One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary
Bohdan Kopytko
,Roman Shevchuk
Journal of Applied Mathematics and Computational Mechanics |
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@article{Kopytko_2016, doi = {10.17512/jamcm.2016.4.08}, url = {https://doi.org/10.17512/jamcm.2016.4.08}, year = 2016, publisher = {The Publishing Office of Czestochowa University of Technology}, volume = {15}, number = {4}, pages = {71--82}, author = {Bohdan Kopytko and Roman Shevchuk}, title = {One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary}, journal = {Journal of Applied Mathematics and Computational Mechanics} }
TY - JOUR DO - 10.17512/jamcm.2016.4.08 UR - https://doi.org/10.17512/jamcm.2016.4.08 TI - One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary T2 - Journal of Applied Mathematics and Computational Mechanics JA - J Appl Math Comput Mech AU - Kopytko, Bohdan AU - Shevchuk, Roman PY - 2016 PB - The Publishing Office of Czestochowa University of Technology SP - 71 EP - 82 IS - 4 VL - 15 SN - 2299-9965 SN - 2353-0588 ER -
Kopytko, B., & Shevchuk, R. (2016). One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary. Journal of Applied Mathematics and Computational Mechanics, 15(4), 71-82. doi:10.17512/jamcm.2016.4.08
Kopytko, B. & Shevchuk, R., 2016. One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary. Journal of Applied Mathematics and Computational Mechanics, 15(4), pp.71-82. Available at: https://doi.org/10.17512/jamcm.2016.4.08
[1]B. Kopytko and R. Shevchuk, "One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary," Journal of Applied Mathematics and Computational Mechanics, vol. 15, no. 4, pp. 71-82, 2016.
Kopytko, Bohdan, and Roman Shevchuk. "One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary." Journal of Applied Mathematics and Computational Mechanics 15.4 (2016): 71-82. CrossRef. Web.
1. Kopytko B, Shevchuk R. One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2016;15(4):71-82. Available from: https://doi.org/10.17512/jamcm.2016.4.08
Kopytko, Bohdan, and Roman Shevchuk. "One-dimensional diffusion processes in half-bounded domains with reflection and a possible jump-like exit from a moving boundary." Journal of Applied Mathematics and Computational Mechanics 15, no. 4 (2016): 71-82. doi:10.17512/jamcm.2016.4.08
ONE-DIMENSIONAL DIFFUSION PROCESSES IN HALF-BOUNDED DOMAINS WITH REFLECTION AND A POSSIBLE JUMP-LIKE EXIT FROM A MOVING BOUNDARY
Bohdan Kopytko 1, Roman Shevchuk 2
1 Institute
of Mathematics, Czestochowa University of Technology
Częstochowa, Poland
2 Vasyl Stefanyk Precarpathian National University
Ivano-Frankivsk, Ukraine
bohdan.kopytko@im.pcz.pl, r.v.shevchuk@gmail.com
Received: 01 August
2016; accepted: 31 August 2016
Abstract. By the method of the classical potential theory, we construct the two-parameter Feller semigroup of operators associated with such a diffusion phenomenon on a half-line with a moving boundary where either a reflection or jump phenomenon occurs at a boundary point.
Keywords: diffusion process, parabolic potential, Feller-Wentzell boundary condition
1. Introduction
In the theory of stochastic processes while studying the diffusion processes in bounded and half-bounded domains, it occurs the situation in which the continuation of motion of a diffusion particle after it reaches the boundary of the domain is performed by jumps. The question on construction of semigroups of operators associated with the diffusion process with the property of a jump-like exit from the boundary of the domain leads to the statement of a boundary-value problem for a linear parabolic equation of the second order with a nonlocal boundary condition. Since the general form of boundary conditions for a one-dimensional (with respect to spatial variable) time-homogeneous diffusion process was established in the works of W. Feller [1] and A.D. Wentzell [2], these conditions were called Feller- -Wentzell boundary conditions.
In the present paper we consider the
one-dimensional parabolic boundary-value problem of Wentzell (with the combination of the derivative with respect
to spatial variable and the nonlocal term) for the case of an inhomogeneous
diffusion process in domain provided the lateral boundary
satisfies the Hölder condition
with respect to the time variable with exponent >
. This problem is stated
in Section 2 and is solved there by the method of ordinary parabolic
potentials. Using its solution in Section 3, we construct the two-parameter
semigroup
of operators
(
fixed), associated with an inhomogeneous Feller process on
the closure
of
which
coincides in
with
the diffusion process given there and its behaviour at point
is determined
by the Feller-Wentzell boundary condition.
Note that similar problems were considered earlier in [3, 4] for the case of bounded and half-bounded domains with fixed boundary points. We also mention works [5-7] where the related problems were studied by the methods of stochastic analysis.
2. Parabolic boundary-value problem of Wentzell
Consider on plane the set
![]() |
denoting by the
closure of
. Let in
the
parabolic equation
![]() | (1) |
is given. We shall seek a solution of equation (1) satisfying the
“initial” condition
![]() | (2) |
and the Feller-Wentzell boundary condition of the form
![]() | (3) |
().
The main problem is to find the function which belongs to
and which satisfies the equation (1) in
, the “initial” condition (2) and the
boundary condition (3).
In the present paper, the following conditions are supposed to be satisfied:
1. The
coefficients and
are bounded on
,
besides,
there exist positive constants
and
such that
for
all
.
2. For
all the next inequalities hold:
where and
are
positive constants,
.
3. The
curve is Hölder
continuous with exponent
on
.
4. , where
denotes
the Banach space of bounded continuous functions on
with
norm
.
5. The
function is positive and continuous on
.
6. is the nonnegative measure on
such that for any
,
,
where and these integrals are continuous on
as functions of
Denote by the
fundamental solution of equation (1) (
).
Its existence is assured by 1), 2) (see [5, Ch. II, §2], [8, Ch. IV, §11]).
Recall that function
is nonnegative, jointly
continuous, continuously
differentiable with respect to
twice continuously differentiable with
respect
to
and satisfies
the inequality
![]() | (4) |
for all
, where
and
are the nonnegative integers such that
;
is
the partial derivative with respect to
of
order
,
is
the partial derivative with respect to
of
order
; symbols
and
denotes (here and in what follows) any
one of various different positive constants.
Recall also that
![]() | (5) |
where
![]() |
and the function satisfies
inequality
![]() | (6) |
where ,
,
is the constant in 2.
Having the fundamental solution we now define
the parabolic potentials that will be used to solve the problem (1)-(3), namely
the Poisson potential
where is the function in
(2), and the simple-layer potential
with density which
is continuous in
and satisfies the inequality
for any . The
last inequality ensures the validity of the formula on the jump for potential
(see [9, Ch. V, §§2-4])
![]() | (7) |
where
Furthermore, from condition 3 and estimate (6) it follows that
![]() | (8) |
We find the solution of problem (1)-(3) of the form
![]() | (9) |
with the unknown function to be determined.
If we substitute the expression (9) for into (3), we obtain, upon using the
relation (7), the following Volterra integral equation of the second kind
![]() | (10) |
where
In order to solve this integral equation, we
have to study the behavior of function and
kernel
. We begin with estimate for
. Write
in
the form
For function we
have
![]() | (11) |
To estimate ,
apply the Lagrange formula to the integrand
in
its expression. We have
,
where is some real number from interval
. Hence
![]() ![]() | (12) |
The same estimate is also valid for . Indeed, using the triangle inequality,
we obtain
![]() ![]() | (13) |
Combining (11), (12) and (13), we conclude that
![]() | (14) |
where is some positive constant depending on
.
Now consider kernel .
Write it as follows
The first term in square brackets in the above expression is already estimated by (8). The absolute values of the second and fourth terms are bounded, respectively, by
and
which becomes clear, respectively, after using
the inequality (4) with
and after applying the Lagrange formula
to difference
and using the inequality (6)
with
, successively.
It remains to estimate the third term in the
expression for which we
denote by
. Write it in the form
Taking the derivative
and then using
the equality
, we
get
From condition 3 it follows that
Consider . Since
we have
Thus kernel in (9) has strong singularity which is caused by
. Therefore we do not know yet
whether a solution of (10) exists. We
shall see
presently that it is nevertheless possible to obtain the solution of (10) by an ordinary method of successive approximations, i.e.,
![]() | (15) |
where
Let us
prove that the integrals on the right side of expression for exist and the series (15) is convergent in
. To
do this, we first break
the expression for
into two terms
satisfying the estimate (8) with some positive constant
and
having
strong singularity, i.e.,
![]() | (16) |
Next, consider and
represent it as follows
where
In view of estimate (8) (with constant ) for
and
inequality (14), we immediately deduce that
![]() ![]() | (17) |
To estimate ,
write
where and
are constants in 1.
Denote by the
inner integral in the last relation. Write it in the form
Changing the variable of integration into
, we
obtain
![]() ![]() | (18) |
In view of (18), we get
![]() | (19) |
where
![]() |
Combining (17) and (19), we conclude that
![]() |
Choose so
small that
and denote
Proceeding by induction, we derive the
following estimates for terms of series (15)
![]() | (20) |
where
Hence, for , we
have
This implies that series (15) is absolutely
convergent in and therefore the function
exists and satisfies the inequality
![]() | (21) |
We have thus
constructed a solution of the boundary-value problem
(1)-(3) of the form (9). From relations (4)-(6) and (21) it follows that
and
![]() | (22) |
The proof of uniqueness of solution of (1)-(3) is based on the maximum principle for parabolic equations and is a repetition of the proof of the analogous assertion in [3] with obvious changes.
We have proved the following theorem:
Theorem 1. Let
the conditions 1-6 hold. Then the problem (1)-(3) has a
unique
solution in
.
Furthermore, this solution has the form (9) and satisfies the estimate (22).
3. Construction of the Feller semigroup
Consider the following problem: construct the
two-parameter semigroup of
operators which describes the inhomogeneous Feller process
on
connected with (1)-(3). Such a Feller
process coincides in
with the diffusion process
given by (1), (2) (with drift
and diffusion
coefficient
) and its behavior at boundary point
is determined by
the Feller-Wentzell boundary condition (3). Note that the two terms of boundary
condition (3)
and
are supposed to correspond to the reflection
phenomenon and the jump phenomenon on the boundary
We introduce the two-parameter family of
linear operators acting on the space
by the rule
![]() | (23) |
where is the solution of
problem (1)-(3) defined by formulas (9), (15).
Let us
show that the family of operators is the desired semigroup. To do this, we first note that the
operators
have the following property: if the se-
quence
is such that
for all
and
then
for all
The proof of this property is based on well-known assertions of
calculus on the passage of the limit under the summation and integral signs
(here this concerns series (15) and integrals on the right side of the
expression (9)). This property allows us to prove the next properties of the
operator family
, without loss of generality, under the assumption that the function
has a compact support.
The next lemma asserts that the operators are positivity
preserving:
Lemma 1. If and
for all
,
then
for all
Proof. Let be any nonnegative function in
having compact support.
Denote by
the minimum of
in
and
assume that
. From the minimum principle it follows
that there exists
such that
. But then the inequalities
and
hold. Furthermore, Theorem 14 in [9, p. 69] assures us that
Next, since it
becomes clear that the fulfillment of condition (3) is
impossible. The contradiction we arrived at indicates that
This completes
the proof of the lemma.
Another
important property of operators is that they are contractive,
i.e.,
This property follows from Lemma 1 together with
the fact that if 1 then
for
all
.
Finally, we show that operator family has the semigroup property
This property is a consequence of the assertion
of uniqueness of the solution
of the problem (1)-(3). Indeed, to
find when
, we can solve the problem (1)-(3) first in the time interval
with the “initial” function
, and then in the time interval
with the “initial” function
.
In other words,
or
The above properties of operators imply the following assertion (see [10, Ch. II], §1):
Theorem 2. Let
the conditions of Theorem 1 hold. Then the two parameter semigroup of operators defined by (23) describes the inhomoge-
neous Feller process on
which coincides in
with
the diffusion process given by (1),
(2) and its behavior at point
is determined by the Feller-Wentzell boundary
condition (3).
References
[1] Feller W., The parabolic differential equations and associated semi-groups of transformations, Ann. Math. 1952, 55, 468-518.
[2] Wentzell A.D., Semigroups of operators that correspond to a generalized differential operator of second order, Dokl. AN SSSR 1956, 111(2), 269-272 (in Russian).
[3] Shevchuk R.V., Inhomogeneous diffusion processes on a half-line, generated by the differential operator with Feller-Wentzell boundary condition, Math. Bull. NTSH 2011, 8, 243-257 (in Ukrainian).
[4] Kopytko B.I., Shevchuk R.V., Diffusions in one-dimensional bounded domains with reflection, absorption and jumps at the boundary and at some interior point, Journal of Applied Mathematics and Computational Mechanics 2013, 12(1), 55-68.
[5] Portenko M.I., Diffusion Processes in Media with Membranes, Institute of Mathematics of the NAS of Ukraine, Kyiv 1995 (in Ukrainian).
[6] Pilipenko A.Yu., On the Skorokhod mapping for equations with reflection and possible jump-like exit from a boundary, Ukrainian Math. J. 2012, 63(9), 1415-1432.
[7] Anulova S.V., On stochastic differential equations with boundary conditions in a half-plane, Izv. AN SSSR Ser. Mat. 1981, 45(3), 491-508 (in Russian).
[8] Ladyzhenskaya O.A., Solonnikov V.A., Ural’tseva N.N., Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow 1967 (in Russian).
[9] Friedman A., Partial Differential Equations of Parabolic Type, Mir, Moscow 1968 (in Russian).
[10] Dynkin E.B., Markov Processes, Fizmatgiz, Moscow 1963 (in Russian).
