^{1,2}

^{1}

^{2}

Let

Throughout the paper, we will denote, for any enough differentiable function

Let

These well-known connections can be extended to the polyharmonic operator

The covariance function of the process

The covariance fonction of the bridge

The covariance fonction of the bridge

Observe that the differential equations and the boundary conditions at 0 are the same in all cases. Only the boundary conditions at 1 differ. Other boundary value problems can be found in [

We refer the reader to [

The aim of this work is to examine all the possible conditioned processes of

The paper is organized as follows. In Section

We consider an

Let

Assume that the functions

If the problem (

In view of (

Concerning the boundary value conditions, referring to (

In the two next sections, we construct processes connected to the equation

Let

Let

Let us check that conditions (

In this section, we construct various bridges related to

For any

for

for

for

for

In this section, we exhibit several interesting properties of the various processes

Below, we provide a representation of

One has the distributional identity

In the case where

By invoking classical arguments of Gaussian processes theory, we have the distributional identity

We now compute the derivatives of

for

for

for

for

for

for

Let

The covariance function of

We decompose

In this section, we write out the natural boundary value problem which is associated with the process

Let

Recall that

We now check the uniqueness of the solution of (

if

if

The proof of Theorem

We have seen in the previous proof that uniqueness is assured as soon as the boundary value conditions at 1 satisfy

Now, we tackle the problem of the prediction for the process

Fix

The functions

Fix

We easily see that the functions

Here, we have a look on the particular case where

On the other hand, when adding two boundary value conditions at time 1 to the foregoing equation, we find six boundary value problems:

For each process, we provide the covariance function, the representation by means of integrated Brownian motion subject to a random polynomial drift, the related boundary value conditions at 1, and the decomposition related to the prediction problem. Since the computations are straightforward, we will omit them and we only report here the results.

For an account on integrated Brownian motion in relation with the present work, we refer the reader to, for example, [

The process corresponding to the set

The process corresponding to the set

The process corresponding to the set

The process corresponding to the set

(i) The solution of the problem associated with the boundary value conditions

(ii) The solution of the problem associated with the boundary value conditions

In this last part, we address the problem of relating the general boundary value problem

Our aim is to characterize the set of indices

We first write out a representation for the Green function of (

The boundary value problem (

Conditions (

A necessary condition for

The Green function

Here, we have a look on the particular case where

Conditioning set | ||||||||
---|---|---|---|---|---|---|---|---|

Differentiating set |

The Green functions related to the other sets cannot be related to some Gaussian processes. The sets are written in Table

Differentiating set | ||||||
---|---|---|---|---|---|---|

Differentiating set |

For

Let

Conditions (

We look for polynomials

We begin by performing the transformation