The Brownian motion process with generalized diffusion matrix and drift vector
Using the method of the classical potential theory, we have constructed a semigroup of operators that describes a multidimensional process of Brownian motion, for which the drift vector and the diffusion matrix are generalized functions.
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irk-123456789-45532009-12-07T12:00:38Z The Brownian motion process with generalized diffusion matrix and drift vector Kopytko, B.I. Novosyadlo, A.F. Using the method of the classical potential theory, we have constructed a semigroup of operators that describes a multidimensional process of Brownian motion, for which the drift vector and the diffusion matrix are generalized functions. 2008 Article The Brownian motion process with generalized diffusion matrix and drift vector / B.I. Kopytko, A.F. Novosyadlo // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 60–70. — Бібліогр.: 10 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4553 en Інститут математики НАН України |
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Using the method of the classical potential theory, we have constructed a semigroup of operators that describes a multidimensional process of Brownian motion, for which the drift vector and the diffusion matrix are generalized functions. |
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Article |
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Kopytko, B.I. Novosyadlo, A.F. |
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Kopytko, B.I. Novosyadlo, A.F. The Brownian motion process with generalized diffusion matrix and drift vector |
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Kopytko, B.I. Novosyadlo, A.F. |
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Kopytko, B.I. |
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The Brownian motion process with generalized diffusion matrix and drift vector |
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The Brownian motion process with generalized diffusion matrix and drift vector |
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The Brownian motion process with generalized diffusion matrix and drift vector |
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The Brownian motion process with generalized diffusion matrix and drift vector |
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The Brownian motion process with generalized diffusion matrix and drift vector |
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brownian motion process with generalized diffusion matrix and drift vector |
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Інститут математики НАН України |
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2008 |
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The Brownian motion process with generalized diffusion matrix and drift vector / B.I. Kopytko, A.F. Novosyadlo // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 60–70. — Бібліогр.: 10 назв.— англ. |
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AT kopytkobi thebrownianmotionprocesswithgeneralizeddiffusionmatrixanddriftvector AT novosyadloaf thebrownianmotionprocesswithgeneralizeddiffusionmatrixanddriftvector AT kopytkobi brownianmotionprocesswithgeneralizeddiffusionmatrixanddriftvector AT novosyadloaf brownianmotionprocesswithgeneralizeddiffusionmatrixanddriftvector |
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2025-07-02T07:46:10Z |
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Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 60–70
UDC 519.21
BOHDAN I. KOPYTKO AND ANDRIY F. NOVOSYADLO
THE BROWNIAN MOTION PROCESS WITH GENERALIZED
DIFFUSION MATRIX AND DRIFT VECTOR
Using the method of the classical potential theory, we have constructed a semigroup
of operators that describes a multidimensional process of Brownian motion, for which
the drift vector and the diffusion matrix are generalized functions.
1. Introduction and formulation of the problem.
On a Euclidean space Rd, d ≥ 2, let us consider two domains:
Dm = {x : x = (x1, . . . , xd) ∈ Rd, (−1)mxd > 0},m = 1, 2. By Dm and S, we denote
the closure and the boundary of Dm , that is, S = {x : x = (x′, xd) ∈ Rd, xd = 0} =
Rd−1, Dm = Dm ∪ S. Suppose that, in Dm, a diffusion process is considered which is
operated by a generating differential operator with constant coefficients
(1) L =
1
2
d∑
i,j=1
bij
∂2
∂xi∂xj
,
where b = (bij) is a symmetric and positive definite matrix.
We also suppose that the bounded continuous functions q1(x′), q2(x′), βkl(x′), αk(x′),
k, l = 1, . . . , d− 1 are defined on S, which will be used for the description of the process
at the points of the boundary of the domains D1, D2. We assume that β(x′) =
(
βkl(x′)
)
is a symmetric and nonnegative definite matrix.
We pose the problem to describe a general enough class of continuous Feller processes
in Rd, for which the generating differential operator at the points of the domains D1 and
D2 coincides with the operator L, and their behavior at the points of the boundary S
is defined with given conjugation condition of Wentzel [1]. This problem is also called
the problem on the pasting of two diffusion processes (see [2,3]). For its solution, we
use analytical methods. With such an approach, the required class of processes will be
generated by a semigroup of operators we specify by means of a solution of the follow-
ing conjugation problem for a linear parabolic equation with the second-order partial
derivatives:
∂u
∂t
= Lu, (t, x) ∈ (0,∞) ×Dm, m = 1, 2,
(2)
u(0, x) = ϕ(x), x ∈ Rd,
(3)
u(t, x′,−0) = u(t, x′,+0), (t, x′) ∈ (0,+∞) × Rd−1,
(4)
2000 AMS Mathematics Subject Classification. Primary 60J60.
Key words and phrases. Brownian motion process, generalized diffusion, analytical methods.
60
THE BROWNIAN MOTION PROCESS 61
L0u ≡ 1
2
d−1∑
k,l=1
βkl(x′)
∂2u(t, x′, 0)
∂xk∂xl
+
d−1∑
k=1
αk(x′)
∂u(t, x′, 0)
∂xk
− q1(x′)
∂u(t, x′,−0)
∂xd
+ q2(x′)
∂u(t, x′,+0)
∂xd
= 0,
(t, x′) ∈ (0,∞) × Rd−1.
(5)
Note that equality (4) means that the required process will be a Feller one, and relation
(5) corresponds to the general Wentzel boundary condition for the multi-dimensional
diffusion processes.
We are interested in the classical solution of problem (2)-(5) that is determined by
a function u(t, x) continuous in the domain (t, x) ∈ [0,∞) × Rd bounded at infinity by
the space variable x, has continuous derivatives ∂u
∂t ,
∂u
∂xi
, ∂2u
∂xi∂xj
(i, j = 1, . . . , d) at the
points of the domains (t, x) ∈ (0,∞)×Dm, m = 1, 2 , and satisfies Eq. (2) and the initial
condition (3) in these domains and conditions (4) and (5) at the points of the boundary
S. Moreover, as follows from (5), the derivatives ∂u
∂xi
, ∂2u
∂xi∂xj
(i, j = 1, . . . , d − 1)
have to exist and be continuous at all points of the domain (t, x) ∈ (0,∞) × Rd. The
existence of such a solution of problem (2)-(5) was first obtained by us using the method
of boundary integral equations with the use of the common potential of a simple layer. In
addition, we will prove that the Markov process constructed with the use of the solution
of problem (2)-(5) can be interpreted as a generalized diffusion process in the sense of
N.I. Portenko [2]. Recall that a similar problem was studied earlier by using analytical
methods in [3], where it was used the construction of the special parabolic potential
of a simple layer in an integral representation of the required semigroup. Moreover, the
initial-boundary-value problem for the common second-order parabolic equation with the
Wentzel boundary condition was considered in [4] and was analyzed by a normalization
method. We mention also papers [5,6], where a problem of construction of the generalized
diffusion was studied by methods of stochastic analysis.
2. Solution of the parabolic problem of conjugation using analytical meth-
ods.
We will construct the classical solution of problem (2)-(5) using the following assump-
tions for the parameters from condition (5):
a) q1, q2, αk, βkl ∈ Hλ(Rd−1), λ ∈ (0, 1), where Hλ(Rd−1) is a Hölder space (see [7]),
moreover q1(x′) ≥ 0, q2(x′) ≥ 0, x′ ∈ Rd−1, inf
x′∈Rd−1
(
q1(x′) + q2(x′)
)
> 0;
b) there exist positive constants β1 and β2 such that, for all x′ ∈ Rd−1 and for any
real vector Θ′ ∈ Rd−1,
β1|Θ′|2 ≤ (β(x′)Θ′,Θ′) ≤ β2|Θ′|2 .
By g(t, x, y) (t > 0, x, y ∈ Rd), we denote a fundamental solution (f.s.) of Eq. (2)
(see [7]). In this case, the function g(t, x, y) is specified by the formula
g(t, x, y) = g(t, x− y) = (2πt)−d/2(det b)−1/2 exp
{
− 1
2t
(
b−1(y − x), y − x
)}
,
where b−1 is the matrix inverse to b, and
(
b−1(y − x), y − x
)
means the scalar product
of the vectors b−1(y − x) and (y − x) in Rd.
Theorem 1. Let the matrix b from (1) be symmetric and positive definite; elements of
the symmetric matrix β and the functions αk, k = 1, . . . , d − 1, q1, q2 from (5) satisfy
conditions a), b), and the initial function ϕ from (3) is twice continuously differentiable
62 BOHDAN I. KOPYTKO AND ANDRIY F. NOVOSYADLO
and bounded together with its derivatives on Rd. Then problem (2)-(5) has a unique
classical solution, for which the estimation
(6)
∣∣u(t, x)
∣∣ ≤ C ||ϕ||
holds at (t, x) ∈ [0, T ] × Rd (T > 0 – fixed), where
||ϕ|| = sup
x∈Rd
|ϕ(x)| + sup
x∈Rd
d∑
i=1
∣∣∣∣∂ϕ(x)
∂xi
∣∣∣∣ + sup
x∈Rd
d∑
i,j=1
∣∣∣∣∂2ϕ(x)
∂xi∂xj
∣∣∣∣ ,
and C is some constant finite for T <∞.
Proof. We will find a solution of problem (2)-(5) in the form
u(t, x) = u0(t, x) + u1(t, x), t > 0, x ∈ Dm, m = 1, 2,(7)
where
u0(t, x) =
∫
Rd
g(t, x, y)ϕ(y)dy,
u1(t, x) =
∫ t
0
dτ
∫
Rd−1
g(t− τ, x, y′)V (τ, y′)dy′,
V (t, x′) (t > 0, x′ ∈ Rd−1) is an unknown function. In the theory of parabolic equa-
tions, the functions u0 and u1 are called the Poisson potential and the potential of a
simple layer respectively. For any bounded and measurable function V , the function u
from (7) satisfies Eq. (2) and conditions (3) and (4). This follows from the conditions of
the theorem and the properties of potentials (see [7]). In addition, the condition
(8) u0 ∈ C1,2
t,x
(
[0,∞) × Rd
)
and the estimation ( (t, x) ∈ [0, T ] × Rd)
(9)
∣∣Dr
t D
p
x u0(t, x)
∣∣ ≤ C ||ϕ||, 2r + p ≤ 2,
hold, where r and p are nonnegative and integer, Dr
t andDp
x are, respectively, the symbols
of partial derivatives with respect to t of order r and with respect to x of order p, and C
is a constant.
Thus, for the solution u of the problem, we need to choose V such that Eq. (5),
inequality (6), and the other properties of the defined classic solution hold.
We suppose a priori that the unknown density V is continuous in the domain (t, x′) ∈
[0,∞) × Rd−1. Also we suppose that V is bounded and continuously differentiable with
respect to variable x′ for t > 0, x′ ∈ Rd−1. In addition, Dx′V (t, x′) is a Hölder function
of the same variable. To find V , we use the condition of conjugation (5). We separate
the conormal derivative in the representation for L0u and, after simple transformations,
obtain the equation
L
′
0u ≡ 1
2
d−1∑
k,l=1
β
(0)
kl (x′)
∂2u(t, x′, 0)
∂xk∂xl
+
d−1∑
k=1
α
(0)
k (x′)
∂u(t, x′, 0)
∂xk
−
− u(t, x′, 0) = Θ(0)(t, x′), t > 0, x′ ∈ Rd−1,(10)
THE BROWNIAN MOTION PROCESS 63
where
β
(0)
kl (x′) =
√
bdd
q1(x′) + q2(x′)
βkl(x′), α
(0)
k (x′) =
√
bdd
q1(x′) + q2(x′)
αk(x′) − q(x′)√
bdd
bkd,
k, l = 1, . . . , d− 1,
Θ(0)(t, x′) =
1
2
1 − q(x′)√
bdd
∂u(t, x′,−0)
∂N(x′)
− 1
2
1 + q(x′)√
bdd
∂u(t, x′,+0)
∂N(x′)
− u(t, x′, 0),
q(x′) =
q2(x′) − q1(x′)
q1(x′) + q2(x′)
, |q(x′)| ≤ 1,
N(x′) = bν(x′)
(
ν(x′) = (0, . . . , 0, 1) ∈ Rd
)
is the conormal vector.
In view of (7) and the relation from a corollary of the theorem on a jump of the
conormal derivative of the potential of a simple layer (see [2,7]), we can write the function
Θ(0) as
(11)
Θ(0)(t, x′) =
V (t, x′)√
bdd
−
∫ t
0
dτ
∫
Rd−1
g(t− τ, x′, y′)V (τ, y′)dy′
− q(x′)√
bdd
∂u0(t, x′, 0)
∂N(x′)
− u0(t, x′, 0).
Then we will consider equality (10) as an autonomous elliptical equation for u(t, x′, 0)
on Rd−1.
At first, we note that the conditions of Theorem 1 guarantee the existence of the main
f.s. Γ(x′, y′) for the operator L
′
0 (see [8–10]) that can be described in our case by the
formula
Γ(x′, y′) =
∫ ∞
0
e−sG(s, x′, y′)ds,
where G(s, x′, y′) (s > 0, x′, y′ ∈ Rd−1) is a f.s. of the uniformly parabolic operator
with Hölder coefficients
1
2
d−1∑
k,l=1
β
(0)
kl (x′)
∂2
∂xk∂xl
+
d−1∑
k=1
α
(0)
k (x′)
∂
∂xk
− ∂
∂s
.
The matrix with elements β(0)
kl (x′), k, l = 1, . . . , d− 1, and the vector, whose compo-
nents are the functions α(0)
k (x′), k = 1, . . . , d − 1, are denoted by β(0)(x′) and α(0)(x′),
respectively.
We note some known properties of the f.s. G (see [2,7]):
1) the function G(s, x′, y′) is nonnegative, continuous in all variables, and is repre-
sented by the formula
(12) G(s, x′, y′) = G0(s, x′, y′) +G1(s, x′, y′), s > 0, x′, y′ ∈ Rd−1,
where
G0(s, x′, y′) = G
(y′)
0 (s, x′ − y′)
= (2πs)−
d−1
2
(
detβ0(y′)
)−1/2 exp
{
− 1
2s
(
β−1
0 (y′)(y − x), y − x
)}
,
and G1(s, x′, y′) can be written with the use of an integral operator with kernel G0 and
density Φ0 that is defined from some integral equation;
2) the functions G,G0, G1, as functions of the arguments t and x, are continuously
differentiable with respect to t, twice continuously differentiable with respect to x′ (the
64 BOHDAN I. KOPYTKO AND ANDRIY F. NOVOSYADLO
function G0 is infinitely continuously differentiable with respect to the mentioned vari-
ables), and satisfy the inequalities
∣∣Dr
sD
p
x′G(s, x′, y′)
∣∣ ≤ Cs−
(d−1)+2r+p
2 exp
{
−c |x
′ − y′|2
s
}
,(13)
∣∣Dr
sD
p
x′G1(s, x′, y′)
∣∣ ≤ Cs−
(d−1)+2r+p−λ
2 exp
{
−c |x
′ − y′|2
s
}
,(14) ∣∣Dr
sD
p
z′G
(y′)
0 (s, z′) −Dr
sD
p
z′G
(ỹ′)
0 (s, z′)
∣∣
≤ C
∣∣y′ − ỹ′
∣∣γs− (d−1)+2r+p
2 exp
{
−c |z
′|2
s
}
, 0 < γ ≤ λ,
(15)
when ever 2r+ p ≤ 2, s ∈ [0, T ], x′, y′, ỹ′, z′ ∈ Rd−1 with positive constants C and c. We
note also that inequality (15) is true for all nonnegative integers r and p, and the constant
C in inequalities (13) and (14) depends, generally speaking, on T . However, in the case
where the function G and its derivatives are estimated together with a coefficient of the
form e−μs, where μ is a positive number, we can always assume that, in inequalities
(13),(14), the constant C does not depend on T . Such consequences are also true for
other functions of such a type;
3) ∫
Rd−1
G(s, x′, y′)dy′ = 1 for all s > 0, x′ ∈ Rd−1;
4)∫
Rd−1
G(s, x′, z′)G(t, z′, y′)dz′ = G(s+ t, x′, y′) for s > 0, t > 0, x′, y′ ∈ Rd−1;
5) for all s ≥ 0, x′ ∈ Rd−1, Θ′ ∈ Rd−1, the equalities∫
Rd−1
G(s, x′, y′)(y′ − x′,Θ′)dy′ =
∫ s
0
dτ
∫
Rd−1
G(τ, x′, y′)(α0(y′),Θ′)dy′,∫
Rd−1
G(s, x′, y′)(y′ − x′,Θ′)2dy′ =
∫ s
0
dτ
∫
Rd−1
G(τ, x′, y′)(β0(y′)Θ′,Θ′)dy′+
+ 2
∫ s
0
dτ
∫
Rd−1
G(τ, x′, y′)(α0(y′),Θ′)(y′ − x′, θ′)dy′
hold.
Let us consider the right-hand side of Eq. (10). We can assume that Θ0(t, x′) (t ≥
0, x′ ∈ Rd−1) from (11) is continuous in two variables and is continuously differentiable
at t > 0 with respect to the variable x′ (x′ ∈ Rd−1) and bounded together with its
derivative. This corollary can be obtained by using conditions of Theorem 1, an a priori
assumption concerning V , and the properties of parabolic potentials. Then (see [9, Ch.
III, §20]) the unique solution of Eq. (10) is represented by the formula
(16)
u(t, x′, 0) = −
∫
Rd−1
Γ(x′, z′)Θ(0)(t, z′)dz′
= −
∫ ∞
0
e−sds
∫
Rd−1
G(s, x′, z′)Θ(0)(t, z′)dz′, t > 0, x′ ∈ Rd−1.
THE BROWNIAN MOTION PROCESS 65
Equating the right-hand sides of relations (7) (where we need to put xd = 0) and (16),
we obtain the required equation for V :
(17)∫ t
0
dτ
∫
Rd−1
K0(t− τ, x′, y′) V (τ, y′)dy′ +
∫ ∞
0
e−sds
∫
Rd−1
G(s, x′, z′)
V (t, z′)√
bdd
dz′ =
= ψ0(t, x′), t > 0, x′ ∈ Rd−1,
where
K0(t− τ, x′, y′) = g(t− τ, x′, y′) −
∫ ∞
0
e−sds
∫
Rd−1
G(s, x′, z′) g(t− τ, z′, y′)dz′,
ψ0(t, x′) =
∫ ∞
0
e−sds
∫
Rd−1
G(s, x′, z′)
(
q(z′)√
bdd
∂u0(t, z′, 0)
∂N(z′)
+ u0(t, z′, 0)
)
dz′
− u0(t, x′, 0).
Using properties of the f.s. G and the Poisson potential, we analyzed the function
ψ0(t, x′). We got that it is continuous, twice continuously differentiable with respect
to x′ at t ≥ 0, x′ ∈ Rd−1, infinitely continuously differentiable with respect to t at
t > 0, x′ ∈ Rd−1, and the following estimations hold for it:∣∣Dp
x′ψ(t, x′)
∣∣ ≤ C||ϕ||, p ≤ 2, (t, x′) ∈ [0, T ] × Rd−1∣∣Dr
tψ(t, x′)
∣∣ ≤ C||ϕ|| t−
2r−1
2 , r ≥ 1, (t, x′) ∈ (0, T ] × Rd−1∣∣Dr
tD
p
x′ψ(t, x′)
∣∣ ≤ C||ϕ|| t−(r−1)− p
2 , r ≥ 1, p = 1, 2, (t, x′) ∈ (0, T ] × Rd−1.
As we can see, Eq. (17) is a first-kind integral equation of the Volterra–Fredholm
type. With the purpose to normalize the equation, we consider an integro-differential
operator E that acts by the rule
(18)
E(t, x′)ψ0
=
√
2/π
{
∂
∂t
∫ t
0
(t− τ)−1/2dτ
∫
Rd−1
ψ0(τ, y′)dy′
×
[
h(t̂− τ, x′, y′)
+
∫ ∞
0
(1 − u
t− τ
)e−
u2
2(t−τ) du
∫
Rd−1
h(t̂− τ, x′, v′)G(u, v′, y′)dv′
]}∣∣∣∣
t̂=t
,
t > 0, x′ ∈ Rd−1,
where h(t, x′, y′) (t > 0, x′, y′ ∈ Rd−1) is an f.s. of a parabolic operator with constant
coefficients
∂
∂t
− 1
2
d−1∑
i,j=1
b̃ij
∂2
∂xi∂xj
, b̃ij = bij −
bid bjd
bdd
, i, j = 1, . . . , d− 1.
Properties 1)-5) can be easily extended to the f.s. h with obvious changes.
Applying the operator E to both sides of Eq. (17) leads to the equivalent second-kind
Volterra integral equation
(19) V (t, x′) =
∫ t
0
dτ
∫
Rd−1
K(t− τ, x′, y′)V (τ, y′)dy′ + ψ(t, x′), t > 0, x′ ∈ Rd−1,
66 BOHDAN I. KOPYTKO AND ANDRIY F. NOVOSYADLO
where
ψ(t, x′) = (bdd)1/2E(t, x′)ψ0,
K(t− τ, x′, y′) =
1
π
{
∂
∂t
∫ t
τ
(t− s)−3/2(s− τ)−1/2ds
∫ ∞
0
ue−
u2
2(t−s) du
×
∫
Rd−1
h(t̂− s, x′, v′)dv′
×
∫
Rd−1
(
G(u, v′, z′) −G
(y′)
0 (u, v′ − z′)
)
h(s− τ, z′, y′)dz′
}∣∣∣∣
t̂=t
−
√
2
π
∫ ∞
0
∂
∂t
(
(t− τ)−1/2e−
u2
2(t−τ)
)
du
×
∫
Rd−1
h(t− τ, x′, z′)G1(u, z′, y′)dz′
= K1(t− τ, x′, y′) −K2(t− τ, x′, y′),
in addition, the kernel K(t− τ, x′, y′) at 0 ≤ τ < t ≤ T, x′, y′ ∈ Rd−1 and some positive
constants C and c allow the estimation
(20)
∣∣∣K(t−τ, x′, y′)
∣∣∣ ≤ C(t−τ)−
3
2+ λ
4
∫ ∞
0
e−c u2
t−τ (t−τ+u)−
d−1
2 exp
{
− c
|x′ − y′|2
t− τ + u
}
du.
We prove the implementation of inequality (20) with an example of estimation of the
function K2 that is included in the formula for the kernel K. On the basis of estimations
(13) and (14) applied to the f.s. h and G1 and the obvious inequality
(21)
∣∣∣∣ ∂∂t (t− τ)−1/2e−
u2
2(t−τ)
∣∣∣∣ ≤ C(t− τ)−3/2e−c1
u2
t−τ ,
we have (0 ≤ τ < t ≤ T, x′, y′ ∈ Rd−1)∣∣∣K2(t− τ, x′, y′)
∣∣∣ ≤ C
∫ ∞
0
(t− τ)−
3
2 e−c u2
t−τ du×
×
∫
Rd−1
(t− τ)−
d−1
2 exp
{
− c
|x′ − z′|2
t− τ
}
u−
(d−1)−λ
2 exp
{
− c
|z′ − y′|2
u
}
dz′.
Since ∫
Rd−1
exp
{
− c
|x′ − z′|2
t− τ
}
exp
{
− c
|z′ − y′|2
u
}
dz′
=
(π
c
)− d−1
2
(
(t− τ)u
t− τ + u
) d−1
2
exp
{
− c
|x′ − y′|2
t− τ + u
}
,
we have∣∣∣K2(t− τ, x′, y′)
∣∣∣ ≤ C(t− τ)−3/2
∫ ∞
0
u
λ
2 e−c u2
t−τ (t− τ + u)−
d−1
2 exp
{
− c
|x′ − y′|2
t− τ + u
}
du.
Using estimations (13)-(15), we can obtain the same inequality also for the function
K1(t−τ, x′, y′) from the kernel K. For this, we disclose a representation for the derivative
of the function K1 with respect to the variable t.
THE BROWNIAN MOTION PROCESS 67
We explore the function ψ(t, x′) from (19). For this purpose, we use the representation
(22)
ψ(t, x′) =
√
2
π
{
∂
∂t
∫ t
0
(t− τ)−1/2dτ
∫ ∞
0
e−
u2
2(t−τ) du
∫
Rd−1
q(y′)
∂u0(τ, y′, 0)
∂N(y′)
dy′
×
∫
Rd−1
h(t̂− τ, x′, v′)G(u, v′, y′)dv′
}∣∣∣∣
t̂=t
+
√
2bdd
π
{
∂
∂t
∫ t
0
dτ
∫ ∞
0
u
(t− τ)3/2
e−
u2
2(t−τ) du
∫
Rd−1
dy′
×
∫
Rd−1
h(t̂− τ, x′, v′)G(u, v′, y′)
(
u0(τ, y′, 0) − u0(τ, v′, 0)
)
dv′
}∣∣∣∣
t̂=t
= ψ1(t, x′) + ψ2(t, x′),
that is obtained after the substitution of the representation for ψ0 from (17) in (18), by
using elementary transformations and properties of the f.s. h and G.
We prove that the function ψ(t, x′) is continuous and continuously differentiable with
respect to the variable x′ ∈ Rd−1 at t > 0. Moreover, in every domain (t, x′) ∈ (0, T ] ×
Rd−1, the estimation
(23)
∣∣∣Dp
x′ψ(t, x′)
∣∣∣ ≤ C ||ϕ|| t−
p
4 , p = 0, 1,
holds.
At first, we estimate the function ψ1 from (22). We represent it by the formula
(24) ψ1(t, x′) =
√
2
π
∫ t
0
dτ
∫ ∞
0
∂
∂t
(
(t− τ)−1/2e−
u2
2(t−τ)
)
du×
×
[∫
Rd−1
q(y′)
∂u0(τ, y′, 0)
∂N(y′)
dy′
∫
Rd−1
h(t− τ, x′, v′)
(
G(u, v′, y′) −G(u, x′, y′)
)
dv′+
+
∫
Rd−1
G(u, x′, y′)
(
q(y′)
∂u0(τ, y′, 0)
∂N(y′)
− q(x′)
∂u0(τ, x′, 0)
∂N(x′)
)
dy′
]
+ q(x′)
∂u0(t, y′, 0)
∂N(x′)
.
As a result of the estimation of all summands from the right-hand side of (24), we prove
inequality (23) at p = 0 for ψ1. In order to obtain the same estimation for ψ2 from (22)
and consequently for ψ, we have to use the representation
√
π
2bdd
ψ2(t, x′)
(25)
=
∫ t
0
dτ
∫ ∞
0
∂
∂t
(
u
(t− τ)3/2
e−
u2
2(t−τ)
)
du
×
[∫
Rd−1
(
ũ0(τ, y′, x′) − ũ0(t, y′, x′)
)
dy′
∫
Rd−1
h(t− τ, x′, v′)G(u, v′, y′)dv′
+
∫
Rd−1
ũ0(t, y′, x′)dy′
∫
Rd−1
h(t− τ, x′, v′)
×
(
G(u, v′, y′) −G(u, x′, y′) −
(
∇′
x′G(u, x′, y′), v′ − x′
))
dv′
+
∫
Rd−1
G(u, x′, y′)
(
ũ0(t, y′, x′) −
(
∇′u0(t, x′, 0), y′ − x′
))
dy′
]
68 BOHDAN I. KOPYTKO AND ANDRIY F. NOVOSYADLO
+
∫ ∞
0
u
t3/2
e−
u2
2t du
∫ u
0
ds
∫
Rd−1
(
α0(y′),∇′u0(t, x′, 0)
)
dy′
+
1
2
∫ t
0
(t− τ)−3/2dτ
×
∫
Rd−1
h(t− τ, x′, v′)
(
ũ0(τ, v′, y′) −
(
∇′u0(τ, x′, 0), v′ − x′
))
dv′,
where ũ0(τ, y′, x′) = u0(τ, y′, 0) − u0(τ, x′, 0), ∇′ =
(
∂
∂x1
, . . . , ∂
∂xd−1
)
. Here, as in the
previous case, we skip the detailed expositions.
Using formula (22) and the scheme of establishing estimation (23) at p = 0, we prove
similarly that the function ψ(t, x′) is differentiable with respect to the space variable,
inequality (23) is true at p = 1 for it, and the derivative Dx′ψ(t, x′) has the Hölder
property at x′.
We are going back to the integral equation (19). We note that its kernel, as a result
of estimation (20), has an integrable singularity. Applying the method of progressive
approximations to this equation, we find V . In addition, we prove that the obtained
solution of Eq. (19) has the same properties as the right-hand side of this equation that
is then the function ψ(t, x′).
Applying estimations (9) and (23) for the functions u0 and V , respectively, and taking
into account the Hölder property by the variable x′ of the density of Eq. (19) guarantee
us the existence of the desired solution of problem (2)-(5) and the implementation of
inequality (6) for it.
Now we prove the statement of the theorem concerning the uniqueness of the found
solution. It is enough only to notice that the solution of problem (2)-(5) constructed by
formulas (7) and (19) for each domain t > 0, x ∈ Dm, m = 1, 2, can be considered as a
classical solution of the parabolic first-boundary problem
Dtu = Lu, (t, x) ∈ (0,∞) ×Dm, m = 1, 2 ,
u(0, x) = ϕ(x), x ∈ Dm, m = 1, 2 ,
u(t, x′, 0) = v(t, x′), (t, x′) ∈ [0,+∞) × Rd−1,
where the function v is defined, by using relation (16).
Theorem 1 is proved.
3. Construction of process.
By B(Rd), we denote the Banach space of all real bounded measurable functions on
Rd with the norm ‖ϕ‖ = sup
x∈Rd
|ϕ(x)| . We prove that the solution of problem (2)-(5)
constructed by formulas (7) and (19) determines a family of linear operators (Tt)t>0 that
acts in the space B(Rd). Consequently, let ϕ ∈ B(Rd). Then the existence of the first
summand in (7) follows from the obvious inequality
(26)
∣∣u0(t, x)
∣∣ ≤ C||ϕ||
that is true with some constant C for t > 0, x ∈ Rd. Now we consider the integral
equation (19) and estimate its right-hand side that is the function ψ(t, x′). For this, it
is possible to use representations (22), (24), and (25) again. By estimating, we need to
take into account that, under the condition ϕ ∈ B(Rd), the derivatives of the second or
higher orders with respect to x and those of the first or higher orders with respect to
t of the function u0(t, x) have a nonintegrable singularity by t. This follows from the
inequality (t ∈ (0, T ], x ∈ Rd)
(27)
∣∣Dr
t D
p
x u0(t, x)
∣∣ ≤ C ||ϕ|| t−
2r+p
2 ,
THE BROWNIAN MOTION PROCESS 69
where r and p are positive and integer, and C is a constant.
We estimate, for example, the last summand from (25). We denote it by I and write
it in the form
I =
1
2
∫ t/2
0
(t−τ)−3/2dτ
∫
Rd−1
h(t−τ, x′, v′)
(
ũ0(τ, v′, y′)−
(
∇′u0(τ, x′, 0), v′−x′
))
dv′+
+
1
2
∫ t
t/2
(t− τ)−3/2dτ
∫
Rd−1
h(t− τ, x′, v′)
(
ũ0(τ, v′, y′) −
(
∇′u0(τ, x′, 0), v′ − x′
))
dv′.
Estimating the right-hand side of the last equality with the use of (13) and (27), we
obtain (t ∈ (0, T ], x′ ∈ Rd−1)
|I| ≤ C‖ϕ‖
[∫ t/2
0
(t− τ)−3/2τ−1/2dτ
∫
Rd−1
(t− τ)−
d−1
2 exp
{
−c |x
′ − v′|2
t− τ
}∣∣v′ − x′
∣∣dv′
+
∫ t
t/2
(t− τ)−1/2τ−1dτ
∫
Rd−1
(t− τ)−
d−1
2 exp
{
−c |x
′ − v′|2
t− τ
}∣∣v′ − x′
∣∣2dv′]
≤ C‖ϕ‖ t−1/2.
Similar inequalities are true for other summands included in the right-hand sides of
formulas (24) and (25). Uniting these inequalities, we obtain that the estimation∣∣ψ(t, x′)
∣∣ ≤ C ||ϕ|| t−1/2,
is true at t ∈ (0, T ], x′ ∈ Rd−1.
The last inequality and (20) lead to the estimation for the solution of Eq. (19):
(28)
∣∣V (t, x′)
∣∣ ≤ C ||ϕ|| t−1/2.
Here, t ∈ (0, T ], x′ ∈ Rd−1, and C is some constant.
Estimation (28) yields the existence of the potential of a simple layer in (7) and the
implementation of inequality (26) for it and, consequently, for the function Ttϕ(x).
As was done in work [3], we prove that the family of operators (Tt)t>0 constructed
by formulas (7) and (19) creates a semigroup that determines some homogeneous non-
precipice Feller process on Rd. We denote its transition probability by P (t, x, dy), so
Ttϕ(x) =
∫
Rd
ϕ(y)P (t, x, dy).
At the end of our researches, we prove, by direct calculations, that the transition
probability P (t, x, dy) has the properties:
a)
(29) sup
x∈Rd
∫
Rd
|y − x|6 P (t, x, dy) ≤ C t3/2 , t ∈ (0, T ],
where C is some positive constant;
b) for any finitary continuous function ϕ(x), x ∈ Rd, and all Θ ∈ Rd,
lim
t↓0
∫
Rd
ϕ(x)
[
1
t
∫
Rd
(y − x,Θ)P (t, x, dy)
]
dx =
∫
Rd−1
ϕ(x′, 0)
(
ᾱ(x′),Θ
)
dx′,
lim
t↓0
∫
Rd
ϕ(x)
[
1
t
∫
Rd
(y − x,Θ)2P (t, x, dy)
]
dx = (bΘ,Θ)
∫
Rd
ϕ(x)dx+
+
∫
Rd−1
ϕ(x′, 0)
(
β̄(x′)Θ,Θ
)
dx′,(30)
70 BOHDAN I. KOPYTKO AND ANDRIY F. NOVOSYADLO
where
ᾱ(x′) =
bdd
q1(x′) + q2(x′)
(
α1(x′), . . . , αd−1(x′), q2(x′) − q1(x′)
)
∈ Rd,
β̄(x′) =
(
β̄ij(x′)
)d
i,j=1
,
βij(x′) =
{
bdd
q1(x′)+q2(x′) βij(x′) if i, j = 1, . . . , d− 1 ,
0 if i = d or j = d.
From inequality (29), it follows that the trajectories of the constructed process are
continuous, and relations (30) show that this process can be interpreted as a generalized
diffusion in the sense of N.I. Portenko [2]. Here, the drift vector and its diffusion matrix
equal, respectively,
α(x′) δS(x) and b+ β(x′) δS(x) ,
where δS(x) is a generalized function on Rd concentrated on the surface S.
Therefore, we proved such a theorem.
Theorem 2. Let the conditions of Theorem 1 be true for the coefficients of the operators
L and L0 from (1) and (5). Then the solution of problem (2)-(5) determines a semigroup
of operators that describes a generalized diffusive process in Rd with the characteristic
expressed by formulas (30).
Bibliography
1. A.D.Wentzel, On boundary conditions for multidimensional diffusion process, Probab. Theory
Appl. 15 (1959), no. 2, 172-185. (in Russian)
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ics of NAS of Ukraine, Kiev, 1995. (in Ukrainian)
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