C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds
We obtain the applications of approach [2, 5, 6] to the high order regularity of solutions to the parabolic Cauchy problem with globally non-Lipschitz coeffcients growing at the in nity of a noncompact manifold. In comparison to [2], where the semigroup properties were studied by application of nonl...
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irk-123456789-1242612017-09-24T03:03:15Z C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds Antoniouk, A.Val. We obtain the applications of approach [2, 5, 6] to the high order regularity of solutions to the parabolic Cauchy problem with globally non-Lipschitz coeffcients growing at the in nity of a noncompact manifold. In comparison to [2], where the semigroup properties were studied by application of nonlinear estimates on variations with use of local arguments of [11], i.e. for manifolds with the C² metric distance function, the developed below approach works for the general noncompact manifold with possible non-unique geodesics between distant points. 2008 Article C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds / A.Val. Antoniouk // Нелинейные граничные задачи. — 2008. — Т. 18. — С. 174-194. — Бібліогр.: 13 назв. — англ. 0236-0497 MSC (2000): 35K05, 47J20, 53B21, 58J35, 60H07, 60H10,60H30 http://dspace.nbuv.gov.ua/handle/123456789/124261 en Інститут прикладної математики і механіки НАН України |
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We obtain the applications of approach [2, 5, 6] to the high order regularity of solutions to the parabolic Cauchy problem with globally non-Lipschitz coeffcients growing at the in nity of a noncompact manifold. In comparison to [2], where the semigroup properties were studied by application of nonlinear estimates on variations with use of local arguments of [11], i.e. for manifolds with the C² metric distance function, the developed below approach works for the general noncompact manifold with possible non-unique geodesics between distant points. |
| format |
Article |
| author |
Antoniouk, A.Val. |
| spellingShingle |
Antoniouk, A.Val. C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds |
| author_facet |
Antoniouk, A.Val. |
| author_sort |
Antoniouk, A.Val. |
| title |
C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds |
| title_short |
C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds |
| title_full |
C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds |
| title_fullStr |
C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds |
| title_full_unstemmed |
C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds |
| title_sort |
c∞-regularity of non-lipshitz heat semigroups on noncompact riemannian manifolds |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/124261 |
| citation_txt |
C∞-regularity of non-Lipshitz heat semigroups on noncompact Riemannian manifolds / A.Val. Antoniouk // Нелинейные граничные задачи. — 2008. — Т. 18. — С. 174-194. — Бібліогр.: 13 назв. — англ. |
| work_keys_str_mv |
AT antonioukaval cregularityofnonlipshitzheatsemigroupsonnoncompactriemannianmanifolds |
| first_indexed |
2025-07-09T01:08:54Z |
| last_indexed |
2025-07-09T01:08:54Z |
| _version_ |
1837129620376256512 |
| fulltext |
174 Нелинейные граничные задачи 18, 174-193 (2008)
c©2008. A.Val. Antoniouk
C∞-REGULARITY OF NON-LIPSCHITZ HEAT
SEMIGROUPS ON NONCOMPACT RIEMANNIAN
MANIFOLDS
We obtain the applications of approach [2, 5, 6] to the high order regularity
of solutions to the parabolic Cauchy problem with globally non-Lipschitz coef-
ficients growing at the infinity of a noncompact manifold.
In comparison to [2], where the semigroup properties were studied by appli-
cation of nonlinear estimates on variations with use of local arguments of [11], i.e.
for manifolds with the C2 metric distance function, the developed below approach
works for the general noncompact manifold with possible non-unique geodesics
between distant points.
Keywords and phrases: heat parabolic equations, nonlinear diffusions on
manifolds, regular properties
MSC (2000): 35K05, 47J20, 53B21, 58J35, 60H07, 60H10,60H30
1.Introduction.
Below we discuss C∞-regularity with respect to the space para-
meter x for solutions of the second order heat parabolic equation
∂u(t, x)
∂t
=
{
1
2
d∑
σ=1
(
〈Aσ(x),
∂
∂x
〉
)2
+ 〈A0(x),
∂
∂x
〉
}
u(t, x),
(t, x) ∈ IR+ ×M
(1)
with initial data u(0, x) = f(x). The nonlinear, growing on the
infinity coefficients Aσ, A0 represent C∞ smooth vector fields on the
noncompact oriented C∞-smooth complete connected Riemannian
manifold without boundary M .
To obtain C∞-regularity properties of solutions to heat equations
(1) in the spaces of continuously differentiable functions we use that
the solution to (1) can be represented
u(t, x) = (Ptf)(x) = Ef(yx
t ) (2)
as a mean E of solution to the heat diffusion equation in the Ito-
Research is partially supported by grants of National Committee on Science
and Technolopgy Ukraine
C∞-regularity of heat semigroups on Riemannian manifolds 175
Stratonovich form
δyx
t = A0(y
x
t )dt+
d∑
σ=1
Aσ(yx
t )δW σ
t , yx
0 = x. (3)
Here W σ denote the independent Wiener processes on IRd, the mean
E is taken with respect to the corresponding Wiener measure on
space Ω = C0([0,∞), IRd). Under solutions to (3) it is understood a
continuous adapted integrable random process IR+ ×M 3 (t, x) →
yx
t ∈M such that ∀ f ∈ C∞
0 (M)
f(yx
t ) = f(x) +
∫ t
0
(A0f)(yx
s )ds+
d∑
σ=1
∫ t
0
(Aσf)(yx
s )δW σ
s . (4)
Since f(yx
t ) and (A·f)(yx
t ) are IR1-valued process, (4) represents Stra-
tonovich equation on real line IR1.
The approach of diffusion processes relates the C∞ properties of
semigroup Pt with the regularity properties of process yx
t since
1. from representation (2) and majorant theorem it is evident that
for continuous with respect to the initial data process x → yx
t
from f ∈ Cb(M) follows that the solution u(t, x) = (Ptf)(x) to
(1) is also a continuous bounded function u(t, x) ∈ Cb(M) for
all t > 0
and
2. due to (2) the derivatives of solution u(t, x) with respect to
the space parameter x ∈ M can be directly expressed via the
derivatives of initial function f and derivatives of process yx
t
with respect to the initial data x.
The already known approaches to the high order regularity of
solutions to (1), e.g. [8] and references within, were based on the
interpretation of the high order derivatives of process x → yx
t with
respect to the initial data x as elements y
(n)
t (x) = T (n)yx
t of the
high order tangent bundles T nM . However, due to the complicate
structure of the high order tangent bundles, the actual study of
the high order regularity of process yx
t and its semigroup Pt was
conducted in the local coordinate vicinities of manifold, leading to
the globally Lipschitz assumptions on the coefficients Aσ, A0 and
boundedness of curvature and all their derivatives. In comparison to
176 A.Val. Antoniouk
the case of linear manifold M = IRn, when under some monotone
assumptions on the coefficients of diffusion equation [10, 12] it is
possible to have a nonlinear growth of coefficients on the infinity, the
elaborated approach to the C∞-regularity of diffusions on noncompact
manifolds has not permitted to single out some kind of monotone
conditions.
In [2, 3] we noticed that, since, evolving in time, process yx
t
travels through different coordinate vicinities of manifold, its first
order derivative with respect to the initial data
∂yx
t
∂x
=
{
∂(yx
t )m
∂xk
}dimM
m,k=1
represents a covector field with respect to the local coordinates (xk) in
the vicinity of initial point x and becomes a vector field with respect
to the local coordinates (ym) in the vicinity where now travels process
yx
t , i.e.
∂yx
t
∂x
∈ T 1,0
yx
t
M ⊗ T 0,1
x M .
To preserve this tensorial invariance property with respect to the
coordinate systems (ym) in the image of flow x → yx
t , the new high
order variations
{
∇∇(n)yx
t
}
of process yx
t were introduced
1st variation.
{
∇∇(1)yx
t
}m
k
=
∂(yx
t )m
∂xk
,
high variations.
{
∇∇(n+1)yx
t
}m
k,j1,...,jn
=
= ∇x
k
{
∇∇(n)yx
t
}m
j1,...,jn
+ Γ m
p q(y
x
t )
{
∇∇(n)yx
t
}p
j1,...,jn
∂(yx
t )q
∂xk
,
(5)
where ∇x
k
{
∇∇(n)yx
t
}m
j1,...,jn
denotes a classical covariant derivative on
variable x
∇x
k
{
∇∇(n)yx
t
}m
γ
= ∂x
k
{
∇∇(n)yx
t
}m
γ
−
∑
j∈γ
Γ h
k j(x)
{
∇∇(n)yx
t
}m
γ|j=h
(6)
and
{
∇∇(n)yx
t
}m
γ|j=h
means substitution of index j in multi-index γ =
{j1, ..., jn} by h. Above indexes m, p, q correspond to the coordinates
in the vicinities, where travels process yx
t , indexes k, j, h to the coordi-
nate vicinities of initial data x.
In comparison to the approach of high order tangent bundles
and due to the presence of additional connection term Γ(yx
t ) in (5),
the nth order variation now represents a vector field with respect to
C∞-regularity of heat semigroups on Riemannian manifolds 177
variable yx
t and nth order covariant field with respect to variable x
∇∇(n)yx
t = {∇∇γ(y
x
t )m}|γ|=n ∈ Tyx
t
M ⊗ (T ∗
xM)⊗n
i.e. it is understood as a tensor and does not belong to the high order
tangent bundle T (n)M .
In this article we are going to apply results of [5, 6] to the
high order regularity of semigroup. The main tool for research gives
the following relation between high order covariant derivatives of
semigroup and initial function [2]
∇(n)
x Ptf(x) =
∑
j1+...+j`=n,`≥1
E 〈∇
(`)
yx
t
f(yx
t ),∇∇
(j1)yx
t ⊗ ...⊗∇∇(j`)yx
t 〉T 0,`
yx
t
M ,
(7)
where arise variations ∇∇(n)
x yx
t .
Unfortunately, the approach of nonlinear estimates on variations
[2] works only for manifolds with smooth Riemannian structures, in
particular, when the metric distance function is twice continuously
differentiable. It is not so for many C∞ manifolds with non-unique
geodesics between distant points, therefore the approach of [2] should
be modified.
The main result is the following. Suppose that the coefficients of
diffusion equation (3) and curvature of manifolds fulfill the following
conditions:
• coercitivity: ∃ o ∈ M such that ∀C ∈ IR+ ∃KC ∈ IR1 such
that ∀x ∈M
〈Ã0(x),∇
xρ2(o, x)〉 + C
d∑
σ=1
‖Aσ(x)‖2 ≤ KC(1 + ρ2(o, x)), (8)
where ρ(x, o) denotes the shortest geodesic distance between
points x, o ∈M ;
• dissipativity: ∀C,C ′ ∈ IR+ ∃KC ∈ IR1 such that ∀x ∈M ,
∀h ∈ TxM 〈∇Ã0(x)[h], h〉 + C
∑d
σ=1 ‖∇Aσ(x)[h]‖
2−
−C ′
∑d
σ=1〈Rx(Aσ(x), h)Aσ(x), h〉 ≤ KC‖h‖
2,
(9)
where Ã0 = A0 + 1
2
d∑
σ=1
∇AσAσ and
[R(A,B)C]m = R m
i jkA
iBjCk
178 A.Val. Antoniouk
denotes curvature operator, related with curvature tensor
R m
i jk =
∂Γ m
i j
∂xk
−
∂Γ m
i k
∂xj
+ Γ `
i jΓ
m
` k − Γ `
i kΓ
m
` j.
Due to the presence of additional curvature term conditions (8)-
(9) generalize the classical dissipativity and coercitivity condi-
tions [10, 12] from the linear Euclidean space to manifold and
represent a kind of monotonicity condition for diffusion processes.
• nonlinear behaviour of coefficients and curvature: for
any n there are constants k• such that for all j = 1, .., n and
∀ x ∈M
‖(∇)jÃ0(x)‖ ≤ (1 + ρ(x, o))k0 ,
‖(∇)jAα(x)‖ ≤ (1 + ρ(x, o))kα ,
‖(∇)jR(x)‖ ≤ (1 + ρ(x, o))kR.
(10)
Let ~qk = (q0, q1, ..., qn), qi ≥ 1 be a family of monotone functions
on IR+ of polynomial behaviour, that fulfill hierarchy
∀ i ≥ 1 qi(b)(1 + b)k/2 ≤ qi+1(b) ∀ b ≥ 0, (11)
related with some parameter k.
Denote by Cn
~q(k)
(M) the space of n-times continuously covariantly
differentiable functions on M , equipped with a norm
‖f‖Cn
~q(k)
(M) = max
i=0,...,n
sup
x∈M
‖(∇x)if(x)‖
qi(ρ2(x, o))
. (12)
Theorem 1. Under conditions (8)-(10) for any n ∈ IN there is k
such that the scale of spaces Cn
~q(k)
(M) is preserved under the action
of semigroup
∀ t ≥ 0 Pt : Cn
~q(k)
(M) → Cn
~q(k)
(M)
and there are constants K,M such that
∀ f ∈ Cn
~q (M) ‖Ptf‖Cn
~q
(M) ≤ KeMt‖f‖Cn
~q
(M). (13)
Proof is conducted in the following Sections.
C∞-regularity of heat semigroups on Riemannian manifolds 179
2. Preliminary study of C∞-regularity of process
yx
t with respect to the initial data x.
Since by (7) the existence of the high order derivatives of semi-
group Pt is related with C∞-regularity of process yx
t with respect to
x, we first discuss the necessary regular properties of variations.
Let us introduce a necessary definition for the parallel transport
h,yh
t
TT
b
a of high order variations. It is specially designed in order to
preserve the tensorial transformation law of T 1,0
y
h(z)
t
M⊗T 0,n
h(z)M -tensors
in the both domain and image of diffusion flow x → yx
t , when such
tensors move along random path [a, b] 3 z → (h(z), y
h(z)
t ) ∈M ×M .
Definition 2. The parallel transport of tensor u
(h(a),y
h(a)
t )
∈ T p,q
h(a)M⊗
T r,s
y
h(a)
t
M from point (h(a), y
h(a)
t ) along path (h(·), y
h(·)
t ) ∈ Lip([a, b],M)
represents a T p,q
h(z)M ⊗ T r,s
y
h(z)
t
M -tensor at each point (h(z), y
h(z)
t ), z ∈
[a, b] of this path. It is denoted by
h,yh
t
TT
(h(z),y
h(z)
t )
(h(a),y
h(a)
t )
u
(h(a),y
h(a)
t )
= Ψ(z) and
for its absolute derivative
ID
IDz
Ψ
(i/α)
(j/β)(z)
def
=
∂
∂z
Ψ
(i/α)
(j/β)(z)+
+
i∑
s=1
Γ is
k `(h(z)) Ψ
i1,..,is−1,k,is+1,..,ip/α
(j/β) (z)[h′(z)]`−
−
j∑
s=1
Γ k
js `(h(z)) Ψ
(i/α)
j1,..,js−1,k,js+1,.,jq/β(z)+
+
r∑
`=1
Γ α`
m n(y
h(z)
t )Ψ
(i/α1,...,α`−1,m,α`+1,...,αr)
(j/β) (z)
∂(y
h(z)
t )n
∂h(z)k
[h′(z)]k−
−
s∑
`=1
Γ m
β` n(y
h(z)
t )Ψ
(i/α)
(j/β1,...,β`−1,m,β`+1,...,βs)
(z)
∂(y
h(z)
t )n
∂h(z)k
[h′(z)]k
the norm ||
ID
IDz
Ψ(z)||T p,q
h(z)
M⊗T r,s
y
h(z)
t
M = 0 vanishes in L∞([a, b]) for
a.e. random ω ∈ Ω. Here multi-indexes (i) = (i1, ..., ip), (j) =
(j1, ..., jq) correspond to the T p,q
h(z)-tensorness of Ψ(z), correspondingly
(α) = (α1, ..., αr), (β) = (β1, ..., βs) to T r,s
y
h(z)
t
-tensorness of Ψ(z) in the
domain and image of mapping x→ yx
t .
180 A.Val. Antoniouk
Remark that the first two lines in the definition of the absolute
derivative
ID
IDz
Ψ
(i/α)
(j/β)(z) along path {h(z), y
h(z)
t }z∈[a,b] correspond to
the classical absolute derivative
D
Dz
Ψ
(i/α)
(j/β)(z) along path {h(z)}z∈[a,b].
The remaining two lines make the resulting expression to become the
invariantly defined tensor with respect to the coordinate transfor-
mations in vicinities, where travels process {y
h(z)
t }z∈[a,b].
Using the autoparallel property of the Riemannian connection
∂kgij(x) = Γ `
k i(x)g`j(x) + Γ `
k j(x)gi`(x), (14)
it is easy to check that the derivative of scalar product of T p,q
x M ⊗
T r,s
yx
t
M -tensors can be expressed in terms of the new type absolute
derivatives
d
dz
〈u(h(z)), v(h(z))〉T p,q
h(z)
M⊗T r,s
y
h(z)
t
M =
= 〈
ID
IDz
u(h(z)), v(h(z))〉T p,q
h(z)
M⊗T r,s
y
h(z)
t
M+
+〈u(h(z)),
ID
IDz
v(h(z))〉T p,q
h(z)
M⊗T r,s
y
h(z)
t
M (15)
and parallel transport operator
h,yh
t
TT constitutes a group ∀ c, d, e ∈
[a, b]
h,yh
t
TT
e
d
h,yh
t
TT
d
c =
h,yh
t
TT
e
c.
Then, taking any mixed tensor ψh(a) ∈ T p,q
h(a) ⊗ T r,s
y
h(a)
t
at point
(h(a), y
h(a)
t ) we have
d
dz
〈ψh(a),
h,yh
t
TT
a
zuh(z)〉T p,q
h(a)
⊗T r,s
y
h(a)
t
=
d
dz
〈
h,yh
t
TT
z
aψh(a), uh(z)〉T p,q
h(z)
⊗T r,s
y
h(a)
t
=
= 〈
h,yh
t
TT
z
aψh(a),
ID
IDz
uh(z)〉T p,q
h(z)
⊗T r,s
y
h(a)
t
= 〈ψh(a),
h,yh
t
TT
a
z
[
ID
IDz
uh(z)
]
〉T p,q
h(a)
⊗T r,s
y
h(a)
t
,
where we used that the derivative of parallel transport vanishes
ID
IDz
h,yh
t
TT
z
aψh(a) = 0.
Integrating on variable z ∈ [a, b] we obtain
< ψh(a),
∫ b
a
h,yh
t
TT
a
z
[
ID
IDz
uh(z)
]
dz >=
∫ b
a
d
dz
< ψh(a),
h,yh
t
TT
a
zuh(z) > dz =
C∞-regularity of heat semigroups on Riemannian manifolds 181
=< ψh(a),
h,yh
t
TT
a
zuh(z) >
z=b
z=a
=< ψh(a),
h,yh
t
TT
a
buh(b) − uh(a) > .
Since ψh(a) was arbitrary, this implies the invariant formula for the
increment of mixed tensors along Lipschitz paths
h,yh
t
TT
a
buh(b) − uh(a) =
∫ b
a
h,yh
t
TT
a
z
[
IDuh(z)
IDz
]
dz (16)
and, in particular, recovers a sense of the new type mixed absolute
derivative of T p,q
h ⊗ T r,s
yh
t
-tensors
d
dz
h,yh
t
TT
a
zuh(z) =
h,yh
t
TT
a
z
[
IDuh(z)
IDz
]
or
IDuh(z)
IDz
=
h,yh
t
TT
z
a
[
d
dz
h,yh
t
TT
a
zuh(z)
]
, z ∈ [a, b].
(17)
Therefore, since the high order variations ∇∇(n)yx
t represent a
particular case of T 0,n
x ⊗T 1,0
yx
t
-tensors, they should be related by similar
to (16) formulas. To find the sufficient monotone conditions on the
existence of the high order derivatives ∇∇(n)yx
t of process x → yx
t we
first construct the solutions y
(n)
t,x of the associated with (3) variational
system and then verify that they represent the high order ∇∇-deriva-
tives: y
(n)
t,x = ∇∇(n)yx
t , ∀n ∈ IN .
The main result about the C∞-regularity of process yx
t follows.
Here we also precise the influence of nonlinearity parameter k (10)
on the growth of high order derivatives.
Lemma 3. Under the conditions of Theorem 1 the new type variations
are related by a.e. integral formulas ∀ f ∈ C∞
0 (M), ∀n ∈ IN
f(y
h(b)
t ) − f(y
h(a)
t ) =
∫ b
a
< ∇f(y
h(z)
t ),∇∇y
h(z)
t [h′(z)] >T
y
h(z)
t
dz, (18)
∇∇(n)y
h(b)
t −
h,yh
t
TT
b
a
[
∇∇(n)y
h(a)
t
]
=
∫ b
a
h,yh
t
TT
b
z
[ [
∇∇(n+1)y
h(z)
t
]
[h′(z)]
]
dz
(19)
for any Lipschitz continuous path h ∈ Lip([a, b],M).
Moreover, they fulfill estimates
∀n ∈ IN ∃Mn E||∇∇(n)yx
t ||
2q ≤ e2qMnt(1 + ρ2(x, o))q(n−1)k. (20)
182 A.Val. Antoniouk
Remark that estimate (20) actually replaces the tool of nonlinear
estimate on variations, discussed e.g. in [2], for manifolds with not
everywhere C2-smooth square of metric distance function ρ2(x, z),
(x, z) ∈M ×M .
Proof. First note that under conditions of Theorem 1 there is a
unique strong solution yx
t to equation (3), which fulfills estimates on
the boundedness and continuity: ∃M ∀ q ≥ 1
[5, Th.5]: E (1 + ρ2(yx
t , o))
q
≤ eMqt (1 + ρ2(x, o))
q
,
[6, Th.6]: Eρ2q(yx
t , y
z
t ) ≤ eMqtρ2q(x, z).
(21)
Moreover, relation (18) was proved in [6, Th.8].
It remains to demonstrate (19) and estimate (20). Remark that
estimate (20) for i = 1 gives an alternative proof of [6, Th.7].
Recall that the differential equations on variations have form [2,
Th.9]
δ([∇∇yx
t ]mγ ) = −Γ m
p q(y
x
t ) [∇∇yx
t ]pγ δy
q +M m
γ αδW
α +Nm
γ dt (22)
with coefficients M m
γ α, Nm
γ , determined by
1. recurrence base for |γ| = 1, γ = {k}:
Mm
k α = ∇`A
m
α (yx
t )∇∇ky
`, Nm
k = ∇`A
m
0 (yx
t )∇∇ky
`; (23)
2. recurrence step
M m
γ∪{k} α = ∇∇kM
m
γ α +R m
p `q(∇∇γy
p)(∇∇ky
`)Aq
α, (24)
Nm
γ∪{k} = ∇∇kN
m
γ +R m
p `q(∇∇γy
p)(∇∇ky
`)Aq
0. (25)
The unique strong solution of variational system (22) can be const-
ructed either by gluing together the solutions of variational equations,
localized to the local coordinate vicinities of U ⊂ M on the random
time intervals of entering and leaving such vicinities, or with the use
of monotone approximations of system (22), similar to [1].
Taking the differential of norm of variational process we have [2,
Lemma 10]
d‖∇∇(i)yx
t ‖
2 = gγε(x) { gmn(∇∇γy
mM n
ε α + ∇∇εy
nM m
γ α)dW α+
+gmn(∇∇γy
mNn
ε + ∇∇εy
nNm
γ +M m
γ αM
n
ε α)dt+
C∞-regularity of heat semigroups on Riemannian manifolds 183
+
1
2
gmn(∇∇γy
m P n
ε + ∇∇εy
n Pm
γ )dt } (26)
with |γ| = |ε| = i and expressions Pm
γ are recurrently defined by
Pm
k = ∇y
`∇AαA
m
α · ∇∇ky
` − R(Aα,∇∇ky)Aα; (27)
Pm
γ∪{k} = ∇∇kP
m
γ + 2R m
p `qM
p
γ α(∇∇ky
`)Aq
α+
+(∇sR
m
p `q)(∇∇γy
p)(∇∇ky
`)Aq
αA
s
α +R m
p `q(∇∇γy
p)(∇∇kA
`
α)Aq
α+ (28)
+R m
p `q(∇∇γy
p)(∇∇ky
`)(∇AαAα).
Since in (28) Pm
γ∪{k} = ∇∇kP
m
γ + ..., the high order coefficient
permits representation
Pm
γ = ∇`∇AαA
m
α · ∇∇γy
` − R(Aα,∇∇γy)Aα+
+
∑
β1∪..∪βs=γ, s≥2
Kβ1,..,βs(∇∇β1y, ...,∇∇βsy)
with coefficients Kβ1,...,βs, depending on A0, Aα, R and their covariant
derivatives.
In the same way, due to (23)-(25), we have similar asymptotic
M m
γ α = ∇y
`A
m
α [∇∇γy
`]+
∑
β1∪...∪βs=γ, s≥2
K ′
β1,...,βs
(∇∇β1y, ...,∇∇βsy); (29)
N m
γ = ∇y
`A
0
α[∇∇γy
`] +
∑
β1∪...∪βs=γ, s≥2
K ′′
β1,...,βs
(∇∇β1y, ...,∇∇βsy)
with multilinear coefficients K ′, K ′′, depending on A0, Aα, R and their
covariant derivatives.
Therefore from (26) the principal part of differential is
d‖∇∇(i)yx
t ‖
2 = 2〈∇∇(i)y,∇y
`Aα[∇∇(i)y`]〉dW α+
+{2〈∇∇(i)y,∇y
` Ã0[∇∇
(i)y`]〉 +
d∑
α=1
‖∇Aα[∇∇(i)y] ‖2−
−
d∑
α=1
〈R(Aα,∇∇
(i)y)Aα,∇∇
(i)y〉 }dt+
+
∑
j1+...+js=i, s≥2
〈∇∇(i)y, {K1
j1,...,js,α(∇∇(j1)y, ...,∇∇(js)y)dW α+
+K2
j1,...,js
(∇∇(j1)y, ...,∇∇(js)y)dt}〉, (30)
i.e. the dissipativity condition arises in the principal part. Like before
the coefficients K1, K2 depend on covariant derivatives of A0, Aα, R.
184 A.Val. Antoniouk
Using asymptotic (30) we come to the dissipativity condition (9)
in principal part and additional terms with lower order variations
h(t) = E||∇∇(i)yx
t ||
2q ≤ h(0)+
+KE
∫∫ t
0
‖∇∇(i)yx
t ‖
2(q−1){dissipativity}C,C′(∇∇(i)yx
t ,∇∇
(i)yx
t )dt+
+
∑
j1+...+js=i, s≥2
E
∫ t
0
‖∇∇(i)yx
t ‖
2(q−1)〈∇∇(i)y,Kj1,...,js(∇∇
(j1)y, ...,∇∇(js)y)〉dt.
(31)
By inequality |xq−1y| ≤ |x|q/q + (q − 1)|y|q/q and (10)
E‖∇∇(i)y‖2(q−1)
∣∣∣Ki;j1,...,js(∇∇
(i)y;∇∇(j1)y, ...,∇∇(js)y)
∣∣∣ ≤
≤ E(1 + ρ2(o, yx
t ))
k/2‖∇∇(i)y‖2q−1‖∇∇(j1)y‖...‖∇∇(js)y‖ ≤
≤ CE‖∇∇(i)y‖2q + C ′
E(1 + ρ2(o, yx
t ))
qk‖∇∇(j1)y‖2q...‖∇∇(js)y‖2q
with k determined by nonlinearity parameters (10).
To transform the last term let us use the inductive assumption
(20) for lower order variations. By Gronwall-Bellmann and Hölder
inequalities (31) implies
h(t) ≤ eCth(0) +
∑
j1+...+js=i, s≥2
C ′
∫ t
0
eC(t−s)
E(1 + ρ2(o, yx
t ))
qk×
×||∇∇(j1)yx
t ||
2q · ... · ||∇∇(js)yx
t ||
2q ≤
≤ eCth(0) +
∑
j1+...+js=i, s≥2
e(C+C′)t sup
s∈[o,t]
(
E(1 + ρ2(o, yx
t ))
qkr0
)1/r0
×
×
s∏
p=1
(
E||∇∇(jp)yx
t ||
2qrp
)1/rp
≤
≤ e(C+C′+2qM)t
∑
j1+...+js=i, s≥2
(1 + ρ2)qk
s∏
p=1
(1 + ρ2)q(jp−1)k ≤
≤ e2qM ′t(1 + ρ2(o, yx
t ))
q(i−1)k,
(32)
which leads to (20).
C∞-regularity of heat semigroups on Riemannian manifolds 185
Finally, let us show how to prove (19). Making assumption that
the differential equation on the parallel transport
h,yh
t
TT
z
ay
(n)
t,h(a) of the
high order variation has similar to (22) form:
δ
[
h,yh
t
TT
z
a y
(n)
t,h(a)
]
=−Γ
(
h,yh
t
TT
z
a y
(n)
t,h(a), δy
h(b)
t
)
+
∑
α
K(n)
α (z)δW α+L(n)(z)dt,
(33)
the following relations are found: ∀ z ∈ [a, b]
ID
IDz
Kz
α = R(Ψz, Aα(y
h(z)
t ))y
(1)
t,h(z)[h
′(z)];
ID
IDz
Lz = R(Ψz, A0(y
h(z)
t ))y
(1)
t,h(z)[h
′(z)].
(34)
with the initial data K
(n)
α (a) = M
(n)
α , L(n)(a) = N (n) defined in (22)
due to
h,yh
t
TT
a
a = Id. These relations are proved in analogue to the
proof of [3, Th.7]. Indeed, taking the integral version of the parallel
transport equation
ID
IDz
(
h,yh
t
TT
z
ay
(n)
t,h(a)) = 0, the expression
∂
∂z
(
h,yh
t
TT
z
ay
(n)
t,h(a))
is written via the connection terms. The further application of Newton-
Leibnitz formula gives the local increments of
h,yh
t
TT
z
ay
(n)
t,h(a) − y
(n)
t,h(a) as
the integrals on [a, z] of these connection terms. Finally, calculating
the Stratonovich differential of these integral formulas, comparing
them with the representation (33) and proceeding further by scheme
[2, (3.11)-(3.19)] the relation (34) is found.
After that the application of (16) to (34) leads to
K
(n)
α (z) =
h,yh
t
TT
z
aM
(n)
α +
+
z∫∫
a
h,yh
t
TT
z
u
{
R
y
h(u)
t
(
h,yh
t
TT
u
a y
(n)
t,h(a), Aα(y
h(u)
t )
)
y
(1)
t,h(u)[h
′(u)]
}
du;
L(n)(z) =
h,yh
t
TT
z
aN
(n)+
+
z∫∫
a
h,yh
t
TT
z
u
{
R
y
h(u)
t
(
h,yh
t
TT
u
a y
(n)
t,h(a), A0(y
h(u)
t )
)
y
(1)
t,h(u)[h
′(u)]
}
du
(35)
186 A.Val. Antoniouk
To obtain relation (19), by schemes of [1] and [7, Sect.4.4-4.5] the
following two estimates on the continuity and regularity of variations
are required: for any Lipschitz continuous path h ∈ Lip([a, b],M)
E|| y
(n)
t,h(b)−
h,yh
t
TT
b
a y
(n)
t,h(a) ||
p
T 1,0
y
h(b)
t
⊗T 0,n
h(b)
≤ |b− a|p||h′||pL∞([a,b],TM)e
Kp,nt
×polp,n
(
1 + ρ(h(a), o) + |b− a| · ||h′||L∞([a,b],TM)
)
; (36)
E|| y
(n)
t,h(b)−
h,yh
t
TT
b
a y
(n)
t,h(a) − y
(n+1)
t,h(b)
[∫ b
a
h
T
b
z h
′(z)dz
]
||p
T 1,0
y
h(b)
t
⊗T 0,n
h(b)
≤
≤ |b− a|2p||h′||2p
L∞([a,b],TM)e
Kp,nt
×polp,n
(
1 + ρ(h(a), o) + |b− a| · ||h′||L∞([a,b],TM)
) (37)
with some polynomials polp,n (·), depending on the order of nonlinearity
k (10),
h
T
b
z denoting the classical parallel transport of tensor along
path h from h(z) to h(b).
By the theory of absolute continuous functions, estimate (36)
leads to the existence of derivative
ID
IDz
h,yh
t
TT
b
zy
(n)
t,h(z) and estimate (37)
calculates this derivative, leading to (19).
To obtain estimate (36), let us first note, that by (29)
M m
γ α(b)−
h,yh
t
TT
b
aM
m
γ α(a) = ∇y
`A
m
α (y
h(b)
t )[∇∇γy
`
t,h(b)]−
−
h,yh
t
TT
b
a
(
∇y
`A
m
α (y
h(a)
t )[∇∇γy
`
t,h(a)]
)
+
+
∑
β1∪...∪βs=γ, s≥2
{
K
′,h(b)
β1,...,βs
(∇∇β1yt,h(b), ...,∇∇βsyt,h(b)) −
−
h,yh
t
TT
b
a
(
K
′,h(a)
β1,...,βs
(∇∇β1yt,h(a), ...,∇∇βsyt,h(a))
)}
=
= ∇y
`A
m
α (y
h(b)
t )[∇∇γy
`
t,h(b)−
h,yh
t
TT
b
a∇∇γy
`
t,h(a)] +
+
{
∇y
`A
m
α (y
h(b)
t )−
h,yh
t
TT
b
a
[
∇y
`A
m
α (y
h(a)
t )
]}
[
h,yh
t
TT
b
a∇∇γy
`
t,h(a)]+
C∞-regularity of heat semigroups on Riemannian manifolds 187
+
∑
β1∪...∪βs=γ, s≥2
{
K
′,h(b)
β1,...,βs
−
h,yh
t
TT
b
aK
′,h(a)
β1,...,βs
}
(∇∇β1yt,h(b), ...,∇∇βsyt,h(b))+
+
∑
β1∪...∪βs=γ, s≥2
s∑
j=1
[
h,yh
t
TT
b
aK
′,h(a)
β1,...,βs
]
(
h,yh
t
TT
b
a∇∇β1yt,h(a), ...,
h,yh
t
TT
b
a∇∇βj−1
yt,h(a),
∇∇βj
yt,h(b)−
h,yh
t
TT
b
a∇∇βj
yt,h(a),∇∇βj+1
yt,h(b), ...,∇∇βsyt,h(b)
)
.
Due to (16) and the first order regularity of process yx
t on initial
data (18), multiples ∇y
`A
m
α (y
h(b)
t )−
h,yh
t
TT
b
a∇
y
`A
m
α (y
h(a)
t ) and K
′,h(b)
β1,...,βs
−
h,yh
t
TT
b
aK
′,h(a)
β1,...,βs
are represented as integrals on [a, b] with linear dependence
on factor h′. Thus, by equations (22), (29), (33) and (35), the principal
parts of equations on the continuity difference ε
(n)
t = y
(n)
t,h(b)−
h,yh
t
TT
b
a y
(n)
t,h(a) has form
δ(ε
(n)
t ) = −Γ(ε
(n)
t , δy
(h(b)
t )+
+
∑
α
{
∇Aα[ε
(n)
t ] + P
(n)
α (Aα, R, ε
(1), ..., ε(n−1))
}
δW α+
+
{
∇A0[ε
(n)
t ] + P
(n)
α (Aα, R, A0 ε
(1), ..., ε(n−1))
}
dt,
with linear with respect to factor h′ and integral on [a, b] terms
P
(n)
α , P
(n)
0 , depending in the polynomial way of coefficients Aα, A0,
curvature R and their covariant derivatives.
Therefore, proceeding like in the previous part of the proof (30)-
(32), singling out the dissipativity condition and using e
(n)
0 = 0, the
inequality (36) is proved in the inductive on the order of variation
way.
Similar, but more bookkeeping arguments work for the diffe-
rentiability difference ∆
(n)
t =y
(n)
t,h(b)−
h,yh
t
TT
b
a y
(n)
t,h(a)−y
(n+1)
t,h(b)
[∫ b
a
h
T
b
z h
′(z)dz
]
in (37), however there are applied relation like
Sβ1,...,βs(y
h(b)
t )−
h,yh
t
TT
b
aSβ1,...,βs(y
h(b)
t )−
−∇∇ySβ1,...,βs(y
h(b)
t )
[
∫ b
a
h,yh
t
TT
b
ay
(1)
t,h(z)[h
′(z)]dz
]
=
=
∫ b
a
∇∇ySβ1,...,βs(y
h(z)
t )
[
h,yh
t
TT
b
ay
(1)
t,h(z)[h
′(z)]
]
dz−
188 A.Val. Antoniouk
−∇∇ySβ1,...,βs(y
h(b)
t )
[
∫ b
a
h,yh
t
TT
b
ay
(1)
t,h(z)[h
′(z)]dz
]
=
=
∫ b
a
dz
∫ b
z
du
(
h,yh
t
TT
z
u∇∇
y∇∇ySβ1,...,βs(y
h(u)
t )
)
×
×
[
h,yh
t
TT
z
u
(
y
(1)
t,h(u)[h
′(u)]
)
, y
(1)
t,h(z)[h
′(z)]
]
.
to conclude that the differentials of difference expressions have form
δ(∆
(n)
t ) = −Γ(ε
(n)
t , δy
h(b)
t )+
+
∑
α
{
∇Aα[∆
(n)
t ] +Q
(n)
α (Aα, R, ∆(1), ...,∆(n−1))
}
δW α+
+
{
∇A0[ε
(n)
t ] +Q
(n)
α (Aα, R, A0 ∆(1), ...,∆(n−1))
}
dt
with quadratic with respect to factor h′ and integral on [a, b]2 multiples
Q
(n)
α , Q
(n)
0 . Due to ∆
(n)
0 = 0 this leads to (37) with powers 2p in the
r.h.s.
3.Proof of C∞-regularity of semigroup Pt
(Theorem 1).
First we are going to obtain the representation formula for deri-
vatives of semigroup via new type variations (7).
Theorem 4. For any f ∈ Cn
~q (M) the semigroup Ptf is n-times
continuously differentiable on x for any t ≥ 0. Its high order deriva-
tives are defined by (7).
Proof. Introduce notations
δm(f, x, t) =
∑
j1+...+j`=m,`≥1
E 〈∇
(`)
yx
t
f(yx
t ),∇∇(j1)yx
t ⊗ ...⊗∇∇(j`)yx
t 〉T 0,`
yx
t
M
(38)
for the left hand sides of (7). First we are going to demonstrate that
for any f ∈ Cn
~q (M) expressions δm(f, x, t) ∈ T 0,m
x M are continuous
on x ∈M for any m = 1, ..., n, t ≥ 0.
Let h ∈ Lip([a, b],M) be any Lipschitz path. Let’s apply (20) to
find majorant function for terms under expectation E in [a, b] 3 z →
δm(f, h(z), t). From (18) and ||∇xρ(x, o)|| ≤ 1 follows estimate
ρ(o, y
h(z)
t ) ≤ ρ(o.y
h(a)
t ) +
∫ z
a
||∇
y
h(θ)
t
ρ(o, y
h(θ)
t )|| · ||
dy
h(θ)
t
dθ
||dθ ≤
C∞-regularity of heat semigroups on Riemannian manifolds 189
≤ ρ(o, y
h(a)
t ) +
∫ b
a
||∇∇(1)y
h(θ)
t || · ||h′(θ)||dθ.
Due to f ∈ Cn
~q (M) it leads to
||∇(`)f(y
h(z)
t )|| ≤ ||f ||Cn
~q
p`(ρ
2(o, y
h(z)
t )) ≤
≤ Kp`
||f ||Cn
~q
(
1 + ρ(o, y
h(z)
t )
)2deg(p`)
≤
≤ Kp`
||f ||Cn
~q
(
1 + ρ(o, y
h(a)
t ) + ||h′||L∞[a,b]
∫ b
a
||∇∇(1)y
h(θ)
t ||dθ
)2deg(p`)
(39)
and the last expression provides uniform on z ∈ [a, b] majorant, which
is integrable due to estimates (20) and (21).
In a similar way we find majorant for variational processes in
expression δm(f, h(z), t), z ∈ [a, b]. Due to (19)
∀ z ∈ [a, b] || ∇∇(j)y
h(z)
t ||
y
h(z)
t
≤
≤ ||∇∇(j)y
h(a)
t ||
y
h(a)
t
+ ||h′||L∞[a,b]
∫ b
a
|| ∇∇(`+1)y
h(θ)
t ||
y
h(θ)
t
dθ
(40)
and the right hand side of (40) is integrable in any power due to (20).
Property (19) and majorants (39),(40) lead to a.e. continuity on
parameter z ∈ [a, b] of expressions under expectation E
in δm(f, h(z), t), m = 0, ..., n for f ∈ Cn
~q (M). The further application
of Lebesgue majorant theorem demonstrates the continuity of map-
pings
[a, b] 3 z → δm(f, h(z), t), m = 0, ..., n,
for any Lipschitz path h ∈ Lip([a, b],M) and f ∈ Cn
~q (M).
Since such continuity along paths h represents one of possible
characterizations of continuous mappings, we conclude the a.e. conti-
nuity of expressions δm
mapping M 3 x → δm(f, x, t) ∈ T 0,mM is continuous
for any f ∈ Cn
~q (M) and t ≥ 0, m = 0, ..., n.
Now we can recurrently prove the required relation ∇(m)Ptf(x) =
δm(f, x, t).
Base of recurrence (m = 1). Using representation Ptf(x) =
Ef(yx
t ) and (41) for ` = 0 we obtain
Ptf(h(b)) − Ptf(h(a)) = E
[
f(y
h(b)
t ) − f(y
h(a)
t )
]
=
190 A.Val. Antoniouk
= E
∫ b
a
< ∇f(y
h(z)
t ),∇∇(1)y
h(z)
t [h′(z)] > dz.
Due to the existence of majorants (39) and (40) for ` = 1, the
expectation and integral can be changed in order. We obtain that
for any h ∈ Lip([a, b],M)
Ptf(h(b)) − Ptf(h(a)) =
∫ b
a
E < ∇f(y
h(z)
t ),∇∇(1)y
h(z)
t [h′(z)] > dz
and by the theory of absolutely continuous functions conclude the
existence of derivative
dPtf(h(z))
dz
= E〈∇f(y
h(z)
t ),∇∇(1)y
h(z)
t [h′(z)]〉 = 〈δ1(f, h(z), t), h
′(z)〉.
Since δ1(f, x, t) is continuous on x, this leads to the existence of
continuous first order derivative ∇Ptf(x) and identity ∇xPtf(x) =
δ1(f, x, t).
Recurrence step. Suppose that we already proved relation
∇
(`)
x Ptf(x) = δ`(f, x, t) for any ` = 0, ..., m < n. Let us show it
for m + 1.
First note that from property
dy
h(z)
t
dz
= ∇∇(1)y
h(z)
t [h′(z)] (18) and
a.e. relations (19) follows a.e. relation
∀ ` = 0, n− 1 ∇(`)f(y
h(b)
t )−
h,yh
t
TT
b
a
[
∇(`)f(y
h(a)
t )
]
=
=
∫ b
a
h,yh
t
TT
b
z
(
∇(`+1)f(y
h(z)
t )
[
∇∇(1)y
h(z)
t [h′(z)]
])
dz
(41)
for any f ∈ Cn
0 (M). Taking cutoffs fχU with χU |U = 1, χU ∈
C∞
0 (M, [0, 1]) and tending U ↗M , representation (41) can be closed
to any f ∈ Cn
~q (M).
Consider the corresponding difference
∇(m)Ptf(h(b))−
h
T
b
a
[
∇(m)Ptf(h(a))
]
=
= E
∑
j1+...+j`=m, `≥1
[
〈∇(`)f(y
h(b)
t ), [∇∇(j1)y
h(b)
t ] ⊗ ...⊗ [∇∇(j`)y
h(b)
t ]〉T 0,`
y
h(b)
t
−
−
h
T
b
a〈∇
(`)f(y
h(a)
t ), [∇∇(j1)y
h(a)
t ] ⊗ ...⊗ [∇∇(j`)y
h(a)
t ]〉T 0,`
y
h(a)
t
]
.
C∞-regularity of heat semigroups on Riemannian manifolds 191
Relations (41) and (19) lead to
∑
j1+...+j`=m, `≥1
[
〈∇(`)f(y
h(b)
t ), [∇∇(j1)y
h(b)
t ] ⊗ ...⊗ [∇∇(j`)y
h(b)
t ]〉−
−
h
T
b
a〈∇
(`)f(y
h(a)
t ), [∇∇(j1)y
h(a)
t ] ⊗ ...⊗ [∇∇(j`)y
h(a)
t ]〉
]
=
=
∫ b
a
∑
j1+...+j`=m+1, `≥1
h
T
b
z
[
〈∇(`)f(y
h(z)
t ), [∇∇(j1)y
h(z)
t ] ⊗ ...
⊗[∇∇(j`)y
h(z)
t ]〉[h′(z)]
]
dz,
i.e. recover the structure of integrand in (38).
The existence of majorants (39) and (40) permits to change the
order of integration and expectation, leading to
∇(m)Ptf(h(b))−
h
T
b
a
[
∇(m)Ptf(h(a))
]
=
∫ b
a
h
T
b
z [δm+1(f, h(z), t)[h
′(z)]] dz.
Therefore the mapping [a, b] 3 z →
h
T
b
z
[
∇(m)Ptf(h(z))
]
is absolutely
continuous with derivative
d
h
T
b
z
[
∇(m)Ptf(h(z))
]
dz
=
h
T
b
z [δm+1(f, h(z), t)[h
′(z)]] .
Since δm+1(f, x, t) is continuous on x, we conclude that the (m+1)th
derivative of semigroup is represented by δm+1(f, x, t).
The final step of the proof of Theorem 4 lies in the verification
of estimate (13). It follows the scheme of [2, Th.15] with application
of estimates (20) instead of nonlinear estimates on variations.
Theorem 5. Under conditions of Theorem 1 estimate (13) holds.
Proof. We apply (20) and (21) to estimate the corresponding
192 A.Val. Antoniouk
seminorms
‖(∇x)iPtf(x)‖
T
(0,i)
x
qi(ρ2(x, o))
≤
≤
∑
j1+...+j`, `≥1
‖E 〈(∇y)`f(yx
t ) ,∇∇(j1)yx
t ⊗ ...⊗∇∇(j`)yx
t 〉T (0,i)
y
‖
T
(0,i)
x
qi(ρ2(x, o))
≤
≤
∑
j1+...+j`, `≥1
(
sup
yx
t ∈M
‖(∇y)`f(yx
t )‖
T
(0,`)
y
q`(ρ2(yx
t , o))
)
×
×
Eq`(ρ
2(yx
t , o))‖∇∇
(j1)yx
t ‖...‖∇∇
(j`)yx
t ‖
qi(ρ2(x, o))
≤ ‖f‖Cn
~q
×
×
∑
j1+...+j`, `≥1
(
Eq`+1
` (ρ2(yx
t , o))
)1/(`+1) ∏̀
m=1
(
E‖∇∇(jm)yx
t ‖
`+1
)1/(`+1)
qi(ρ2(x, o))
≤
≤ K2eM ′t‖f‖Cn
~q
∑
j1+...+j`, `≥1
q`(ρ
2(x, o))
∏̀
m=1
(1 + ρ2(x, o))k(jm−1)/2
qi(ρ2(x, o))
≤
≤ K2eM ′t‖f‖Cn
~q
∑
j1+...+j`, `≥1
q`(ρ
2(x, o))(1 + ρ2(x, o))k(i−`)/2
qi(ρ2(x, o))
,
leading to hierarchy (11). Above we also applied that for qi ≥ 1 of
polynomial behaviour there is K such that 1
K
(1 + b)deg(qi) ≤ qi(b) ≤
K(1 + b)deg(qi), so from (21) follows
E
[
qi(ρ
2(o, yx
t ))
]n
≤ Kn
E
[
1 + ρ2(o, yx
t )
]n·deg(qi) ≤
≤ Knen·deg(qi)Mt
[
1 + ρ2(o.x)
]n·deg(qi) ≤ K2nen·deg(qi)Mtqi(ρ
2(o, x)).
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Department of Nonlinear Analysis, Institute of
Mathematics NAS Ukraine,
Tereschenkivska str. 3, 01601 MSP Kiev-4, Ukraine
antoniouk@imath.kiev.ua
Received 15.05.07
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