On some Gaussian Bernstein processes in RN and the periodic Ornstein-Uhlenbeck process

In this article we prove new results regarding the existence of Bernstein processes associated with the Cauchy problem of certain forward-backward systems of decoupled linear deterministic parabolic equations defined in Euclidean space of arbitrary dimension N, whose initial and final conditions are positive measures. We concentrate primarily on the case where the elliptic part of the parabolic operator is related to the Hamiltonian of an isotropic system of quantum harmonic oscillators. In this situation there are many Gaussian processes of interest whose existence follows from our analysis, including N-dimensional stationary and non-stationary Ornstein-Uhlenbeck processes, as well as a Bernstein bridge which may be interpreted as a Markovian loop in a particular case. We also introduce a new class of stationary non-Markovian processes which we eventually relate to the N-dimensional periodic Ornstein-Uhlenbeck process, and which is generated by a one-parameter family of non-Markovian probability measures. In this case our construction requires an infinite hierarchy of pairs of forward-backward heat equations associated with the pure point spectrum of the elliptic part, rather than just one pair in the Markovian case. We finally stress the potential relevance of these new processes to statistical mechanics, the random evolution of loops and general pattern theory.


Introduction and outline
Let us consider the two adjoint parabolic Cauchy problems and v(x;T ) = T (x); x 2 R N ; 1 where T 2 (0; +1) is arbitrary. In these equations, x denotes Laplace's operator with respect to the spatial variable, V is a real-valued function while ' 0 and T are positive measures on R N . Both (1) and (2) can then be looked upon as de…ning a forward-backward system of decoupled linear deterministic heat equations in Euclidean space. To wit, the potential solutions to (1) wander o¤ to the future, whereas those of (2) evolve into the past. In Section 2 below we show that under very general conditions on V; ' 0 and T , we can associate with (1)-(2) a class of the so-called Bernstein or reciprocal processes, henceforth denoted by Z 2[0;T ] . These are processes that constitute a generalization of Markov processes which have played an increasingly important rôle in various areas of mathematics and mathematical physics over the years (see, e.g., [1], [3]- [5], [10], [18], [21] and the many references therein for a history and ealier works on the subject). In the case of (1)-(2) the state space of the processes is the entire Euclidean space, and their construction requires a transition function as well as a joint probability distribution for Z 0 and Z T , which we denote respectively by P and in the sequel. While P depends exclusively on the parabolic Green function associated with (1)- (2), the measure involves both Green's function and ' 0 ; T , or more generally a statistical mixture of ' 0 and T . It is P and that allow one eventually to write out all the …nite-dimensional distributions of Z 2[0;T ] , on which the remaining part of this article is based. In Section 3 we concentrate primarily on certain Markovian Gaussian Bernstein processes associated with a particular case of (1)- (2), namely, @ t u(x; t) = 1 2 x u(x; t) 2 2 jxj 2 u(x; t); (x; t) 2 R N (0; T ] ; u(x; 0) = ' 0 (x); x 2 R N and @ t v(x; t) = 1 2 x v(x; t) 2 2 where > 0 and j:j denotes the Euclidean norm in R N , whose right-hand side is the Hamiltonian of an isotropic system of quantum harmonic oscillators, up to a sign (see, e.g., [15]). There we apply the results of Section 2 in order to construct and analyze three processes of interest, by determining explicitly in each case the …nite-dimensional projections of the corresponding Gaussian measures. The …rst one is stationary and thereby esssentially a N -dimensional Ornstein-Uhlenbeck process, while the second one is a non-stationary Ornstein-Uhlenbeck process conditioned to start at the origin of R N . The third one is a process that shares many of the properties of a bridge, which we call a Bernstein bridge and which we may identify with a Markovian loop in a particular case. The measures we need for the construction of each one of these are intimately tied up with a very speci…c choice of initial-…nal data in (3)- (4). The situation is quite di¤erent in Section 4, where we construct a new family of stationary non-Markovian Bernstein processes that are related to the N -dimensional periodic Ornstein-Uhlenbeck process. There, we prove that the relevant non-Markovian probability measures disintegrate into statistical mixtures of the form where each m is a measure related to initial-…nal conditions ' m;0 and m;T in and respectively. In other words, we show there that the construction of the 's given by (5) requires the consideration of a hierarchy of in…nitely many pairs of problems of the form (3)-(4) associated with the whole pure point spectrum of the elliptic operator on the right-hand side, rather than just one pair in the Markovian case. Finally, we also point out the potential applications of those new processes to statistical mechanics, to the problem of random evolution of loops in space and to general pattern theory, to name only three. In an appendix we also prove an important series expansion for the Green function associated with(3)-(4).

A class of Bernstein processes in R N
Generally speaking Bernstein processes can take values in any topological space countable at in…nity, and there are several equivalent ways to characterize them (see, e.g., [10]). However, the following de…nition will be su¢ cient for our purposes: De…nition 1. Let N 2 N + and T 2 (0; +1) be arbitrary. We say the R N -valued process Z 2[0;T ] de…ned on the complete probability space ( ; F; P) is a Bernstein process if for every bounded Borel measurable function f : R N 7 ! R, and for all r; s; t satisfying r 2 (s; t) [0; T ]. In (8), the -algebras are and where B N stands for the Borel -algebra on R N .
The dynamics of such a process at any time r 2 (s; t) are, therefore, solely determined by the properties of the process at times s and t, irrespective of its behavior prior to instant s and after instant t. Of course, it is this fact that generalizes the Markov property.
In order to associate a mere Bernstein process with (1)-(2) we now impose the following hypothesis, which regards Green's function alone: (H1) The measurable function V : R N 7 ! R is such that the parabolic Green function g associated with (1)-(2) is jointly continuous in all variables and satis…es g(x; t; y) > 0 for all x; y 2 R N and every t 2 (0; T ]. Let us now write M R N R N ; C for the space of measures we are interested in, namely, the topological dual of the Fréchet space of all complex-valued, compactly supported functions on R N R N endowed with the usual locally convex topology (see, e.g., [19]). Having (11) at our disposal, let us introduce the functions p (x; t; z; r; y; s) := g(x; t r; z)g(z; r s; y) g(x; t s; y) and P (x; t; E; r; y; s) := Z E dzp (x; t; z; r; y; s) for every E 2 B N , both being well de…ned and positive for all x; y; z 2 R N and all r; s; t satisfying r 2 (s; t) [0; T ]. For every F 2 B N B N , let us also consider a positive measure 2 M R N R N ; C such that de…nes a probability measure on B N B N , thus satisfying Z The knowledge of both (13) and (14) then makes it possible to associate a Bernstein process with (1)- (2). The precise statement is the following: Theorem 1. Assume that Hypothesis (H1) holds, and let P and be given by (13) and (14)- (15), respectively. Then, there exists a probability space ( ; F; P ) supporting an R N -valued Bernstein process Z 2[0;T ] such that the following properties are valid: 4 (a) The function P is the transition function of Z 2[0;T ] in the sense that P (Z r 2 E jZ s ; Z t ) = P (Z t ; t; E; r; Z s ; s) for each E 2 B N and all r; s; t satisfying r 2 (s; t) [0; T ] : (b) For every n 2 N + , the …nite-dimensional distributions of the process are given by P (Z t1 2 E 1 ; ::: for all E 1 ; :::; E n 2 B N and all t 0 = 0 < t 1 < ::: < t n < T , where x 0 = x. In particular we have for each E 2 B N and every t 2 (0; T ). Moreover, and for each E 2 B N .
(c) P is the only probability measure leading to the above properties.
Proof. Up to minor technical details, a direct adaptation of the method developed in Section 2 of [21] leads to P (Z 0 2 E 0 ; Z t1 2 E 1 ; ::: for all E 0 ; :::; E T 2 B N and all t 0 = 0 < t 1 < ::: < t n < T , where x 0 = x. In particular we have for all E 0 ; E T 2 B N , that is, (14) is the joint probability distribution of Z 0 and Z T . Now, from (12) we obtain after n 1 cancellations of factors in order to obtain the second equality, so that (16) follows by choosing E 0 = E T = R N in (20). Relations (18) and (19) are just a particular case of (21).
Of course, we can say more about Z 2[0;T ] if we know more about . First, in which case we also say that is Markovian. This result can be traced back to the more general Theorem 3.1 in [10], and allows us to make a closer connection between Z 2[0;T ] and (1)-(2) provided we impose the following hypothesis: (H2) The measures ' 0 ; T 2 M R N ; C in (1)-(2) are positive, and there exist a unique classical positive solution to (1) and a unique classical positive solution to (2), namely, and respectively.
We then have the following consequence of Theorem 1: for every integer n 2, all E 1 ; :::; E n 2 B N and all t 0 = 0 < t 1 < :: for each E 2 B N and every t 2 (0; T ). Finally, and Proof. We …rst rewrite (16) as P (Z t1 2 E 1 ; ::: and then substitute the choice of the measure into (19) and (30), using (24)-(25) along with the symmetry property of g with respect to the spatial variables.

Remarks.
(1) In the Markovian case, we have thus exhibited a very general class of initial-…nal conditions in (1)-(2) which allows us to determine the processes Z 2[0;T ] completely, including their marginal distributions (28)-(29). The converse point of view was developed in [2] and its references, where it was shown instead that it is a general class of marginal distributions which determines the initial-…nal data of the relevant partial di¤erential equations, through a system of nonlinear integral equations. However, the resulting initial-…nal conditions of [2] belong to the class of positive continuous functions, and not to the larger class of positive measures as is the case in this article.
(2) A glance at (25) shows that it is su¢ cient to use elementary time reversal in Green's function to obtain the solution to (2). Although the situation is not always that simple, particularly when the given parabolic equations are nonautonomous, it is still possible to de…ne a quite appropriate probabilistic notion of time symmetry in general. We refer the reader to [21] for further details.
Relations (16) and (26) are the fundamental relations that will allow us to construct the Gaussian processes associated with (3)-(4) in the next sections.

Two Ornstein-Uhlenbeck processes and a Bernstein bridge
In the remaining part of this article we denote by (:; :) R N the Euclidean inner product in R N , and by L 2 R N ; C the usual Lebesgue space of all complexvalued, square-integrable functions on R N . We begin by considering the forwardbackward system (3)-(4) with centered Gaussian initial-…nal data, namely, and thereby identifying the measure ' 0; = T; with its Gaussian density relative to the Lebesgue measure in R N . Let us recall that the self-adjoint realization in L 2 R N ; C of the elliptic operator on the right-hand side of these equations has a pure point spectrum. More speci…cally, for every m 2 N let be the one-dimensional, suitably scaled Hermite functions where and where the H m 's are the Hermite polynomials 8 The following spectral result regarding the operator on the right-hand side of (31) is well-known and in any case can be veri…ed directly by means of an explicit computation: where the m j 's vary independently on N provide an orthonormal basis of L 2 R N ; C , and moreover the eigenvalue equation holds for every x 2R N .
In particular, by reference to (34) and (35) whose associated eigenvalue is E = N 2 , so that (37) corresponds to the initial-…nal conditions in (31)-(32) up to a normalization factor chosen in such a way that (15) holds for the measure is the N -dimensional version of Mehler's kernel for t 2 (0; T ] (see the appendix for details). In this way, the unique classical positive solutions to (31)-(32) satisfying the requirement Z respectively. Then the following result holds, where E stands for the expectation functional on ( ; F; P ): Proposition 1. The Bernstein process Z 2[0;T ] associated with (31)-(32) in the sense of Corollary 1 is a Gaussian and Markovian process such that Furthermore, the components of Z 2[0;T ] satisfy the relation for all s; t 2 [0; T ] and all i; j 2 f1; :::; N g. Thus, Z 2[0;T ] identi…es in law with a process whose components are all independent, one-dimensional and stationary Ornstein-Uhlenbeck processes.
Proof. The fact that Z 2[0;T ] is Gaussian and Markovian satisfying (42) follows from Corollary 1 with (38) and (40)-(41) plugged into (26) and (27), so that the density of the probability distribution for Z t1 ; ::: The nN nN corresponding covariance matrix is then of the form C I N with I N the identity matrix in R N , where C 1 is tridiagonal and obtained by identi…cation of the quadratic form in the argument of the above exponentials when N = 1. This gives for k = 2; :::; n 1; e (tn t n 1 ) sinh( (tn tn 1)) for k = n (the second line not being there when n = 2), and for k = 2; :::; n: Consequently, using for instance the analytical inversion formulae in [9], or by direct veri…cation, we obtain for all k; l 2 f1; :::; ng ; so that (43) eventually holds. Now, let us consider the forward Itô integral equation where W 2[0;T ] is a given Wiener process in R N , and where Z 0 is distributed according to (42) and independent of W 2[0;T ] . It is well known that the solution process X 2[0;T ] to (44) is centered Gaussian with covariance (43) (see, e.g., Section 5.6 in Chapter 5 of [11]), so that Ornstein-Uhlenbeck process with the stated properties.
Next, we show that if we require instead Z 0 to be a given point in R N , the solution process to (44) is also a particular Bernstein process. We do this in the simplest case where Z 0 is the origin, by considering the forward-backward system and with the Dirac measure. The corresponding Markovian measure (14) is then determined by respectively. We then have the following result: The Bernstein process Z 2[0;T ] associated with (45)-(46) in the sense of Corollary 1 is a Gaussian and Markovian process such that Furthermore we have for all s; t 2 [0; T ] and all i; j 2 f1; :::; N g. Thus, Z 2[0;T ] identi…es in law with a process whose components are all independent, one-dimensional non-stationary Ornstein-Uhlenbeck processes conditioned to start at the origin.
Proof. As in the proof of Proposition 1, the …rst part of the statement including (49)-(51) follows from the appropriate substitutions into the formulae of Corollary 1. Furthermore, the matrix C 1 resulting from the identi…cation of the quadratic form in the Gaussian density of Z t1 ; :::; Z tn is the same as that in Proposition 1, with the exception of C 1 1;1 which now reads : Inverting again we eventually obtain for all k; l 2 f1; :::; ng ; so that (52) holds. Therefore, Z 2[0;T ] is indeed a Ndimensional Ornstein-Uhlenbeck process with the stated properties. Remarks.
(1) It is equally easy to condition the Ornstein-Uhlenbeck process so that it ends at the origin of R N when t = T . The underlying Bernstein procesŝ Z 2[0;T ] is then simply determined by swaping the initial-…nal conditions in (45)  (2) Whereas the Bernstein process of Proposition 1 is stationary, that of Proposition 2 is not. This is intuitively understandable, as some kind of non trivial dynamics ought to be necessary to steer the process from a deterministic state at t = 0 to a Gaussian distribution at the end of its journey.
Of course, there is an almost unlimited number of possibilities of getting various Bernstein processes in the above manner, just by choosing ' 0; and T; appropriately. We complete this section by constructing yet another process which shares many properties of a Markovian bridge. For this, we consider the forward-backward system (3)-(4) with Dirac measures concentrated at the origin and at a given point a 2 R N as initial-…nal data, namely, and where m = 2 sinh ( T ) and v (x; t) = n exp and Then the following result is valid: Proposition 3. The Bernstein process Z 2[0;T ] associated with (53)-(54) in the sense of Corollary 1 is a Gaussian and Markovian process such that for each t 2 (0; T ) and every E 2 B N , where and Proof. We begin by proving (60). Using (55), (56) and (59) we …rst have after regrouping terms, and furthermore from (58). The substitution of (66)-(67) into (65) then leads to Now, for the numerator of the argument in the second exponential of (68) we have by virtue of (61). Therefore, taking (62) and (69) into account in (68) we get following the cancellation of two exponential factors, which gives the desired result according to (27).
As for the proof of (63), we remark that (28) and (29) lead to for every E 2 B N , respectively, which immediately imply the claim. We now turn to the proof of (64), by noticing that in this case the …nitedimensional density in R nN is Therefore, for the tridiagonal matrix C 1 corresponding to the quadratic part when N = 1 we obtain for k = 2; :::; n 1; sinh( (T tn 1)) sinh( (T tn)) sinh( (tn tn 1)) for k = n (the second line still not being there if n = 2), and C 1 ;k;k 1 = C 1 ;k 1;k = sinh ( jt k t k 1 j) for k = 2; :::; n: Consequently, inverting again and using numerous relations among hyperbolic functions we eventually get for k l; which leads to (64) by standard arguments. Finally, we note that the curve : [0; T ] 7 ! R + 0 given by (62) is concave aside from satisfying (0) = (T ) = 0, and that it takes on the maximal value Thus, the process Z 2[0;T ] is indeed non-stationary with the stated properties.
Remark. We may dub the process Z 2[0;T ] of the preceding corollary a Bernstein bridge, which represents a random curve whose ends are pinned down at speci…ed points in space. We remark that the corresponding Gaussian law is no longer centered, unless a = o in which case the process materializes a Markovian loop which retains the main features of a Brownian loop. In fact, Z 2[0;T ] does reduce to a Brownian bridge in the limit ! 0 + since In the next section we introduce a new class of Bernstein processes which we can eventually relate to the so-called periodic Ornstein-Uhlenbeck process, and which is generated by a one-parameter family of non-Markovian probability measures.

A family of non-Markovian Bernstein processes and the periodic Ornstein-Uhlenbeck process
The Bernstein processes of this section are still de…ned from measures which are intimately tied up with problems of the form (3)-(4), but their …nite-dimensional distributions will be determined exclusively from Theorem 1. Instead of considering just one pair of parabolic problems such as (3)-(4), we …rst introduce an in…nite hierarchy of forward-backward systems of the form u(x; 0) = ' m;0; (x) = e 1 2 ( namely, one such pair for each m j 2 N and every j, where the h mj ; 's are the Hermite functions of Lemma 1. Accordingly, this means that we are considering as many pairs of such systems as is necessary to take into account the whole pure point spectrum of the elliptic operator on the right-hand side. We remark that (70)-(71) constitutes a generalization of (31)-(32), the latter system being associated with the bottom of the spectrum where m j = 0 for each j. Whereas the associated measures remain suitably normalized, the drawback is that they are no longer positive according to the following result: Proof. According to Proposition A.1 of the appendix we have for Mehler's kernel (38), where the series converges absolutely and uniformly for all x; y 2 R N . Moreover, since the tensor products N j=1 h mj ; provide an orthonormal basis of The fact that the above measures are signed prevents one from applying directly the general results of Section 2 to (70)-(71) with a …xed m 6 = 0, as it would prevent one from applying the main result of [2] brie ‡y discussed at the end of Section 2. However, we can still save the day by constructing a oneparameter family of bona …de probability measures from all the m; 's. Indeed the following result is valid, where we call a measure non-Markovian whenever (22) does not hold: and which disintegrate into a statistical mixture of the m; 's. In other words, for each > 0, each > 0 and every m 2 N N there exist numbers p m; ; > 0 such that^ In fact, it is su¢ cient to takê ; (x; y) = (2 (cosh( ( + 1) T ) 1)) N 2 g (x; T; y)g (x; T; y): Proof. From the series expansion (75) and the orthogonality relations (76) we have Z by summing the underlying geometric series, from which we obtain (77) as a consequence of (78) and the identity 4 sinh 2 ( + 1) T 2 = 2 (cosh( ( + 1) T ) 1) : As for the second statement of the lemma, let us de…ne the sequence of numbers p m; ; := (2 (cosh( ( + 1) T ) 1)) Summing as above we get X according to (78), which is the desired conclusion.
Remark. Strictly speaking, the measure^ ; does not exist for = 0 but the limit ;+ (x; y) := lim !0+^ ; (x; y) = (2 (cosh( T ) 1)) N 2 g (x; T; y) (x y) (80) does, by virtue of the fact that g is Green's function associated with the partial di¤erential equation in (31). Said di¤erently, Lemma 3 and its proof remain valid for the measure^ ;+ associated with (80) since we have T ] associated with the measure^ ;+ identi…es in law with a stationary process whose components are all independent, one-dimensional and periodic Ornstein-Uhlenbeck processes.

22
for all k; l 2 f1; :::; ng or, equivalently, so that (83) eventually holds. Let us now consider the case = 0, namely, the case corresponding to the measure^ ;+ de…ned from (80), for which (82) for all i; j 2 f1; :::; N g, respectively. Let us also consider the forward Itô integral equation with periodic boundary conditions rather than (44). It is known from Theorem 2.1 in [14] (see also [17] or Section 5 in [18] for the case N = 1) that the solution to (86) can be written out explicitly and de…nes a non-Markovian centered Gaussian stationary process, namely, the so-called periodic Ornstein-Uhlenbeck process given by whose variance and covariance are given by (84) and (85), respectively. Therefore, Z ;+ 2[0;T ] identi…es in law with that process. Remarks.
(1) It is of course also a posteriori clear that the processes of Proposition 4 are not Markovian since the covariances (83) do not factorize as the product of a function of s times a function of t, in contrast to all the cases investigated in Section 3. Furthermore, a very di¤erent way of understanding such factorization properties in the Markovian case was put forward in Section 6 of [13], where the covariances were written as the product of two linearly independent solutions to some suitable Sturm-Liouville problems.
(2) Problem (86) is part of a more general class of linear stochastic di¤erential equations that were investigated by several authors, including [14], [16]- [18] and some of the references therein. In this context we ought to mention an analysis of the law of the solution to (86) in one dimension carried out in Section 5 of [18], which establishes a relation with Bernstein processes whose state space is one-dimensional. The main tool used there is a formula of integration by parts proved directly on the underlying in…nite-dimensional path space by means of Malliavin's calculus. This is in sharp contrast to the method we have developed in this section, as we have …rst constructed a one-parameter family of non-Markovian, N -dimensional Bernstein processes associated with the in…nite system (70)-(71), which we have only a posteriori identi…ed with the solution to (86) when = 0.
(3) While the processes Z ; >0 2[0;T ] materialize a one-parameter family of non-Markovian random curves in R N , they might also occur naturally in completely di¤erent contexts, as the limiting process Z ;+ 2[0;T ] does. Thus, this process is quite relevant to the mathematical investigation of certain quantum systems in equilibrium with a thermal bath since it identi…es with the Gaussian process of mean zero used in Theorem 2.1 of [8] to compute the expectations of some relevant physical quantities in statistical mechanics. Indeed, their covariances coincide since the equality 5 Appendix: a series expansion for Mehler' s Ndimensional kernel The notation in this appendix is, of course, the same as in the preceding sections. The expansion in question is the following: Proposition A1. We have where the series converges absolutely for every t 2 (0; T ] and uniformly for all x; y 2 R N . Furthermore, (88) is indeed Green's function associated with the partial di¤ erential equations in (31).
Proof. We …rst prove the result for the one-dimensional case N = 1, namely, 2 sinh ( t) valid for every 2 ( 1; +1), which allows one to express the probability density of two jointly Gaussian variables as a power series in the correlation parameter (see, e.g., [20]). By using the scaled Hermite functions (33) instead, we obtain so that in order to prove (89) we only need to identify . To this end we …rst compare the argument of the exponential on the left-hand side of (89) with that of (90), which gives the two conditions The substitution of (93) into (90) then gives (89) after some simple algebraic manipulations. Now, using (89) we obtain X in the sense of distributions since the N j=1 h mj ; 's constitute a complete orthonormal system in L 2 R N ; C . Consequently, g is indeed Green's function associated with the partial di¤erential equation in (31).
Remark. Of course, the semi-group composition law g (x; s + t; y) = Z R N dzg (x; s; z)g (z; t; y) valid for all x; y 2 R N is inherent in the fact that g is Green's function for (31), but follows most directly from the series expansion (88) and the orthogonality relations Z Furthermore, we also observe that (88) brings out the entire pure point spectrum of Lemma 1 in the argument of the exponential, a fact that is crucial in our construction of Section 4.