BOUNDARY CONDITIONS FOR ONE-DIMENSIONAL BIHARMONIC PSEUDO PROCESS

We study boundary conditions for a stochastic pseudo processes corresponding to the biharmonic operator. The biharmonic pseudo process ( BPP for short). is composed, in a sense, of two di(cid:11)erent particles, a monopole and a dipole. We show how an initial-boundary problems for a 4-th order parabolic di(cid:11)erential equation can be represented by BPP with various boundary conditions for the two particles: killing, reﬂection and stopping.


Introduction
We will call − 2 ≡ −∂ 4 x a biharmonic operator. It plays an important role in the theory of elasticity and fluid dynamics. For instance the parabolic differential equation is closely related to the Kuramoto-Sivashinsky equation and the Cahne-Hilliard equation (see [17,Chapter III §4]). It is easy to see that the fundamental solution p(t, x) to (1.1) is given by This p(t, x) takes negative values (see Hochberg [8]), so no stochastic process corresponds to (1.1) in the usual sense.
Several attempts have been made to relate the biharmonic operator to random processes. Krylov [9] considered a stochastic pseudo process whose "transition probability density" was p(t, x) as in (1.2), despite its taking negative values. We will call this pseudo process a biharmonic pseudo process or BP P for short (it is occasionally called the Krylov motion).
Following Krylov's idea, Hochberg [8] started a systematic study of BP P . His article included a definition of a stochastic integral with respect to BP P . Nishioka [14] calculated the joint distribution of the first hitting time and place for BP P hitting the boundary of a half-line. This was extended by Nakajima and Sato [10] to the space-time case. Some new developments in this direction can be found in a paper by Beghin, Hochberg and Orsingher [1].
Other attempts included a paper by Funaki [7] who introduced the concept of iterated Brownian motion. After his pioneering work, Burdzy [2] began to study path properties of iterated Brownian motion; a large number of related papers followed. Burdzy and Madrecki [3,4] modified Funaki's model in a different way; they defined a stochastic integral with respect to the their process.
This paper is devoted to the process BP P on a half-line. We will study various boundary conditions for the probabilistic model and relate them to their analytic counterparts. We will motivate our results by first reviewing the classical case of the standard Brownian motion on (0, ∞).
Consider the following initial-boundary value problem for the heat equation: where may take values 0, 1 or 2. Solutions to this initial-boundary value problem may be represented using Brownian motion on (0, ∞) which is killed, reflected, or stopped at the boundary point 0, depending on the value of . The relationship can be summarized as follows: We note that there is no analytic difference between Brownian transition probabilities on the open interval (0, ∞) in the cases when the Brownian particle is killed or stopped at the boundary but we make the distinction in Table 1.1 to emphasize the analogy with some results on BP P .
In this article, we will establish a relationship analogous to Table 1.1, between BP P and several initial-boundary value problems for (1.1).
We start with a few heuristic ideas. A big difference between a Brownian motion and BP P , proved in [13], is that the Brownian motion is a model for the motion of a single particle but when the BP P leaves the interval (0, ∞), it appears to consist of two types of particles. We will call these particles a monopole and a dipole. The names are justified by a result on the conservation of charges given in the last section of this article. The particles behave independently at the boundary. Therefore we need two different boundary conditions in order to solve the initialboundary value problem for (1.1). In a sense, each boundary condition controls the behavior of a monopole or a dipole, although the exact relationship, summarized in Table 1.2 below, is more complicated than that.
We proceed with a more formal presentation of the main results although the fully rigorous statements are postponed until later in the paper. Consider the following initial-boundary value problems for (1.1), where f is a given bounded smooth function on [0, ∞) and and m can take the values = 0, . . . , 4 and m = 0, . . . , 5. The following two tables summarize our main results.
For some values of and m, the initial boundary value problem has a probabilistic representation using BP P with killing, reflection or stopping for the monopole and dipole. The correspondence between the various values of and m and boundary conditions for BP P is coded in Tables 1.2 and 1.3 in the self-evident manner. In Table 1.2, the symbol "×" means that (1.4) is not well-posed with such analytic boundary conditions (see Example 7.9). The symbol "H" means such analytic boundary conditions are not related to BP P with killing, reflection or stopping at the boundary. However, these analytic conditions can be related to BP P . This requires a new idea of a "higher order reflecting boundary" which will be discussed in a forthcoming article. The diagonal entries in Table 1.2 and those below the diagonal are irrelevant for obvious reasons and so they are marked with "−−". The same symbol in Table 1.3 means that we have not found a relationship between this set of analytic conditions and BP P .
In §2, we review results from [13,14] on the joint distribution of the first hitting time and place for BP P and the concepts of a monopole and dipole. In §3 through §8, we construct and study BP P 's with various boundary conditions.
The construction of the BP P in the case when we have killing or stopping of the monopole at the boundary is similar in a sense to the Brownian motion case. Only slight modifications are needed when the same boundary conditions are prescribed to the dipole.
When we set the reflecting boundary for a monopole or dipole, we cannot apply a direct analogy with the reflecting Brownian motion (see §5.2). We will construct a BP P with the reflection as the limit of a suitable sequence of approximations.
In §9, we verify the law of the conservation of charges, and conclude that the total amount of charges is conservative if and only if there is no killing for a monopole.
We will use some basic results on solutions of 4-th order PDE's, Laplace and Fourier-Laplace transforms through the paper. These can be found in [5,6,16] for instance.
Acknowlegement The author would like to thank the referee made many important and precise suggestions for improvements of the paper.

Preliminaries
We start with a review of several function spaces used in the article.
Let X denote either the one dimensional Euclidean space R 1 or the half-line [0, ∞). B b (X) will stand for the space of all bounded measurable functions defined on X. C(X) will be the space of all continuous functions on X while C b (X) will be the space of all bounded functions in C(X). C 1 (X) will denote the space of all continuously differentiable functions on X and C 1 b (X) will be the space of all bounded functions in C 1 (X). Dirac's delta function will be denoted by δ(x) and δ a (db) will be the delta measure with the unit mass at a point {a}, so we may use the convention δ a (db) = δ(a − b) db.
For a fixed positive t, the function p(t, x) of (1.2) is an even function of x and it belongs to S.
For positive t and s, the following formulae hold Note that p(t, x) can take negative values. Hochberg [8] proved that p(1, |x|) = a|x| −1/3 exp{−b|x| 4/3 } cos c|x| 4/3 + a lower order term for large |x|, where a, b, and c are positive constants. It follows that Using this p(t, x), Krylov [9] defined a finitely additive signed measure P x on cylinder sets in where 0 ≤ t 1 < · · · < t n and B k 's are Borel sets in R 1 . We put If we fix 0 ≤ t 1 < · · · < t n , then this P x is a σ-additive finite measure on R n , but it is not a σ-additive measure on the smallest σ-field which includes all cylinder sets in R [0,∞) . We say that a function f defined on for a Borel function g on R n and 0 ≤ t 1 < · · · < t n . For each tame function f , we define its expectation in the usual way-if f is as in (2.5), then we set if the right hand side exists.
We extend the expectation to other functions as follows. Let n and N be natural numbers. For We define the expectation of an admissible function F (ω) by The expectation is unique if it exists, due to (iii) of Definition 2.1 (see [14]). Let A be a subset in R [0,∞) . When the characteristic function I A (ω) is admissible, we let For a positive number α, H α [0, ∞) will denote the space of all functions defined on [0, ∞) which satisfy Hölder's condition of order α. According to Krylov [9], This implies that the total mass of |P x | is concentrated on C[0, ∞). However for a technical reason related to Definition 2.1 (we need to deal with ω N n ), we will take a larger space D[0, ∞) as the path space of the pseudo process corresponding to P x . From now on, we will identify P x and E x with their restrictions to D[0, ∞).

The first hitting time distribution, monopole and dipole
We will review results from [13,14] in this section..

The distribution of the first hitting time and place
Given ω ∈ D[0, ∞), the random time is called the first hitting time of the interval (−∞, 0). It can be proved that the function is admissible, and we can calculate its expectation. For each λ > 0 and β ∈ R 1 , where θ 1 ≡ exp{πi/4} and θ 2 ≡ exp{3πi/4}.
In the classical probability setting, one can derive the joint distribution of the first hitting time and place from (3.2), using Bochner's theorem. However the theorem cannot be applied to BP P , since (3.2) is not positive definite in β, and so we have to extend the notion of the "joint distribution" itself.
Given functions φ and ϕ in S, it can be proved that the function is admissible and its expectation defines a continuous bilinear functional on S ×S. This fact is directly implied by Schwartz's result on Fourier-Laplace transforms of his temperate distributions.
The following lemma is a slight modification of his result.

Lemma 3.1 ([15]). Let c be a positive number and U (λ, β) be a complex-valued function of a real variable β and a complex variable λ with λ > c. Assume that for each real β, U is a holomorphic function in λ with λ > c and it satisfies
where C and k are non-negative constants. Then U is the Fourier-Laplace transform of a Schwartz temperate distribution whose support lies in [0, ∞) × R 1 .
We will always take the principal value for λ 1/4 . If λ > c for any positive number c, then −π/8 < arg λ 1/4 < π/8. This implies that (3.2) satisfies (3.4). Hence there exists a Schwartz temperate distribution q(t, a; x) such that holds for all φ and ϕ in S.
In the classical probability theory, distributions of real-valued random variables may be thought of as non-negative continuous linear functionals on the function space C b (R 1 ). We define distributions of functions of BP P to be continuous linear functionals on a function space which includes S.

Definition 3.2.
We call Schwartz's temperate distribution q(t, a; x) in (3.5) the density of the distribution (3.3), and let which is a continuous bilinear functional on S × S for each x ≥ 0.

Proposition 3.3 ([14])
. Let x ≥ 0. Then in the distribution sense, where δ (a) is the first derivative of Dirac's delta function δ(a), Moreover, the support of the distribution in (3.6 The explicit formula (3.6) makes it possible to extend (3.3) to a continuous bilinear functional on a larger space than S × S.

Corollary 3.5 ([14]). The distribution (3.3) can be extended to a continuous bilinear functional on
The strong Markov property holds for BP P at time τ 0 (ω).

Monopole and dipole
In physics a particle is called a dipole when it carries two charges of equal magnitude but opposite signs. A small magnet is a typical example of a dipole. Heuristically, one may represent a dipole by As → 0, (3.9) converges to −δ (a) in the distribution sense. Therefore we will call −δ (a) a dipole. For the Brownian motion B(t), it is well known that Comparing this with (3.6), we see that in a sense, BP P behaves as a mixture of two particles of different types when it hits an interval.
Informally speaking, a particle of the first type is represented by δ(a) and carries a charge of a single sign just as a Brownian particle does. The second particle is represented by −δ (a) and corresponds to a dipole. The following informal definition will enable us to present our results using intuitive notation.
Definition 3.7. We will call a particle represented by δ(a) a monopole and a particle represented by −δ (a) a dipole. We define distributions of the first hitting time and place for the monopole and dipole as follows.

The initial particle
When BP P starts with an initial distribution µ(dy), its distribution at time t is given by Recall from Definition 3.7 that δ(y) stands for the monopole and the dipole is represented by −δ (y). Therefore when a monopole starts from a point x, the distribution of the process at time t is On the other hand, when a dipole starts from a point x, the distribution at time t is We have a similar formula for the joint distribution of the first hitting time and place. When a monopole starts from a point x, the distribution is

is a monopole and in da]
= same as the right hand side of (3.10). (3.12) On the other hand when a dipole starts from x, then the distribution is is a monopole and in da ] (3.13) Remark 3.9. Informally speaking, (3.13) shows that a dipole generates both a monopole and a dipole at the hitting time of (−∞, 0), just like a monopole does.

Some Laplace transforms
For future reference, we list some Laplace and Fourier-Laplace transforms of K(t, x), J(t, x) and p(t, x). Let λ > 0 and x ≥ 0. From Proposition 3.3, we havê (3. 16) In addition, we define a new function

is a monopole and in da]
It is elementary to check that (3.20)

is a monopole and in da]
(4.1) We will refer to {P 00 x [ω(t) ∈ db] : x ≥ 0} as a BP P with killing boundaries for both particles.

Remark 4.2.
Here is an intuitive description of BP P with killing boundaries for both particles.
(ii) The following linear functionals are identical and continuous on B b [0, ∞), Proof. We easily obtain (4.2) and (4.3) when we recall the definitions of the terms on the right hand side of (4.1) from Section 3.2. It is straightforward to check that (3.20) is the Fourier-Laplace transform of (4.2) in the usual sense. Part (ii) follows from the fact that p(t, x) ∈ S and from the estimates stated in Remark 3.4.
(i) If f is a bounded continuous function on [0, ∞), then As expected, the distribution P 00 x [ω(t) ∈ db] solves (1.4) with the Dirichlet boundary condition.

Reflection for the monopole and killing of the dipole
The construction of the process BP P with the boundary conditions specified above will use an approximating sequence. We will prove existence of a family { P r0
Assume for the moment that P r0 x [ω(t) ∈ db] is a Schwartz temperate distribution and denote its Fourier-Laplace transform by Apply the Fourier-Laplace transform to the both sides of (5.1) to obtain Lemma 5.2. (i) When > 0, the following is a solution of (5.2):

4) for any c > 0, and it is the Fourier-Laplace transform of a Schwartz temperate distribution
Proof. (i) Put x = in (5.2) to see that (ii) Let λ > c > 0. We take the principal value for λ 1/4 and recall that −π/8 < arg λ 1/4 < π/8. After applying this fact to (3.14) through (3.17), we see that (3.4) holds forK,Ĵ , andĜ 00 , and also for the fraction on the right hand side of (5.3) when > 0 is small. Now Lemma 3.1 implies our assertion.

Definition 5.4. This above limit will be denoted by {P
.
where p 00 is the function in (4.2) and Q r0 is defined in (5.6). Then we have and ∂ x U r0 (λ, 0, β) = 0 and ∂ 2 x U r0 (λ, 0, β) = 0. Since the Fourier-Laplace transform is unique for our p r0 , these prove the first part of the theorem. The second part is immediate from the same argument as in the proof of Corollary 4.6.

An analogy to the classical reflecting Brownian motion
For the Brownian motion B(t), it is well known that the following two process X 1 and X 2 have the same distributions. They are referred to as reflecting Brownian motions. Let τ 0 be the first hitting time of the point x = 0.
It is not difficult to check that similarly defined transformations of BP P do not have identical distributions. Keeping this fact in mind, we define a new BP P as follows. In [14], the following was proved. For b ≥ 0, where Q r0 is given in (5.6). We record this partial analogy with the classical reflected Brownian motion in the next theorem.
Proof. The result follows easily from (5.11) and Theorem 5.5.

Reflecting boundaries for both particles
The construction of BP P with these boundary conditions requires an approximation procedure similar to that in the previous section. We will prove that there exists a family { P rr x [ω(t) ∈ db] : x ≥ 0} satisfying the following equation. Assuming for the moment existence of P rr x [ω(t) ∈ db], we denote by U rr its Fourier-Laplace transform. Recall (3.13) and note that (5.12) is transformed into the following equation. For

is a monopole and in da]
(5.13) Lemma 5.11. (i) For > 0, the following is a solution of (5.13): Proof. (i) Substitute for x in (5.13). Then differentiate both sides of (5.13) with respect to x and then again substitute for x. As a result, we obtain the following equations, , β), We can solve the equations for U rr (λ, , β) and ∂ x U rr (λ, , β) and substitute the results into (5.13). Then (5.14) easily follows. Part (ii) can be proved the same way as part (ii) of Lemma
Proof. If we let → 0 in (5.14) then we obtain .
The right hand side equals to U r0 in the proof of Lemma 5.4, so it is the Fourier-Laplace transform of P r0

Killing of monopole and reflection for dipole
Once again, we start with an approximating sequence. We will find a solution { P 0r
x ≥ 0, } solving (6.1) exists. We denote the Fourier-Laplace transform of P 0r x [ω(t) ∈ db] by U 0r (λ, x, β). Then we apply the Fourier-Laplace transform to (6.1). Using (3.13), we obtain for x > 0, The following lemma can be proved the same way as Lemma 5.11 so we omit its proof.

Lemma 6.2. (i)
For > 0, the following is a solution of (6.2): (ii) For small > 0, this U 0r satisfies (3.4) with any c > 0, and it is the Fourier-Laplace transform of a Schwartz temperate distribution P or x [ω(t) ∈ db]. Lemma 6.3. In the sense of convergence of distributions, P 0r x [ω(t) ∈ db] converges to a limit as → 0. Definition 6.4. We denote this limit by {P 0r x [ω(t) ∈ db] : x ≥ 0}, and call it the distribution of BP P with the killing boundary for the monopole and the reflecting boundary for the dipole.
When we take the principal value of λ 1/4 , U 0r satisfies (3.4), and Lemma 3.1 implies our assertion.
Theorem 6.5. For x ≥ 0, b ≥ 0, and t > 0, we define a function where Q 0r is the function in (6.6). Let {P 0r x [ω(t) ∈ db] : x ≥ 0} denote the distribution in Definition 6.4. Then we have Proof. The theorem is immediate from Lemma 6.2 and (6.6).
The following theorem is easily proved using (6.4) in the same way as Theorem 5.11 so we omit its proof.

Theorem 6.6. (i) Let f be a bounded smooth function on the interval
Then in the classical sense, this v satisfies (1.4) with = 0 and m = 2.

is a monopole and in da]
The family of distributions given in the last definition represents a BP P whose both particles are stopped after exiting (0, ∞). An intuitive description of the process is the following.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability density p(t, x) until it hits the interval (−∞, 0).
Then we have P ss x [ω(t) ∈ db] = p ss (t, x, b) db. Proof. We apply the Fourier-Laplace transform to P ss x [ω(t) ∈ db], and denote it by U ss (λ, x, β). x U ss (λ, 0, β) = 0 and ∂ 5 x U ss (λ, 0, β) = 0. This proves part (i) of the theorem, since the Fourier-Laplace transform is unique. For (ii), it is sufficient to note that ∂ x p ss (t, x, b) is a linear functional on C 1 b [0, ∞) and the left hand side of (4.5) is well-defined with p 00 (t, x, b) replaced by ∂ x p ss (t, x, b).

Stopped monopole and killed dipole
x ≥ 0} be given by Remark 7.6. A heuristic view of BP P with the boundary behavior specified in the section title is this.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability density p(t, x) until it hits the interval (−∞, 0).
The first assertion of the theorem follows from this and from uniqueness of the Fourier-Laplace transform. The proof of the second is the same as in Theorem 7.3.
We present an example of a "not well-posed problem" for (1.4): These v 1 and v 2 are classical solutions of (1.4) with = 1. We consider their Laplace transforms , j = 1, 2. Since they are smooth and both satisfy ∂ x V j (λ, 0) = 0, we have This fact implies that solutions to (1.4) are not unique when = 1 and m = 5. Note that an analogous argument also holds when = 0 and m = 4.

Stopped monopole and reflected dipole
This case calls for an approximation procedure similar to those discussed in earlier sections. We are looking for a family { P sr
Let U sr (λ, x, β) denote the Fourier-Laplace transform of P sr x , if it exists. Recall (3.13) and note that (7.11) can be transformed to The following lemma is similar to several lemmas presented earlier in the paper so we omit its proof.
These prove (i). Note that ∂ x p 0s (t, x, b) is a linear functional on C 1 b [0, ∞). The second part of the theorem can be proved by arguments used earlier in the paper.

Stopped dipole and reflected monopole
For the last time in this paper we use an approximation procedure to construct a process. The first step is to find { P rs x [ω(t) ∈ db] : x ≥ 0} such that
Assuming existence, we denote by U rs (λ, x, β) the Fourier-Laplace transform of P rs x . Recall (3.13). We see that (8.6) is transformed into the following equation: U rs (λ, x, β) =Ĝ 00 (λ, x, β) +K(λ, x) U sr (λ, , β) +Ĵ (λ, x) iβ λ . (8.7) We omit the proof of the following result. We have just proved that if > 0, then there exist distributions P rs x [ω(t) ∈ db] satisfying equations (8.6). However the following proposition asserts that we cannot pass to 0 with . Hence we cannot construct a BP P with reflection for the monopole and stopped dipole, at least not using our method.  diverges as → 0, and so (8.8) also diverges.

Conservation of charge
Let {Q x [ω(t)] : x ≥ 0} be the generic notation for the distribution of BP P on [0, ∞) with any boundary conditions for the monopole and dipole. Assume that ≥ 1 and m ≥ 1 in (1.4). Since ρ(0, x) = 1 in view of (9.1), we see that ρ is a solution of (1.4) with the constant initial function f (x) ≡ 1.
The constant function v(t, x) ≡ 1 is a solution of (1.4) with f ≡ 1, if ≥ 1 and m ≥ 1. Now uniqueness of the solution implies that ρ(t, x) ≡ 1, and Q x [ω(t) ∈ db] is conservative.
On the other hand, if = 0, then the constant v(t, x) ≡ 1 is not a solution of (1.4) with the constant initial function f ≡ 1, and uniqueness of the solution implies that ρ(t, x) = 1 for some (t, x). In this case, Q x [ω(t) ∈ db] is not conservative.