Pasting of two diffusion processes on a line with nonlocal boundary conditions
In this paper, we obtain an integral representation of an operator semigroup that describes the Feller process on a line that is a result of pasting together two diffusion processes with a nonlocal condition of conjugation.
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irk-123456789-45522009-12-07T12:00:27Z Pasting of two diffusion processes on a line with nonlocal boundary conditions Kononchuk, P. In this paper, we obtain an integral representation of an operator semigroup that describes the Feller process on a line that is a result of pasting together two diffusion processes with a nonlocal condition of conjugation. 2008 Article Pasting of two diffusion processes on a line with nonlocal boundary conditions / P. Kononchuk // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 52–59. — Бібліогр.: 9 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4552 519.21 en Інститут математики НАН України |
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In this paper, we obtain an integral representation of an operator semigroup that describes the Feller process on a line that is a result of pasting together two diffusion processes with a nonlocal condition of conjugation. |
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Article |
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Kononchuk, P. |
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Kononchuk, P. Pasting of two diffusion processes on a line with nonlocal boundary conditions |
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Kononchuk, P. |
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Kononchuk, P. |
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Pasting of two diffusion processes on a line with nonlocal boundary conditions |
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Pasting of two diffusion processes on a line with nonlocal boundary conditions |
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Pasting of two diffusion processes on a line with nonlocal boundary conditions |
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Pasting of two diffusion processes on a line with nonlocal boundary conditions |
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Pasting of two diffusion processes on a line with nonlocal boundary conditions |
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pasting of two diffusion processes on a line with nonlocal boundary conditions |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/4552 |
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Pasting of two diffusion processes on a line with nonlocal boundary conditions / P. Kononchuk // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 52–59. — Бібліогр.: 9 назв.— англ. |
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AT kononchukp pastingoftwodiffusionprocessesonalinewithnonlocalboundaryconditions |
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2025-07-02T07:46:07Z |
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2025-07-02T07:46:07Z |
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1836520431922708480 |
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Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 52–59
UDC 519.21
PAVLO KONONCHUK
PASTING OF TWO DIFFUSION PROCESSES ON
A LINE WITH NONLOCAL BOUNDARY CONDITIONS
In this paper, we obtain an integral representation of an operator semigroup that
describes the Feller process on a line that is a result of pasting together two diffusion
processes with a nonlocal condition of conjugation.
1. Introduction and formulation of the problem. Let D1 = {x ∈ R1 : x < 0}
and D2 = {x ∈ R1 : x > 0} be domains on R1, S = {0} is boundary of the domain Di,
Di = Di ∪{0} is the closure of Di, i = 1, 2. Assume that Li is a second-order differential
operator that operates on the set C2
K(Di) of all twice continuously differentiable functions
with compact supports:
(1) Liϕ(x) =
1
2
bi(x)
d2ϕ(x)
dx2
+ ai(x)
dϕ(x)
dx
,
where bi(x) and ai(x), i = 1, 2, are continuous and bounded on Di, and also bi(x) ≥ 0.
Let us also assume that, at the point x = 0, the following integral operator is defined:
(2) L0ϕ(0) =
∫
R1
(ϕ(0) − ϕ(y))μ(dy),
where μ(·) is a nonnegative Borel measure on R1, μ(R1) > 0.
We assume that operator (2) is a part of the general Feller–Wentzell boundary operator
([1], [2]) that corresponds to jumps of the process after its reach to the boundary S.
Let us consider the following problem: to construct an operator semigroup Tt that
generates the Feller process on R1 such that it coincides with a diffusion process controlled
by the operator Li, i = 1, 2, and its behavior at the point x = 0 is determined by the
boundary condition
(3) L0ϕ(0) = 0.
The formulated problem is often called the problem of pasting of diffusion processes
on a line (e.g., see [3], [4]).
Note that the problem of pasting of two diffusion processes on a line was previously
considered in the most general formulation in [5]. Also, the problem of existence of a Feller
semigroup that describes a diffusion process on a domain with boundary conditions of
kind (2) has been considered in [6]. In the papers mentioned above, solutions of respective
problems were found, by using methods of functional analysis.
In this article, the desired semigroup will be constructed by analytical methods using
a solution of the corresponding conjugation problem for a second-order parabolic linear
2000 AMS Mathematics Subject Classification. Primary 60J60.
Key words and phrases. Diffusion process, nonlocal boundary conditions, analytical methods.
52
KNOTTING OF DIFFUSION PROCESSES ON A LINE 53
equation with discontinuous coefficients. The problem is to find a function u(t, x) (t >
0, x ∈ R1 that satisfies the following conditions:
∂u
∂t
= Liu, t > 0, x ∈ D1 ∪ D2,(4)
u(0, x) = ϕ(x), x ∈ D1 ∪ D2,(5)
u(t,−0) = u(t,+0), t > 0,(6) ∫
R1
(u(t, 0) − u(t, y))μ(dy) = 0, t > 0.(7)
Note that, in problem (4)–(7), relationship (6) represents the Feller property for the
process, and equality (7) represents boundary condition (3).
For solving problem (4)–(7), we will use the classical potential method. This approach
allows us to obtain an integral representation for the concerned semigroup.
2. Basic notations. Let Dr
t and Dp
x be symbols of partial derivative with respect to t
of order r and with respect to x of order p, respectively, where r and p are nonnegative
integers; B(R1) is the Banach space of all measurable bounded real-valued functions
on R1 with the norm ‖ϕ‖ = supx |ϕ(x)|; T is a fixed number; R2
∞ = (0,∞) × R1;
R2
T = (0, T ) × R1; Ω is a domain in R2∞ or in R2
T ; C(Ω) (C(Ω)) is a set of functions
continuous in Ω (Ω) with continuous derivatives Dt, Dp
x, p = 1, 2 on Ω (Ω); Hα(R1),
α ∈ (0, 1), denotes the Hölder space as in [7, p.16]; and Dδ = {x ∈ R1 : |x| > δ > 0}; C,
c are positive constants not depending on (t, x), whose exact values are irrelevant.
3. Solution of the parabolic conjugation problem using analytical meth-
ods. Additionally, we assume that L1, L2 from (1) and the measure μ(·) from (2) satisfy
the following conditions:
a) functions bi(x), ai(x), i = 1, 2, are defined on R1, and bi, ai ∈ Hα(R1);
b) there exist constants b0i, b1i, i = 1, 2, such that 0 < b0i ≤ b1i, i = 1, 2, and
b0i ≤ bi(x) ≤ b1i, i = 1, 2;
c) there exists Δ > 0 such that, for 0 < δ < Δ and for all functions ϕ from B(R1),∣∣∣∣∫Dδ
ϕ(y)μ(dy)
∣∣∣∣ ≤ C1‖ϕ‖,(8) ∣∣∣∣∣
∫
R1\Dδ
ϕ(y)μ(dy)
∣∣∣∣∣ ≤ C2(δ)‖ϕ‖,(9)
where C1 > 0 does not depend on δ, and C2(δ) → 0 as δ → 0.
Remark 1. From a) and b), it follows (see [7]) that there exists a fundamential solution
(f.s.) of Eq. (4) that will be denoted by gi(t, x, y) (t > 0, x, y ∈ R1), i = 1, 2.
Remark 2. c) implies that μ{0} = 0 and μ(R1) < ∞. Without loss of generality, we
consider μ(R1) = 1.
Let us recall some known properties of f.s. gi, i = 1, 2 that we will use further in our
paper:
1) the function gi(t, x, y) is nonnegative continuous in all variables and is expressed
by the formula
(10) gi(t, x, y) = gi0(t, x, y) + gi1(t, x, y), t > 0, x, y ∈ R1, i = 1, 2,
where
gi0 = (2πbi(y)t)−
1
2 exp
(
− (x− y)2
2bi(y)t
)
,
54 PAVLO KONONCHUK
gi1(t, x, y) is in the form of an integral operator with a kernel gi0 and a density Φi0 that
is defined from some integral equation (gi1 ≡ 0 when t ≤ 0);
2) the function gi(t, x, y), i = 1, 2, as a function of arguments t and x, is continuously
differentiable with respect to t and twice continuously differentiable with respect to x,
and
|Dr
tD
p
xgi(t, x, y)| ≤ Ct−
1+2r+p
2 exp
(
−c |x− y|2
t
)
,(11)
|Dr
tD
p
xgi1(t, x, y)| ≤ Ct−
1+2r+p−α
2 exp
(
−c |x− y|2
t
)
,(12)
where 2r + p ≤ 2, 0 < t ≤ T.
We now establish a classical solvability of problem (4)–(7) on the space of all functions
continuous and bounded in the variable x.
Theorem 1. Assume that conditions a)-c) hold for the coefficients of the operators L1
and L2 from (1) and for the measure μ(·) from (2). Then, for every continuous function
ϕ ∈ B(R1) from (5), problem (4)–(7) has a unique solution
(13) u ∈ C1,2((0,∞) ×Di) ∩ C((0,∞) × R1), i = 1, 2,
for which (t ∈ (0, T ], x ∈ R1)
(14) |u(t, x)| ≤ C‖ϕ‖,
and this solution can be obtained in the form
(15) u(t, x) =
∫
R1
gi(t, x, y)ϕ(y)dy +
∫ t
0
gi(t− τ, x, 0)Vi(τ)dτ, t > 0, x ∈ Di, i = 1, 2,
where Vi(t) (t > 0) is a solution of some system of second-kind Volterra integral equations.
Proof. According to the statement of Theorem 1, we will find a solution of problem (4)–
(7) as (15), where Vi, i = 1, 2, are the unknown functions that will be defined from the
conjugation conditions (6) and (7). To this end, we denote the first and second terms on
the right-hand side of Eq. (15) by ui0(t, x) and ui1(t, x) i = 1, 2, respectively. Substitut-
ing the expression for u(t, x) in conditions (6),(7), we obtain a system of equations for
Vi, i = 1, 2,
(16)
∫ t
0
gi(t− τ, 0, 0)Vi(τ)dτ −
2∑
j=1
∫ t
0
(∫
Dj
gj(t− τ, y, 0)μ(dy)
)
Vj(τ)dτ = Φi(t),
t > 0, i = 1, 2,
where
Φi(t) =
2∑
j=1
∫
Dj
uj0(t, y)μ(dy) − ui0(t, 0), i = 1, 2.
One can see that the system of equations (16) is a system of first-kind Volterra integral
equations. Using the Holmgren hold (e.g., see [8]), we transform it to an equivalent system
of second-kind Volterra integral equations. For this, we define the operator
E(t)Φ =
√
2
π
d
dt
∫ t
0
(t− s)−
1
2 Φ(s)ds, t > 0,
KNOTTING OF DIFFUSION PROCESSES ON A LINE 55
and apply it to both sides of the system of equations (16). Taking into account properties
1), 2) for gi and condition c), we obtain, after easy reductions, the equalities
(17) Vi(t) =
2∑
j=1
∫ t
0
Kij(t− τ)Vj(τ)dτ + Ψi(t), t > 0, i = 1, 2,
where
Kii(t− τ) =
√
2bi(0)
π
d
dt
∫ t
τ
(t− s)−
1
2
(∫
Di
gi1(s− τ, y, 0)μ(dy) − gi1(s− τ, 0, 0)
)
ds−
− bi(0)
∫
Di
∂gi0(t− τ, y, 0)
∂y
μ(dy), i = 1, 2,
Kij(t− τ) =
√
2bi(0)
π
d
dt
∫ t
τ
(t− s)−
1
2
(∫
Dj
gj1(s− τ, y, 0)μ(dy)
)
ds
−
√
bi(0)bj(0)
∫
Dj
∂gj0(t− τ, y, 0)
∂y
μ(dy), i, j = 1, 2,
Ψi(t) =
√
bi(0)E(t)Φi, i = 1, 2, i �= j.
Equations (17) form a system of second-kind Volterra integral equations for Vi, i = 1, 2.
In this system of equations, the first terms that are a part of expressions for kernels
Kij(t− τ) (we will denote them K
(1)
ij (t− τ)) and Ψi(t) can be estimated as∣∣∣K(1)
ij (t− τ)
∣∣∣ ≤ √
bi(0)CT (t− τ)−1+ α
2 , i, j = 1, 2,(18)
|Ψ(t)| ≤
√
bi(0)KT ‖ϕ‖t−
1
2 , i = 1, 2,(19)
which holds in every domain of the form 0 ≤ τ < t ≤ T and 0 < t ≤ T , respectively,
with some constants CT and KT . Estimations (18) and (19) have similar proofs. For
example, we will prove estimation (19). By differentiating the integral in the expression
for Ψi(t), we obtain the formula
(20) Ψi(t) =
√
2bi(0)
π
(
1
2
∫ t
0
(t− s)−
3
2 (Φi(t) − Φi(s))ds+ t−
1
2 Φi(t)
)
, i = 1, 2.
After that, estimating the right-hand side of (20), using inequality (11), and the formula
of finite growth for the difference Φi(t) − Φi(s), we obtain (0 < t ≤ T )
|Φi(t)| ≤ C‖ϕ‖, |Φi(t) − Φi(s)| ≤ C‖ϕ‖s−1(t− s),
thus
|Ψi(t)| ≤
√
bi(0)C‖ϕ‖
(∫ t
2
0
(t− s)−
3
2 ds+
∫ t
t
2
(t− s)−
1
2 s−1ds+ t−
1
2
)
≤
√
bi(0)KT ‖ϕ‖t−
1
2 .
Similarly we obtain estimation (18). Concerning the functions that determine the
second term in the expression for Kij(t−τ) (denote them by K(2)
ij ), the direct application
of inequality (11) to the derivatives ∂gj0
∂y , j = 1, 2, results in a nonintegrable singularity
for these functions at t = τ . Despite that, we will prove that the general method of
56 PAVLO KONONCHUK
successive approximations is applicable to the system of equations (17). This means that
we may try to find s solution of the system of equations (17) in the form of a series
(21) Vi(t) =
∞∑
k=0
V
(k)
i (t), i = 1, 2,
where
V
(0)
i (t) = Ψi(t),
V
(k)
i (t) =
2∑
j=1
∫
0
Kij(t− τ)V (k−1)
j (τ)dτ, k = 1, 2, . . .
Estimate V (1)
i (t). For this, we use the equality
(22) V
(1)
i =
2∑
l,j=1
∫ t
0
K
(l)
ij (t− τ)V (0)
j (τ)dτ =
2∑
l,j=1
V
(1l)
ij (t), i = 1, 2.
Taking into account (18) and (19), we obtain
(23)
∣∣∣V (11)
ij (t)
∣∣∣ ≤ KT ‖ϕ‖
√
bj(0)bi(0)CT
∫ t
0
(t− τ)−1+ α
2 τ−
1
2 dτ
= KT ‖ϕ‖t−
1
2
√
bi(0)bj(0)
CT Γ(α
2 )Γ(1
2 )
Γ(1+α
2 )
t
α
2 .
Before estimating V (12)
ij , we recall the formula
V
(12)
ij (t) =
√
bi(0)
bj(0)
1√
2πbj(0)
∫
Dj
μ(dy)
∫ t
0
y
(t− τ)
3
2
exp
(
− y2
2bj(0)(t− τ)
)
V
(0)
j (τ)dτ,
i, j = 1, 2.
Using (19), we obtain∣∣∣V (12)
ij (t)
∣∣∣
≤ KT‖ϕ‖
√
bi(0)
∫
Dj
μ(dy)
1√
2πbj(0)
∫ t
0
|y|
(t− τ)
3
2 τ
1
2
exp
(
− y2
2bj(0)(t− τ)
)
dτ.
Since
1√
2πbj(0)
∫ t
0
|y|
(t− τ)
3
2 τ
1
2
exp
(
− y2
2bj(0)(t− τ)
)
dτ = t−
1
2 exp
(
− y2
2bj(0)t
)
,
we have
(24)
∣∣∣V (12)
ij (t)
∣∣∣ ≤ KT ‖ϕ‖
√
bi(0)t−
1
2
∫
Dj
exp
(
− y2
2bj(0)t
)
μ(dy).
From (22)–(24), we obtain the estimation
(25)∣∣∣V (1)
i (t)
∣∣∣ ≤ KT ‖ϕ‖
√
bi(0)t−
1
2
×
⎛⎝ (
√
b1(0) +
√
b2(0))CT Γ(α
2 )Γ(1
2 )
Γ(1+α
2 )
t
α
2 +
2∑
j=1
∫
Dj
exp
(
− y2
2bj(0)t
)
μ(dy)
⎞⎠ ,
i = 1, 2, t ∈ (0, T ].
KNOTTING OF DIFFUSION PROCESSES ON A LINE 57
On the right-hand side of (25), we use the notations
at =
(√
b1(0) +
√
b2(0)
)
CT Γ(α
2 )Γ(1
2 )
Γ(1+α
2 )
t
α
2 , bt =
2∑
j=1
∫
Dj
exp
(
− y2
2bj(0)t
)
μ(dy).
Notice that condition c) guarantees the following inequality for bt:
bt ≤ bT ≤
∫
R1
exp
(
− y2
2b(0)T
)
μ(dy) < 1,
where b(0) = max{b1(0), b2(0)}.
Further, by using the method of induction on k, we establish the following estimation
for V (k)
i (t):
(26)
∣∣∣V (k)
i (t)
∣∣∣ ≤ KT ‖ϕ‖
√
bi(0)t−
1
2
k∑
m=0
Cm
k a
(k−m)
t bmt , i = 1, 2, k = 0, 1, 2, . . . ,
where
a
(m)
t =
((√
b1(0) +
√
b2(0)
)
CT Γ(α
2 )
)m
Γ(1
2 )
Γ(1
2 +mα
2 )
tm
α
2 , m = 0, 1, . . . , k.
Here, CT and KT are the constants from inequalities (18) and (19), respectively. Taking
estimation (26) into account, we obtain
(27)
∞∑
k=0
∣∣∣V (k)
i (t)
∣∣∣ ≤ KT ‖ϕ‖
√
bi(0)t−
1
2
∞∑
k=0
k∑
m=0
Cm
k a
(k−m)
t bmt
= KT ‖ϕ‖
√
bi(0)t−
1
2
∞∑
k=0
a
(k)
t
∞∑
m=0
Cm
k+mb
m
t
= KT ‖ϕ‖
√
bi(0)t−
1
2
∞∑
k=0
a
(k)
t
(1 − bt)k+1
≤ KT ‖ϕ‖
√
bi(0)t−
1
2
∞∑
k=0
(
(
√
b1(0) +
√
b2(0)CT Γ(α
2 )
)k
Γ(1
2 )
Γ(1
2 + kα
2 )(1 − bt)k+1
.
Inequality (27) guarantees the convergence of the series in (21) and gives the estimation
for Vi, i = 1, 2:
(28) |Vi(t)| ≤ C‖ϕ‖t− 1
2 i = 1, 2,
where t ∈ (0, T ], and C is some constant.
So we have constructed a system of integral equations (17) and verified an estimation
for (28). The given estimation and (12) for r = p = 0 ensure the existence of the function
ui1, i = 1, 2 from (15) and inequality (14) for it. It is obvious that the same inequality
holds for the function ui0, i = 1, 2, from (15) that holds also for the function u. This
means that we have proven the existence of a solution of problem (4)–(7).
Remark 3. If we add condition (3) to the statements of Theorem 1, then the obtained
solution of problem (4)–(7) belongs to the space C([0,∞) × R1).
Let us prove the uniqueness of the constructed solution of problem (4)–(7). Assume
that there exist two distinct solutions of the problem that belong to class (13). We denote
them by u(1)(t, x) and u(2)(t, x). Then the function u(t, x) = u(1)(t, x) − u(2)(t, x) is a
solution of problem (4)–(7) when ϕ(x) ≡ 0 which is continuous in the domain [0,∞)×R1,
58 PAVLO KONONCHUK
so its parts in the domains (t, x) ∈ [0,∞) ×D1 and (t, x) ∈ [0,∞) ×D2 are, at the same
time, solutions of the first parabolic boundary-value problem
Dtu = Liu, (t, x) ∈ (0,∞) ×Di, i = 1, 2,(29)
u(0, x) = 0, x ∈ Di, i = 1, 2,(30)
u(t, 0) = v(t), t ≥ 0(31)
where
v(t) =
∫
R1
(
u(1)(t, y) − u(2)(t, y)
)
μ(dy).
Since the function v(t) has the Hölder property when t > 0, the first boundary-value
problem has a unique solution that can be represented as
(32) u(t, x) =
∫ t
0
gi(t− τ, x, 0)Vi(τ)dτ, (t, x) ∈ (0,∞) ×Di, i = 1, 2
(e.g., see [8]). Following the proof of the existence of a solution of problem (4)–(7)
given above, one can notice that the functions V1(t) and V2(t) from (32) are, at the
same time, the unique solutions of the homogeneous system of integral equations (17),
where Ψi(t) ≡ 0, i = 1, 2. So Vi(t) = 0 (i = 1, 2), which yields u(t, x) ≡ 0 and
u(1)(t, x) ≡ u(2)(t, x). Theorem 1 is proved.
4. Construction of a diffusion process. From Theorem 1, it follows that, using the
solution of problem (4)–(7), we can determine the family of linear operators (Tt)t>0 that
acts in the space B(R1). For ϕ ∈ B(R1), we put
(32) Ttϕ(x) =
∫
R1
gi(t, x, y)ϕ(y)dy+
∫ t
0
gi(t−τ, x, 0)Vi(τ, ϕ)dτ, t > 0, x ∈ Di, i = 1, 2,
where Vi(t, ϕ) ≡ Vi(t), i = 1, 2, is a solution of the system of integral equations (17).
We will study properties of the operators {Tt} considering them on the space
M =
{
ϕ ∈ B(R1) ∩ C(R1) : ϕ(0) =
∫
R1
ϕ(y)μ(dy)
}
.
This restriction is related to the facts that we are firstly interested in Feller processes that
are generated by the operators {Tt} and, secondly, one can assert that limt→0 Ttϕ(x) =
ϕ(x) for every x ∈ R1, as it follows from Remark 3 [with fitting condition (3)]. One can
easily prove that M is a closed subspace of the space of all bounded continuous functions
on R1, and the operators {Tt} leave M invariant (that is, TtM ⊂ M for every t ≥ 0).
We will show that the operators {Tt}, t ≥ 0 satisfy the following conditions:
1’) if ϕn ∈ M when n = 1, 2, ..., supn ‖ϕn‖ < ∞, and, for all x ∈ R1, we have
limn→∞ ϕn(x) = ϕ(x), then, for all t > 0, x ∈ R1, the next relations are satisfied:
limn→∞ Vi(t, ϕ) = Vi(t, ϕ), i = 1, 2, and limn→∞ Ttϕ(x) = Ttϕ(x) (the last relation is
obviously satisfied even when t = 0);
2’) Ttϕ(x) ≥ 0 for all t ≥ 0, x ∈ R1, whenever the function ϕ ∈ M satisfies the
property that ϕ(x) ≥ 0 for all x ∈ R1;
3’) for all t1 ≥ 0, t2 ≥ 0, the next relation holds:
Tt1+t2 = Tt1Tt2 ;
4’) ‖Tt‖ ≤ 1 for all t ≥ 0.
Conditions 1’)-4’) can be easily verified. In particular, condition 1’) is a corollary from
properties of the solution of the system of equations (17) (in series (21) that represent
the function Vi(t, ϕn), i = 1, 2, we can take limit term-by-term) and from the Lebesgue
theorem about passing to the limit under the sign of integral. Property 2’) which means
KNOTTING OF DIFFUSION PROCESSES ON A LINE 59
that the operator Tt leaves the cone of nonnegative functions from the space M to be
invariant and property 3’) called a semigroup property are corollaries of the maximum
principle for parabolic equations ([9]) and the statement of Theorem 1 concerning the
uniqueness of a solution of problem (4)–(7), respectively. Finally, to verify property 4’)
which means that, for every t ≥ 0, the operator Tt is a contraction operator, it is enough
to notice (remembering 2’)) that Ttϕ0(x) ≡ 1 for all t ≥ 0, x ∈ R1, if only ϕ0(x) ≡ 1.
Hence, we make conclusion (e.g., see [4]) that the operator semigroup Tt, t ≥ 0,
constructed by formulas (32) and (17) determines some homogeneous Feller process on
R1. Denote its transition probability by P(t, x, dy), so that
Ttϕ(x) =
∫
R1
P(t, x, dy)ϕ(y).
Therefore, we have proved the next theorem.
Theorem 2. Let the coefficients of the operators L1 and L2 from (1) and the measure
μ(·) from (2) satisfy conditions a)-c). Then the solution of problem (4)–(7) constructed
in Theorem 1 uniquely determines the operator semigroup Tt, t ≥ 0 that describes a ho-
mogeneous Feller process on R1 such that its parts at the inner points of the domains
D1 and D2 coincide with the diffusion processes generated by the operators L1 and L2,
respectively, and its behavior at {0} is determined by the nonlocal conjugation condi-
tion (3).
Prentice-Hall, Englewood Cliffs, 1964
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6. A.L. Skubachevkiy, On Feller semigroups for multidimensional diffusion processes, Doklady
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1964.
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