU ≤ 1.
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CONTINUITY WITH RESPECT TO THE INITIAL DATA AND ABSOLUTE-CONTINUITY ... 1305
Similar to [8] (ff. (16) – (18)) the following representation for the left-hand side of
(3.3) is fulfilled ∫
M×M
(L∗ψ(x, z))ρ2(x, z)dσ(x)dσ(z) =
= lim
ε→0+
∫
M×M
ψ(x, z)
{
ρ2(ηε
0(x), η
ε
0(z))− ρ2(x, z)
ε
+
+
1
2
d∑
α=1
ρ2(ηε
α(x), ηε
α(z)) + ρ2(η−ε
α (x), η−ε
α (z))− 2ρ2(x, z)
ε2
}
dσ(x)dσ(z). (3.4)
Here ηε
0, η
ε
α denote the shifts along vector fields A0, Aα and η0
· (x) = x. Operator
L∗ has representation L∗ =
1
2
∑d
α=1
(A∗α)2 + A∗0 in terms of adjoint fields X∗f =
= −(divX)f −Xf to vector field X.
Now let us estimate fractions in the right-hand side of (3.4). In the vicinity of
geodesic γ(`), ` ∈ [0, 1] from γ(0) = x to γ(1) = z that minimizes (1.5) consider
smooth vector field H. Introduce a family of paths
[0, 1]× (−δ, δ) 3 (`, s) → γ(`, s) ∈M
such that at s = 0 path γ(`, s)
∣∣
s=0
= γ(`) gives geodesic γ above. Parameter s
corresponds to the evolution along field H:
∂
∂s
γ(`, s) = H(γ(`, s)). (3.5)
In the following lemma we find estimates on the first- and second-order differences in
(3.4). FieldH will be chosen later to beH(`, s) = A0(γ(`, s)) orH(`, s) = Aα(γ(`, s))
correspondingly.
Lemma 3.1 ([8], Lemma 2). The following estimates on difference operators on
metric function are fulfilled
ρ2(γ(0, ε), γ(1, ε))− ρ2(o, x)
ε
≤
≤
1∫
0
∂
∂s
∣∣∣∣
s=0
∣∣γ̇(`, s)∣∣2d`+
ε∫
0
1∫
0
∣∣∣∣ ∂2
∂s2
∣∣γ̇(`, s)∣∣2∣∣∣∣ d` ds, (3.6)
ρ2(γ(0, ε), γ(1, ε)) + ρ2(γ(0, ε), γ(1,−ε))− 2ρ2(o, x)
ε2
≤
≤
1∫
0
∂2
∂s2
∣∣∣∣
s=0
∣∣γ̇(`, s)∣∣2d`+
1
2
ε∫
0
1∫
0
∣∣∣∣ ∂3
∂s3
∣∣γ̇(`, s)∣∣2∣∣∣∣ d` ds, (3.7)
where we used notation γ̇(`, s) =
∂
∂`
γ(`, s).
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1306 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
The right-hand side terms in (3.6), (3.7) have the following representations in terms
of field H :
∂
∂s
∣∣γ̇(`, s)∣∣2 = 2
〈
γ̇,∇H[γ̇]
〉
, (3.8)
1
2
∂2
∂s2
|γ̇(`, ε)|2 =
∣∣∇H[γ̇]
∣∣2 − 〈
γ̇, R(H, γ̇)H
〉
+
〈
γ̇,∇(∇HH)[γ̇]
〉
. (3.9)
The third derivative has representation
∂3
∂s3
∣∣γ̇(`, s)∣∣2 =
〈
γ̇,D[γ̇]
〉
with operator D that
depends on the field H up to its third-order covariant derivative and on curvature tensor
and its covariant derivative.
Next we use Lemma 3.1 to find weak estimates on operator L. Due to (3.3), this
ends the proof of Theorem 3.1.
Lemma 3.2. Under conditions (2.1), (2.2), and (2.5) there is constant K such that
∀ψ ∈ C∞0,+(M ×M)∫
M×M
(L∗ψ(x, z))ρ2(x, z)dσ(x)dσ(z) ≤ K
∫
M×M
ψ(x, z)ρ2(x, z)dσ(x)dσ(z). (3.10)
Proof. Coincides with the proof of [8] (Lemma 3). It is only necessary to choose
fields H to be H(γ(`, s)) = A0(γ(`, s)) or H(γ(`, s)) = Aα(γ(`, s)) for the first- and
second-order differences in (3.4).
In [8] (Lemma 3) we used additional factor (1−`) in field H, i.e., the choice of field
H was H(γ(`, s)) = (1− `)A0(γ(`, s)) or H(γ(`, s)) = (1− `)Aα(γ(`, s)), this made
point γ(1, s) = z to be the same for all s. Therefore, in calculation [8] (ff. (30) – (33))
does not appear additional multiple ` and it is a little simpler.
4. Estimates on the continuity with respect to the initial data. Now we apply
weak estimates (3.2) to show, similar to [11], that some process on manifold represents
a supermartingale. Thus we overcome the difficulties, related with the direct application
of the Ito formula arguments (1.6), (1.7).
Recall that process Xt is supermartingale with respect to the flow of σ-algebras Ft if
for all 0 ≤ s ≤ t it is fulfilled E(Xt|Fs) ≤ Xs, where E(·|Fs) denotes the conditional
expectation with respect to σ-algebra Fs.
Theorem 4.1. Under conditions (2.1), (2.2), and (2.5) there is an independent of
U⊂M constant K such that process
ρ2(yU
t (x, z))−K
t∫
0
ρ2(yU
s (x, z))ds (4.1)
represents an integrable supermartingale with respect to the canonical flow of σ-algebras
Ft, related with d-dimensional Wiener process Wα
t , α = 1, . . . , d, in (1.1). Notation
ρ2(yU
t (x, z)) means the geodesic distance between first and second components of pro-
cess yU
t (x, z) (3.1) on the product M ×M.
Moreover, the solution of equation (1.1) continuously depends on the initial data,
i.e., estimate (1.3) is true.
Proof follows lines of proofs of Lemma 4 in [8]. Since we work below with the
components of process yU
t (x, z) and have to make several relevant modifications, we
outline its main steps.
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CONTINUITY WITH RESPECT TO THE INITIAL DATA AND ABSOLUTE-CONTINUITY ... 1307
Recall, that semigroup PU
t , generated by localized process yU
t (x, z) (3.1), preserves
the space C∞0,+(M ×M) of non-negative C∞-functions with compact support, so the
integrals below are finite. The application of weak estimate (3.10) leads to
∀ϕ ∈ C∞0,+(M ×M) :
d
dt
∫
M×M
ϕ(x, z)
{
PU
t ρ
2(·, ·)
}
(x, z)dσ(x)dσ(z) =
=
d
dt
∫
M×M
{
[PU
t ]∗ϕ
}
(x, z)ρ2(x, z))dσ(x)dσ(z) =
=
∫
M×M
[LU ]∗
{
[PU
t ]∗ϕ
}
(x, z)ρ2(x, z)dσ(x)dσ(z) =
=
∫
M×M
[L]∗
(
ζU (x)ζU (z)
{
[PU
t ]∗ϕ
}
(x, z)
)
ρ2(x, z)dσ(x)dσ(z) ≤
≤ K
∫
M×M
{
[PU
t ]∗ϕ
}
(x, z)ρ2(x, z)dσ(x)dσ(z) =
= K
∫
M×M
ϕ(x, z)
{
PU
t ρ
2(·, ·))
}
(x, z)dσ(x)dσ(z)
where we applied L1 = 0, used that due to the compactness of support of function
ζU ≥ 0 the integrand ψ = ζU (x)ζU (z)
{
[PU
t ]∗ϕ
}
belongs to space C∞0,+(M ×M),
then applied (3.10) and property ζU ≤ 1.
Therefore for all ϕ ∈ C∞0,+(M ×M) we have estimate∫
M×M
ϕ(x, z)
{
PU
t ρ
2(·, ·)
}
(x, z)dσ(x)dσ(z) ≤
≤
∫
M×M
ϕ(x, z)
ρ2(x, z)) +K
t∫
0
{
PU
s ρ
2(·, ·))
}
(x, z)ds
dσ(x)dσ(z)
and, removing ϕ, conclude its pointwise consequence
{
PU
t ρ
2(·, ·)
}
(x, z) ≤ ρ2(x, z)) +K
t∫
0
{
PU
s ρ
2(·, ·)
}
(x, z)ds. (4.2)
The Markov property of process yU
t (x, z) implies for its semigroup PU
t that
(PU
t f)(yU
s (x, z)) = E
(
f(yU
t+s(x, z))
∣∣Fs
)
, t, s ≥ 0, (4.3)
which permits to substitute instead of x, z initial data yU
τ (x, z) in (4.2). From (4.3) we
have
E
(
ρ2(yU
t+τ (x, z))
∣∣Fτ
)
= (PU
t ρ
2(·, ·))(yU
τ (x, z)) ≤
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 10
1308 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
≤ ρ2(yU
τ (x, z)) +K
t∫
0
{
PU
s ρ
2(·, ·)
}
(yU
τ (x, z))ds =
= ρ2(yU
τ (x, z)) +K E
t+τ∫
τ
ρ2(yU
s (x, z))ds
∣∣∣∣Fτ
, (4.4)
which means that process (4.1) is supermartingale. Indeed, the supermartingale property
E
ρ2(yU
t+τ (x, z))−K
t+τ∫
0
ρ2(yU
s (x, z))ds | Fτ
≤
≤ ρ2(yU
τ (x, z))−K
τ∫
0
ρ2(yU
s (x, z))ds
coincides with (4.4). The integrability of process (4.1) follows from the compactness of
the closure of set
{
x : ζU (x) > 0
}
.
Next suppose that initial data x, z ∈ U. Introduce stopping time
τU (ω) = inf
{
t ≥ 0: yU
t (x, z) 6∈ U × U
}
.
The Doob – Meyer free choice theorem, e.g. [12] (Ch. VI, § 2), permits to substitute
any finite stopping times S, T such that 0 ≤ S ≤ T into the supermartingale property
E(XT |FS) ≤ XS . Let’s apply it with S = 0 and T = t ∧ τU to supermartingale (4.1).
Due to E(·|F0) = E(·) we have
mt = Eρ2(yU
t∧τU (x, z)) ≤ ρ2(x, z) +KE
t∧τU∫
0
ρ2(yU
s (x, z))ds ≤
≤ m0 +KE
t∫
0
ρ2(yU
s∧τU (x, z))ds = m0 +K
t∫
0
msds,
where yU
s∧τU (x, z) = yU
τU (x, z) for s ≥ τU is a stopped process on the boundary of U
and we enlarged the upper limit of integral.
From Gronwall – Bellmann inequality we conclude
Eρ2(yU
t∧τU (x, z))) ≤ eKtρ2(x, z). (4.5)
Let Un denote the open ball at point o with radius n, then for sufficiently large n
points x, z ∈ Un. Consider measurable random set Vn(t) =
{
ω : ∀s ∈ [0, t] yx
t (ω) ∈ Un
and yz
t (ω) ∈ Un
}
that corresponds to paths of processes yx
t (ω), yz
t (ω) (1.1), staying
inside of set Un till time t. Then
(
yx
t (ω), yz
t (ω)
)
= yUn
t∧τUn
(x, z, ω) for all ω ∈ Vn(t)
and (4.5) leads to
E 1Vn(t)ρ
2(yx
t , y
z
t ) = E 1Vn(t)ρ
2(yUn
t∧τUn
(x, z)) ≤
≤ Eρ2(yUn
t∧τUn
(x, z)) ≤ eKtρ2(x, z), (4.6)
with characteristic function 1Vn(t) of set Vn(t).
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CONTINUITY WITH RESPECT TO THE INITIAL DATA AND ABSOLUTE-CONTINUITY ... 1309
Due to the non-explosion lim
n→∞
τUn(ω) = ∞ [8], for a.e. ω both paths
yx
t (ω), yz
t (ω) sooner or later completely lie in all sets Un for n ≥ n0 with sufficiently
large n0. Therefore sequence Vn(t) is increasing to the full probability space, i.e., lower
limit lim
n→∞
1Vn(t)(ω) = 1 a.e. The application of Fatoux lemma
(
i.e., that for fn ≥ 0
the lower limits fulfill
∫
lim
n→∞
fndµ ≤ lim
n→∞
∫
fndµ
)
to the left-hand side of (4.6)
leads to the statement
E ρ2(yx
t , y
z
t ) ≤ lim
n→∞
Eρ2(yUn
t∧τUn
(x, z)) ≤ eKtρ2(x, z). (4.7)
The theorem is proved.
In the following theorem we generalize Theorem 4.1 to the polynomials of metric
function. Remark that the convex function of supermartingale should not be a super-
martingale again, so the use of coercitivity and dissipativity conditions (2.1), (2.2) is
essential for existence of appropriate constant KQ in (4.9).
Theorem 4.2. Let Q be a nonnegative monotone polynomial function on half-line
R+ such that
∃C ∀ z ≥ 0 z Q′(z) ≤ C Q(z), z
∣∣Q′′(z)∣∣ ≤ C Q′(z). (4.8)
Under conditions (2.1), (2.2) and (2.5) there is constant KQ such that uniformly on
vicinity U the process
Q
(
ρ2(yU
t (x, z))
)
−KQ
t∫
0
Q
(
ρ2(yU
s (x, z))
)
ds (4.9)
is an integrable supermartingale.
Moreover, a unique solution yx
t to problem (1.1) fulfills the estimate on the continuity
with respect to the initial data
EQ
(
ρ2(yx
t , y
z
t )
)
≤ eKQtQ
(
ρ2(x, z)
)
. (4.10)
Proof is done in analogue to [8] (Theorem 5).
5. Absolute-continuity approach to the first-order regularity with respect to the
initial data. Consider arbitrary smooth path h ∈ C∞([a, b],M) that starts at point
x = h(a). To obtain the equation on the first variation y(1)
t (x)
[
h′(a)
]
=
d
du
∣∣∣∣
u=0
y
h(u)
t
let us formally differentiate (1.2)〈
∇f(yx
t ), y(1)
t (x)
[
h′(a)
]〉
=
=
〈
∇f(x), h′(a)
〉
+
∑
α
t∫
0
〈
∇(Aαf)(yx
s ), y(1)
s (x)
[
h′(a)
]〉
δWα
s +
+
t∫
0
〈
∇(A0f)(yx
s ), y(1)
s (x)
[
h′(a)
]〉
ds. (5.1)
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1310 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
This leads to the equation on the first-order variation in local coordinates
e.g. [13, 14]
δ
[
y
(1)
t (x)
]m
j
= −Γm
p q(y
x
t )
[
y
(1)
t (x)
]p
j
δ(yx
t )q+
+
∑
α
∇pA
m
α (yx
t )
[
y
(1)
t (x)
]p
j
δWα
t +∇pA
m
0 (yx
t )
[
y
(1)
t (x)
]p
j
dt (5.2)
with initial data y(1)
0 = I.
In next theorem we give sufficient conditions for the solvability of equation (5.2),
see also e.g. [13] (Theorem 15).
Theorem 5.1. Suppose that conditions (2.1), (2.2), and (2.5) are fulfilled. Then
equation (5.2) has unique solution, i.e., exists a continuous adapted process y(1)
t (x)[h′]
with values in Tyx
t
M, such that for any f ∈ C∞0 (M) and h′ ∈ TxM relation (5.1) is
true.
In particular,
∀ p ≥ 1 ∃Kp such that E
∥∥y(1)
t (x)
∥∥p
Tyx
t
⊗T∗x M
≤ eKpt. (5.3)
Proof. Since equation on the first variation is formally calculated as derivative of
(1.1), it also has the following equivalent form to (5.2) (see [13], (3.1)):
δ
([
y(1)(x)m
j
])
=
∑
α
∂Am
α (yx
t )
∂yp
[
y
(1)
t (x)
]p
j
δWα
t +
∂Am
0 (yx
t )
∂yp
[
y
(1)
t (x)
]p
j
dt. (5.4)
Therefore it represents an equation with the locally Lipschitz coefficients. Standard
results about the solvability of finite-dimensional diffusion equations lead to the local
existence and uniqueness of its solutions till the first explosion time, e.g. [2, 4, 6, 11].
The non-explosion of process y(1)
t (x) follows from the following representation of
the local differential of its norm
d‖y(1)
t (x)‖2 = 2
〈
y
(1)
t (x),∇y
`Aα
[[
y
(1)
t (x)
]`
] 〉
dWα+
+
{
2
〈
y
(1)
t (x),∇y
` Ã0
[[
y
(1)
t (x)
]`
] 〉
+
d∑
α=1
∥∥∥∇Aα
[
y
(1)
t (x)
] ∥∥∥2
−
−
d∑
α=1
〈
R(Aα, y
(1)
t (x))Aα, y
(1)
t (x)
〉}
dt,
proved as a base of recurrence (for γ = ∅) in Lemma 13 [13] (see also [13], (4.28) for
i = 1).
Therefore the dissipativity condition (2.2) arises in the second line. Due to the
initial data y(1)
0 (x) =
∂x
∂x
= Id and Gronwall – Bellmann inequality, it leads to the
non-explosion estimate (5.3), i.e., to the existence and uniqueness of solution y(1)
t (x) to
(5.2), (5.4) for all t ≥ 0.
The theorem is proved.
In the following theorem we apply the theory of absolute continuous functions to
show the first-order regularity of diffusion process.
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CONTINUITY WITH RESPECT TO THE INITIAL DATA AND ABSOLUTE-CONTINUITY ... 1311
Theorem 5.2. Under conditions (2.1), (2.2), and (2.5) process yx
t is differentiable
with respect to the initial data. For any Lipschitz continuous path h ∈ Lip([a, b],M) its
derivative
dy
h(z)
t
dz
is represented by solution of the first-order variational equation (5.2)
dy
h(z)
t
dz
= y
(1)
t (h(z))
[
h′(z)
]
and a.e. integral relation (2.7) is fulfilled.
Proof. Let us prove relation (2.7) for functions f with sufficiently small support,
then the use of the decomposition of identity guarantees (2.7) for arbitrary f ∈ C∞0 (M).
The main idea of proof is following: for any small vicinity U ⊂M we are going to
construct the globally defined functions θi ∈ C(M), i = 1, . . . ,dimM, such that
1) the superposition with diffusion θi ◦ yx
t is regular, i.e., exists
∂θi(yx
t )
∂x
(Step 1);
2) any function f ∈ C∞0 (M) with compact support in U has an unique representa-
tion in terms on θi (Step 2), i.e.,
∃f̃ ∈ C∞(Rdim M ,R) such that ∀x ∈ U f(x) = f̃(θ1(x), . . . , θdim M (x)).
These properties guarantee that for any f ∈ C∞0 (U) expression f(yx
t ) =
= f̃
(
θ1(yx
t ), . . . , θdim M (yx
t )
)
is again regular with respect to the initial data.
Finally, in Step 3 we will use the last property to derive the equation on deriva-
tive
∂yx
t
∂x
.
Step 1. Construction of special coordinate system and use of continuity esti-
mates (4.10) to guarantee the existence of derivative with respect to the initial data.
First note that for any point o ∈ M there is a sufficiently small vicinity U =
= U(o) 3 o and points outside of this vicinity oi = oi(o) 6∈ U, i = 1, . . . ,dimM , such
that they generate the smooth local coordinate’s mapping in U
θ(x) =
(
θi(x)
)dim M
i=1
: U → Rdim M by rule θi(x) = ρ(oi, x).
Recall that above θi(x) = ρ(oi, x) denotes the shortest geodesic distance from x ∈ U
to point oi, i = 1, . . . ,dimM.
Vicinity U must be also chosen so that for any point x ∈ U there is no point z ∈ U
such that it has the same coordinates θ(x) = θ(z). Last assumption actually means that
the points oi(o) are sufficiently far from U, so that the “phantom” images of U
Ph(U) =
{
z 6∈ U : θ(z) = θ(x) for some x ∈ U
}
do not intersect with U. Moreover, by varying the size of U(o) and points oi(o) 6∈ U(o)
we can guarantee that “phantom” images of U are far from U :
∃ ε > 0 ∀ o ∈M dist
(
U(o), Ph(U(o))
)
> 2ε. (5.5)
Next, since for sufficiently small U the coordinate mapping θ : U → Rdim M be-
comes bijection with continuous inverse, the set θ(U) =
{
θ(x) : x ∈ U
}
⊂ Rdim M ,
being a preimage of open set U, is open too. Moreover, due to the role of parame-
ter ε (5.5), i.e., the absence of equidistant points, the mapping θ is C∞-smooth in the
ε-vicinity of U with C∞-smooth inverse.
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1312 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
Finally remark that θi(x) are globally defined continuous functions on the manifold
M, but only in the ε-vicinity of U they form the coordinate system. In particular, due
to the triangle inequality ∣∣ρ(oi, x)− ρ(oi, z)
∣∣ ≤ ρ(x, z), (5.6)
functions θi are globally Lipschitz continuous with constant 1.
Introduce processes
θi(yx
t ) = ρ(oi, y
x
t ),
then from (4.10) and (5.6) we have
∀ p ≥ 1, T > 0 : sup
t∈[0,T ]
E
∣∣θi(yx
t )− θi(yz
t )
∣∣p ≤
≤ sup
t∈[0,T ]
E
[
ρ(yx
t , y
z
t )
]p ≤ eKpT ρp(x, z).
Therefore path [a, b] ∈ z → θi(yh(z)
t ) ∈ L∞
(
[0, T ], Lp(Ω,W)
)
is Lipschitz continuous
for Lipschitz continuous h ∈ Lip ([a, b],M):
∀ c, d ∈ [a, b] : sup
t∈[0,T ]
E|θi
(
y
h(c)
t
)
− θi
(
y
h(d)
t
)
|p ≤ |c− d|eKpT ‖h′‖L∞([a,b],TM).
By theory of absolute continuous functions there exists derivative
dθi(yh(z)
t )
dz
∈ L∞
(
[a, b]× [0, T ], Lp(Ω,W)
)
with Lipschitzness constant
sup
z∈[a,b],t∈[0,T ]
E
∥∥∥∥∥dθi(yh(z)
t )
dz
∥∥∥∥∥
p
T
y
h(z)
t
M⊗T∗
h(z)M
≤ eKpT ‖h′‖p
L∞([a,b],TM) (5.7)
and we have a.e. relation
θi
(
y
h(b)
t
)
− θi
(
y
h(a)
t
)
=
b∫
a
dθi(yh(z)
t )
dz
dz. (5.8)
Above by W we denoted Wiener measure, related with process {Wα
t }α.
Step 2. Construction of unique Rdim M -representations of functions in small coor-
dinate vicinities.
Next note that any function f ∈ C∞0 (U) with compact support in U there is a
smooth function f̃ ∈ C∞0 (Rdim M ) that provides its unique coordinate representation in
terms of coordinates θ(x)
f(x) = 1U (x)f̃(θ(x)), (5.9)
where 1U (x) denotes the characteristic function of U.
At the first step, function f̃ is defined as the coordinate version of f in coordinates θ.
Then it is continued to all Rdim M by zero outside of the open set θ(U). The character-
istic 1U (x) is added to avoid the “phantom” copies of function f outside of vicinity U,
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CONTINUITY WITH RESPECT TO THE INITIAL DATA AND ABSOLUTE-CONTINUITY ... 1313
which appear at the equidistant points z such that θi(x) = ρ(oi, x) = ρ(oi, z) = θi(z),
i = 1, . . . ,dimM. Remark also that property (5.9) is not true for all functions f on M
because θ(x) can not form a global coordinate system on M.
However, the presence of factor 1U (x) in (5.9) does not influence further calculations
because the compact support of f completely lies in the open set U. For example, the
differential operations do not feel the factor 1U (x)
(Af)(x) = (∇Af)(x) = 1U (x)
dim M∑
j=1
∂j f̃(θ(x))∇Aθ
j(x) =
= 1U (x)
dim M∑
j=1
(Ãj∂j f̃)(θ(x)) = 1U (x) (Ãf̃)
(
θ(x)
)
with local coordinates Ãj(θ(x)) = Aθj(x) of vector field A in the vicinity U.
Remark that each function Ãj can be uniquely extended to C∞0 -function over all
Rdim M : for points θ(x) ∈ Rdim M with x ∈M such that dist(x,U) ≤ ε it is defined by
formula
Ãj
(
θ(x)
)
= χε
(
dist(x,U)
)
Aθj(x)
and by zero at all other points of Rdim M . A fixed and independent on o ∈ M function
χε ∈ C∞(R+, [0, 1]) is such that χε(0) = 1 and χε(λ) = 0, λ ≥ ε, parameter ε
appeared in (5.5). In other words, we take the coordinates Ãj in the image of ε-vicinity
of U, leave them unchanged in θ(U) and crop them to zero outside of θ-image of the
ε-vicinity of U. Since the mapping θ is C∞-smooth in the ε-vicinity of U with C∞-
smooth inverse, we obtain C∞0 -regularity of Ãj . However, due to f ∈ C∞0 (U) the
function Af ∈ C∞0 (U), i.e., the values of field A outside of U do not play a role.
Remark also that the representation (5.9) is more comfortable than, for example, the
use of embeddings of manifold M into Euclidean spaces M ⊂ Rn of higher dimensions
n > dimM. In this case one should use the global coordinates of Rn instead of the local
coordinates of M, e.g. [6, 11] and references therein. In particular, such approach leads
to complicate work with the continuation of coefficients of equation from embedded
submanifold M ⊂ Rn to Rn and forces to enter additional projectors from Rn to M for
embedded Rn-versions of (1.1).
One more advantage is that for functions inside of U we have their unique represen-
tations f̃ in terms of local coordinates θi(x), i.e., a unique function f̃ on Rdim M . This
property will permit us below to make all calculations for local functions f ∈ C∞0 (U)
in linear coordinate space Rdim M .
Step 3. Derivation of stochastic equation for derivative
dθ(yh(u)
s )
du
and its relation
with the first-order variation process y(1)
s (h(u)).
Since the superposition of smooth finite function and Lipschitz continuous function
is also Lipschitz continuous, we have from (5.8)
f
(
y
h(b)
t
)
− f
(
y
h(a)
t
)
= 1U
(
y
h(z)
t
)
f̃
(
θ(yh(z)
t )
) ∣∣∣∣z=b
z=a
=
=
b∫
a
1U
(
y
h(z)
t
) dim M∑
j=1
∂j f̃
(
θ(yh(z)
t )
) dθj(yh(z)
t )
dz
dz. (5.10)
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1314 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
From another side, the definition of solution yx
t (1.2) leads to the representation for
difference
f
(
y
h(b)
t
)
− f
(
y
h(a)
t
)
= f
(
h(b))− f(h(a)
)
+
+
∑
α
t∫
0
[
(Aαf)
(
yh(b)
s
)
− (Aαf)
(
yh(a)
s
)]
δWα
s +
+
t∫
0
[
(A0f)
(
yh(b)
s
)
− (A0f)
(
yh(a)
s
)]
ds =
= 1U (h(b))f̃
(
θ(h(b))
)
− 1U (h(a))f̃
(
θ(h(b))
)
+
+
∑
α
t∫
0
[
1U (yh(b)
s )
(
Ãαf̃
)(
θ(yh(b)
s )
)
− 1U
(
yh(a)
s
) (
Ãαf̃
)(
θ(yh(a)
s )
)]
δWα
s +
+
t∫
0
[
1U
(
yh(b)
s
) (
Ã0f̃
)(
θ(yh(b)
s )
)
− 1U
(
yh(a)
s
) (
Ã0f̃
)(
θ(yh(a)
s )
)]
ds. (5.11)
Again applying that the superposition of smooth function Ãf̃ ∈ C∞0 (Rd) and Lip-
schitz continuous map [a, b] → θi
(
y
h(z)
s
)
is also Lipschitz continuous, we obtain from
(5.10) and (5.11) that
b∫
a
1U
(
y
h(z)
t
) dim M∑
j=1
∂j f̃
(
θ(yh(z)
t )
) dθj(yh(z)
t )
dz
dz =
=
b∫
a
1U (h(z))
dim M∑
j=1
∂j f̃
(
θ(h(z))
) dθj(h(z))
dz
dz+
+
∑
α
t∫
0
b∫
a
1U
(
yh(z)
s
) dim M∑
j=1
(
∂jÃαf̃
)(
θ(yh(z)
s )
) dθj(yh(z)
s )
dz
dz
δWα
s +
+
t∫
0
b∫
a
1U
(
yh(z)
s
) dim M∑
j=1
(
∂jÃ0f̃
)(
θ(yh(z)
s )
) dθj(yh(z)
s )
dz
dz
ds.
Due to (5.3) and (5.7) the terms under integrals above are in L∞
(
[a, b] × [0, T ],
Lp(Ω,W)
)
, p ≥ 1, T > 0. So the order of integrals
∫ b
a
and
∫ t
0
can be changed. As
h ∈ Lip([a, b],M) and [a, b] were arbitrary, the integrands under
∫ b
a
must coincide: for
almost all z ∈ [a, b]
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CONTINUITY WITH RESPECT TO THE INITIAL DATA AND ABSOLUTE-CONTINUITY ... 1315
1U
(
y
h(z)
t
) dim M∑
j=1
∂j f̃
(
θ(yh(z)
t )
) dθj(yh(z)
t )
dz
=
= 1U (h(z))
dim M∑
j=1
∂j f̃
(
θ(h(z))
) dθj(h(z))
dz
+
+
∑
α
t∫
0
1U
(
yh(z)
s
) dim M∑
j=1
(
∂jÃαf̃
)(
θ(yh(z)
s )
) dθj(yh(z)
s )
dz
δWα
s +
+
t∫
0
1U
(
yh(z)
s
) dim M∑
j=1
(
∂jÃ0f̃
)(
θ(yh(z)
s )
) dθj(yh(z)
s )
dz
ds.
Turning back to the invariant notations and fields on M we obtain〈
∇f
(
y
h(z)
t
)
,
dy
h(z)
t
dz
〉
=
〈
∇f(h(z)),
dh(z)
dz
〉
+
+
∑
α
t∫
0
〈
∇(Aαf)
(
yh(z)
s
)
,
dy
h(z)
s
dz
〉
δWα
s +
t∫
0
〈
∇(A0f)
(
yh(z)
s
)
,
dy
h(z)
s
dz
〉
ds.
(5.12)
Here we used that when y
h(z)
s ∈ U all terms with index j represent coordinates of
corresponding tensor-invariant objects. From another side, when yh(z)
s 6∈ U terms with
index j are no more coordinates, but as the multiple 1U (yh(z)
s ) = 0 and supports of f,
Aαf, A0f lie in U, these terms are also invariant, being defined by zero outside of U.
Relation (5.12) is a further advantage of relation (5.9): it is fulfilled for the process
yx
t that may many times enter and leave vicinity U. Therefore we do not need to restrict
the consideration till the first exit times and use local arguments.
Finally notice that equations (5.12) and (5.1) have the same structure. Due to the co-
inciding initial data
dθj(h(z))
dz
=
[
h′(z)
]j
and the uniqueness of solutions for equation
(5.1), the first variation coincides with the derivative with respect to the initial data
y
(1)
t (h(z))
[
h′(z)
]
=
dy
h(z)
t
dz
.
After substitution of this relation into (5.10) we come to (2.7).
The theorem is proved.
1. Belopolskaja Y. I., Daletskii Y. L. Stochastic equations and differential geometry. – Berlin: Kluwer
Acad. Publ., 1996. – 294 p.
2. Elworthy K. D. Stochastic differential equations on manifolds // London Math. Soc. Lect. Note Ser.
– 1975. – 70.
3. Grigor’yan A. Analytic and geometric background of recurrence and non-explosion of the Brownian
motion on Riemannian manifolds // Bull. Amer. Math. Soc. – 1999. – 36, № 2. – P. 135 – 249.
4. Hsu E. P. Stochastic analysis on manifolds // Grad. Stud. Math. – 2002. – 38.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 10
1316 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
5. Ikeda N., Watanabe S. Stochastic differential equations and diffusion processes. – Dordrecht: North-
Holland Publ., 1981.
6. Kunita H. Stochastic flows and stochastic differential equations. – Cambridge Univ. Press, 1990.
7. Antoniouk A. Val. Upper bounds on second order operators, acting on metric function // Ukr. Math.
Bull. – 2007. – 4, № 2. – P. 7.
8. Antoniouk A. Val., Antoniouk A. Vict. Non-explosion for nonlinear diffusions on noncompact mani-
folds // Ukr. Math. J. – 2007. – 59, № 11. – P. 1454 – 1472.
9. Krylov N. V., Rozovskii B. L. On the evolutionary stochastic equations // Contemp. Probl. Math. –
1979. – 14. – P. 71 – 146.
10. Pardoux E. Stochastic partial differential equations and filtering of diffusion processes // Stochastics.
– 1979. – 3. – P. 127 – 167.
11. Stroock D. An introduction to the analysis of paths on a Riemannian manifold // AMS Math. Surv.
and Monogr. – 2002. – 74.
12. Meyer P. A. Probability and potentials. – Blaisdell Publ. Co., 1966.
13. Antoniouk A. Val, Antoniouk A. Vict. Regularity of nonlinear flows on noncompact Riemannian
manifolds: differential vs. stochastic geometry or what kind of variations is natural? // Ukr. Math.
J. – 2006. – 58, № 8. – P. 1011 – 1034.
14. Antoniouk A. Val., Antoniouk A. Vict. Nonlinear calculus of variations for differential flows on
manifolds: geometrically correct introduction of covariant and stochastic variations // Ukr. Math.
Bull. – 2004. – 1, № 4. – P. 449 – 484.
Received 20.06.07
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