On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space
We consider some the following differential equation with interaction governed by a generalized function/ The conditions that guarantee the existence and uniqueness of a solution when
 mapping a belongs to some Sobolev space are obtained.
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2005
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| Cite this: | On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space / V.B. Brayman // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 29–41. — Бібліогр.: 5 назв.— англ. |
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| author | Brayman, V.B. |
| author_facet | Brayman, V.B. |
| citation_txt | On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space / V.B. Brayman // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 29–41. — Бібліогр.: 5 назв.— англ. |
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| description | We consider some the following differential equation with interaction governed by a generalized function/ The conditions that guarantee the existence and uniqueness of a solution when
mapping a belongs to some Sobolev space are obtained.
|
| first_indexed | 2025-12-07T16:37:39Z |
| format | Article |
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Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 29–41
UDC 519.21
VOLODYMYR B. BRAYMAN
ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION OF A
DIFFERENTIAL EQUATION WITH INTERACTION GOVERNED BY
GENERALIZED FUNCTION IN ABSTRACT WIENER SPACE
We consider the following differential equation with interaction governed by a gener-
alized function 0:
dx(u, t)
dt
= a(x(u, t), t), x(u, 0) = u, t = 0 ◦ x(·, t)−1.
The conditions that guarantee the existence and uniqueness of a solution when
mapping a belongs to some Sobolev space are obtained.
1. Introduction
Let (X, H, μ) be an abstract Wiener space, i.e. X is a real separable Banach space
and μ is a Gaussian measure on X with the Cameron–Martin space H (cf. [1]). Consider
the evolution of a material system in X in the case where the behaviour of each particle
depends not only on the position of this particle but also on some characteristic of the
whole system represented by a generalized function. Examples of such characteristics are
mass distributions at some surfaces, their derivatives, ets.
Let W k
p = W k
p (X, H, μ), k ∈ N, p ≥ 1, be the Sobolev space (cf. [1], the precise
definition will be given later), and let W−k
q = (W k
p )∗, where 1
p + 1
q = 1, be the space
of generalized functions equipped with ∗-weak topology. Denote, by x(u, t), the position
of the particle starting from u at time t. Assume that the characteristic of the material
system at time t is κt ∈ W−k
q , and the evolution of the system is described by the
differential equation with interaction
(1)
{
dx(u,t)
dt = a(x(u, t), κt)
x(u, 0) = u, κt = κ0 ◦ x(·, t)−1, t ≥ 0.
where a : X × W−k
q → H is a measurable transformation. Here, the generalized func-
tion κt = κ0 ◦ x(·, t)−1 is said to be the image of the generalized function κ0 under
transformation x(·, t) if, for every test function f ∈ W k
p , we have f ◦ x(·, t) ∈ W k
p and
〈f, κt〉 = 〈f ◦ x(·, t), κ0〉. Note that if κ0 is a measure on X , then the definition of κt
coincides with the standard definition of the image of a measure.
2000 AMS Mathematics Subject Classification. Primary 60H07,28C20.
Key words and phrases. Differential equation with interaction, Sobolev space, generalized function.
This research has been partially supported by the Ministry of Education and Science of Ukraine,
project N GP/F8/0086.
29
30 VOLODYMYR B. BRAYMAN
Definition 1. Measurable mapping x : X ×R → X is said to be a solution of Eq. (1) if
1) for every t ≥ 0, the generalized function κt = κ0 ◦ x(·, t)−1 belongs to W−k
q .
2) for μ-almost all u,
x(u, t) = u +
∫ t
0
a(x(u, s), κs)ds holds for all t ≥ 0;
3) for every t ≥ 0, the measure μ ◦x(·, t)−1 is absolutely continuous with respect to μ;
Remark. Condition 3) in Definition 1 provides that the solution does not depend on the
particular choice of a modification of a.
In this article, we obtain some sufficient conditions for the existence and uniqueness
of solution of (1). To formulate them, we need to recall some standard constructions and
notations from the Malliavin calculus (cf. [1]).
For any separable Hilbert space E, we denote, by FC∞(X, E), a set of smooth cylindi-
cal functions, i.e. functions of the form
f(u) =
m∑
l=1
ϕl(〈y1, u〉, . . . , 〈yn, u〉)el,
where y1, . . . , yn ∈ X∗, ϕ1, . . . , ϕm ∈ C∞
b (Rn) and e1, . . . , em ∈ E. The derivative ∇
along H is defined, for f ∈ FC∞(X, E), by
∇f(u) =
m∑
l=1
n∑
i=1
∂ϕl
∂xi
(〈y1, u〉, . . . , 〈yn, u〉)j∗yi ⊗ el ∈ FC∞(X, E1),
where E1 = H(H, E) is the space of Hilbert–Schmidt operators from H to E equipped
with the Hilbert–Schmidt norm. Define higher order derivatives on FC∞(X, E) itera-
tively by setting E0 = E, ∇0 = 1IFC∞(X,E) and, for k ∈ N,
Ek = H(H, Ek−1), ∇k = ∇ ◦∇k−1 : FC∞(X, E) → FC∞(X, Ek).
Note that Ek can be identified with the space of k-linear Hilbert–Schmidt operators on
H with range in E.
For any k ∈ N and p ∈ [1, +∞), the operator ∇k is closable under the norm ‖f‖p,k =∑k
i=0 ‖∇if‖Lp(X,Ei,μ).
The completion of FC∞(X, E) under this norm is a Sobolev space W k
p (X, E, μ) ⊂
Lp(X, E, μ). The extensions Dk : W k
p (X, E, μ) → Lp(X, E, μ) of derivatives ∇k to W k
p
are called stochastic derivatives. By δ : D(δ) ⊂ Lq(X, H, μ) → Lq(X, R, μ), 1
p + 1
q = 1,
we denote the divergence operator, i.e. the operator adjoint to D. Denote, by ‖ · ‖H, the
Hilbert–Schmidt norm in each of Ek, k ≥ 1, and, by ‖ · ‖op, the operator norm in L(H).
Now we can formulate the results.
Theorem 1. Let a : X × W−k
q → H be such that
1) ∃p0 ≥ 1 ∀κ ∈ W−k
q a(·, κ) ∈ W k
p0
(X, H, μ);
2) c0 = sup
u∈X
κ∈W−k
q
‖a(u, κ)‖H < ∞, ∀1 ≤ l ≤ k cl = sup
u∈X
κ∈W−k
q
‖Dla(u, κ)‖H < ∞;
3) ∀c > 0 θ(c) = sup
κ∈W−k
q
∫
X
exp(c|δa(u, κ)|)μ(du) < ∞;
4) if {κ, κn, n ≥ 1} ⊂ W−k
q and κn → κ, n → ∞, ∗-weakly in W−k
q , then a(u, κn) →
a(u, κ), n → ∞, in measure μ.
Suppose that
∃ε > 0 κ0 ∈ W−k
q+ε.
ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION 31
Then Eq. (1) has a solution on [0, +∞).
Theorem 2. Let a, κ0 satisfy the conditions of Theorem 1 and, moreover, ∃L > 0 ∃q1 <
q ∀u ∈ X ∀h ∈ H ∀κ1, κ2 ∈ W−k
q
‖a(u, κ1) − a(u + h, κ2)‖H ≤ L(‖h‖H + ‖κ1 − κ2‖q1,−k−1),
∀1 ≤ l ≤ k
‖Dla(u, κ1) − Dla(u + h, κ2)‖H ≤ L(‖h‖H + ‖κ1 − κ2‖q1,−k−1),
where ‖κ‖q,−k = sup
f∈W k
p
‖f‖p,k≤1
|〈f, κ〉|. Then Eq. (1) has a unique solution on [0, +∞).
Remark. If the transformation a in (1) depends only on the first argument, i.e. a(u, μ) =
a0(u), then Eq. (1) turns to be an ordinary differential equation
(1’)
{ dx(u,t)
dt = a0(x(u, t)),
x(u, 0) = u.
It is well known that Eq. (1′) has a unique solution if the transformation a is Lipschitz-
ian. The sufficient conditions for the existence and uniqueness of solution of (1′) were
studied in [2–4] in the case where the transformation a belongs to some Sobolev space,
instead of being Lipschitzian. In particular, it was proved in [3] that if a0 ∈ W 1
p (X, H, μ0)
and exp(|δa0|) ∈ Lc(X, H, μ0), exp(‖Da0‖op) ∈ Lc(X, H, μ0) for some c > 0, then Eq.
(1′) has a unique solution.
2. The space of generalized functions W−k
q−
We shall prove Theorem 1 in a slightly different form involving other spaces of general-
ized functions. Note that if p̃ > p, 1
p+ 1
q = 1 and 1
p+ 1
q = 1, then W k
p ⊂ W k
p , W−k
q ⊂ W−k
q .
Denote W k
p+ =
⋃
p>p W k
p , W−k
q− =
⋂
q<q W−k
q .
The elements of W−k
q− are linear functionals on W k
p+. Define the topology τ on W−k
q− as
τ = C(W k
p+, W−k
q− ). Then κn
τ→ κ, n → ∞, in W−k
q− means by definition that, for every
p̃ > p, the sequence κn, n ≥ 1 converges to κ ∗-weakly in W−k
p , i.e., for every p̃ > p and
for every test function f ∈ W k
p , we have 〈f, κn〉 → 〈f, κ〉, n → ∞.
We now can formulate the result in terms of the spaces W−k
q− .
Theorem 1′. Let a : X × W−k
q− → H be such that
1) ∃p0 ≥ 1 ∀κ ∈ W−k
q− a(·, κ) ∈ W k
p0
(X, H, μ);
2) c0 = sup
u∈X
κ∈W−k
q−
‖a(u, κ)‖H < ∞, ∀1 ≤ l ≤ k cl = sup
u∈X
κ∈W−k
q−
‖Dla(u, κ)‖H < ∞;
3) ∀c > 0 θ(c) = sup
κ∈W−k
q−
∫
X exp(c|δa(u, κ)|)μ(du) < ∞;
4) if {κ, κn, n ≥ 1} ⊂ W−k
q− and κn
τ→ κ, n → ∞, in W−k
q− then a(u, κn) →
a(u, κ), n → ∞, in measure μ.
Then, for every κ0 ∈ W−k
q− , Eq. (1) has a solution on [0, +∞) such that κt ∈
W−k
q− , t ≥ 0.
Remark. Condition 1) in Definition 1 is caused by the fact that the transformation a is
defined on X × W−k
q . Hence, it is reasonable to replace it by κt ∈ W−k
q− , t ≥ 0 in the
setup of Theorem 1′.
32 VOLODYMYR B. BRAYMAN
Theorem 1 follows from Theorem 1′ immediately since W−k
q+ε− ⊂ W−k
q and κ0 ∈
W−k
q+ε ⊂ W−k
q+ε− imply κt ∈ W−k
q+ε− ⊂ W−k
q , t ≥ 0.
Theorem 1′ shows that the solution of Eq. (1) preserves the space W−k
q− , i.e. if the
initial value κ0 belongs to W−k
q− , then the images κt, t ≥ 0, remain the elements of the
same space. The following example shows that the solution of Eq. (1) does not preserve
the space W−k
q , hence the condition κ0 ∈ W−k
q+ε in Theorem 1 cannot be replaced by
κ0 ∈ W−k
q . Therefore, the spaces W−k
q− are more natural when dealing with Eq. (1) than
usual spaces W−k
q .
Example 1. Let X = R, μ(du) = 1√
2π
e−u2/2du, and let a generalized function κ0 be
defined by 〈f, κ0〉 =
∫
R
f(u)1I{u≥0}eu2/2q−√
uμ(du), f ∈ W 1
p . Since∫
R
(1I{u≥0}eu2/2q−√
u)qμ(du) =
1√
2π
∫ ∞
0
eu2/2−√
u · e−u2/2du =
1√
2π
∫ ∞
0
e−q
√
udu < ∞,
we get κ0 ∈ Lq(dμ) ⊂ W−1
q (dμ).
Let a(u, κ) ≡ 1. Then Eq. (1) turns out to be an ordinary differential equation which
has a unique solution x(u, t) = u + t. We have
〈f, κt〉 =
∫
R
f(u + t)1I{u≥0}eu2/2q−√
uμ(du) =
1√
2π
∫ ∞
t
f(v)e(v−t)2/2q−√
v−t·
·e−(v−t)2/2dv =
∫
R
f(v)1I{v≥t}ev2/2q+(q−1)vt/q−(q−1)t2/2−√
v−tμ(dv).
Hence, κt must be a regular generalized function, but∫
R
(
1I{v≥t}ev2/2q+(q−1)vt/q−(q−1)t2/2−√
v−t
)q
μ(dv) =
=
1√
2π
∫ ∞
t
e(q−1)vt−(q−1)t2/2−q
√
v−tdv = ∞
and κt /∈ W−1
q , t > 0. On the other hand, for every q̃ < q, we have κt ∈ Lq(dμ) ⊂
W−1
q (dμ). Therefore, κt ∈ W−1
q− .
3. The proof of Theorem 1′
Note that it is sufficient to obtain the existence of a solution on [0, 1]. Really, since
κ1 ∈ W−k
q− , one can determine the solution on [k, k + 1], k ≥ 0, in succession solving (1)
on [0, 1] with the initial value κk instead of κ0.
Let g : [0, 1] → W−k
q− be some measurable mapping. By [3, Theorem 5.3.1], the
equation
xg(u, t) = u +
∫ t
0
a(xg(u, s), g(s))ds
has a unique (up to μ-equivalence) solution xg(u, t), t ∈ [0, 1] and, moreover, for every
t ∈ [0, 1], the measure μ ◦ xg(·, t)−1 is absolutely continuous with respect to μ. In the
next section, we will verify that, for every t ∈ [0, 1] and f ∈ W k
p+, the function f ◦xg(·, t)
belongs to W k
p+, and there exists the image of the generalized function κ0 ◦ xg(·, t)−1,
i.e. an element of W−k
q− such that, for every f ∈ W k
p+, we have 〈f, κ0 ◦ xg(·, t)−1〉 =
〈f ◦ xg(·, t), κ0〉 (see Proposition 1).
Define
(2) F (g)(t) = κ0 ◦ xg(·, t)−1, t ∈ [0, 1].
ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION 33
Then if the function g satisfies F (g)(t) = κ0 ◦ xg(·, t)−1 = g(t), t ∈ [0, 1], then xg(u, t)
solves (1).
Thus, the solutions of (1) correspond to fixed points of the transformation F. To prove
the existence of a fixed point, we apply the Schauder theorem.
Theorem 3 [5, Theorem 2(3.XVI)]. Let Y0 be a closed convex subset of a linear normed
space Y , and let F : Y0 → Y0 be a continuous transformation such that F (Y0) is relatively
compact. Then F has a fixed point.
Set Y = C([0, 1], W−k
q− ), where W−k
q− is equipped with a metric λ defined as follows.
Fix a sequence 1 < q1 < . . . < qn < qn+1 < . . . < q such that qn → q, n → ∞, and,
for every n ≥ 1, find the family of functions {fnm, m ≥ 1} dense in W k
pn
, where pn is
determined by the condition 1
pn
+ 1
qn
= 1. For every κ1, κ2 ∈ W−k
qn
, we set
λn(κ1, κ2) =
∑
m≥1
1
2m
(1 ∧ |〈fnm, κ1〉 − 〈fnm, κ2〉|).
Then λn is a metric in W−k
qn
, because κ1 �= κ2 implies λn(κ1, κ2) �= 0. Therefore,
λ(κ1, κ2) =
∑
n≥1
1
2n λn(κ1, κ2), κ1, κ2 ∈ W−k
q− is a metric in W−k
q− . Without loss of
generality, we may assume that each of fnm, n, m ≥ 1, belongs to FC∞(X, R) and has
bounded derivatives of any orders.
Remark. Let us compare the convergence in the metric λ with convergence in the topol-
ogy τ in W−k
q− . The sequence of elements of W−k
qn
converges ∗-weakly if and only if it is
bounded in the norm of W−k
qn
and converges in the metric λn. Hence, the sequence of
elements of W−k
q− converges in the topology τ if and only if it is bounded in the norms of
W−k
qn
, n ≥ 1, and converges in the metric λ.
In the next section, we will verify that, for every g ∈ Y , the function F (g) defined by
(2) belongs to Y , and the set F (Y ) is relatively compact. Also we will find a closed convex
set Y0 such that F (Y ) ⊂ Y0 ⊂ Y and F : Y0 → Y0 is continuous. Then the conditions
of the Schauder theorem are valid, and the transformation F has a fixed point, which
proves Theorem 1′. �
4. Properties of the transformation F
Proposition 1. For every g ∈ Y, t ∈ [0, 1], the generalized function F (g)(t) belongs to
W−k
q− .
Proof. Fix g ∈ Y, t ∈ [0, 1]. First, we prove that there exist constants c̃l, 1 ≤ l ≤ k, which
depend on a but do not depend on g ∈ Y and t ∈ [0, 1] such that
(3) esssup
u∈X
‖Dxg(u, t) − 1IH‖H ≤ c̃1 and esssup
u∈X
‖Dlxg(u, t)‖H ≤ c̃l, 2 ≤ l ≤ k.
Since, for μ-almost all u, the derivative Dxg(u, t) satisfies the equation
Dxg(u, t) = 1IH +
∫ t
0
Da(xg(u, s), g(s))Dxg(u, s)ds for every t ≥ 0
(cf. [4, Lemma 5.17]), we have
‖Dxg(u, t)‖op ≤ 1 +
∫ t
0
‖Da(xg(u, s), g(s))‖H‖Dxg(u, s)‖opds ≤
≤ 1 + c1
∫ t
0
‖Dxg(u, s)‖opds
34 VOLODYMYR B. BRAYMAN
for μ-almost all u. By the Gronwall inequality, this implies ‖Dxg(u, t)‖op ≤ ec1t ≤ ec1 =
c̃op for μ-a.a u ∈ X, t ∈ [0, 1]. Hence, for μ-almost all u and for every t ∈ [0, 1],
‖Dxg(u, t) − 1IH‖H ≤
∫ t
0
‖Da(xg(u, s), g(s))‖H‖Dxg(u, s)‖opds ≤ c1c̃opt ≤ c1c̃op = c̃1.
Now
D2xg(u, t) =
∫ t
0
Da(xg(u, s), g(s))D2xg(u, s)ds +
∫ t
0
D2a(xg(u, s), g(s))(Dxg(u, s))2ds
implies
‖D2xg(u, t)‖H ≤ c1
∫ t
0
‖D2xg(u, s)‖Hds + c2c̃
2
op for μ-a.a u, t ∈ [0, 1],
and, by the Gronwall inequality,
∃c̃2 > 0 ∀t ∈ [0, 1] ∀g ∈ Y esssup
u∈X
‖D2xg(u, t)‖H ≤ c̃2.
Similar calculations prove (3) in succession for every l ≤ k.
Fix any n ≥ 1. Let us verify that F (g)(t) ∈ W−k
qn
, i.e. F (g)(t) is a linear continuous
functional on W k
pn
, where p1 > . . . > pn > pn+1 > . . . are taken from the definition of
the metric λ. We prove that, for every f ∈ W k
pn
, the function f ◦xg(·, t) belongs to W k
pn+1
and
(4) ‖f ◦ xg(·, t)‖pn+1,k ≤ c̃‖f‖pn,k, f ∈ W k
pn
.
To simplify notations, we denote here and thereafter, by c̃, any constants which depend
on a, pn, and pn+1 and do not depend on f, g, and t.
Denote, by Lg
t , the density of the measure μ ◦ xg(·, t)−1 with respect to μ. Then, for
every f ∈ Lpn(X, R, μ) we have, by the Hölder inequality,∫
X
|f ◦ xg(u, t)|pn+1μ(du) =
∫
X
|f(u)|pn+1Lg
t (u)μ(du) ≤
≤
(∫
X
|f(u)|pnμ(du)
) pn+1
pn ·
(∫
X
(Lg
t (u))
pn
pn−pn+1 μ(du)
) pn−pn+1
pn
.
By [3, Theorem 5.12] for every c > 1 and 0 ≤ t ≤ 1, we have∫
X
(Lg
t (u))cμ(du) ≤
⎛⎝1 +
c − 1
c
sup
κ∈W−k
q−
∫
X
exp(c|δa(u, κ)|)μ(du)
⎞⎠ e1/c,
hence
∀f ∈ Lpn(X, R, μ) f ◦ xg(·, t) ∈ Lpn+1(X, R, μ) and ‖f ◦ xg(·, t)‖Lpn+1
≤ c̃‖f‖Lpn
.
For every f ∈ W 1
pn
similarly to [4, Corollary 5.6], we have∫
X
‖D(f ◦ xg(u, t))‖pn+1
H μ(du) =
∫
X
‖Df ◦ xg(u, t)Dxg(u, t)‖pn+1
H μ(du) ≤
≤
∫
X
‖Df ◦ xg(u, t)‖pn+1
H · c̃pn+1
op μ(du) = c̃
pn+1
op
∫
X
‖Df(u)‖pn+1
H Lg
t (u)μ(du) ≤
≤ c̃
(∫
X
‖Df(u)‖pn
H μ(du)
) pn+1
pn
,
ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION 35
hence
∀f ∈ W 1
pn
f ◦ xg(·, t) ∈ W 1
pn+1
and ‖f ◦ xg(·, t)‖pn+1,1 ≤ c̃‖f‖pn,1.
Similarly for every f ∈ W k
pn
and for every 2 ≤ l ≤ k,∫
X
‖Dl(f ◦ xg(u, t))‖pn+1
H μ(du) ≤ c̃
l∑
i=1
∫
X
‖Dif ◦ xg(u, t)‖pn+1
H μ(du) ≤
≤ c̃
l∑
i=1
(∫
X
‖Dif(u)‖pn
H μ(du)
) pn+1
pn
.
Therefore,
∀f ∈ W k
pn
f ◦ xg(·, t) ∈ W k
pn+1
and ‖f ◦ xg(·, t)‖pn+1,k ≤ c̃‖f‖pn,k,
and (4) is proved. Hence, for every f ∈ W k
pn
, we can define 〈f, F (g)(t)〉 = 〈f ◦xg(·, t), κ0〉
because of f ◦ xg(·, t) ∈ W k
pn+1
and κ0 ∈ W−k
q− ⊂ W−k
qn+1
. Moreover,
|〈f, F (g)(t)〉| = |〈f ◦ xg(·, t), κ0〉| ≤ ‖κ0‖qn+1,−k · ‖f ◦ xg(·, t)‖pn+1,k ≤
≤ c̃‖κ0‖qn+1,−k‖f‖pn,k = Rn‖f‖pn,k.
Thus, we have F (g)(t) ∈ ∩n≥1W
−k
qn
= W−k
q− . Proposition 1 is proved.
Proposition 2. For every g ∈ Y = C([0, 1], W−k
q− ), the function F (g) belongs to Y and,
moreover, the family of functions {F (g), g ∈ Y } is equicontinuous.
Proof. By Proposition 1 for every g ∈ Y , the function F (g) maps [0, 1] to W−k
q− . Hence,
the first assertion of Proposition 2 follows from the second one.
We have to check that
∀ε > 0 ∃δ > 0 ∀g ∈ Y ∀t1, t2 ∈ [0, 1] |t1 − t2| < δ ⇒ λ(F (g)(t1), F (g)(t2)) < ε.
Since
λ(F (g)(t1), F (g)(t2)) ≤
≤
N∑
n=1
1
2n
(
1 ∧
N∑
m=1
1
2m
(1 ∧ |〈fnm, F (g)(t1)〉 − 〈fnm, F (g)(t2)〉|)
)
+
1
2N−1
,
it is sufficient to prove that
∀n, m ∈ N ∀ε > 0 ∃δ = δnm > 0 ∀g ∈ Y ∀t1, t2 ∈ [0, 1]
(5) |t1 − t2| < δ ⇒ |〈fnm, F (g)(t1)〉 − 〈fnm, F (g)(t2)〉| < ε.
Fix n, m ∈ N and f = fnm. By definition of the metric λ, we have f ∈ FC∞(X, R) ⊂
W k
pn−1
. Then
∀t ∈ [0, 1] f ◦ xg(·, t) ∈ W k
pn
and
∀t1, t2 ∈ [0, 1] |〈f, F (g)(t1)〉 − 〈f, F (g)(t2)〉| = |〈f ◦ xg(·, t1) − f ◦ xg(·, t2), κ0〉| ≤
≤ ‖f ◦ xg(·, t1) − f ◦ xg(·, t2)‖pn,k · ‖κ0‖qn,−k.
Since f has bounded derivatives of any orders, we have∫
X
|f ◦ xg(·, t1) − f ◦ xg(·, t2)|pnμ(du) ≤
≤ c̃f
∫
X
‖xg(·, t1) − xg(·, t2)‖pn
H μ(du).
36 VOLODYMYR B. BRAYMAN
Here and thereafter, we denote, by c̃f , any constants which depend on f but do not
depend on g ∈ Y and t1, t2 ∈ [0, 1]. Also we have∫
X
‖D(f ◦ xg(u, t1)) − D(f ◦ xg(u, t2))‖pn
H μ(du) ≤
≤ 2pn−1
∫
X
‖Df ◦ xg(u, t1) − D(f ◦ xg(u, t2)‖pn
H ·
·‖Dxg(u, t1)‖pn
opμ(du)+2pn−1
∫
X
‖Df ◦xg(u, t1)‖pn
H · ‖Dxg(u, t1)−Dxg(u, t2)‖pn
H μ(du) ≤
≤ c̃f
(∫
X
‖xg(u, t1) − xg(u, t2)‖pn
H μ(du) +
∫
X
‖Dxg(u, t1) − Dxg(u, t2)‖pn
H μ(du)
)
and similarly
∀l ≤ k
∫
X
‖Dl(f ◦ xg(u, t1)) − Dl(f ◦ xg(u, t2))‖pn
H μ(du) ≤
≤ c̃f
(∫
X
‖xg(u, t1) − xg(u, t2)‖pn
H μ(du) +
l∑
i=1
∫
X
‖Dixg(u, t1) − Dixg(u, t2)‖pn
H μ(du)
)
.
It remains to check that
(6) ∀ε > 0 ∃δ > 0 ∀g ∈ Y ∀t1, t2 ∈ [0, 1] |t1 − t2| < δ ⇒ ‖xg(·, t1) − xg(·, t2)‖pn,k < ε.
Let 0 ≤ t1 < t2 ≤ 1. Then xg(u, t2) = xg(u, t1) +
∫ t2
t1
a(xg(u, s), g(s))ds implies, for
μ-almost all u, esssupu∈X ‖xg(u, t2) − xg(u, t1)‖H ≤ (t2 − t1)c0,
Dxg(u, t2) = Dxg(u, t1) +
∫ t2
t1
Da(xg(u, s), g(s))Dxg(u, s)ds
implies, for μ-almost all u,
esssup
u∈X
‖Dxg(u, t2) − Dxg(u, t1)‖ ≤ (t2 − t1)c1c̃op,
and similarly
∀l ≤ k esssup
u∈X
‖Dlxg(u, t2) − Dlxg(u, t1)‖H ≤ |t2 − t1 |̃c̃l,
where ˜̃cl is a function of the constants c0, . . . , cl from condition 2) of Theorem 1′ and of
the constants c̃op, c̃1, . . . , c̃l defined in the proof of Proposition 1.
Therefore, ‖xg(·, t1) − xg(·, t2)‖pn,k ≤ ˜̃c|t1 − t2|, where ˜̃c does not depend on g ∈ Y
and t1, t2 ∈ [0, 1]. Thus, (6) holds true, and the equicontinuity of the family of functions
F (Y ) is proved.
Proposition 3. The set F (Y ) is relatively compact.
Proof. Let us verify the conditions of the Arzela–Ascoli theorem. Since, by Proposition
2, the family of functions F (Y ) is equicontinuous, it remains to check that, for every
t ∈ [0, 1], there exists a compact set Kt ⊂ W−k
q− such that
∀g ∈ Y F (g)(t) ∈ Kt.
It was obtained at the end of the proof of Proposition 1 that
(7) ∀n ≥ 1 ∃Rn > 0 ∀f ∈ W k
pn
∀g ∈ Y ∀t ∈ [0, 1] |〈f, F (g)(t)〉| ≤ Rn‖f‖pn,k.
ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION 37
Set K(n) = {κ ∈ W−k
qn
∣∣‖κ‖qn,−k ≤ Rn}. Then, by the Banach–Alaoglu theorem,
K(n) is a ∗-weak compact in W−k
qn
. Therefore, K(n) is a compact in W−k
qn
with the
metric λn. Let K = ∩n≥1K(n). Then (7) implies that, for every t ∈ [0, 1], we have
{F (g)(t), g ∈ Y } ⊂ K. Moreover, K is a compact in W−k
q− . Really, fix any sequence
{κm, m ≥ 1} ⊂ K. For every n ≥ 1, there exist a subsequence which converges in W−k
qn
since K ⊂ K(n) and K(n) is a compact in W−k
qn
. Then, by applying the diagonal method,
we can find a subsequence which converges in each of W−k
qn
. Hence, this subsequence
converges in W−k
q− . Thus, K is a compact in W−k
q− . The Arzela–Ascoli theorem implies
that F (Y ) is relatively compact.
Set Y0 = C([0, 1], K), where K is a compact constructed in the proof of Proposition 3.
Then F (Y ) ⊂ Y0 ⊂ Y , and Y0 is a closed convex subset of Y. It is evident that F maps
Y0 to Y0.
Proposition 4. The transformation F is continuous on Y0.
Proof. Let gn → g0, n → ∞, in Y0. We have to check that F (gn) → F (g0), n → ∞,
in Y0. Since F (Y0) is relatively compact, there exist a subsequence {ni, i ≥ 1} such
that F (gni) converges in Y0 as i → ∞. Thus, it is sufficient to verify that if gn → g0
and F (gn) → g̃ in Y0, n → ∞, then g̃ = F (g0). We prove that, for every f from the
definition of the metric λ and for every t ∈ [0, 1], 〈f, F (g0)(t)〉 = 〈f, g̃(t)〉. This will imply
g̃ = F (g0). Fix any f from the definition of the metric λ for t ∈ [0, 1].
Since 〈f, F (gn)(t)〉 → 〈f, g̃(t)〉, n → ∞, it is sufficient to check that 〈f, F (gn)(t)〉 →
〈f, F (g0)(t)〉, n → ∞, or, equivalently 〈f ◦ xgn(·, t) − f ◦ xg0 (·, t), κ0〉 → 0, n → ∞. Fix
any m ∈ N. We will verify that
(8) ∀κ ∈ W−k
qm
〈f ◦ xgn(·, t) − f ◦ xg0 (·, t), κ〉 → 0, n → ∞.
Since f ∈ FC∞(X, R) ⊂ W k
pm−1
, we get f ◦ xgn(·, t) ∈ W k
pm
and ‖f ◦ xgn(·, t)‖pm,k
≤
c̃‖f‖pm−1,k
, n ≥ 0, where c̃ is a constant which depends on pm−1, pm but does not depend
on n. It is sufficient to check (8) for κ from a dense subset of W−k
qm
, for example for
regular generalized functions κ of the form
(9) 〈f, κ〉 =
∫
X
f(u)ρ(u)μ(du), where ρ ∈ Lqm(X, R, μ).
Fix any κ defined by (9). We have
|〈f ◦ xgn(·, t) − f ◦ xg0 (·, t), κ〉| =
∣∣∣∣∫
X
(f ◦ xgn(·, t) − f ◦ xg0(·, t))ρ(u)μ(du)
∣∣∣∣ ≤
≤ c̃f
∫
X
‖xgn(u, t) − xg0 (u, t)‖H |ρ(u)|μ(du) ≤
(10) ≤ c̃f
(∫
X
‖xgn(u, t) − xg0 (u, t)‖pm
H μ(du)
)1/pm
(∫
X
|ρ(u)|qmμ(du)
)1/qm
,
where c̃f is a constant which depends only on f.
Similar to the proof of Theorem 5.21 in [3], it can be checked that∫
X
‖xgn(u, t) − xg0 (u, t)‖pm
H μ(du) ≤
≤ c
(∫
X
∫ 1
0
‖a(u, gn(s)) − a(u, g0(s))‖pm−1
H μ(du)ds
) pm
pm−1
,
where c is a constant which depends only on c1 and θ(c) defined in conditions 2), 3) of
Theorem 1′.
38 VOLODYMYR B. BRAYMAN
Note that gn(t) ∈ K, n ≥ 1. Hence, the sequence {gn(t), n ≥ 1} is bounded in the
norms W−k
qm
, m ≥ 1. Moreover, by the remark after the definition of the metric λ, the
convergence gn(t) → g0(t), n → ∞, in W−k
q− with the metric λ implies the convergence
gn(t) τ→ g0(t), n → ∞. By condition 4) of Theorem 1′, this implies
∀t ∈ [0, 1] a(u, gn(t))
μ−→ a(u, g0(t)), n → ∞,
and, by the Lebesgue dominated convergence theorem,∫
X
∫ 1
0
‖a(u, gn(s)) − a(u, g0(s))‖pm−1
H μ(du)ds → 0, n → ∞.
Therefore, (10) proves (8) for any regular generalized function κ ∈ W−k
qm
. Since regular
generalized functions are dense in W−k
qm
, (8) is proved for every κ ∈ W−k
qm
. In particular,
(8) holds for κ = κ0. Then F (g0)(t) = g̃(t), t ∈ [0, 1], and the continuity of F is proved.
5. The proof of Theorem 2
Assume that Eq. (1) has solutions x(u, t) and y(u, t). Then
x(u, t) = u +
∫ t
0
a(x(u, s), κx
s )ds, t ≥ 0,
y(u, t) = u +
∫ t
0
a(y(u, s), κy
s )ds, t ≥ 0,
where κ
x
s = κ0 ◦ x(·, s)−1 ∈ W−k
q , κ
y
s = κ0 ◦ y(·, s)−1 ∈ W−k
q , s ≥ 0. We will find t0 > 0
which depends only on c0, . . . , ck, L and ‖κ0‖q,−k from the conditions of Theorems 1 and
2 such that x(u, s) = y(u, s), s ≤ t0 for μ-almost all u. This implies the uniqueness of
the solution.
Set
Δlx(t) = esssup
u∈X
sup
s≤t
‖Dlx(u, s) − Dly(u, s)‖H, 0 ≤ l ≤ k,
Δκ(t) = sup
s≤t
‖κ
x
s − κ
y
s ‖q1,−k−1 ,
where q1 < q is defined in the formulation of Theorem 2.
Note that Δκ(t) is correctly defined since
κ
x
s − κ
y
s ∈ W−k
q ⊂ W−k−1
q1
, q1 < q.
We have ‖x(u, t) − y(u, t)‖H ≤ L
(∫ t
0
‖x(u, t) − y(u, t)‖Hds +
∫ t
0
‖κ
x
s − κ
y
s ‖q1,−k−1ds
)
,
thus
(11) Δ0x(t) ≤ LtΔ0x(t) + LtΔκ(t).
Let us estimate Δκ(t). We have
‖κ
x
s − κ
y
s ‖q1,−k−1 = sup
‖f‖p1,k+1≤1
|〈f ◦ x(·, s) − f ◦ y(·, s), κ0〉| ≤
≤ ‖κ‖q,−k sup
‖f‖p1,k+1≤1
‖f ◦ x(·, s) − f ◦ y(·, s)‖p,k,
where 1
p1
+ 1
q1
= 1, since, for every f ∈ W k+1
p1
⊂ W k
p1
, we have f ◦x(·, s) ∈ W k
p , f ◦y(·, s) ∈
W k
p .
ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION 39
Set
Δlκ(t) =
= sup
s≤t
sup
‖f‖p1,k+1≤1
(∫
X
‖Dl(f ◦ x(u, s)) − Dl(f ◦ y(u, s))‖p
Hμ(du)
)1/p
, 0 ≤ l ≤ k.
Then
(12) Δκ(t) ≤ ‖κ0‖q,−k
k∑
l=0
Δlκ(t).
By (3), there exists a constant c̃ which depends only on a and is such that
esssup
u∈X
sup
t≤1
(
‖x(u, t) − u‖H + ‖Dx(u, t) − 1IH‖H +
k∑
l=2
‖Dlx(u, t)‖H
)
≤ c̃,
esssup
u∈X
sup
t≤1
(
‖y(u, t)− u‖H + ‖Dy(u, t) − 1IH‖H +
k∑
l=2
‖Dly(u, t)‖H
)
≤ c̃.
Similarly to the proof of Proposition 5.2.1 in [3], it can be checked that∫
X
|f ◦ x(u, s) − f ◦ y(u, s)|pμ(du) ≤
≤ c̃
(∫
X
‖Df(u)‖p1
Hμ(du)
)p/p1
(∫ s
0
∫
X
‖a(u, κx
r ) − a(u, κy
r )‖p2
H μ(du)dr
)p/p2
≤
≤ c̃Δκ(t)tp/p2 , s ≤ t,
where c̃ is a constant which depends only on a and p2 > p. Thus, Δ0κ(t) ≤ c̃Δκ(t)t1/p2 .
Let us estimate Δ1κ(t). By the chain rule, we have
(13)
∫
X
‖D(f ◦ x(u, s)) − D(f ◦ y(u, s))‖p
Hμ(du) ≤
≤ 2p−1
∫
X
‖Df ◦ x(u, s)−Df ◦ y(u, s)‖p
Hμ(du)·
·
(
esssup
u∈X
sup
s≤t
‖Dx(u, s)‖op
)p
+2p−1
(∫
X
‖Df ◦ y(u, s)‖p
Hμ(du)
)p/p
·
·
(∫
X
‖Dx(u, s) − Dy(u, s)‖q
Hμ(du)
)p/q
,
where 1
p + 1
q = 1
p and p < p̃ < p1.
Denote, by Ly
s , the density of the measure μ ◦ y(·, s)−1 with respect to μ. Then∫
X
‖Df ◦ y(u, s)‖p
Hμ(du) =
∫
X
‖Df(u)‖p
HLy
s(u)μ(du) ≤
≤
(∫
X
‖Df(u)‖p1
Hμ(du)
)p/p1 (∫
X
(Ly
s(u))q1μ(du)
)p/q1
,
where 1
p1
+ 1
q1
= 1
p .
Note that y(u, t) satisfie the equation
y(u, t) = u +
∫ t
0
ay
s(y(u, s))ds
40 VOLODYMYR B. BRAYMAN
with ay
s = a(·, κy
s ). Then, by [3, Theorem 5.1.2], there exists a constant c̃ > 0 which
depends only on a such that∫
X
(Ly
s(u))q1μ(du) ≤ c̃, 0 ≤ s ≤ t.
Also we have
‖Dx(u, t) − Dy(u, t)‖H ≤
∫ t
0
‖Da(x(u, s), κx
s )‖H · ‖Dx(u, s) − Dy(u, s)‖Hds+
+
∫ t
0
‖Da(x(u, s), κx
s ) − Da(y(u, s), κy
s )‖H · ‖Dy(u, s)‖opds ≤
≤ c̃
(∫ t
0
(‖Dx(u, s) − Dy(u, s)‖H + ‖x(u, s) − y(u, s)‖H + ‖κ
x
s − κ
y
s ‖q1,−k−1) ds
)
for μ-almost all u, where c̃ depends only on a and L.
Similarly to the proof of Proposition 5.2.1 in [3], we get∫
X
‖Df ◦ x(u, s) − Df ◦ y(u, s)‖p
Hμ(du) ≤ c̃
(∫
X
‖D2f(u)‖p1
H
)p/p1
·
·
(∫ s
0
‖Da(u, κx
r ) − Da(u, κy
r )‖p2
Hμ(du)dr
)p/p2
≤ c̃Δκ(t)tp/p2 , s ≤ t,
where c̃ is a constant which depends only on a. Hence, (13) implies
Δ1κ(t) ≤ c̃(Δκ(t)t1/p2 + (Δ1x(t) + Δ0x(t) + Δκ(t))t).
Similarly,
Δlκ(t) ≤ c̃ sup
s≤t
sup
‖f‖p1,k+1≤1
⎛⎝∑
i≤l
∫
X
‖Dif ◦ x(u, s) − Dif ◦ y(u, s)‖p
Hμ(du)
⎞⎠1/p
+
+c̃
∑
i≤l
Δix(t),∫
X
‖Dif ◦ x(u, s) − Dif ◦ y(u, s)‖p
Hμ(du) ≤ c̃tp/p2Δκ(t), s ≤ t
and
(14) Δix(t) ≤ c̃t
⎛⎝∑
j≤i
Δjx(t) + Δκ(t)
⎞⎠ .
Hence,
(15) Δlκ(t) ≤ c̃
⎛⎝Δκ(t)t1/p2 +
⎛⎝∑
i≤l
Δix(t) + Δκ(t))t
⎞⎠⎞⎠ .
By (11), (12), (14), and (15), we have
Δκ(t) +
∑
i≤k
Δix(t) ≤ ct1/p2
⎛⎝Δκ(t) +
∑
i≤k
Δix(t)
⎞⎠ , t < 1,
where c depends only on a, ‖κ0‖q,−k, and L. Thus, for 0 < t0 < 1 such that ct
1/p2
0 < 1, we
have Δ0x(t0) = 0. That is, for μ-almost all u for every s ≤ t0, we have x(u, s) = y(u, s).
The uniqueness is proved. �
ON THE EXISTENCE AND UNIQUENESS OF THE SOLUTION 41
Consider an example which shows that one actually needs Lipschitzian conditions with
the ‖ · ‖q1,−k−1-norm in Theorem 2, although generalized functions belong to W−k
q .
Example 2. Let X = R, μ(du) = 1√
2π
e−u2/2du, and the generalized function κ0 is the
delta-function δ0, i.e. 〈f, κ0〉 = 〈f, δ0〉 = f(0). Fix 1 < p < 2. By the Sobolev embedding
theorem, W 1
p (R, μ) ⊂ C(R). Hence, δ0 ∈ W−1
q , where 1
p + 1
q = 1. Set f(x) =
√|x|, x ∈ R.
Then f ∈ W 1
p for 1 < p < 2 and 〈f, κ〉, κ ∈ W−1
q , is correctly defined. Let ϕ ∈ C∞
0 (R)
be such that |x| ≤ 1
2 implies ϕ(x) = x, |x| ≥ 1 implies ϕ(x) = 0, maxx∈R |ϕ(x)| ≤ 1,
and maxx∈R |ϕ′(x)| ≤ 2. Let a(u, κ) = a(κ) = ϕ(〈f, κ〉), κ ∈ W−1
q . Then Eq. (1) has the
form { dx(u,t)
dt = ϕ(〈f, κt〉) = ϕ(〈f ◦ x(·, t), κ0〉) = ϕ(
√|x(0, t)|),
x(u, 0) = u.
Consider x1(0, t) = 0, x2(0, t) = 1
4 t2, t ∈ [0, 1]. Then x1,2(u, t) = u + x1,2(0, t) are two
distinct solutions of (1) on [0, 1]. It is straightforward to verify that the conditions of
Theorem 1 are valid. Moreover, we have
∀κ1, κ2 ∈ W−1
q |a(κ1) − a(κ2)| = |ϕ(〈f, κ1〉) − ϕ(〈f, κ2〉)| ≤
≤ 2|〈f, κ1〉 − 〈f, κ2〉| ≤ 2‖f‖p,1‖κ1 − κ2‖q,−1,
but the solution of (1) is not unique.
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(New York) 87 (1997), no. 4, 3577–3731.
2. A.B. Cruzeiro, Equations differentielles sur l’espace de Wiener et formules de Cameron-Martin
non-lineaires. (French), J. Funct. Anal. 54 (1983), no. 2, 206–227.
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E-mail : braym@imath.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4423 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T16:37:39Z |
| publishDate | 2005 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Brayman, V.B. 2009-11-09T15:30:16Z 2009-11-09T15:30:16Z 2005 On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space / V.B. Brayman // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 29–41. — Бібліогр.: 5 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4423 519.21 We consider some the following differential equation with interaction governed by a generalized function/ The conditions that guarantee the existence and uniqueness of a solution when
 mapping a belongs to some Sobolev space are obtained. This research has been partially supported by the Ministry of Education and Science of Ukraine, project N GP/F8/0086. en Інститут математики НАН України On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space Article published earlier |
| spellingShingle | On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space Brayman, V.B. |
| title | On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space |
| title_full | On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space |
| title_fullStr | On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space |
| title_full_unstemmed | On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space |
| title_short | On the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space |
| title_sort | on the existence and uniqueness of the solution of a differential equation with interaction governed by generalized function in abstract wiener space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4423 |
| work_keys_str_mv | AT braymanvb ontheexistenceanduniquenessofthesolutionofadifferentialequationwithinteractiongovernedbygeneralizedfunctioninabstractwienerspace |