Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition
Using analytical methods, we consider the problem of constructing a nonhomogeneous multidimensional diffusion process in a halfspace with given diffusion characteristics at the inner points and general Wentzel boundary conditions.
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| Цитувати: | Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition / Z.Ya. Tsapovska // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 145–154. — Бібліогр.: 12 назв.— англ. |
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irk-123456789-45602009-12-07T12:00:35Z Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition Tsapovska, Z.Ya. Using analytical methods, we consider the problem of constructing a nonhomogeneous multidimensional diffusion process in a halfspace with given diffusion characteristics at the inner points and general Wentzel boundary conditions. 2008 Article Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition / Z.Ya. Tsapovska // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 145–154. — Бібліогр.: 12 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4560 519.21 en Інститут математики НАН України |
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Using analytical methods, we consider the problem of constructing a nonhomogeneous multidimensional diffusion process in a halfspace with given diffusion characteristics at the inner points and general Wentzel boundary conditions. |
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Tsapovska, Z.Ya. |
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Tsapovska, Z.Ya. Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition |
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Tsapovska, Z.Ya. |
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Tsapovska, Z.Ya. |
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Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition |
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Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition |
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Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition |
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Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition |
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Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition |
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nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general wentzel boundary condition |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/4560 |
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Nonhomogeneous diffusion processes in a halfspace whose behaviour on the boundary is described by general Wentzel boundary condition / Z.Ya. Tsapovska // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 145–154. — Бібліогр.: 12 назв.— англ. |
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AT tsapovskazya nonhomogeneousdiffusionprocessesinahalfspacewhosebehaviourontheboundaryisdescribedbygeneralwentzelboundarycondition |
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2025-07-02T07:46:28Z |
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2025-07-02T07:46:28Z |
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1836520455434928128 |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 145–154
UDC 519.21
ZHANNETA YA. TSAPOVSKA
NONHOMOGENEOUS DIFFUSION PROCESSES IN A
HALFSPACE WHOSE BEHAVIOUR ON THE BOUNDARY IS
DESCRIBED BY GENERAL WENTZEL BOUNDARY CONDITION
Using analytical methods, we consider the problem of constructing a nonhomogeneous
multidimensional diffusion process in a halfspace with given diffusion characteristics
at the inner points and general Wentzel boundary conditions.
1. Introduction. In this paper, we consider the problem of constructing a multi-
plicative operator family that describes a multidimensional nonhomogeneous diffusion
process in a domain with general Wentzel boundary conditions [1]. Analytical methods
are used to solve this problem. Within these methods, the desired operator family can be
determined using the solution of the corresponding boundary-value problem, where the
boundary conditions as well as the equation in the domain is described by a second-order
parabolic linear partial differential equation with a term that contains the directional de-
rivative on the normal to the boundary. In turn, we have obtained the classical solvability
of the Wentzel parabolic problem by using a simple-layer potential [2]. Here, we will con-
fine ourselves to a model problem, where the diffusion process is given in a domain that
is the upper half-space in an Euclidian space. Note that the stated problem with its
special cases previously was studied using different approaches mostly for homogeneous
processes [3–7]. As for the Wentzel parabolic boundary problem, it was studied, besides
[2], also in [8–10], by using different methods.
2. Basic notations and definitions. Let Rn, n ≥ 2, be the n-dimensional Euclidian
space; Rn+1
t = [0, t) × Rn, 0 < t ≤ T , T > 0 is fixed; Rn
t = [0, t) × Rn−1; x =
(x′, xn) = (x1, . . . , xn−1, xn) a point in Rn; x′ = (x1, . . . , xn−1) a point in Rn−1; (s, x)
a point in Rn+1
t ; (s, x′) a point in Rn
t ; |x|2 =
n∑
i=1
x2
i ; |x′|2 =
n−1∑
i=1
x2
i ; (x, y) =
n∑
i=1
xiyi;
(x′, y′) =
n−1∑
i=1
xiyi.
We will use the following notations for the differential operators: Dr
s and Dp
x are the
partial derivatives with respect to s of order r and any partial derivative with respect
to x of order p, respectively, where r and p are integer nonnegative numbers; D1
t = Dt,
Di ≡ ∂
∂xi
, Dij = Dji ≡ ∂2
∂xi∂xj
, i, j = 1, . . . , n; ∇′ = (D1, . . . , Dn−1).
Similarly as in [11, p. 16], H
k+λ
2 , k+λ
s x (B) ≡ H
k+λ
2 , k+λ(B) (k = 0, 1, 2, λ ∈ (0, 1), B is
a domain in the space Rn+1
t or Rn
t , B is the closure of B) mean the respective Hölder
spaces; H
0
k+λ
2 , k+λ(B) is a set of functions from H
k+λ
2 , k+λ(B) that (in case k = 2 also
with the derivative with respect to s) vanishes when s = t.
2000 AMS Mathematics Subject Classification. Primary 60J60.
Key words and phrases. Nonhomogeneous diffusion process, general Wentzel boundary condition,
analytical method, parabolic potentials.
145
146 ZHANNETA YA. TSAPOVSKA
By ||w||
H
k+λ
2 , k+λ(B)
, we denote the norm of a function w in H
k+λ
2 , k+λ(B). Also we use
the Hölder spaces Hk+λ(Rn), Hk+λ(Rn−1), aggregates of continuous functions Ck(B),
C1, 2(B), and the Banach space of bounded measurable functions B(Rn) with the norm
||ϕ|| = sup
x∈Rn
|ϕ(x)|.
In Rn, we consider the domain D = Rn
+ = {x ∈ Rn |xn > 0} with the boundary
S = Rn−1, and, in R
n+1
t , we consider Ωt = {(t, x) ∈ Rn+1
t |xn > 0} with the side
boundary Σt = R
n
t . By ν(x′) = (0, . . . , 0, 1) ∈ Rn, we denote the inner unit normal
vector to S at x′ ∈ Rn−1. Everywhere below, C and c are positive constants that do not
depend on (s, x), and their specific values are not interesting for us.
3. Parabolic potentials. Regularizer. In a layer Rn+1
T , let us consider a second-order
uniformly parabolic operator with real coefficients,
(1)
Lu ≡ 1
2
n∑
i,j=1
bij(s, x)Diju(s, x) +
n∑
i=1
ai(s, x)Diu(s, x) +Dsu(s, x), (s, x) ∈ Rn+1
t .
Assume that the coefficients of the operator L are defined in R
n+1
T , and the following
assumptions are true:
A1)
n∑
i,j=1
bij(s, x)ξiξj ≥ δ0|ξ|2, bij = bji, δ0 > 0, ∀(s, x) ∈ R
n+1
T , ∀ξ ∈ Rn;
A2) bij , ai ∈ H
λ
2 , λ(R
n+1
t ), i, j = 1, . . . , n.
Assumptions A1), A2) guarantee the existence of a fundamental solution (f.s.)
g(s, x, t, y), 0 ≤ s < t ≤ T, x, y ∈ Rn,
for the operator L ([4, 11]):
(2) g(s, x, t, y) = g0(s, x, t, y) + g1(s, x, t, y),
where
g0(s, x, t, y) = g
(t,y)
0 (t− s, x− y)
= (2π(t− s))−n/2(det b(t, y))−1/2 exp
{
− (b−1(t, y)(y − x), y − x)
2(t− s)
}
,
b(t, y) = (bij(t, y))n
i,j=1, b−1(t, y) = (bij(t, y))n
i,j=1 is the inverse matrix to b(t, y), g1 is
the integral term with a ”weaker” singularity than that of g0 at s→ t, and g ≡ 0 if s ≥ t.
A function g(s, x, t, y) is nonnegative continuous with respect to the aggregate of the
variables, with fixed t ∈ (0, T ], y ∈ Rn, as a function of the arguments (s, x) ∈ [0, t)×Rn,
it satisfies the equation Lu = 0 and the condition
(3) lim
s↑t
∫
Rn
g(s, x, t, y)ϕ(y)dy = ϕ(x),
for every t ∈ (0, T ], x ∈ Rn, and a bounded continuous function ϕ(x) on Rn.
Among other properties of f.s. g, we also note the following ones:
1)
∫
Rn
g(s, x, t, y) dy = 1 for all 0 ≤ s < t ≤ T , x ∈ Rn;
2)
∫
Rn
g(s, x, t, y)g(t, y, u, z) dy = g(s, x, u, z) for all 0 ≤ s < t ≤ T , x ∈ Rn, y ∈ Rn;
3) for 0 ≤ s < t ≤ T , x ∈ Rn, Θ ∈ Rn, the next assumptions hold:∫
Rn
(y − x,Θ)g(s, x, t, y)dy =
t∫
s
dτ
∫
Rn
(a(τ, y),Θ)g(s, x, t, y)dy,
NONHOMOGENEOUS DIFFUSION PROCESSES IN A HALFSPACE 147
∫
Rn
(y − x,Θ)2g(s, x, t, y)dy =
t∫
s
dτ
∫
Rn
g(s, x, τ, z)(b(τ, z)Θ,Θ)dz+
+ 2
t∫
s
dτ
∫
Rn
g(s, x, τ, z)(a(τ, z),Θ)(z − x,Θ)dz;
4) there exist the positive constants C and c such that, for the functions g and g1 with
0 ≤ s < t ≤ T , x, y ∈ Rn, the estimations
(4) |Dr
sD
p
xg(s, x, t, y)| ≤ C(t− s)−
n+2r+p
2 exp
{
− c
|x− y|2
t− s
}
, 2r + s ≤ 2,
(5) |Dr
sD
p
xg1(s, x, t, y)| ≤ C(t− s)−
n+2r+p−λ
2 exp
{
− c
|x− y|2
t− s
}
, 2r + s ≤ 2,
hold.
Let us consider the integral representing a parabolic simple-layer potential:
(6) u1(s, x, t) =
t∫
s
dτ
∫
Rn−1
g(s, x, τ, z′)V (τ, z′)dz′,
where 0 ≤ s < t ≤ T , x ∈ Rn, V (t, z′) is a bounded measurable function given on
Σ = R
n
T . From properties of f.s. g and (4), it follows that u1(s, x, t), as a function of the
arguments (s, x) ∈ [0, t) × Rn, is continuous for 0 ≤ s < t ≤ T , x ∈ Rn and satisfies the
equation Lu1 = 0 in the domain (s, x) ∈ [0, t) × (Rn\S) and the zero initial condition
lim
s↑t
u(s, x, t) = 0.
Let, for (s, x′) ∈ R
n
T , the conormal vector N(s, x′) = (N1(s, x′), . . . , Nn(s, x′)),
Ni(s, x′) =
n∑
j=1
bij(s, x′)νj(x′), i = 1, . . . , n,
be defined. Then if V ∈ H
0
λ
2 , λ(R
n
t ), then u1 ∈ H
0
1+λ
2 , 1+λ(R
n
t ) [12]; and, for the conormal
derivative of the function u1, a jump formula holds ([4, p. 59], [11, p. 459]):
(7)
∂u1(s, t, x′)
∂N(s, x′)
= lim
x→x′
x∈Rn
+
∂u1(s, t, x)
∂N(s, x′)
= −V (s, x′) +
t∫
s
dτ
∫
Rn−1
∂g(s, x′, τ, z′)
∂N(s, x′)
V (τ, z′)dz′,
(s, x′) ∈ Rn
t .
The integral on the right-hand side of (7) is called the direct value of the conormal
derivative of a simple-layer potential. Its existence follows from the inequality (0 ≤ s <
t ≤ T , x′, z′ ∈ Rn−1)
(8)
∣∣∣∣∂g(s, x′, τ, z′)
∂N(s, x′)
∣∣∣∣ ≤ C(τ − s)−
n+1−λ
2 exp
{
− c
|x′ − z′|2
τ − s
}
.
Now we will define a boundary operator E that further will be used as the regularizer
of a first-kind Volterra integral equation equivalent in some sense to the boundary-value
problem formulated below in Section 4. To this end, in Rn
T , we consider the parabolic
operator
L′ ≡ 1
2
n−1∑
i,j=1
hij(s, x′)Dij −Dt, (s, x′) ∈ Rn
T ,
148 ZHANNETA YA. TSAPOVSKA
whose coefficients are defined by the relation
hij = bij −
binbjn
bnn
, i, j = 1, . . . , n− 1.
From assumptions A1), A2), it follows that the operator L′ is uniformly parabolic, the
functions hij(s, x′) are Hölder with respect to two variables. Moreover, for the operator
L′, there exists f.s.
(9) h(s, x′, t, y′) = h0(s, x′, t, y′) + h1(s, x′, t, y′), 0 ≤ s < t ≤ T, x′, y′ ∈ Rn−1,
where h0 and h1, as in (2), mean, respectively, the main and additional (integral) terms
of f.s. h.
For f.s. h, properties 1)–4) of f.s. g from (2) hold with obvious changes.
Note also a connection between the functions g(s, x′, t, y′) and h(s, x′, t, y′). Using
formula (2) and the definition of f.s. h, we easily obtain the equality
g(s, x′, t, y′) = (2πbnn(t, y′)(t− s))−1/2 [h(s, x′, t, y′) − h1(s, x′, t, y′)] + g1(s, x′, t, y′),
0 ≤ s < t ≤ T, x′, y′ ∈ Rn−1.(10)
Consider an integro-differential operator E : Ψ0 → EΨ0 that acts on functions Ψ0 from
Rn
T in the following way:
E(s, x′, t)Ψ0 =
√
2
π
{
∂
∂s
∫ t
s
(τ − s)−1/2dτ
∫
Rn−1
h(ŝ, x′, τ, y′)Ψ0(τ, y′)dy′
}∣∣∣∣
ŝ=s
,
(s, x′) ∈ [0, t) × Rn−1.(11)
From results obtained in [12], it follows that E is a linear bounded operator that acts
from H
0
1+λ
2 , 1+λ(R
n
t ) into H
0
λ
2 , λ(R
n
t ), for which there exists an inverse operator E−1. Ad-
ditionally, we determine that the operator E converts the functions from H
0
2+λ
2 , 2+λ(R
n
t )
into H
0
1+λ
2 , 1+λ(R
n
t ). In addition, E is a regularizer in the case of the first boundary-
value problem [12]. Namely, by using relations (10), (11), and (3), properties 1)–3), and
inequalities (4) and (5), we obtain
(12)
E(s, x′, t)u1 = −Ṽ (s, x′) +
t∫
s
dτ
∫
Rn−1
R(s, x′, τ, y′)V (τ, y′)dy′, ∀ V ∈ H
0
λ
2 , λ(R
n
T ),
where Ṽ (s, x′) = (bnn(s, x′))−1/2V (s, x′), and estimation (8) holds for the kernel R.
Using f.s. g from (2), we can determine two more potentials that are used to solve
the Cauchy problem for a generalized second-order parabolic equation. There are the
Poisson potential
(13) u2(s, x, t) =
∫
Rn
g(s, x, t, y)ϕ(y)dy, (s, x) ∈ Rn
t
, and the volume potential
u3(s, x, t) =
t∫
s
dτ
∫
Rn
g(s, x, τ, z)f(τ, z)dz, (s, x) ∈ Rn
t ,
where ϕ(y), y ∈ Rn, and f(τ, z), (τ, z) ∈ R
n+1
T , are the given functions.
Assume that ϕ is bounded and continuous in Rn, and f ∈ H
λ
2 , λ(R
n+1
t ). Then one
can affirm (see [11, Ch. IV, §14]) that the functions ui, i = 2, 3, are continuous in
R
n+1
t and satisfy the equation Lu2 = 0, Lu3 = −f in the domain (s, x) ∈ [0, t) × Rn
NONHOMOGENEOUS DIFFUSION PROCESSES IN A HALFSPACE 149
also as the initial conditions lim
s↑t
u2(s, x, t) = ϕ(x), lim
s↑t
u3(s, x, t) = 0, x ∈ Rn. Also
u3 ∈ H
2+λ
2 , 2+λ(R
n+1
t ) and, in the case where ϕ ∈ H2+λ(Rn), u2 ∈ H
2+λ
2 , 2+λ(R
n+1
t ).
4. Problem statement and its solution. Assume that, in D = R
n
+, a generating
second-order differential operator of some nonhomogeneous diffusion process that acts
on the set of all twice continuously differentiable functions with compact carriers C2
k(D)
is given as
(14) Lϕ(x) =
1
2
n∑
i,j=1
bij(s, x)Dijϕ(x) +
n∑
i=1
ai(s, x)Diϕ(x),
where bij(s, x), ai(s, x) are bounded continuous functions on ΩT , b(s, x) = (bij(s, x))n
i,j=1
is a symmetric nonnegative definite matrix. Assume also that the Wentzel boundary
operator [1] is given, i.e., a mapping from C2
k(D) into the space of all continuous functions
on ΣT = R
n
T has the form
L0ϕ(x) =
1
2
n−1∑
k,l=1
βkl(s, x)Dklϕ(x) +
n−1∑
k=1
αk(s, x)Dkϕ(x) + q(s, x)
∂ϕ(x)
∂xn
− ρ(s, x)Lϕ(x),
(s, x) ∈ ΣT ,(15)
where βkl(s, x), αk(s, x), q(s, x), ρ(s, x) are bounded continuous functions on ΣT such
that β(s, x) = (βkl(s, x))n−1
k,l=1 is a symmetric nonnegative definite matrix, q(s, x) ≥ 0,
ρ(s, x) ≥ 0.
We will set up the problem to construct a multiplicative operator family Tst, 0 ≤ s <
t ≤ T , that describes a continuous nonbreaking Feller process at D such that it coincides
with a diffusion process controlled by the operator L at the inner points of D, and its
behaviour on the boundary S is determined by the boundary condition
(16) L0ϕ(x) = 0, x ∈ S.
We will use analytical methods to solve the problem. It means (see [3, 4]) that the
required operator family will be determined using a solution of the following parabolic
boundary-value problem with respect to Tstϕ(x) = u(s, x, t):
(17) Lu = Lu+Dsu = 0, (s, x) ∈ Ωt,
(18) lim
s↑t
u(s, x, t) = ϕ(x), x ∈ R
n
+,
(19)
L0u =
1
2
n−1∑
k,l=1
βkl(s, x)Dklu+
n−1∑
k=1
αk(s, x)Dku+ q(s, x)Dnu+ ρ(s, x)Dsu = 0,
(s, x) ∈ Rn
t .
Solving the Wentzel parabolic boundary-value problem using the potential
method. We will study the classical solvability of problem (17)–(19) assuming that, for
the coefficients of the operator L from (17), assumptions A1), A2) hold. Moreover, for
the coefficients of the operator L0 from (19), excepting the mentioned assumptions, the
following assumptions hold:
B1)
n−1∑
k,l=1
βkl(s, x′)ξkξl ≥ μ0|ξ′|2, μ0 > 0, ∀ (s, x′) ∈ R
n
T , ∀ ξ′ ∈ Rn−1;
B2) βkl, αk, q ∈ H
λ
2 , λ(R
n
T ), ρ ≡ 1, inf
s,x′
q(s, x′) > 0.
150 ZHANNETA YA. TSAPOVSKA
Also we assume that the function ϕ from (18) is smooth enough and satisfies the fitting
condition
1
2
n−1∑
k,l=1
βkl(s, x′)Dklϕ(x′) +
n−1∑
k=1
αk(s, x′)Dkϕ(x′) + q(s, x′)Dnϕ(x′) − Lϕ(x′) = 0,
s = t, x′ ∈ Rn−1.(20)
Theorem 1. Let, for the coefficients of the operators L and L0 from (1) and (19),
conditions A1), A2) and B1), B2) hold, respectively. Then, for every function ϕ ∈
H2+λ(Rn) from (18) that satisfies the fitting condition (20), problem (17)–(19) has the
unique solution
(21) u ∈ H
2+λ
2 , 2+λ(Ωt),
and the estimation
(22) ||u||
H
2+λ
2 , 2+λ(Ωt)
≤ C||ϕ||H2+λ(Rn)
is true.
Proof. We will look for a solution of problem (17)–(19) of the form
(23) u(s, x, t) = u1(s, x, t) + u2(s, x, t),
where the functions u1 and u2 are defined by formulas (6) and (13), respectively, and the
density V from the simple-layer potential (6) is unknown. In order to find it, we will use
the boundary condition (19). Consider V to be a function of the arguments s, x, t and
also consider a priori that, as a function dependent on (s, x), it belongs to the Hölder
class H
0
λ
2 , λ(R
n
t ). Separating the conormal derivative in the formula for L0u in (19) and
executing simple transformations, we obtain the equality
(24)
L′
0u =
n−1∑
k,l=1
βkl(s, x′)Dklu+
n−1∑
k=1
α
(0)
k (s, x′)Dku+Dsu = −Θ(0)(s, x′, t), (s, x′) ∈ Rn
t ,
where
α
(0)
k (s, x′) = αk(s, x′) − q(s, x′)
bkn(s, x′)
bnn(s, x′)
, k = 1, . . . , n− 1,
Θ(0)(s, x′, t) =
q(s, x′)
bnn(s, x′)
∂u(s, x′, t)
∂N(s, x′)
=
=
q(s, x′)
bnn(s, x′)
⎡⎣∂u2(s, x′, t)
∂N(s, x′)
− V (s, x′, t) +
t∫
s
dτ
∫
Rn−1
∂g(s, x′, τ, z′)
∂N(s, x′)
V (τ, z′, t)dz′
⎤⎦ .
Further we will consider equality (24) to be an independent parabolic equation in Rn
t
for the function u(s, x′, t). As follows from the conditions of Theorem 1, additional as-
sumption about V , and mentioned properties of the potentials (see Section 3) in this equa-
tion, its coefficients and the right-hand side Θ(0) belong to the Hölder class H
λ
2 , λ(R
n
t ).
The conditions of the theorem guarantee the existence of f.s. for the operator L′
0. We
will denote it by Γ(s, x′, t, y′) (0 ≤ s < t ≤ T , x′, y′ ∈ Rn−1). Hence, we may conclude
that there exists a unique classical solution of Eq. (24) that satisfies the boundary-initial
condition lim
s↑t
u(s, x′, t) = ϕ(x′), x′ ∈ Rn−1. In addition,
(25) u ∈ H
2+λ
2 , 2+λ(R
n
t ),
NONHOMOGENEOUS DIFFUSION PROCESSES IN A HALFSPACE 151
and this solution can be written in the form
(26)
u(s, x′, t) =
∫
Rn−1
Γ(s, x′, t, y′)ϕ(y′)dy′ +
t∫
s
dτ
∫
Rn−1
Γ(s, x′, τ, z′) Θ(0)(τ, z′, t)dz′,
(s, x′) ∈ Rn
t .
Thus, we have two different expressions for the function u(s, x′, t): relation (23), where
one must substitute (s, x) = (s, x′) ∈ Rn
t , and relation (26). Equating the right-hand
sides of these equations, we obtain the integral equation for V ,
(27)
t∫
s
dτ
∫
Rn−1
[g(s, x′, τ, z′) +K0(s, x′, τ, z′)] V (τ, z′, t)dz′ = Ψ0(s, x′, t), (s, x′) ∈ Rn
t ,
where
Ψ0(s, x′, t) =
∫
Rn−1
Γ(s, x′, t, y′)ϕ(y′)dy′ −
∫
Rn
g(s, x′, t, y)ϕ(y)dy+
+
t∫
s
dτ
∫
Rn−1
Γ(s, x′, τ, z′)
q(τ, z′)
bnn(τ, z′)
∂u2(τ, z′, t)
∂N(τ, z′)
dz′.
For the kernel K0, whose explicit form can be easily calculated when 0 ≤ s < t ≤ T ,
x′, z′ ∈ Rn−1, the estimation
(28) |K0(s, x′, τ, z′)| ≤ C(τ − s)−
n−1
2 exp
{
− c
|x′ − z′|2
τ − s
}
.
holds.
Considering the expression for the function Ψ0 from (27), relation (20), and properties
of parabolic potentials (see Section 3), we establish that Ψ0 ∈ H
0
2+λ
2 , 2+λ(R
n
t ).
Equation (27) is a Volterra equation of the first kind. Applying the operator E from
(11) to both sides of it and taking (12) and (28) into account, one can easily ascertain that
this equation will transform into an equivalent second-type integral Volterra equation of
the form
(29) V (s, x′, t) +
t∫
s
dτ
∫
Rn−1
K(s, x′, τ, z′)V (τ, z′, t)dz′ = Ψ(s, x′, t), (s, x′) ∈ Rn
t ,
where Ψ(s, x′, t) = (bnn(s, x′))1/2E(s, x′, t)Ψ0, therewith EΨ0 ∈ H
0
1+λ
2 , 1+λ(R
n
t ), Ψ ∈
H
0
λ
2 , λ(R
n
t ), and the kernel K satisfies inequality (8), when 0 ≤ s < t ≤ T , x′, z′ ∈ Rn−1.
Solving Eq. (29) by the convergence method, we find V . At this, we check that
V ∈ H
0
λ
2 , λ(R
n
t ), and estimation (22) holds for the norm ||V ||
H
0
λ
2 , λ(R
n
t )
.
It remains only to validate condition (21) for the constructed solution and the state-
ment of Theorem 1 considering the uniqueness. To prove this, it is enough to notice that
the solution of problem (17)–(19) constructed with the use of formulas (23) and (29) can
be interpreted as a solution of the first boundary-value parabolic problem
Lu(s, x, t) = 0, (s, x) ∈ Ωt,
lim
s↑t
u(s, x, t) = ϕ(x), x ∈ R
n
+,
u(s, x, t) = v(s, x, t), (s, x) ∈ Σt,
152 ZHANNETA YA. TSAPOVSKA
under the fitting condition
v(s, x, t)|s=t = ϕ(x), Dsu(s, x, t)|s=t = Dsv(s, x, t)|s=t, x ∈ Rn−1,
where the function v(s, x, t), (s, x) ∈ Σt is defined using relation (26). Then (see, e.g.,
[11, Ch. IV, §5]) the conditions of Theorem 1 together with conditions (20) and (25)
guarantee the existence of the unique solution of the problem that belongs to class (21)
and satisfies estimation (22). Theorem 1 is proved.
Remark.
If it is unnecessary to satisfy the fitting condition (20) for initial function ϕ from (18)
in the statement of Theorem 1, then the solution of problem (17)–(19) determined by
formulas (23) and (29) satisfies the condition
u ∈ C1, 2(Ωt) ∩ C(Ωt),
therewith the function u with its possible derivatives are bounded with respect to the
variable x.
Construction of the process. From Theorem 1, it follows that, over functions ϕ from
the class H2+λ(Rn), a multiplicative operator family Tst, 0 ≤ s < t ≤ T is defined, whose
analytical representation is given by the formula
(30) Tstϕ(x) = T
(1)
st ϕ(x) + T
(2)
st ϕ(x), (s, x) ∈ R
n+1
t ,
where T (1)
st ϕ(x) = u1(s, x, t), T (2)
st ϕ(x) = u2(s, x, t), u1 and u2 are defined by formulas
(6) and (13), respectively, the function V (s, x, t) is from the simple-layer potential; and
T
(1)
st ϕ(x) is a solution of the integral equation (29). Notice that the fulfilment of the
property
(31) Tst = TsuTut (s < u < t)
for the operators Tst is a simple corollary from the statements of Theorem 1 considering
the uniqueness of a solution of problem (17)–(19).
Now we prove that the operators Tst can be applied to functions ϕ from the space
B(Rn). Let ϕ ∈ B(Rn). Then the existence of the second addend in (30) follows from
the obvious inequality
(32) |T (2)
st ϕ(x)| ≤ C ||ϕ||
that holds in every domain of the form 0 ≤ s < t ≤ T , x ∈ Rn with some constant C.
To prove the existence of the function T
(1)
st ϕ(x) from (30), we will study, at first, the
integral equation (29). In this equation, its right-hand side Ψ(s, x′, t) up to the factor
(bnn(s, x′))1/2 is a result of the action of the operator E from (11) on the function Ψ0
from (27). Combining the first two terms in the expression for Ψ0 into a single one, we
denote it by Ψ01. Then, using properties 1)–3) applied to f.s. h from (9), we can easily
obtain the formula
(
π
2bnn(s, x′)
)1/2
Ψ(s, x′, t)
(33)
=
1
2
t∫
s
(τ − s)−3/2dτ
∫
Rn−1
h(s, x′, τ, y′)[Ψ01(τ, y′, t) − Ψ01(τ, x′, t)
− (y′ − x′,∇′
x′Ψ01(τ, x′, t))]dy′
+
1
2
t∫
s
(τ − s)−3/2[Ψ01(τ, x′, t) − Ψ01(s, x′, t)]dτ + (t− s)−1/2Ψ01(s, x′, t)
NONHOMOGENEOUS DIFFUSION PROCESSES IN A HALFSPACE 153
+
t∫
s
dα
∫
Rn−1
q(α, z′)
bnn(α, z′)
∂u2(α, z′, t)
∂N(α, z′)
dz′
×
⎧⎨⎩ ∂
∂s
α∫
s
(τ − s)−1/2dτ
∫
Rn−1
h(ŝ, x′, τ, y′)Γ(τ, y′, α, z′) dy′
⎫⎬⎭
∣∣∣∣∣∣
ŝ=s
.
In turn, the expression in the brackets in the last term can be written in the form
similar to the representation of the function E(s, x′, t)Ψ01. Consequently, using the basic
inequalities for f.s. g, Γ, and h, we obtain the estimate
(34) |Ψ(s, x, t)| ≤ C ||ϕ||(t− s)−1/2, 0 ≤ s < t ≤ T, x′ ∈ Rn−1.
It is obvious that the same estimate also holds for the solution of Eq. (29) found by
the convergence method. Applying inequalities (4) and (34) to f.s. g and the density V ,
respectively, we prove the existence of the function T (1)
st ϕ(x) and, therefore, the existence
of the function Tstϕ(x), as well as the fulfilment of estimation (32) for them.
Further, with regard for relation (33), it is easy to verify that if lim
n→∞ϕn(x) = ϕ(x) for
all x ∈ Rn and sup
n,x
|ϕn(x)| < ∞), then lim
n→∞Ψ(s, x′, t, ϕn) = Ψ(s, x′, t, ϕ), consequently
lim
n→∞V (s, x′, t, ϕn) = V (s, x′, t, ϕ) for all 0 ≤ s < t ≤ T, x′ ∈ Rn−1. Together with
estimation (34) for V , it allows us to state that
(35) lim
n→∞Tstϕn(x) = Tstϕ(x).
Using (35), the necessary properties of the operator Tst can be verified only for smooth
ϕ. In particular, it is easy to conclude by using (35) that the operators Tst have property
(31) also in the case where they act on functions ϕ from the class B(Rn).
Now we will prove that the operator family (Tst) defined by formulas (29) and (30)
has the property such that it transforms the nonnegative functions into nonnegative
functions.
Lemma. If properties A1), A2) and B1), B2) hold for coefficients of the operators L
and L0 from (14) and (15), then, for every nonnegative function ϕ ∈ B(Rn), the function
Tstϕ is nonnegative.
Proof. Taking (35) into account, it is enough to prove Lemma only in the case where a
function ϕ belongs to the class H2+λ(Rn) and is finite. From Theorem 1, it follows that,
in this case, the function Tstϕ(x) satisfies Eq. (17) in the domain (s, x) ∈ [0, t) × Rn
+,
initial condition (18), and boundary condition (19). Set some 0 < t ≤ T and denote,
by γ, the minimum of the function Tstϕ in a domain (s, x) ∈ [0, t] × R
n
+. Assume that
γ < 0. Since ϕ(x) ≥ 0 and Tstϕ → 0 when |x| → ∞ and due to the maximum principle
(see [13, Ch. II]), there exist s0 ∈ [0, t) and x′0 ∈ Rn−1 such that Ts0tϕn(x′0) = γ.
Because the function Tstϕ(x) obviously is not a constant, there exists a neighborhood
U of the point (s0, x′0) such that Ts0tϕ(x′0) > γ for (s, x) ∈ U ∩ {[0, t) × Rn
+}. But
then, as it follows from Theorem 14 [13, p. 69], the inequality DnTstϕ(x′0) > 0 holds
at the point (s0, x′0). In addition, at the point (s0, x′0), the equalities DiTs0tϕ(x′0) = 0,
i = 1, . . . , n− 1, DsTstϕ(x′0)|s=s0 = 0 and the inequality
n−1∑
k,l=1
βkl(s0, x′0)DklTs0tϕ(x′0) ≥ 0
obviously hold. Consequently, we obtain that, at the point (s0, x′0), the function Tstϕ(x)
does not satisfy the boundary condition (19). So, our assumption that γ < 0 is false.
Lemma is proved.
154 ZHANNETA YA. TSAPOVSKA
Noticing that V (s, x′, t, ϕ0) ≡ 0 for the function ϕ0(y) ≡ 1 and, consequently,
Tstϕ0(x) ≡ 1,
we conclude that the operator family (Tst)0≤s<t≤T determines some nonhomogeneous
nonbreaking Feller process on R
n
+. If we will denote its transition probability by
P (s, x, t, dy),
then we can write the relation
Tstϕ(x) =
∫
Rn
P (s, x, t, dy)ϕ(y), 0 ≤ s < t ≤ T, x ∈ R
n
+.
Finally, we prove that trajectories of the constructed process are continuous. This state-
ment is a corollary from the inequality
sup
x∈R
n
+
∫
Rn
|y − x|4P (s, x, t, dy) ≤ C(t− s)2, s ∈ [0, t),
which can be verified by straight computations.
Hence, the following theorem holds.
Theorem 2. Let conditions A1), A2) and B1),B2) hold for coefficients of the opera-
tors L from (14) and L0 from (15). Then there exists a multiplicative operator family
(Tst)0≤s<t≤T that is defined by formulas (30), (29) and describes a diffusion process in a
closed domain R
n
+ such that, in inner points of the domain, it is controlled by the operator
L, and its behavior on the boundary is determined by the boundary condition (16).
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