1. Introduction
The connection between parabolic equations and diffusion processes is well understood; the same cannot be said for ultraparabolic equations and ultradiffusion processes. Until recently, theoretical results have been fairly limited relative to the existence and uniqueness of solutions to ultraparabolic equations, deriving from two methodologies. In one, the analysis is affected along the characteristic of the first-order temporal operator, requiring that the speed of propagation varies only spatially. Such an approach was developed by Piskunov [
1] in the classical case and extended by Lions [
2] to the generalized sense. The second approach is based on the method of fundamental solutions and was implemented by Il’in [
3] for the classical Cauchy problem and extended to more general domains via convolution by Vladimirov and Drožžinov [
4], albeit at the expense of necessitating constant coefficients in the operator. Recently, however, using energic techniques Marcozzi [
5] has established the well-posedness and Galerkin approximation of the generalized solution (strong and weak) to the terminal value problem for square integrable data on bounded temporal and spatial domains. We extend here the results of [
5] to linear ultraparabolic terminal value/infinite-horizon temporal problems posed on unbounded spatial domains. We then provide a probabilistic interpretation of the solution in terms of the expectation of an associated ultradiffusion process.
Historically, the connection between the expectation of ultradiffusion processess and the solution to ultradiffusion equations arose from the work of Kolmogorov [
6,
7] and Uhlenbeck and Ornstein [
8] in relation to Brownian motion in phase space—the same with respect to Chandrasekhar [
9] in the context of boundary layers and Marshak [
10] relative to the Bolzmann equation. A contemporary example may be found in the formulation of so-called Asian options from mathematical finance (cf. [
11]), which obtains theoretical context with the present results. The paper is organized as follows. In
Section 2, we consider deterministic aspects of the problem, while, in
Section 3, the probabilistic interpretation is presented.
Appendix A introduces certain regularity results, which, while essential for the analysis, are too extensive to prove in full. In
Appendix B, we show formally that the ultraparabolic/ultradiffusion association is locally that of a parameterized parabolic/diffusion.
2. Approximation Solvability
We consider here the existence, uniqueness and approximation of the terminal value/infinite horizon problem on unbounded spatial domains for the linear ultraparabolic equations. To this end, let
,
,
, and finally
, for some finite
. The functional setting will be the weighted Sobolev spaces defined as follows. Spatially, we let
such that
and
with their respective norms
for all
, and
for all
. The relation “
” constitutes an
evolution triple.
Temporally, let
and
such that
which we equip with the norm
for all
. We associate with
the dual space
and the norm
, for all
. In addition, let
where
, which we associate with the norm
for all
. Finally, we define
We consider the
ultraparabolic t-terminal value/infinite
-horizon problem for
satisfying the evolutionary equation
subject to the terminal condition
where
for given
for some sufficiently large
.
The
generalized problem associated with (1)–(2) is: supposing (3)–(7), find
satisfying
for almost all
, such that
where
is the scalar product canonically defined on the Hilbert space
X,
denotes the value of the linear functional
at
,
and
for all
and
. In Equation (
8), the expressions
and
denote generalized derivatives on
; that is, Equation (
8) means explicitly
for all test functions
.
For
, the mapping
is bilinear and bounded; we likewise assume that
is strongly positive;
Remark 1. We note that Equations (1)–(2) is an infinite horizon problem in ϑ. That is, the far-field behavior of ϑ is implicitly defined relative to the weight γ.
Remark 2. In general, the validity of (11) will be problem dependent, predicated upon the spatial asymptotic behavior of u.
For
, we define the operator
such that
from which it follows that
is linear, continuous, and strongly monotone by (11). In particular, we have
and
for all
and
.
Lemma 1. Given (3)–(7), the formulations (1)–(2) and (8)–(9) are equivalent.
Proof of Lemma 1. By integration by parts and the density of test functions in
, we have
for all
and almost all
. From (8), (12) and (15), we deduce that
for all
and almost all
, in which case (1) follows. The converse derives from (1) and
, which imply (8). □
Proposition 1. Uniqueness. We suppose (3)–(7) and (11); let . Then, there exists at most one solution to (8)–(9).
Proof of Proposition 1. We consider (8)–(9) with
and
; setting
in (8), we obtain
or
from (11), in which case
Integrating over the domain
, for some
, and applying Green’s Theorem, it follows that
and so
However,
is not summable on
, which contradicts the condition
, from which it follows that
. □
We consider the regularization of (1)–(2) to domains of finite extent. To this end, it suffices for
to have an extension to, or to be of compact support in,
. Without loss of generality, we may assume that
. For
, let
and
There exists a unique
satisfying the ultraparabolic terminal value problem (cf. [
5])
subject to the terminal conditions
and boundary conditions
We denote by the extension of by zero to the compliment of .
Lemma 2. We suppose (3)–(7), (11), and
; then,for all . Proof of Lemma 2. Taking the inner product of (16) with
, we have
or
Integrating the above over
, it follows that
or
and so
in which case
or
such that
therefore,
for all
. □
We obtain a supplementary estimate on .
Lemma 3. We suppose (3)–(7), (11), (20), and , then Proof of Lemma 3. We consider the parabolic regularization with respect to
of (16)–(19). To this end, let
; we define the space of test functions on
such that
in which case we obtain the evolution triple “
”. We denote
and
for brevity and equip
with the norm
Let
where
and
, which we equip with the norm
The perturbation problem associated with (16)–(19) is: for any
, we seek
satisfying the parabolic equation
where
, subject to the terminal condition
and boundary conditions
The problem (23)–(27) is well-posed, noting in particular the necessity of the auxiliary boundary condition (27).
We denote by
the extension of
by zero to the compliment of
. Taking the inner product of (23) with
, it follows that
where
and
in which case
where
Integrating the above in time, we have that
With (21), we proceed as per Lemma 2 to obtain
which is valid for all
; passing to the limit, we obtain
which holds for all
m. We determine the estimate in
analogously. □
In the following result, we establish the existence of the solution to (8)–(9) as well as its approximation by the regularization (16)–(19).
Proposition 2. Existence. We suppose (3)–(7), (11), (20), and ; then, there exists a satisfying (8)–(9). Moreover, the sequence converges such that in
andas , where , for any (fixed) . Proof of Proposition 2. From the estimates (21) and (22), it follows that, possibly after extracting a subsequence, in , in and in , where u satisfies (8)–(9).
In order to show convergence of the regularizations
, we have from (8) that
Multiplying the above by
and applying the Green’s formula over
, we obtain
and the result follows from (11) and
in
. □
Let
denote a basis in
. We set
where
and
, such that
,
and
as
. The
Galerkin equations associated with (16)–(19) are defined
for
, such that
for
, where
for all
and
,
is the inner product on the Hilbert space
, and
is the value of the linear functional
at
.
We immediately obtain the constructive approximation of (8)–(9)) by the Galerkin procedure (28)–(33) from ([
5], Propositions 4 and 5) and Proposition 2.
Proposition 3. Galerkin Approximation. We suppose (3)–(7), (11), (20), , and . Let be the -Galerkin approximation to defined by (28)–(33) and
u the solution to (8)–(9), then in andas , where , for any (fixed) . Remark 3. Propositions 2 and 3 likewise hold with , where we imply the Galerkin approximation per the proof of Lemma 3.
3. Probabilistic Interpretation
In order to provide a probabilistic interpretation of the solution to (1)–(2), we make the additional assumptions that
We likewise suppose the existence of a function
such that
and
satisfies the same assumptions as
f. From Proposition 2 and
Appendix A, we allow that there exists a unique solution
to the problem (1)–(2).
We let
(or
) and define
and
in which case
for
sufficiently large (cf. (7)). In particular,
,
a and
b are elements of
. Moreover, by extending the functions
a,
b, and
outside of
, we may assume that
as well as
for all
.
We now seek a probabilistic interpretation of the function
u satisfying (1)–(2) by constructing a stochastic differential equation for which the trajectories
are the characteristics of
. To this end, we take a probability space
, an increasing family of sub-
-algebras
of
, and a
-valued standardized Wiener process
, which is an
martingale. We can then consider, on an arbitrary finite interval, the stochastic differential equation
where
x and
are fixed and non-random; the solution of (51)–(54) is unique.
Proposition 4. The assumptions of Proposition 2, as well as (34) through (46); the solution of (1)–(2) is given by Proof of Proposition 4. The proof relies on the existence and uniqueness of the regular solution to the ultraparabolic terminal value problem (1)–(2). With this exception, the result is standard such that we will provide only a brief exposition, deferring to e.g., ([
12], Chapter 2, Theorem 7.4). For the process
we have that
for all
and all
, in which case the right-hand side of (55) is well-defined.
We shall now prove (55) in the case
. We set
Differentiating the functional
, applying Ito’s formula to
, and integrating from
t to
T, we obtain
From (56) with
and the assumptions (44) on the growth of
, we have that the expectation of the stochastic integral is defined and is equal to zero. We therefore have that
in which case (55) is identical to
and so the problem reduces to proving (55) with
, with
f replaced by
, and with
u replaced by
and a solution to (1)–(2) corresponding to data
and 0.
We therefore assume
; we prove that
We start by considering the bounded case. We approximate
f by
defined by
Since
and
,
,
, we can uniquely define
as the solution of
such that
and
We note that
is bounded. This follows as per ([
5], Prop. 2’).
To this end, let
and
be the exit time form
of the process
. We can suppose that the (fixed) initial data
of (51)–(54) belongs to
, for
R that is sufficiently large. That is, we have, from the continuity of the process a.s.
for some
with
, in which case
for
. As above, with the use of Ito’s formula applied to
between the instants
t and
, taking
, and using the continuity of
on
, we have
where
However, from (63), we have
for
, and so
as
. Application of Lesbesque’s theorem then provides the result (62).
From the estimates
and
it follows that
lies in a bounded subset of
and we obtain (59) by proceeding to the limit successively in
M and
N. □