Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds
We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee non-explosion of solutions in a finite time. This property leads to the existence and uniqueness of solutions for corresponding stochastic differential equation with globally non-Lipschitz coe...
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509488259268608 |
|---|---|
| author | Antoniouk, A. Val. Antoniouk, A. Vict. Антонюк, О. Вал. Антонюк, О. Вік. |
| author_facet | Antoniouk, A. Val. Antoniouk, A. Vict. Антонюк, О. Вал. Антонюк, О. Вік. |
| author_sort | Antoniouk, A. Val. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:53:28Z |
| description | We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee non-explosion of solutions in a finite time.
This property leads to the existence and uniqueness of solutions for corresponding stochastic differential equation with globally non-Lipschitz coefficients.
Proposed approach is based on the estimates on diffusion generator, that weakly acts on the metric function of manifold.
Such estimates enable us to single out a manifold analogue of monotonicity condition on the joint behaviour of the curvature of manifold and coefficients of equation. |
| first_indexed | 2026-03-24T02:41:54Z |
| format | Article |
| fulltext |
UDC 519.217.4, 517.955.4, 517.956.4
A. Val. Antoniouk, A. Vict. Antoniouk (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
NON-EXPLOSION AND SOLVABILITY
OF NONLINEAR DIFFUSION EQUATIONS
ON NONCOMPACT MANIFOLDS∗
ВIДСУТНIСТЬ ВИБУХУ ТА IСНУВАННЯ РОЗВ’ЯЗКIВ
ДЛЯ НЕЛIНIЙНИХ ДИФУЗIЙНИХ РIВНЯНЬ
НА НЕКОМПАКТНИХ БАГАТОВИДАХ
We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee
non-explosion of solutions in a finite time. This property leads to the existence and uniqueness of solutions
for corresponding stochastic differential equation with globally non-Lipschitz coefficients.
Proposed approach is based on the estimates on diffusion generator, that weakly acts on the metric
function of manifold. Such estimates enable us to single out a manifold analogue of monotonicity condition
on the joint behaviour of the curvature of manifold and coefficients of equation.
Знайдено достатнi умови на коефiцiєнти дифузiйного рiвняння на некомпактному багатовидi, за
яких розв’язки не вибухають у скiнченний промiжок часу. Ця властивiсть приводить до iснування
та єдиностi розв’язкiв вiдповiдних стохастичних рiвнянь з глобально нелiпшицевими коефiцiєнтами.
Запропонований пiдхiд спирається на оцiнки на генератор дифузiї, що слабко дiє на метричну
функцiю багатовиду. Використання таких оцiнок дозволяє знайти узагальнення умови монотонностi
на випадок багатовиду, що поєднує поведiнку кривини багатовиду та коефiцiєнтiв рiвняння.
1. Introduction. A rigorous procedure for the construction of solutions of diffusion
equations on manifold was suggested a long ago, see [1 – 3] and references therein. In
comparison to the stochastic differential equation on linear space when the solution can
be constructed in a global coordinate system, the main difficulty with manifold was that
it does not have global coordinate system.
Let A0, Aα be C∞-smooth vector fields, globally defined on the oriented smooth
complete connected Riemannian manifold M without boundary and δWα denote Strato-
novich differentials of independent one-dimensional Wiener processesWα
t , α = 1, . . . , d,
The diffusion, written in Ito – Stratonovich form
δyxt = A0(yxt )dt+
d∑
α=1
Aα(yxt )δW
α
t , yx0 = x, (1)
can be correctly defined in any local coordinate vicinity U of manifold with the use of
integral equations on random intervals t ∈ (τin, τout)
yit∧τout(x) = yiτin(x) +
t∧τout∫
τin
Ai0(y
x
s )ds+
d∑
α=1
t∧τout∫
τin
Aα(yxs )δW
α
s , yx0 = x. (2)
Above τin, τout denote the times when process yxt enters and leaves vicinity U , i.e.,
yxt ∈ U for all t ∈ (τin, τout).
∗ Partially supported by grants of State Committee on Research and Technology.
c© A. VAL. ANTONIOUK, A. VICT. ANTONIOUK, 2007
1454 ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1455
The global solution to (1) is first constructed starting from some vicinity of initial
point x. Then it is extended in further domains, where comes process yxt , with help
of (2). This procedure of gluing together the local solutions on random intervals into
global solution may be correctly done for diffusion equations with locally Lipschitz
smooth coefficients [1 – 4]. The resulting solution is well-defined on some random
interval [0, τ∞(ω)), but not necessarily for all t ≥ 0.
Since compact manifolds always have a finite covering by local coordinate vicinities,
it can be shown that random time τ∞ = ∞ in compact case and the solution of diffusion
equation (1) exists for all t ≥ 0. On the contrary, because Wiener process Wt leaves any
bounded ball in Rd with non-zero probability, in the noncompact case the explosion may
occur. Depending on properties of coefficients of (1) and geometry of manifold there
may exist a finite explosion time τ∞ such that process yxt leaves any bounded vicinity U
of manifold M at time τ∞: ∀U⊂M yxτ∞ 6∈ U . Then the solution yxt could be correctly
defined only till the explosion time τ∞(ω).
In the case of non-explosion the probability of set {ω : τ∞(ω) <∞} is zero and the
unique global solution yxt of equation (1) may be defined for all t ≥ 0. It represents a
continuous adapted locally integrable process which fulfills an independent on particular
coordinate vicinities variant of (1): for any function with compact support f ∈ C∞0 (M)
f(yxt ) = f(x) +
t∫
0
(A0f)(yxs )ds+
d∑
α=1
t∫
0
(Aαf)(yxs )δW
α
s . (3)
Since f(yxt ) and (A·f)(yxt ) are R1-valued processes, equation (3) represents Stratonovich
equation on real line R1. Using functions f that coincide with the local coordinates
f(x) = xi in the coordinate vicinities U of manifold, it is possible to localize equation (3)
back to (2).
Moreover, in contrary to the explosion case, when there is no sense of yxt (ω) for ω
such that t ≥ τ∞(ω), in non-explosion case the diffusion semigroup Ptf(x) = E f(yxt )
has sense and the applications to the heat parabolic Cauchy problems on manifolds
become possible. Due to (3) semigroup Pt is generated by the second order operator of
Hörmander type
Lf =
1
2
d∑
α=1
Aα(Aαf) +A0f. (4)
The known conditions for non-explosion and, hence, for the existence and uniqueness
of global solutions yxt to problem (1), lie in the global Lipschitz assumptions on the
coefficients of equation A0, Aα and boundedness of the curvature of manifold, e.g.
[1, 2, 4, 5]. Further research of non-explosion for stochastic equations on manifolds
mainly concerned geometric properties of manifold, like its Brownian or martingale
completeness, see, for example, [6, 7] and references therein.
However, in the case of noncompact spaces with zero curvature (like Rd) it is also
possible to prove non-explosion for a wide class of equations with nonlinear globally
non-Lipschitz coefficients, that fulfill a kind of monotone conditions of dissipativity and
coercitivity [8, 9]. Arises a natural question whether the global Lipschitzness assumption
on the coefficients of equation can be avoided in the case of noncompact manifold.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
1456 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
In this article we discuss conditions on the joint behaviour of coefficients A0, Aα
of equation (1) and curvature tensor R of noncompact manifold M that guarantee non-
explosion, i.e., the existence and uniqueness of solutions to (1). The found conditions
generalize the dissipativity and coercitivity conditions from linear space to the manifold
and permit to work with essentially nonlinear diffusion with globally non-Lipschitz
coefficients.
The main idea is that the following estimate on the metric distance function ρ on
process yxt :
Eρ2(o, yxt ) ≤ eKt(1 + ρ2(o, x)), (5)
leads to the non-explosion: for all t ≥ 0 ρ(o, yxt ) <∞ almost everywhere. Here o ∈M
is some fixed point of manifold and ρ(o, x) denotes the shortest geodesic distance
between points o and x.
At the first look, estimate (5) can be obtained from the formal application of Ito –
Stratonovich formula (3) to the metric function f(yxt ) = ρ2(o, yxt )
Eρ2(o, yxt ) = ρ2(o, x) +
t∫
0
E
{
AII0 +
1
2
d∑
α=1
(AIIα )2
}
ρ2(o, yxs )ds, (6)
where we used notation AII for vector field A acting on the second variable x of function
ρ(o, x): AIIρ2(o, x) = 〈A(x),∇x〉ρ2(o, x).
Then, if operator L fulfills the following estimate on function ρ(o, x):
∃K : LIIρ2(o, x) =
{
AII0 +
1
2
d∑
α=1
(AIIα )2
}
ρ2(o, x) ≤ K(1 + ρ2(o, x)), (7)
formula (6) leads to
Eρ2(o, yxt ) ≤ ρ2(o, x) +K
t∫
0
E(1 + ρ2(o, yxs ))ds
and gives non-explosion estimate (5).
However, in comparison to the Euclidean case with smooth metric ρ2(o, x) = ‖o−
− x‖2, for the case of general manifold M function ρ2(o, x) may be non-differentiable
for points x ∈ N from some hypermanifold N ⊂M of lower dimension. Then operator{
AII0 +
1
2
∑d
α=1
(AIIα )2
}
in the right-hand side of (6) is bad defined on it and formal
reasoning (5) – (7) does not work.
One more problem in estimation of (7) is that metric ρ2(o, x) does not have a direct
representation, as in the linear case ‖o−x‖2. It is a minimum of length functional along
paths from o to x
ρ2(o, x) = inf
1∫
0
∣∣γ̇(`)∣∣2d`, γ(0) = o, γ(1) = x
, where γ̇(`) =
∂
∂`
γ(`),
(8)
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1457
therefore it is hard to provide the implicit representations for arbitrary differential
operators, acting on it. The known approaches were mainly adapted for Laplace –
Beltrami ∆ or similar operators and related with the use of geodesic deviations formulas
and Jacobi fields, e.g. [10, 11], survey [7] and references therein.
In [12] it is found a way to obtain upper bound (7) on the generator L, acting on
the metric function at points of its C2-regularity. Since in general situation the metric
function is not everywhere twice differentiable, results of [12] are not directly applicable
to the study of non-explosion.
The article consists of two parts. First, in Lemma 1 we develop upper bounds of
[12] outside of geodesic between points o and x and estimate difference approximations
of second order operators. Then, in Lemma 2, we prove estimates on operator L that
weakly acts on metric function.
These weak estimates are used in Lemma 3 to demonstrate, in analogue to arguments
of [5], that process 1 + ρ2(o, yxt )−K
∫ t
0
{1 + ρ2(o, yxs )}ds represents supermartingale
for sufficiently large K. This leads to moment estimate (5) and, in fact, replaces the Ito
formula arguments (5) – (7).
Finally, in Theorem 2 estimates (5) are extended from ρ2(o, x) to the polynomials
of metric function.
2. Main results. Let us implement the following conditions on coefficients A0, Aα
and curvature R. In particular, they generalize the classical dissipativity and coercitivity
conditions [8, 9] from the linear Euclidean space to manifold:
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)); (9)
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, (10)
where Ã0 = A0 +
1
2
∑d
α=1
∇AαAα and [R(A, h)A]m = R m
p `qA
pA`hq denotes the
curvature operator, related with (1, 3) curvature tensor with components
R 2
1 34 =
∂Γ 2
1 3
∂x4
− ∂Γ 2
1 4
∂x3
+ Γ j
1 3Γ
2
j 4 − Γ j
1 4Γ
2
j 3. (11)
For simplicity of further calculations we denoted by numbers the positions of correspon-
ding indexes. By tradition the repeating indexes mean silent summations.
Notation ∇H[h] means the directional covariant derivative, defined by
(∇H(x)[h])i = ∇jH
i(x) · hj . (12)
Main result of article is the following:
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
1458 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
Theorem 1. Suppose that conditions (9), (10) are fulfilled. Then equation (1) has
a unique solution that does not explode in a finite time and fulfills estimate (5).
Proof. To localize equation (1) consider open set U⊂M with compact closure U
and function ζU with compact support such that
√
ζU ∈ C∞0 (M, [0, 1]) and ζU (z) = 1
for z ∈ U , 0 ≤ ζU < 1 outside of U . Introduce operator
LUf = ζULf =
1
2
d∑
α=1
√
ζUAα(
√
ζUAαf) + ζUA0f −
1
2
d∑
α=1
√
ζU (Aα
√
ζU )Aαf,
that corresponds to the localized Stratonovich diffusion
δyUt (x) =
(
ζUA0 −
1
2
d∑
α=1
√
ζU (Aα
√
ζU )Aα
)
(yxs )ds+
+
d∑
α=1
√
ζU (yxt )Aα(yxt )δW
α
t , yx0 = x. (13)
Equation (13) has globally Lipschitz coefficients with all bounded derivatives, therefore
it has a unique solution which is C∞-differentiable on the initial data x [1, 2, 4, 5].
Since for initial data x outside of support ζU we have yUt (x) = x for all t ≥ 0, its
diffusion semigroup (PUt f)(x) = Ef(yUt (x)) preserves the space C∞0,+(M) of non-
negative continuously differentiable functions with compact support.
Now let us prepare the independent on U weak estimates on generators LU : ∃K
∀ζU ∈ C∞0 (M, [0, 1]), ζU
∣∣
U
= 1 ∀ϕ ∈ C∞0,+(M):
∫
M
(
[LU ]∗ϕ
)
ρ2(o, x)dσ(x) ≤ K
∫
M
ϕ(x)(1 + ρ2(o, x))dσ(x), (14)
where dσ denotes the Riemannian volume on M .
As [LU ]∗ = [ζUL]∗ = L∗ζU , estimate (14) follows from the weak estimate on
operator L
∃K ∀ψ ∈ C∞0,+(M) :∫
M
(L∗ψ(x))ρ2(o, x)dσ(x) ≤ K
∫
M
ψ(x)(1 + ρ2(o, x))dσ(x) (15)
if one substitutes first ψ = ζUϕ and then applies 0 ≤ ζU ≤ 1. Here L∗ =
=
1
2
∑d
α=1
[A∗α]2 + A∗0 with the adjoint field X∗ to vector field X defined by X∗f =
= −(divX)f −Xf .
To prove (15) let us first note that for any smooth vector field X in a vicinity of
some point z of manifold N and smooth function f on N representations are satisfied
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1459
Xf(z) = lim
ε→0
1
ε
ε∫
0
Xf(zs)ds = lim
ε→0
1
ε
ε∫
0
d
ds
f(zs)ds = lim
ε→0
f(zε)− f(z)
ε
,
(16)
X(Xf)(z) = lim
ε→0
1
ε2
ε∫
0
ds
s∫
−s
X(Xf)(z`)d` =
= lim
ε→0
1
ε2
ε∫
0
ds
s∫
−s
d
d`
(Xf)(z`)d` =
= lim
ε→0
1
ε2
ε∫
0
{(Xf)(zs)− (Xf)(z−s)}ds = lim
ε→0
1
ε2
ε∫
0
d
ds
{f(zs) + f(z−s)}ds =
= lim
ε→0
f(zε) + f(z−ε)− 2f(z)
ε2
.
Here we used notation zε for the differential flow along field X: zε = z+
∫ ε
0
X(zs)ds.
Therefore, due to the compactness of support of function ψ in (15), the following
representation of the left-hand side of (15) is valid∫
M
(L∗ψ(x))ρ2(o, x)dσ(x) = lim
ε→0+
∫
M
ψ(x)
{
ρ2(o, zε0(x))− ρ2(o, z0
0(x))
ε
+
+
1
2
d∑
α=1
ρ2(o, zεα(x)) + ρ2(o, z−εα (x))− 2ρ2(o, z0
α(x))
ε2
}
dσ(x). (17)
Here zε0(x), z
ε
α(x) denote the shifts along vector fields A0, Aα with initial data z0
0(x) =
= x, z0
α(x) = x.
Representation (17) follows from (16) and form of adjoint field X∗, because due to
the Stokes formula
∫
∂D
X · dS =
∫
D
divX dσ the increment of volume along field X
is equal to
d
dε
∣∣∣∣
ε=0
dσ(zεX(x))
dσ(x)
= (divX)(x). Indeed, for ϕ,ψ ∈ C∞0 (M) one has
∫
M
(L∗ψ)ϕdσ =
∫
M
ψ(Lϕ) dσ =
= lim
ε→0+
∫
M
ψ(x)
{
ϕ(zε0(x))− ϕ(z0
0(x))
ε
−
−1
2
d∑
α=1
ϕ(zεα(x)) + ϕ(z−εα (x))− 2ϕ(z0
α(x))
ε2
}
dσ(x) =
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
1460 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
= lim
ε→0+
∫
M
{
1
ε
[
ψ(z−ε0 (x))
dσ(z−ε0 (x))
dσ(x)
− ψ(x)
]
+
+
1
2ε2
d∑
α=1
[
ψ(z−εα (x))
dσ(z−εα (x))
dσ(x)
+ ψ(zεα(x))
dσ(zεα(x))
dσ(x)
−
−2ψ(z0
α(x))
dσ(z0
α(x))
dσ(x)
]}
ϕ(x)dσ(x), (18)
where to get the last line with ϕ(x) we shifted back along fields {−A0,−Aα}. For
ψ ∈ C∞0 (M) and vector fields A0, Aα expression in figure brackets in (18) converges
to L∗ψ uniformly on M . Due to the compactness of support of ψ we can close
(18) from ϕ(x) to ρ2(o, x) and, making a reverse shift along fields A0, Aα, obtain
representation (17).
Now let us estimate fractions in the right-hand side of (17).
In the vicinity of geodesic γ(`), ` ∈ [0, 1] from γ(0) = o to γ(1) = x that minimizes
(8) 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 from o to x and parameter
s corresponds to the evolution along H:
∂
∂s
γ(`, s) = H(γ(`, s)). (19)
Note that for s 6= 0 each path {γ(`, s), ` ∈ [0, 1]} must not be geodesic, unlike
in formulas for geodesic deviations. Later we will choose field H to be H(`, s) =
= `2A0(γ(`, s)) or H(`, s) = `Aα(γ(`, s)) for the first and second order differences
in (17).
Lemma 1. 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, (20)
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, (21)
where we used notation γ̇(`, s) =
∂
∂`
γ(`, s).
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1461
The right-hand side terms in (20), (21) have the following representations in terms
of field H:
∂
∂s
|γ̇(`, s)|2 = 2〈γ̇,∇H[γ̇]〉, (22)
1
2
∂2
∂s2
|γ̇(`, ε)|2 = |∇H[γ̇] |2 − 〈γ̇, R(H, γ̇)H〉+ 〈γ̇,∇(∇HH)[γ̇]〉. (23)
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.
Proof. Let’s apply (16) with N = M ×M , X = HI ⊗HII and function f(z) =
= ρ(o, x) for z = (o, x). Using the minimal property of geodesic, i.e., that the path
γ(`, s) is longer than geodesic from γ(0, s) to γ(1, s), we can estimate terms with ε in
(17) from above and obtain
ρ2(γ(0, ε), γ(1, ε))− ρ2(o, x)
ε
≤
∫ 1
0
|γ̇(`, ε)|2d`−
∫ 1
0
|γ̇(`, 0)|2d`
ε
, (24)
ρ2(γ(0, ε), γ(1, ε)) + ρ2(γ(0, ε), γ(1,−ε))− 2ρ2(o, x)
ε2
≤
≤
∫ 1
0
|γ̇(`, ε)|2d`+
∫ 1
0
|γ̇(`,−ε)|2d`− 2
∫ 1
0
|γ̇(`, 0)|2d`
ε2
. (25)
Above we actually get rid of a problem of implicit representations for operators on
metric functions (8). Remark also that the only term with s = 0, i.e., ρ2(o, x), was
written exactly along geodesic γ(`, 0) from (8).
Let h(s) =
∫ 1
0
|γ̇(`, s)|2d`. Then estimates
h(ε)− h(0) =
ε∫
0
h′(s)ds = εh′(0) +
ε∫
0
[h′(s)− h′(0)] ds =
= εh′(0) +
ε∫
0
s∫
0
h′′(τ)dτ
ds = εh′(0) +
ε∫
0
ε∫
ε−τ
h′′(τ)ds
dτ =
= εh′(0) +
ε∫
0
τh′′(τ)dτ ≤ ε
h′(0) +
ε∫
0
|h′′(τ)|dτ
,
h(s) + h(−s)− 2h(0) =
ε∫
0
h′(s)ds−
0∫
−ε
h′(s)ds =
ε∫
0
(h′(s)− h′(−s))ds =
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1462 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
=
ε∫
0
s∫
−s
h′′(τ)dτ
ds = ε2h′′(0) +
ε∫
0
s∫
−s
[
h′′(τ)− h′′(0)
]
dτ
ds ≤
≤ εh′′(0) +
ε∫
0
s∫
−s
τ∫
0
|h′′′(δ)|dδ
dτ
ds ≤ ε2
h′′(0) +
1
2
ε∫
0
|h′′′(δ)|dδ
,
lead, due to (24), (25), to the statements (20), (21).
Now let us find expressions for
∂m
∂sm
∣∣γ̇(`, s)∣∣2 in (20) and (21).
Let us use that by continuity arguments, for any ` and sufficiently small δ(`) the
path
{
γ(`, z)
}
z∈(−δ(`),δ(`)) completely lies in some coordinate vicinity (xi). In this
coordinate system relation (19) has integral form
γi(`, s) = γi(`) +
s∫
0
Hi(γ(`, z))dz (26)
with point γ(`) on initial geodesic from o to x. Therefore
γ̇i(`, s) = γ̇i(`) +
s∫
0
∂kH
i(γ(`, z))γ̇k(`, z)dz,
and
∂
∂s
γ̇i(`, s) = ∂kH
i(γ(`, s))γ̇k(`, s) = (∇kH
i − Γ i
k hH
h)γ̇k(`, s), (27)
where we changed to the covariant derivatives. In particular, from above formula and
(19) it follows commutation
∂
∂s
∂
∂`
γi(`, s) =
∂
∂`
∂
∂s
γi(`, s).
Relation (27) and autoparallel property of Riemannian connection
∂kgmn(x) = ghnΓ h
k m + gmhΓ h
k n (28)
lead to relation (22):
∂
∂s
|γ̇(`, s)|2 =
∂
∂s
[
gij(γ(`, s))γ̇i(`, s)γ̇j(`, s)
]
=
= ∂kgij
∂
∂s
γk · γ̇iγ̇j + 2gij γ̇i
∂
∂s
γ̇j =
= 2gij γ̇i(∇kH
j)γ̇k = 2〈γ̇,∇H[γ̇]〉.
In a similar way
1
2
∂2
∂s2
|γ̇(`, ε)|2 =
∂
∂s
〈γ̇(`, s),∇H[γ̇(`, s)]〉 =
∂
∂s
{
gij(γ)γ̇i[∇kH
j(γ)]γ̇k
}
=
= ∂mgij(γ)Hmγ̇i[∇kH
j(γ)]γ̇k + gij
{
(∇mH
i − Γ i
m h)γ̇
m
}
[∇kH
j(γ)]γ̇k+
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NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1463
+gij γ̇i
[
∂m∇kH
j(γ) ·Hm(γ)
]
γ̇k + gij(γ)γ̇i[∇kH
j(γ)]
{
(∇mH
k − Γ k
m h)γ̇
m
}
,
where, after the differentiation of product, we substituted relations (19) and (27).
Using property (28), transforming partial derivative ∂m∇kH
j to covariant∇m∇kH
j
and contracting the terms with connection Γ we have
1
2
∂2
∂s2
|γ̇(`, ε)|2 = gij(∇mH
i)γ̇m(∇kH
j)γ̇k +
+ gij γ̇
i(∇m∇kH
j)Hmγ̇k + gij(γ)γ̇i(∇kH
j)(∇mH
k)γ̇m.
Next commute the covariant derivatives in the second term∇m∇kH
j = ∇k∇mH
j+
+R j
h kmH
h to obtain
1
2
∂2
∂s2
|γ̇(`, ε)|2 = |∇H[γ̇] |2+
+gij γ̇i(∇k∇mH
j +R j
h kmH
h)Hmγ̇k + gij γ̇
i(∇kH
j)(∇mH
k)γ̇m =
= |∇H[γ̇] |2 − 〈γ̇, R(H, γ̇)H〉+ gij γ̇
i(∇k∇mH
j)Hmγ̇k+
+gij γ̇i(∇kH
j)(∇mH
k)γ̇m
with curvature operator R(H, γ̇). Redenoting indexes m↔ k in the third term we have
3rd + 4th terms = gij γ̇
i(∇m∇kH
j)Hkγ̇m + gij γ̇
i(∇kH
j)(∇mH
k)γ̇m =
= gij γ̇
i(∇m{Hk∇kH
j})γ̇m
which leads to (23).
By similar calculation the third derivative
∂3
∂s3
|γ̇(`, s)|2 depends on the field H and
its covariant derivatives up to the third order and on the curvature tensor R and its first
order covariant derivative.
The lemma is proved.
Now we apply Lemma 1 to find estimates on difference approximation (17) of
operator L.
Lemma 2. Under coercitivity and dissipativity assumptions (9), (10)
∃K ∀ψ ∈ C∞0,+(M) :
∫
M
(L∗ψ(x))ρ2(o, x)dσ(x) ≤ K
∫
M
ψ(x)(1 + ρ2(o, x))dσ(x).
(29)
Proof. Let us make a particular choice H0(`, s) = `2A0(γ0(`, s)) and Hα(`, s) =
= `Aα(γα(`, s)) in (20), (21) with γ0(`, s), γα(`, s) generated by H0, Hα. Then due to
H(0, s) = 0 the point γ(0, s) = o for all s ∈ [−ε, ε] and we have from (22), (23){
ρ2(o, γ0(1, ε))− ρ2(o, x)
ε
+
+
1
2
d∑
α=1
ρ2(o, γα(1, ε)) + ρ2(o, γα(1,−ε))− 2ρ2(o, x)
ε2
}
≤
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1464 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
≤
1∫
0
I(γ̇(`, 0))d`+
ε∫
0
1∫
0
J(γ̇(`, s))d` ds (30)
where terms at s = 0 are equal to
I(γ̇) = 2
〈
∇
(
`2A0 +
1
2
d∑
α=1
∇`Aα [`Aα]
)
[γ̇], γ̇
〉
+
+
d∑
α=1
{
|∇(`Aα)[γ̇] |2 − 〈R(`Aα, γ̇)`Aα, γ̇〉
}
and rest terms have form
J(γ̇(`, s)) =
∣∣∣∣ ∂2
∂s2
|γ̇0(`, s)|2
∣∣∣∣+ 1
2
d∑
α=1
∣∣∣∣ ∂3
∂s3
|γ̇(`, s)|2
∣∣∣∣ .
Using that ∇`[γ̇] =
∂`
∂`
= 1 and ∇Aα` =
∂`
∂s
= 0, which leads to
∇(∇`Aα [`Aα])[γ̇] = ∇γ̇(`2∇AαAα) = `2∇(∇AαAα)[γ̇] + 2`∇AαAα,
we can further rewrite term I(γ̇)
I(γ̇) = 2`2〈∇A0[γ̇], γ̇〉+ 4`〈A0, γ̇〉+
+
d∑
α=1
{
`2|∇Aα[γ̇] |2 + 2`〈Aα,∇Aα[γ̇]〉+ |Aα|2 − `2〈R(Aα, γ̇)Aα, γ̇〉+
+`2〈∇(∇AαAα)[γ̇], γ̇〉+ 2`〈∇AαAα, γ̇〉
}
. (31)
Using estimate ∣∣∣〈∇Aα[γ̇], Aα
〉∣∣∣ ≤ `
2
∣∣∇Aα[γ̇]
∣∣2 +
1
2`
|Aα|2
we find
I(γ̇) ≤ `2
(
2
〈
∇Ã0[γ̇], γ̇
〉
+ 2
d∑
α=1
∣∣∇Aα[γ̇]
∣∣2 − d∑
α=1
〈
R(Aα, γ̇)Aα, γ̇
〉)
+
+4`〈Ã0(γ), γ̇〉+ 2
d∑
α=1
|Aα|2. (32)
Using that
∇γ(`)ρ2(o, γ(`)) = 2ρ(o, γ(`))∇γ(`)ρ(o, γ(`)) = 2`ρ(o, x)
γ̇(`)
ρ(o, x)
= 2`γ̇,
we have
2`〈Ã0(γ), γ̇〉 =
〈
Ã0(γ),∇γ(`)ρ2(o, γ(`))
〉
.
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NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1465
Finally, applying the coercitivity and dissipativity assumptions (9), (10) to (32), we
conclude
1∫
0
I(γ̇)d` ≤
1∫
0
{
2KC`
2|γ̇|2 +KC′(1 + ρ2(o, γ(`)))
}
d` ≤ K(1 + ρ2(o, x)), (33)
where we also used that ` ≤ 1 and path γ(`, 0) = γ(`) realizes the geodesic between o
and x.
Due to (22), (23) and analogous representation of the third derivative, the rest terms
J(γ̇) in (30) are estimated by
J(γ̇) ≤ T0 |γ̇0(`, s)|2 +
d∑
α=1
Tα |γ̇α(`, s)|2
with some functions T0, Tα depending on the coefficients of equation and curvature
tensor and their covariant derivatives up to the third order. Since in (17) the support of
ψ is compact and the limits are taken in some δ-vicinity of point x, the possible paths
γ(`, s) belong to the bounded set
Zψ,o,δ =
{
y ∈M : y lies on some geodesics from o to x ∈ B(suppψ, δ)
}
.
Therefore
1∫
0
J(γ̇)d` ≤ sup
z∈Zψ,o,δ
∣∣{T0, Tα}(z)
∣∣ ·
1∫
0
|γ̇0(`, s)|2d`+
d∑
α=1
1∫
0
|γ̇α(`, s)|2d`
.
Due to (22) the integrals vs =
1∫
0
|γ̇(`, s)|2d` are estimated in the following way
vs ≤ v0 +
s∫
0
v′sds = ρ2(o, x) +
s∫
0
1∫
0
〈
∇H[γ̇(`, s)], γ̇(`, s)
〉
d`
ds ≤
≤ ρ2(o, x) + sup
y∈Zψ,o,δ
|∇H(y)| ·
s∫
0
h(s)ds
which gives
vs ≤ ρ2(o, x) exp
{
s sup
y∈Zψ,o,δ
|∇H|(y)
}
.
We come to
ε∫
0
1∫
0
J(γ̇)d` ds ≤
≤ ερ2(o, x) · sup
y∈Zψ,o,δ
M(A0,∇A0,∇2A0, Aα,∇Aα,∇2Aα,∇3Aα, R,∇R)
with a finite resulting constant supM due to the compactness of set Zψ,o,δ .
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1466 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
Combining the above estimate with (33) and (17), and taking limit lim
ε→0+
we have (29).
The lemma is proved.
Let us recall that we developed weak estimates on L because metric ρ2(o, x) may be
non-differentiable at all points x and Ito formula arguments were not applicable. Now,
similar to [5], we replace the Ito formula approach to non-explosion estimate (5) – (7)
by weak estimates (29) and a statement that some process on manifold represents a
supermartingale. By definition, process Xt is supermartingale with respect to the flow
of σ-algebras Ft if for all 0 ≤ s ≤ t it is satisfied E(Xt|Fs) ≤ Xs. Here E(·|Fs)
denotes the conditional expectation with respect to σ-algebra Fs.
Lemma 3. Under coercitivity and dissipativity conditions (9), (10) there is an
independent on sets U⊂M constant K such that process
[1 + ρ2(o, yUt (x))]−K
t∫
0
[
1 + ρ2(o, yUs (x))
]
ds (34)
is 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).
Proof. First recall, that semigroup PUt , generated by localized process yUt (x) (13),
preserves the space C∞0,+(M) of nonnegative continuously differentiable functions with
compact support. Therefore the integrals below are finite and weak estimate (29) implies
∀ϕ ∈ C∞0,+(M) :
d
dt
∫
M
ϕ(x)
{
PUt (1 + ρ2(o, ·))
}
(x)dσ(x) =
=
d
dt
∫
M
{
[PUt ]∗ϕ
}
(x)(1 + ρ2(o, x))dσ(x) =
=
∫
M
[LU ]∗
{
[PUt ]∗ϕ
}
(x) · (1 + ρ2(o, x))dσ(x) =
=
∫
M
[L]∗
(
ζU (x)
{
[PUt ]∗ϕ
}
(x)
)
· (1 + ρ2(o, x))dσ(x) ≤
≤ K
∫
M
{
[PUt ]∗ϕ
}
(x) · (1 + ρ2(o, x))dσ(x) = K
∫
M
ϕ(x)
{
PUt (1 + ρ2(o, ·))
}
(x)dσ(x),
where we used that due to the compactness of support of function ζU ≥ 0 the integrand
ψ = ζU (x)
{
[PUt ]∗ϕ
}
∈ C∞0,+(M), then applied (29) and property ζU ≤ 1. To come to
the last line we also applied that L1 = 0.
Therefore for all ϕ ∈ C∞0,+(M) we have estimate∫
M
ϕ(x) ·
{
PUt (1 + ρ2(o, ·))
}
(x)dσ(x) ≤
≤
∫
M
ϕ(x) ·
(1 + ρ2(o, x)) +K
t∫
0
{
PUs (1 + ρ2(o, ·))
}
(x)ds
dσ(x)
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NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1467
and its pointwise consequence
{
PUt (1 + ρ2(o, ·))
}
(x) ≤ (1 + ρ2(o, x)) +K
t∫
0
{
PUs (1 + ρ2(o, ·))
}
(x)ds. (35)
Next we use the Markov property of process yUt (x). In particular, for semigroup PUt
it gives
(PUt f)(yUs (x)) = E(f(yUt+s(x)) | Fs), t, s ≥ 0, (36)
which permits to substitute process yxt instead of initial data x. Property (36) can be
checked by taking qτ = E([PUt−τf ](yUs+τ ) | Fs) and using Ito formula for depending on
time functions to get q′s = 0, s ∈ [0, t]. Therefore q0 = qt and (36) is true. After that
(36) should be closed from C2 to continuous functions.
Let us substitute instead of x initial data yUτ (x) in (35) to obtain from (36) for
function h(x) = 1 + ρ2(o, x) that
E(h(yUt+τ (x))|Fτ ) = (PUt h)(y
U
τ (x)) ≤ h(yUτ (x)) +K
t∫
0
{
PUs h
}
(yUτ (x))ds =
= h(yUτ (x)) +K E
t+τ∫
τ
h(yUs (x))ds |Fτ
. (37)
Inequality (37) actually means that the process (34) is supermartingale. Indeed, the
supermartingale property
E
h(yUt+τ (x))−K
t+τ∫
0
h(yUs (x))ds
∣∣∣∣∣∣ Fτ
≤ h(yUτ (x))−K
τ∫
0
h(yUs (x))ds
coincides with (37). The integrability of process (34) follows from the compactness of
the closure of set {x : ζU (x) > 0}.
The lemma is proved.
End of proof of Theorem 1. Suppose that initial data x ∈ U . Introduce stopping
time
τU (ω) = inf{t ≥ 0: yxt 6∈ U}.
The Doob – Meyer free choice theorem, e.g. [13], permits to substitute any finite
stopping times 0 ≤ S ≤ T into the supermartingale property E(XT |FS) ≤ XS . Let’s
apply it with S = 0 and T = t ∧ τU to supermartingale (34). Due to E(·|F0) = E(·)
we have
mt = E(1 + ρ2(o, yUt∧τU (x))) ≤ (1 + ρ2(o, x)) +KE
t∧τU∫
0
(1 + ρ2(o, yUs (x)))ds ≤
≤ m0 +KE
t∫
0
(1 + ρ2(o, yUs∧τU (x)))ds = m0 +K
t∫
0
msds,
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1468 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
where yUs∧τU (x) = yUτU (x) for s ≥ τU is a stopped process on the boundary of U and
we enlarged the upper limit of integral.
Gronwall – Bellmann inequality implies that
E(1 + ρ2(o, yUt∧τU (x))) ≤ eKt(1 + ρ2(o, x)). (38)
Choose now a sequence of balls Un = {z ∈ M : ρ(o, z) < n}, then after number
n0 such that ρ(o, x) > n0, the sequence of stopping times τUn is monotone increasing.
Due to (38)
E 1{ω:t≥τUn (ω)} · (1 + ρ2(o, yUt∧τU (x))) ≤
≤ E(1 + ρ2(o, yUt∧τU (x))) ≤ eKt(1 + ρ2(o, x))
with characteristic function 1A of set A.
Since for t ≥ τUn ρ(o, yxt ) = n, we have
E 1{ω:t≥τUn (ω)} ≤
eKt(1 + ρ2(o, x))
1 + n2
→ 0, n→∞,
and almost everywhere
τ∞ = lim
n→∞
τUn = ∞. (39)
As ζU
∣∣
U
= 1, the processes yUnt (x) and yUmt (x) coincide till the first exit time
from vicinity Un∧m. Therefore the unique solution yxt to problem (1) equals to solutions
yUnt (x) till the first exit time t ≤ τUn .
Property (39) implies that for coercitive and dissipative coefficients in (1) the limit
process yxt = lim
n→∞
yUnt (x) is correctly defined for all t ≥ 0 as a unique solution to (1).
In particular it does not explode in a finite time.
The theorem is proved.
In next theorem we generalize statement of Lemma 3 to the polynomials of metric
function. Remark that the convex function of supermartingale should not be a supermar-
tingale again, therefore the application of coercitivity and dissipativity conditions (9),
(10) is necessary to find appropriate constant KP in (41).
Theorem 2. Let P be a positive monotone polynomial function on half-line R+
such that
∃C ∀z ≥ 0: (1 + z)P ′(z) ≤ C P (z), (1 + z)|P ′′(z)| ≤ C P ′(z). (40)
Under coercitivity and dissipativity assumptions (9), (10) there is constant KP such that
for any vicinity U the process
P (ρ2(o, yUt (x)))−KP
t∫
0
P (ρ2(o, yUs (x)))ds (41)
is integrable supermartingale.
Moreover, a unique solution yxt to problem (1) fulfills
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NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1469
EP (ρ2(o, yxt )) ≤ eKP tP (ρ2(o, x)) (42)
and process
P (ρ2(o, yxt ))−KP
t∫
0
P (ρ2(o, yxs ))ds (43)
represents supermartingale.
Proof. This statement is verified like in the previous theorem, the only difference is
that due to the monotonicity of P the first order estimate (20) transforms to
P (ρ2(γ(0, ε), γ(1, ε)))− P (ρ2(o, x))
ε
≤
≤
P
(∫ 1
0
|γ̇(`, ε)|2d`
)
− P
(∫ 1
0
|γ̇(`, 0)|2d`
)
ε
≤
≤ ∂
∂s
∣∣∣∣
s=0
P
1∫
0
|γ̇(`, s)|2d`
+
ε∫
0
∣∣∣∣∣∣ ∂
2
∂s2
P
1∫
0
|γ̇(`, s)|2d`
∣∣∣∣∣∣ ds =
= P ′(ρ2(o, x))
1∫
0
∂
∂s
∣∣∣∣
s=0
|γ̇(`, s)|2d`+
ε∫
0
JP (γ̇s)ds (44)
with JP (γ̇s) =
∣∣∣∣ ∂2
∂s2
P
(∫ 1
0
|γ̇(`, s)|2d`
)∣∣∣∣.
Similarly, the second order estimate (21) becomes
P (ρ2(γ(0, ε), γ(1, ε))) + P (ρ2(γ(0, ε), γ(1,−ε)))− 2P (ρ2(o, x))
ε2
≤
≤
P
(∫ 1
0
|γ̇(`, ε)|2d`
)
+ P
(∫ 1
0
|γ̇(`,−ε)|2d`
)
− 2P
(∫ 1
0
|γ̇(`, 0)|2d`
)
ε2
≤
≤ ∂2
∂s2
∣∣∣∣
s=0
P
1∫
0
|γ̇(`, s)|2d`
+
1
2
ε∫
0
∣∣∣∣∣∣ ∂
3
∂s3
P
1∫
0
|γ̇(`, s)|2d`
∣∣∣∣∣∣ ds =
= P ′(ρ2(o, x))
1∫
0
∂2
∂s2
∣∣∣∣
s=0
|γ̇(`, s)|2d`+
+P ′′(ρ2(o, x))
1∫
0
∂
∂s
∣∣∣∣
s=0
|γ̇(`, s)|2d`
2
+
ε∫
0
NP (γ̇s)ds, (45)
with Np(γ̇s) =
1
2
∣∣∣∣ ∂3
∂s3
P
(∫ 1
0
|γ̇(`, s)|2d`
)∣∣∣∣.
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
1470 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
Therefore we have additional term with P ′′ in comparison to (21). Its multiple is
treated in a similar way
1∫
0
∂
∂s
∣∣∣∣
s=0
|γ̇(`, s)|2d` = 2
1∫
0
〈∇(`Aα)[γ̇], γ̇〉d` =
= 2
1∫
0
〈Aα + `∇Aα[γ̇], γ̇〉d` ≤ 2
1∫
0
∣∣Aα + `∇Aα[γ̇]
∣∣2d`
1/2 1∫
0
|γ̇|2d`
1/2
.
Due to
∫ 1
0
|γ̇(`, o)|2d` = ρ2(o, x) we have
P ′′(ρ2(o, x))
1∫
0
∂
∂s
∣∣∣∣
s=0
|γ̇(`, s)|2d`
2
≤
≤ 4|P ′′(ρ2(o, x))|ρ2(o, x)
1∫
0
∣∣Aα + `∇Aα[γ̇]
∣∣2d` ≤
≤ 8C P ′(ρ2(o, x))
1∫
0
(
|Aα|2 + `2|∇Aα[γ̇]|2
)
d`.
This leads to additional terms in the right-hand side of (31) and, due to the coercitivity
and dissipativity assumptions (9), (10), gives estimate on all terms in lines (44) and (45)
line (44) + line (45) ≤ P ′(ρ2(o, x)) ·K(1 + ρ2(o, x)) +
ε∫
0
{
JP (γ̇s) +NP (γ̇s)
}
ds ≤
≤ KC P (ρ2(o, x)) +
ε∫
0
{
JP (γ̇s) +NP (γ̇s)
}
ds.
Therefore (33) transforms to{
P (ρ2(o, γ0(1, ε)))− P (ρ2(o, x))
ε
+
+
1
2
d∑
α=1
P (ρ2(o, γα(1, ε))) + P (ρ2(o, γα(1,−ε)))− 2P (ρ2(o, x))
ε2
}
≤
≤ KC P (ρ2(o, x)) + ε sup
γs⊂Zψ,o,δ
{
JP (γ̇s) +NP (γ̇s)
}
. (46)
Like in the proof of Lemma 2, the rest terms with JP , NP vanish for ε → 0+.
Therefore (29) adopts form
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
NON-EXPLOSION AND SOLVABILITY OF NONLINEAR DIFFUSION EQUATIONS ... 1471∫
M
(L∗ψ(x))P (ρ2(o, x))dσ(x) ≤ KP
∫
M
ψ(x)P (ρ2(o, x))dσ(x). (47)
Proceeding further like in Lemma 3, we obtain that (41) is a supermartingale.
In particular, an analogue of estimate (38) is true
EP (ρ2(o, yUt∧τU (x))) ≤ eKP tP (ρ2(o, x)). (48)
Next consider measurable random set Vn(t) =
{
ω : ∀s ∈ [0, t] yxt (ω) ∈ Un
}
that
corresponds to paths of process yxt (ω), staying inside of set Un till time t. Then
yxt (ω) = yUn
t∧τUn (x, ω) for all ω ∈ Vn(t) and (48) leads to
E 1Vn(t)P
(
ρ2(o, yxt (x))
)
≤ EP
(
ρ2(o, yUn
t∧τUn (x))
)
≤ eKP tP (ρ2(o, x)). (49)
Due to non-explosion lim
n→∞
τUn(ω) = ∞, each path yxt (ω) completely lies in some
Un for sufficiently large n. Therefore sequence Vn(t) is increasing to the full probability
space and 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 (49):
EP
(
ρ2(o, yxt (x))
)
≤ lim
n→∞
EP
(
ρ2(o, yUn
t∧τUn (x))
)
≤ eKP tP (ρ2(o, x))
leads to the statement (42).
To check that (43) represents supermartingale, let us apply Doob – Meyer free choice
theorem with S = s ∧ τU and T = t ∧ τU to supermartingales (41). It follows that
processes
θnt = P
(
ρ2(o, yUn
t∧τUn )
)
−KP
t∧τUn∫
0
P
(
ρ2(o, yUn
s∧τUn )
)
ds
represent supermartingales, i.e., for all 0 ≤ s ≤ t and A ∈ Fs
Eθnt 1A ≤ Eθns 1A. (50)
Since for all ω ∈ Vn(t) and s ∈ [0, t] process θnt coincides with the limit process
θns (ω) = θ∞s (ω) df= P (ρ2(o, yxt (ω)))−KP
s∫
0
P (ρ2(o, yxs (ω)))ds,
we can replace θns by θ∞s , s ∈ [0, t], on set Vn(t) in the calculation below
E
(
1Vn(t)θ
∞
t + (1− 1Vn(t))θnt
)
1A =
= Eθnt 1A ≤ Eθns 1A = E
(
1Vn(t)θ
∞
s + (1− 1Vn(t))θns
)
1A. (51)
Here we also applied property (50).
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
1472 A. VAL. ANTONIOUK, A. VICT. ANTONIOUK
Then notice that the independent on n estimate is true
sup
n≥1
sup
s∈[0,t]
E[θns ]2 <∞
due to (48) applied to function P 2 instead of P (this function again fulfills (40)). Then,
because (1− 1Vn(t))1A → 1A a.e. for n→∞, the terms with (1− 1Vn(t)) in (51) tend
to zero.
Moreover, due to estimate (42), process |θ∞s | is integrable sup
s∈[0,t]
E|θ∞s | < ∞ and
gives an integrable majorant for 1Vn(t)θ
∞
s 1A for s ∈ [0, t]. Using that 1Vn(t) → 1 a.e.
one takes a limit in (51) to get the supermartingale property ∀ 0 ≤ s ≤ t Eθ∞t 1A ≤
≤ Eθ∞s 1A for process θ∞t (43).
Acknowledgement. Authors wish to express their gratitude for referee comments,
that significally improved a general presentation of subject for the reader.
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3. Ito K., McKean H. P. Diffusion processes and their sample paths. – Springer, 1965.
4. Kunita H. Stochastic flows and stochastic differential equations. – Cambridge Univ. Press, 1990.
5. Stroock D. An introduction to the analysis of paths on a Riemannian manifold // Math. Surv. and
Monogr. – 2002. – 74.
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Received 22.09.06
ISSN 1027-3190. Укр. мат. журн., 2007, т. 59, № 11
|
| id | umjimathkievua-article-3404 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:41:54Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ee/b58b46ab378d7e8bbd2e312e8a5c38ee.pdf |
| spelling | umjimathkievua-article-34042020-03-18T19:53:28Z Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds Відсутність вибуху та існування розв'язків для нелініних дифузійних рівнянь на некомпактних багатовидах Antoniouk, A. Val. Antoniouk, A. Vict. Антонюк, О. Вал. Антонюк, О. Вік. We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee non-explosion of solutions in a finite time. This property leads to the existence and uniqueness of solutions for corresponding stochastic differential equation with globally non-Lipschitz coefficients. Proposed approach is based on the estimates on diffusion generator, that weakly acts on the metric function of manifold. Such estimates enable us to single out a manifold analogue of monotonicity condition on the joint behaviour of the curvature of manifold and coefficients of equation. Знайдено достатні умови на коєФіцієнти дифузійного рівняння на некомпактному багаroвидi, за яких розв'язки не вибухають у скінченний проміжок часу. Ця властивість приводить до існування та єдиності розв'язків відповідних стохастичних рівнянь з глобально неліпшицевими коефіцієнтами. Запропонований підхід спирається на оцінки на генератор дифузії, що слабко діє на метричну функцію багатовиду. Використання таких оцінок дозволяє знайти узагальнення умови монотонності на випадок багатовиду, що поєднує поведінку кривини багатовиду та коефіцієнтів рівняння. Institute of Mathematics, NAS of Ukraine 2007-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3404 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 11 (2007); 1454–1472 Український математичний журнал; Том 59 № 11 (2007); 1454–1472 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3404/3551 https://umj.imath.kiev.ua/index.php/umj/article/view/3404/3552 Copyright (c) 2007 Antoniouk A. Val.; Antoniouk A. Vict. |
| spellingShingle | Antoniouk, A. Val. Antoniouk, A. Vict. Антонюк, О. Вал. Антонюк, О. Вік. Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds |
| title | Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds |
| title_alt | Відсутність вибуху та існування розв'язків для нелініних дифузійних рівнянь на некомпактних багатовидах |
| title_full | Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds |
| title_fullStr | Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds |
| title_full_unstemmed | Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds |
| title_short | Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds |
| title_sort | nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3404 |
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