Martin Larsson
email: mar-tin.larsson@math.ethz.ch

Sergio Pulido
email: ser-gio.pulidonino@ensiie.fr.

Polynomial diffusions on compact quadric sets *

Keywords: Polynomial diffusion, sums of squares, biquadratic forms, stochastic invariance, pathwise uniqueness, smooth densities. MSC2010 subject classifications: 60J60, 60H10, 11E25

Polynomial processes are defined by the property that conditional expectations of polynomial functions of the process are again polynomials of the same or lower degree. Many fundamental stochastic processes, including affine processes, are polynomial, and their tractable structure makes them important in applications. In this paper we study polynomial diffusions whose state space is a compact quadric set. Necessary and sufficient conditions for existence, uniqueness, and boundary attainment are given. The existence of a convenient parameterization of the generator is shown to be closely related to the classical problem of expressing nonnegative polynomials-specifically, biquadratic forms vanishing on the diagonal-as a sum of squares. We prove that in dimension d ≤ 4 every such biquadratic form is a sum of squares, while for d ≥ 6 there are counterexamples. The case d = 5 remains open. An equivalent probabilistic description of the sum of squares property is provided, and we show how it can be used to obtain results on pathwise uniqueness and existence of smooth densities.

Introduction

Many fundamental stochastic processes appearing in probability theory are polynomial, meaning that conditional expectations of polynomial functions of the process again have polynomial form; see (1.4) below. Examples include Brownian motion, geometric Brownian motion, Ornstein-Uhlenbeck processes, squared Bessel processes, Jacobi processes, Lévy processes, as well as large classes of multidimensional generalizations, such as affine processes and Fisher-Wright diffusions. Polynomial processes have been studied in various degrees of generality by several authors, for instance Wong (1964), [START_REF] Mazet | Classification des semi-groupes de diffusion sur R associé à une famille de polynômes orthogonaux[END_REF], [START_REF] Forman | The Pearson diffusions: a class of statistically tractable diffusion processes[END_REF], [START_REF] Cuchiero | Affine and Polynomial Processes[END_REF], [START_REF] Cuchiero | Polynomial processes and their applications to mathematical finance[END_REF], [START_REF] Bakry | Symmetric diffusions with polynomial eigenvectors[END_REF], [START_REF] Bakry | Orthogonal polynomials and diffusion operators[END_REF], [START_REF] Filipović | Polynomial diffusions and applications in finance[END_REF]. Polynomial processes are also important in applications. In mathematical finance, the polynomial property is useful to build tractable and flexible pricing models in a variety of situations; see for instance [START_REF] Zhou | Itô conditional moment generator and the estimation of short-rate processes[END_REF], [START_REF] Delbaen | An interest rate model with upper and lower bounds[END_REF], [START_REF] Larsen | Diffusion models for exchange rates in a target zone[END_REF], [START_REF] Gourieroux | Multivariate Jacobi process with application to smooth transitions[END_REF], [START_REF] Cuchiero | Polynomial processes and their applications to mathematical finance[END_REF], Filipović et al. (2016[START_REF] Filipović | Linear-rational term structure models[END_REF], [START_REF] Ackerer | The Jacobi stochastic volatility model[END_REF], and [START_REF] Filipović | Polynomial diffusions and applications in finance[END_REF]. Every affine diffusion is polynomial; however, non-deterministic affine diffusions do not admit compact state spaces, see [START_REF] Krühner | Classification of affine processes with compact state space[END_REF], which may be a drawback in applications. In the present paper we consider polynomial diffusions on (possibly solid) compact quadric sets. This is a natural class of simple state spaces that nonetheless exhibit a rich mathematical structure.

The state space E is defined as follows. Fix d ∈ N and let

E = {x ∈ R d : p(x) ≥ 0} or E = {x ∈ R d : p(x) = 0}
for some p ∈ Pol 2 such that E is compact and nondegenerate (nonempty and not a point).

Here Pol k denotes the vector space of polynomial functions on R d of total degree less than or equal to k. Up to an affine transformation, E is either the closed unit ball B d or the unit sphere S d-1 ; see Section 2. Next, consider continuous maps a : R d → S d and b : R d → R d such that a ij ∈ Pol 2 and b i ∈ Pol 1 for all i, j ∈ {1, . . . , d},

where S d is the set of d × d symmetric matrices. We are interested in E-valued weak solutions to stochastic differential equations of the form

dX t = b(X t ) dt + σ(X t ) dW t , (1.2)
where σ : R d → R d×n for n ∈ N is a continuous map with σσ ⊤ = a on E, and W is an ndimensional Brownian motion. In particular, we are interested in how probabilistic properties of (1.2) are connected to algebraic properties of the coefficients a, b and the state space E. Diffusions whose coefficients satisfy (1.1) are called polynomial. The reason is that (1.1) holds if and only if the associated generator G , given by

G f (x) = 1 2 Tr(a(x)∇ 2 f (x)) + b(x) ⊤ ∇f (x), (1.3)
preserves polynomials in the sense that G Pol k ⊆ Pol k holds for all k ∈ N. If X is a polynomial diffusion with generator G , a simple argument based on Itô's formula shows that conditional expectations of polynomials have a particularly simple form: For any q ∈ Pol k , one has

E[q(X T ) | X t ] = H(X t ) ⊤ e (T -t)G k q, (1.4)
where H(x) = (h 1 (x), . . . , h N k (x)) with N k = dim Pol k is a basis for Pol k , and G k ∈ R N k ×N k and q ∈ R N k are the corresponding coordinate representations of G | Pol k and q, respectively;

see Filipović and Larsson (2016, Theorem 3.1) or Cuchiero et al. (2012, Theorem 2.7) for details.

Let us summarize the main results of the present paper. Conditions for existence and uniqueness in law of solutions to (1.2) can be extracted from known results in the literature, and we briefly review these issues in Section 2. A related question in the case E = B d is whether the interior of the state space is stochastically invariant, to which we provide a complete answer; see Proposition 2.2. Next, in Section 3 we undertake a more detailed investigation of how to parameterize all weak solutions to (1.2). Theorem 3.3 gives a general representation, modulo the requirement that the diffusion matrix a(x) be positive semidefinite on the state space. Characterizing positivity is a difficult problem, which translates into the algebraic question of describing when certain biquadratic forms BQ(x, y) are nonnegative. This is closely related to the representability of BQ(x, y) as a sum of squares of polynomials. Such a representation always exists if d ≤ 4, but not if d ≥ 6; see Theorem 3.5. The case d = 5 remains open. In Section 4 we study the probabilistic consequences of the existence of a sum of squares representation. We show that such a representation exists if and only if (1.2) can be replaced by a certain much more structured SDE; see Theorem 4.1. Solutions to this SDE can sometimes be shown to have the pathwise uniqueness property; see Corollary 4.4 and Theorem 4.6. Moreover, it becomes possible to make detailed assertions regarding the existence of smooth densities by relying on the classical Hörmander condition; see Theorem 4.9 and its corollaries. Finally, in Section 5, we collect the algebraic developments needed for the results presented in Section 3. Here an important role is played by the so-called Plücker relations from algebraic geometry.

The following notation will be used throughout this paper: Elements of R d are viewed as column vectors. Hom k denotes the subspace of Pol k consisting of homogeneous polynomials of total degree exactly k. For two matrices A and B of compatible size, we write A, B = Tr(A ⊤ B) for the trace product. The convex cone of positive semi-definite matrices in S d is denoted S d + , and we often write A 0 to signify that the symmetric matrix A is positive semidefinite. The space of skew-symmetric d × d matrices is written Skew(d), and we let Skew(2, d) denote the subset of rank-two elements. Let e i = (0, . . . , 0, 1, 0, . . . , 0) ⊤ be the ith canonical unit vector in R d . We denote by D 1 , . . . , D m , with m = dim Skew(d) = d 2 , the elementary skew-symmetric matrices S ij = e i e ⊤ je j e ⊤ i listed in lexicographic order. That is, (D 1 , . . . , D m ) = (S 12 , . . . , S 1d , S 23 , . . . , S 2d , . . . , S d-1,d ).

(1.5)

Existence, uniqueness, and boundary attainment

We consider polynomial diffusions whose state space is either the unit ball B d or the unit sphere S d-1 . Up to an affine change of coordinates, this covers all nondegenerate compact quadric sets. Indeed, consider a compact set E = {x ∈ R d : p(x) ≥ 0} that is nonempty and not a point, where p ∈ Pol 2 . Then ∇ 2 p is constant and negative definite, so by completing the square one sees that E is an affine transformation of B d . Similarly, the set {x ∈ R d : p(x) = 0} is an affine transformation of S d-1 . Note also that the condition (1.1) is invariant under affine transformations.

The following sets play an important role in the description of polynomial diffusions on B d and S d-1 : C = {c : R d → S d : c ij ∈ Hom 2 for all i, j, and c(x)x ≡ 0},

C + = {c ∈ C : c(x) ∈ S d
+ for all x}.

(2.1)

We have the following crude characterization theorems, most of which follows fairly directly from known results. We start with the unit ball. (ii) The coefficients a and b are of the form

a(x) = (1 -x 2 )α + c(x), b(x) = b + Bx, (2.2) for some α ∈ S d + , c ∈ C + , b ∈ R d , and B ∈ R d×d such that b ⊤ x + x ⊤ Bx + 1 2 Tr(c(x)) ≤ 0 for all x ∈ S d-1 . (2.3)
Proof. It is proved in Filipović and Larsson (2016, Proposition 7.1) that (ii) is equivalent to the conditions

a(x) 0 on E, a(x)x = 0 on S d-1 , G p(x) ≥ 0 on S d-1 , (2.4)
where p(x) = 1-x 2 and G is given by (1.3). That (i) implies (2.4) follows from Theorem 6.1 in [START_REF] Filipović | Polynomial diffusions and applications in finance[END_REF]. The reverse implication follows from Theorem 6.3 and Proposition 7.1 in [START_REF] Filipović | Polynomial diffusions and applications in finance[END_REF] when the inequality in (2.3) is strict. For the general case one can, for instance, invoke Example 2.5 and Theorem 2.2 in Da [START_REF] Da Prato | Stochastic viability of convex sets[END_REF]. Alternatively, one can use (2.4) to verify the positive maximum principle directly, and then apply Ethier and Kurtz (2005, Theorem 4.5.4).

If d = 1, the polynomial diffusions on B 1 = [-1, 1] are simply the Jacobi processes dX t = (b + BX t )dt + σ 1 -X 2 t dW t . In this case C + = C = {0}.
In contrast, for d > 1 the set C + is non-trivial, and its elements c induce diffusive fluctuations tangentially to the sphere due to the property c(x)x ≡ 0. For example, our results imply that for d = 2 one can always represent polynomial diffusions on the unit ball as solutions to the SDE

dX t = (b + BX t )dt + 1 -X 2 1t -X 2 2t σ dW 1 t dW 2 t + ν X 2t -X 1t dW 3 t for some b ∈ R 2 , B ∈ R 2×2 , σ ∈ R 2×2 , and ν ∈ R.
Here the presence of the term involving the third Brownian motion W 3 gives rise to tangential diffusion. In higher dimensions, explicit parameterizations become more difficult to obtain. Indeed, Theorem 2.1 provides no explicit description of C + , and a significant part of the present paper is devoted to studying this set in detail. This is the topic Section 3.

It is frequently of interest to know whether a solution X to (1.2) that starts in the interior of B d remains in the interior. This question of boundary attainment is resolved by the following result, which is a refinement of Theorem 5.7 in [START_REF] Filipović | Polynomial diffusions and applications in finance[END_REF]. In its statement, we assume that a(x) and b(x) satisfy (1.1) along with the two equivalent conditions in Theorem 2.1. In particular, (2.3) is satisfied. Let P x denote the law of the solution X to (1.2) with initial condition x ∈ B d . A measurable subset D ⊆ B d is said to be invariant for X if, for every x ∈ D, one has P x (X t ∈ D for all t ≥ 0) = 1.

Proposition 2.2. The interior

B d \ S d-1 is invariant for X if and only if b ⊤ x + x ⊤ (B + α)x + 1 2 Tr(c(x)) ≤ 0 for all x ∈ S d-1 .
(2.5)

Proof. Let p(x) = 1 -x 2 and note that a(x)∇p(x) = p(x)h(x) where h(x) = -2αx. Furthermore, 2 G p(x) -h(x) ⊤ ∇p(x) = -2p(x) Tr(α) -4 b ⊤ x + x ⊤ (B + α)x + 1 2 Tr(c(x)) .
(2.6) If (2.5) fails, then the right-hand side of (2.6) is strictly negative for some x ∈ S d-1 . Also, G p(x) ≥ 0 due to (2.3). Thus Filipović and Larsson (2016, Theorem 5.7(iii)) implies that B d \ S d-1 is not invariant for X (a look at the proof of that result shows that its assumption that {t : X t ∈ S d-1 } has Lebesgue measure zero is not needed when X 0 / ∈ S d-1 ). Suppose now (2.5) holds, and let

X 0 = x 0 ∈ B d \ S d-1 . Define τ = inf{t : X t ∈ S d-1 }. We must show that P x 0 (τ < ∞) = 0. For t ∈ [0, τ ), Itô's formula yields log p(X t ) = log p(x 0 ) + t 0 2 G p(X s ) -h(X s ) ⊤ ∇p(X s ) 2p(X s ) ds + t 0 ∇p(X s ) ⊤ σ(X s ) p(X s ) dW s .
Suppose we can find a constant κ 1 > 0 such that 2 G p(x)h(x) ⊤ ∇p(x) ≥ -2κ 1 p(x) for all x ∈ B d .

(2.7)

Then log p(X t ) + κ 1 t is a local submartingale on [0, τ ), bounded from above on bounded time intervals. The submartingale convergence theorem then implies that lim t→T ∧τ log p(X t ) exists in R for any deterministic T < ∞, so that T < τ . We deduce that τ = ∞, as desired. This is the same "McKean's argument" as in the proof of Filipović and Larsson (2016, Theorem 5.7(i)-(ii)). A suitable version of the submartingale convergence theorem can found in, for instance, Larsson and Ruf (2014, Lemma 4.14). It remains to argue (2.7). Since Tr(c(x)) is a quadratic form in x, we can find

C ∈ S d such that Tr(c(x)) = x ⊤ Cx. Define Σ = -α -1 2 (B + B ⊤ + C) ∈ S d .
In view of (2.6), condition (2.7) then becomes

x ⊤ Σx -b ⊤ x ≥ -κ(1 -x 2 ) for all x ∈ B d , (2.8)
where the constant κ is related to κ 1 by 2κ = κ 1 -Tr(α). On the other hand, the assumption (2.5) written in terms of Σ becomes

x ⊤ Σx -b ⊤ x ≥ 0 for all x ∈ S d-1 .
(2.9)

We claim that (2.9) in fact implies that (2.8) holds for κ = b . For x = 0 the statement is obvious. For x = 0, define x = x/ x and apply (2.9) to get x ⊤ Σx ≥ b ⊤ x. Multiplying both sides by x 2 , rearranging, and applying the Cauchy-Schwartz inequality gives

x ⊤ Σx -b ⊤ x ≥ -b ⊤ x(1 -x ) ≥ -b x (1 -x ) ≥ -b (1 -x 2 ),
as required.

The case of the sphere is simpler. In particular, the question of boundary attainment becomes vacuous.

Theorem 2.3. Assume a and b satisfy (1.1) and the state space is E = S d-1 . The following conditions are equivalent:

(i) There exists a continuous map σ : R d → R d×d with σσ ⊤ = a on S d-1 such that (1.2) admits an S d-1 -valued weak solution for any initial condition in S d-1 .

(ii) The coefficients a and b satisfy a = c on S d-1 for some c ∈ C + , and b(x) = Bx for some B ∈ R d×d , where 2x ⊤ Bx + Tr(c(x)) ≡ 0.

(2.10)

Proof. Filipović and Larsson (2016, Theorem 6.3) yields that (ii) implies (i). For the reverse implication, one argues as in the proof of Theorem 2.1 that (i) implies Theorem 2.1(ii), but with equality in (2.3). This in turn implies that a(x) = c(x) for x ∈ S d-1 and that b = 0, whence (2.10) holds.

Theorems 2.1 and 2.3 do not make any uniqueness statements. However, since the state space E is compact, the polynomial property implies that uniqueness in law for (1.2) always holds. More precisely, we have the following result: Lemma 2.4. Assume a and b satisfy (1.1), let σ : R d → R d×n , n ∈ N, be a continuous map with σσ ⊤ = a on E, and let x ∈ E. Let X be an E-valued weak solution to (1.2) with initial condition X 0 = x. Then the law of X is uniquely determined by a, b, and x.

Proof. See Filipović and Larsson (2016, Corollary 5.2). The argument relies on the fact that a, b, and x uniquely determine all joint moments of all finite-dimensional marginal distributions of X due to (1.4); see Filipović and Larsson (2016, Corollary 3.2) for details. Due to the compactness of E, this in turn determines the law of X.

Remark 2.5. So far no assertions have been made regarding pathwise uniqueness of solutions to (1.2). Pathwise uniqueness depends on the choice of σ, and the existence of a suitable σ is a delicate question in general. See Spreij and Veerman (2012, Remark 2.3) for an instructive discussion in connection with affine diffusions on parabolic sets. Sometimes pathwise uniqueness is immediate: If a(x) is positive definite on the interior B d \ S d-1 then σ(x) = a(x) 1/2 is locally Lipschitz there, which by standard arguments yields pathwise uniqueness up to the first hitting time of the boundary. In Section 4.2 we discuss other situations where pathwise uniqueness can be established.

Tangential diffusion, biquadratic forms, and sums of squares

In this section we undertake a detailed analysis of the linear space C and the convex cone C + appearing in Theorems 2.1 and 2.3, and whose role is to generate diffusive movements tangentially to S d-1 . It tuns out that a complete description is difficult to obtain in general; see Theorem 3.5 below. Fortunately, a partial characterization suitable for applications is within reach. We start with some general remarks, introducing some concepts and notation along the way.

Any map c : R d → S d whose components lie in Hom 2 induces a map BQ :

R d × R d → R via the expression BQ(x, y) = y ⊤ c(x)y. (3.1)
This map is a biquadratic form: a quadratic form in x for each fixed y, and a quadratic form in y for each fixed x. If c ∈ C + , then additionally BQ(x, y) ≥ 0 and BQ(x, x) = 0 for all (x, y). The following lemma states that the converse is true as well.

Lemma 3.1. The set C + is in one-to-one correspondence with the set of all nonnegative biquadratic forms vanishing on the diagonal. The correspondence is given by (3.1).

Proof. Any biquadratic form BQ(x, y) in d + d variables induces a map c : R d → S d with components in Hom 2 by the formula c ij (x) = ∂ 2 y i y j BQ(x, y) for i = j, and c ii (x) = ∂ 2 y i y i BQ(x, y)/2. Furthermore, if BQ(x, y) ≥ 0 for all (x, y), then c(x) 0 for all x. In this case, BQ(x, x) = 0 implies c(x) 1/2 x 2 = x ⊤ c(x)x = 0, and hence c(x)x = 0, for all x. Thus c ∈ C + .

Remark 3.2. Note that the set C does not correspond one-to-one with the set of all (not necessarily nonnegative) biquadratic forms vanishing on the diagonal. The reason is that x ⊤ c(x)x ≡ 0 does not imply c(x)x ≡ 0 in the absence of positive semidefiniteness. An example is the map

c(x) = -2x 1 x 2 x 2 1 x 2 1 0 , which satisfies x ⊤ c(x)x ≡ 0 but not c(x)x ≡ 0.
It will become apparent that the relevant stepping stone toward an understanding of C + is the set C , not the set of biquadratic forms vanishing on the diagonal.

Our first main result is a characterization of C . Let m = d 2 = dim Skew(d). For any H = (h pq ) ∈ S m , define a map c H by c H (x) = m p,q=1 h pq D p xx ⊤ D ⊤ q , (3.2)
where D 1 , . . . , D m are given by (1.5). It is clear that c H ∈ C , and it turns out that every element of C is in fact of this form. Naively one might then conjecture that the dimension of C equals dim S m = m+1 2 , the number of free parameters h pq appearing in (3.2). However, this turns out to be incorrect. Indeed, there exist linear relations among the maps D p xx ⊤ D ⊤ q in (3.2); to capture them, we consider the linear space

K = {H ∈ S m : c H (x) ≡ 0} . (3.3)
The proof of the following theorem is provided in Section 5.1.

Theorem 3.3. The set C and its dimension are given by

C = {c H : H ∈ S m } and dim C = 2 d 4 + 3 d 3 + d 2 = d 2 (d 2 -1) 12 , (3.4)
where m = d 2 and we set

d k = 0 for k > d. Moreover, dim K = dim S m -dim C = d 4 .
The linear space K has several interesting properties that are used in the proofs of our subsequent results. This is discussed in detail in Section 5.2. In a nutshell, the situation is the following: Each H ∈ S m naturally corresponds to a quadratic form on the space of skewsymmetric matrices, which when evaluated at A = xy ⊤yx ⊤ ∈ Skew(2, d) yields precisely BQ(x, y); see (5.1). Now, it turns out that K admits a set of basis vectors that can be identified with the so-called Plücker polynomials from algebraic geometry. The zero set of these polynomials is isomorphic to Skew(2, d). As a consequence, the quotient S m /K can be identified with the set of quadratic forms acting on Skew(2, d). Thus, in view of Theorem 3.3, each element of C corresponds to a unique quadratic form on Skew(2, d). Finally, we mention that it is possible to explicitly write down a basis for K ; see Remark 5.4.

We now turn to the delicate question of positivity: c(x) 0 for all x, or equivalently BQ(x, y) ≥ 0 for all (x, y). A simple example where this holds is c(x) = A xx ⊤ A ⊤ , where A ∈ Skew(d). More generally, by taking conic combinations, elements of the form

c(x) = A 1 xx ⊤ A ⊤ 1 + • • • + A m xx ⊤ A ⊤ m , A 1 , . . . , A m ∈ Skew(d), (3.5)
lie in C + . The corresponding biquadratic form is given by

BQ(x, y) = (y ⊤ A 1 x) 2 + • • • + (y ⊤ A m x) 2 ,
which has the important property that it can be written as a sum of squares of polynomials (indeed, of quadratic forms). The converse is also true: whenever BQ(x, y) is a sum of squares, the corresponding c(x) is of the form (3.5). Specifically, one has the following:

Lemma 3.4. Let c ∈ C , m = d 2 .
The following conditions are equivalent. (i) BQ(x, y) = y ⊤ c(x)y is a sum of squares of polynomials.

(ii) c = c H for some H ∈ S m + . (iii) c(x) = m p=1 A p xx ⊤ A ⊤ p for some A 1 , . . . , A m ∈ Skew(d).
Proof. (iii) =⇒ (i): Obvious. (i) =⇒ (ii): We first show that (i) implies that there exist m ′ ∈ N and A

1 , . . . , A m ′ ∈ Skew(d) such that y ⊤ c(x)y = m ′ p=1 (y ⊤ A p x) 2 . To this end, suppose y ⊤ c(x)y = f 1 (x, y) 2 + • • • + f m ′ (x,
y) 2 for some polynomials f p , p = 1, . . . , m ′ , where m ′ ∈ N. By setting x = y = 0, we see that f p (0, 0) = 0 for all p. Since also f p (sx, y) 2 ≤ y ⊤ c(sx)y = s 2 y ⊤ c(x)y, each f p is linear in x. Similarly, each f p is linear in y. It follows that f p (x, y) = y ⊤ A p x for some matrix A p ∈ R d×d . Finally, (x ⊤ A p x) 2 ≤ x ⊤ c(x)x ≡ 0, whence A p is skew-symmetric. This proves the claimed representation. Now, since D 1 , . . . , D m form a basis for Skew(d), for each p there exists

u p = (u 1 p , . . . , u m p ) ∈ R m such that A p = m i=1 u i p D i . Thus, y ⊤ c(x)y = y ⊤ m ′ p=1 A p xx ⊤ A ⊤ p y = y ⊤ m i,j=1 h ij D i xx ⊤ D ⊤ j y,
where

H = (h ij ) ∈ S m is given by H = u 1 u ⊤ 1 + • • • + u m ′ u ⊤ m ′
and thus is positive semidefinite. We deduce (ii).

(ii) =⇒ (iii):

Writing H = u 1 u ⊤ 1 + • • • + u m u ⊤ m
for some vectors u p with components (u 1 p , . . . , u m p ) and substituting into (3.2) yields

c H (x) = m p=1 (u 1 p D 1 + • • • + u m p D m ) xx ⊤ (u 1 p D 1 + • • • + u m p D m ) ⊤ , which is of the desired form with A p = m i=1 u i p D i .
Lemma 3.4 shows that those c ∈ C + whose associated biquadratic form is a sum of squares have a nice parameterization in terms of positive semidefinite matrices H. One is thus naturally led to ask whether every nonnegative biquadratic form vanishing on the diagonal can be written as a sum of squares. Our next main result addresses this question. Its proof is provided in Sections 5.3 and 5.4, relying on the material developed in Section 5.2. The proof also depends on a known result (Lemma 5.5) on existence of low-rank elements in the intersection of S m + with certain affine subspaces of S m . (ii) If d ≥ 6, then there exists a nonnegative biquadratic form in d + d variables vanishing on the diagonal that is not a sum of squares. Equivalently, there exists c ∈ C + that is not of the form c = c H for some H ∈ S m + .

Remark 3.6. In [START_REF] László | On sum of squares decomposition for a biquadratic matrix function[END_REF] it was observed that the conclusion of Theorem 3.5(i) holds for d ≤ 3, while the case d ∈ {4, 5} was left open; see also [START_REF] László | Sum of squares representation for the Böttcher-Wenzel biquadratic form[END_REF]. The case d = 3 can also be deduced from [START_REF] Quarez | On the real zeros of positive semidefinite biquadratic forms[END_REF]. We have not been able to determine whether there exists a nonnegative biquadratic form in 5+5 variables, vanishing on the diagonal, that is not a sum of squares. The counterexample that establishes Theorem 3.5(ii) comes from [START_REF] László | On sum of squares decomposition for a biquadratic matrix function[END_REF].

For the benefit of the reader we recap this construction in Section 5.4 using the setting and notation of the present paper. The study of sum of squares representations of nonnegative biquadratic forms (not necessarily vanishing on the diagonal) goes back to [START_REF] Choi | Positive semidefinite biquadratic forms[END_REF].

4 Consequences of the sum of squares property

Throughout this section, we assume that a(x) and b(x) satisfy (2.2)-( 2.3) for some fixed

α ∈ S d + , c ∈ C + , b ∈ R d , B ∈ R d×d
, and let BQ(x, y) = y ⊤ c(x)y be the associated nonnegative biquadratic form. Theorem 2.1 guarantees that a B d -valued weak solution to (1.2) exists for a suitable choice of σ and for any initial condition x ∈ B d . Denote its law by P x , and observe that it is uniquely determined by a, b, and x; see Lemma 2.4. In this section we discuss consequences of the existence of a sum of squares representation of BQ(x, y).

SDE representation

Our first result shows that P x can be represented as the law of the solution to a certain structured SDE if and only if BQ(x, y) is a sum of squares. Before stating it, we give the following definition:

Definition 4.1. A process Y is called a Skew(d)-valued correlated Brownian motion with drift if it is of the form Y t = A 0 t + A 1 W 1 t + • • • + A m W m t , (4.1)
where A 0 , . . . , A m ∈ Skew(d) and W = ( W 1 , . . . W m ) is an m-dimensional Brownian motion.

Theorem 4.2. The following conditions are equivalent:

(i) BQ(x, y) = y ⊤ c(x)y is a sum of squares of polynomials.

(ii) For each x ∈ B d , P x is the law of the unique B d -valued weak solution to the SDE In this case, B is related to B and A 0 , . . . , A m in (4.1) through

dX t = (b + BX t ) dt + 1 -X t 2 α 1/2 dW t + (• dY t )X t (4.2) with initial condition X 0 = x, where -B ∈ S d + , W = (W 1 , . . . , W d ) is a d-dimensional Brownian motion,
1 2 (B -B ⊤ ) = A 0 and 1 2 (B + B ⊤ ) = B - 1 2 m p=1 A ⊤ p A p . (4.3) Proof. (i) =⇒ (ii): Lemma 3.4 implies that c(x) = m p=1 A p xx ⊤ A ⊤ p (4.4)
for some A 1 , . . . , A m ∈ Skew(d). Define A 0 ∈ Skew(d) and B ∈ S d via (4.3). We claim that -B ∈ S d + . Indeed, writing the condition (2.3) in terms of B, and using that x ⊤ A 0 x = 0 by skew-symmetry, yields b ⊤ x + x ⊤ Bx ≤ 0 for all x ∈ S d-1 . (4.5) Inserting x and -x into this inequality and adding, one obtains x ⊤ Bx ≤ 0 for all x ∈ S d-1 , whence -B ∈ S d + as claimed. Next, writing the SDE (4.2) in Itô form yields

dX t = (b + BX t ) dt + σ(X t ) dW t d W t , where σ(x) = 1 -x 2 α 1/2 A 1 x • • • A m x ∈ R d×(d+m) . (4.6)
In view of (4.4) we have σ(x) σ(x (ii) =⇒ (i): Let σ(x) be given by (4.6). Since P x is the uniquely determined law of the solution to (4.2), one obtains

) ⊤ = (1-x 2 )α+c(x) = a(x
a(x) = σ(x) σ(x) ⊤ = (1 -x 2 )α + m p=1 A p xx ⊤ A ⊤ p .
Since a(x) = (1x 2 )α + c(x), it follows that (4.4) holds. Lemma 3.4 then implies (i).

The terms on the right-hand side of (4.2) have natural interpretations. Since B is symmetric with real non-positive eigenvalues, the drift has a mean-reverting component BX t in addition to the constant part b. The term involving W t induces diffusive motion along the eigenvectors of α, with quadratic variation that is scaled by the squared distance 1 -X t 2 to the boundary of B d . Finally, the term (• dY t )X t can be viewed as an infinitesimal random rotation of X t ; see also [START_REF] Gareth | Rolling with "slipping": I[END_REF] and [START_REF] Van Den Berg | Brownian motion on a hypersurface[END_REF]. Heuristically, the Stratonovich increment • dY t is a small random Skew-symmetric matrix that maps X t to an infinitesimal tangent vector of the ball with radius X t . We prefer using the Stratonovich form here, as it allows us to clearly separate tangential motion from radial motion in (4.2). If one were to switch to Itô form, the resulting drift correction would interfere with this interpretation. In particular, even for specifications where b = 0, B = α = 0, for which X is constant, an inward-pointing "drift" would appear in the Itô formulation. See also Corollary 4.4 below.

Remark 4.3. In the setting of Theorem 4.2, (4.3) implies that the necessary and sufficient condition (2.5) for boundary attainment reduces to

b ⊤ x + x ⊤ ( B + α)x ≤ 0 for all x ∈ S d-1 .
If X takes values in S d-1 , the sum of squares property of BQ(x, y) has stronger implications. In particular, X can now be constructed as the unique strong solution to a suitable SDE.

Corollary 4.4. Assume the coefficients a and b satisfy Theorem 2.3(ii). The following conditions are equivalent:

(i) BQ(x, y) = y ⊤ c(x)y is a sum of squares of polynomials.

(ii) For each x ∈ S d-1 , P x is the law of the S d-1 -valued unique strong solution to the SDE

dX t = (• dY t )X t (4.7)
with initial condition X 0 = x, where Y is a Skew(d)-valued correlated Brownian motion with drift.

(iii) The operator G = 1 2 Tr(a ∇ 2 ) + b ⊤ ∇ corresponding to the SDE (1.2) can be expressed in Hörmander form as

G = V 0 + 1 2 m p=1 V 2 p , (4.8)
where for p = 0, . . . , m, V p is the linear vector field given by V p (x) = A p x for some A p ∈ Skew(d).

Proof. (i) ⇐⇒ (ii) is deduced from the corresponding equivalence in Theorem 4.2. Indeed, the forward implication follows since b = 0, α = 0, and B = 0. Here the latter is due to (2.10) and (4.3), which give

x ⊤ Bx = x ⊤ Bx + 1 2 m p=1 x ⊤ A p A ⊤ p x = x ⊤ Bx + 1 2 Tr(c(x)) = 0, x ∈ R d .
Note that the form of c(x) follows from Lemma 3.4. For the converse, simply note that (ii) implies that Theorem 4.2(ii) is satisfied with b = 0, α = 0, and B = 0. This yields (i). The implication (ii) =⇒ (iii) is well-known, and follows from a brief calculation using the identity

V 2 p f (x) = Tr(∇ 2 f (x)A p xx ⊤ A ⊤ p ) -(A ⊤ p A p x) ⊤ ∇f (x). Finally, the implication (iii) =⇒ (i) is again a calculation showing that c(x) = A 1 xx ⊤ A ⊤ 1 + • • • + A m xx ⊤ A ⊤ m , which gives (i) via Lemma 3.4.
Example 4.5. The classical Brownian motion on the sphere is covered by Corollary 4.4. In particular, it is a polynomial diffusion. Indeed, by taking

A p = D p for p = 1, . . . , m, one obtains c(x) = i<j (e i e ⊤ j -e j e ⊤ i )xx ⊤ (e j e ⊤ i -e i e ⊤ j ) = (x ⊤ x)Id -xx ⊤ , so that a(x) = c(x) = Id -xx ⊤ on S d-1
. This is the orthogonal projection onto the tangent space of S d-1 at x. Setting A 0 = 0, we thus recover the following well-known quadratic SDE for Brownian motion on the sphere:

dX t = (Id -X t X ⊤ t ) • dW t .
The linear SDE in Corollary 4.4 is different, and was originally obtained by [START_REF] Gareth | Rolling with "slipping": I[END_REF]; see also [START_REF] Van Den Berg | Brownian motion on a hypersurface[END_REF].

Pathwise uniqueness

It is natural to ask whether polynomial diffusions on the unit ball can be realized as solutions to SDEs for which pathwise uniqueness holds. As mentioned in Remark 2.5, if a(x) is positive definite on the interior of B d , one obtains pathwise uniqueness up to the first hitting time of S d-1 by taking the positive semidefinite square root σ(x) = a(x) 1/2 . If BQ(x, y) is a sum of squares, more can be said. Indeed, in this case Theorem 4.2 yields pathwise uniqueness up to the first hitting time of S d-1 , regardless of whether or not a(x) is nonsingular. Moreover, if the state space is S d-1 , Corollary 4.4 shows that pathwise uniqueness always holds (still under the sum of squares assumption, of course).

Looking at the SDE (4.2) in Theorem 4.2, there may be some hope that pathwise uniqueness holds globally, not just up the first hitting time of S d-1 . We have not been able to prove this in general. Instead, we present a partial result relying on the method developed by DeBlassie ( 2004) for proving pathwise uniqueness for certain SDEs on the unit ball; see also [START_REF] Swart | Pathwise uniqueness for a sde with non-lipschitz coefficients[END_REF]. Specifically, we consider the following special case of (4.2):

dX t = -κX t dt + ν 1 -X t 2 dW t + A 0 X t dt + A 1 X t • d W 1 t + • • • + A m X t • d W m t , (4.9)
where A 0 , . . . , A m ∈ Skew(d), W = (W 1 , . . . , W d ) and W 1 , . . . , W m are independent Brownian motions, and, crucially, κ and ν are positive scalar constants. Within the polynomial class, this is a generalization of the equation considered by [START_REF] Swart | Pathwise uniqueness for a sde with non-lipschitz coefficients[END_REF] and DeBlassie ( 2004), who take the tangential components A 0 , . . . , A m to be zero. Note that even in this case, it is not known whether pathwise uniqueness holds for (4.9) for arbitrary values of κ and ν. The reason why DeBlassie ( 2004) method can be applied in the present situation is contained in the following calculation. Define Y t = 1 -X t 2 . Then, an application of Itô's formula and the change-of-variable formula for the Stratonovich integral yield

dY t = 2κ X t 2 dt -2ν Y t X ⊤ t dW t -d ν 2 Y t dt -2X ⊤ t A 0 X t dt -2 m p=1 X ⊤ t A p X t d W p t .
The skew-symmetry of the A p implies that the quadratic terms X ⊤ t A p X t all vanish, so that

dY t = 2κ X t 2 -d ν 2 Y t dt -2ν Y t X ⊤ t dW t . (4.10)
This no longer involves the matrices A p . Note that this crucially relies on the linearity in X t of the last m + 1 terms of (4.9). This property is specific for the sum of squares case (4.2), and is in general not present in (1.2).

Theorem 4.6. Assume κ/ν 2 > √ 2 -1. Then pathwise uniqueness holds for (4.9).

Proof. Let X and X be two solutions to (4.9), driven by the same Brownian motions and starting from the same starting point x 0 ∈ B d . Pathwise uniqueness clearly holds up to the first time the boundary is attained, so it suffices to take x 0 on the boundary, and prove

X t = X t for all t ≤ τ , where τ = inf{t ≥ 0 : X t ∧ X t ≤ 1 -ε} for some arbitrary ε > 0. Define Y t = 1 -X t 2 and Y t = 1 -X t 2 .
In order to improve readability we often omit time subscripts below. We also omit the details of several cumbersome but straightforward calculations.

We apply the method by [START_REF] Deblassie | Uniqueness for diffusions degenerating at the boundary of a smooth bounded set[END_REF]. Define the function

F (y, y) = (y p -y p ) 2
where p ∈ (1/2, 1) is required to satisfy κ > ν 2 (1p). It then follows from Lemma 2.1 in DeBlassie ( 2004) together with (4.10) that F (Y, Y ) satisfies Itô's formula on the time interval [0, τ ], provided ε > 0 is small enough depending on κ, ν, and p. Writing

F 1 = ∂ y F (Y, Y ), F 2 = ∂ y F (Y, Y ), etc.
, this means that the following calculations are rigorous. First,

dF (Y, Y ) = F 1 dY + F 2 d Y + 1 2 F 11 d Y, Y + F 12 d Y, Y + 1 2 F 22 d Y , Y .
Next, in view of (4.10), a calculation yields

dF (Y, Y ) = (local martingale) + κ(F 1 + F 2 ) + ν 2 (F 11 Y + 2F 12 Y 1/2 Y 1/2 + F 22 Y ) ( X 2 + X 2 ) dt + κ(F 1 -F 2 ) + ν 2 (F 11 Y -F 22 Y ) ( X 2 -X 2 ) dt -2ν 2 F 12 Y 1/2 Y 1/2 X -X 2 dt -d ν 2 (F 1 Y + F 2 Y ) dt = (local martingale) + I 1 ( X 2 + X 2 ) dt + I 2 dt + I 3 dt + I 4 dt.
We now insert the explicit expression for F and its derivatives. Define Z = (Y p-1 -Y p-1 )( Y p -Y p ) and note that Lemma A.1 in DeBlassie ( 2004) yields

(Y p-1/2 -Y p-1/2 ) 2 ≤ (2p -1) 2 4p(1 -p) Z whenever p ∈ ( 1 2 , 1
). Making use of this inequality, one obtains

I 1 = -2pκZ +2pν 2 (1-p)Z +2p 2 ν 2 (Y p-1/2 -Y p-1/2 ) 2 ≤ -2pν 2 κ ν 2 -1 + p - (2p -1) 2 4(1 -p) Z.
The expression in parentheses on the right-hand side is maximized by p = 1 -√ 2/4 ∈ ( 1 2 , 1), which makes its value equal to κ/ν 2 + 1 -√ 2. Note that the hypothesis on κ and ν ensures that κ > ν 2 (1p) holds for this choice of p, as was required above. Thus, on [0, τ ],

I 1 ( X 2 + X 2 ) ≤ -4(1 -ε)pν 2 κ ν 2 + 1 - √ 2 Z.
Next, we have

I 2 = -2pν 2 κ ν 2 -1 + p (Y p-1 + Y p-1 )(Y p -Y p )(Y -Y ) -2p 2 ν 2 (Y 2p-1 -Y 2p-1 )(Y -Y ).
Since κ/ν 2 ≥ 1p, one obtains I 2 ≤ 0. Furthermore, since

-F 12 Y 1/2 Y 1/2 = 2p 2 Y p-1/2 Y p-1/2 ≤ 2p 2 ε 2p-1 for t ≤ τ , we get I 3 ≤ 4p 2 ε 2p-1 ν 2 X -X 2 on [0, τ ]. Finally, I 4 = -2pd ν 2 F ≤ 0.
Putting together these estimates yields, for t ≤ τ ,

F (Y t , Y t ) ≤ (local martingale) -4(1 -ε)pν 2 κ ν 2 + 1 - √ 2 t 0 Z s ds + 4p 2 ε 2p-1 ν 2 t 0 X s -X s 2 ds.
(4.11)

We need the dynamics of X -X 2 . Similar calculations as those leading to (4.10) yield

d X -X 2 = -2κ X -X 2 + d ν 2 (Y 1/2 -Y 1/2 ) 2 dt + 2ν(Y 1/2 -Y 1/2 )(X -X) ⊤ dW.
Moreover, one has the inequality (Y

1/2 -Y 1/2 ) 2 ≤ cε 2-2p Z on [0, τ ],
where c is a constant that only depends on p; see the proof of (DeBlassie, 2004, Lemma 3.6). In conjunction with (4.11) one then obtains, for t ≤ τ ,

F (Y t , Y t ) + X t -X t 2 ≤ (local martingale) + d ν 2 cε 2-2p -4(1 -ε)pν 2 κ ν 2 + 1 - √ 2 t 0 Z s ds + (4p 2 ε 2p-1 ν 2 -2κ) t 0 X s -X s 2 ds.
Since κ/ν 2 + 1 -√ 2 > 0 by assumption, we may choose ε sufficiently small to make the finite variation terms nonpositive. Thus, on [0, τ ], F (Y t , Y t ) + X t -X t 2 is bounded from above by a nonnegative local martingale M t with M 0 = 0. Thus M t = 0 on [0, τ ], implying that the same holds for the left-hand side. This yields X = X on [0, τ ], as desired. The proof is complete.

Remark 4.7. While the restriction on κ and ν excludes some specifications, there is certainly an overlap with the set of parameters for which the boundary is attained. Indeed, by Proposition 2.2 the boundary may be attained if and only if κ/ν 2 < 1.

Existence of smooth densities

If BQ(x, y) is a sum of squares and the state space is the unit sphere, Corollary 4.4(iii) and Hörmander's theorem lead to simple conditions under which the solution to (1.2) possesses a smooth density with respect to a suitable surface area measure. To state the precise result we introduce some notation. Let A 0 , . . . , A m be elements of Skew(d) and fix x 0 ∈ S d-1 . Define

g 0 = {A 1 , . . . , A m } g k = g k-1 ∪ {[B, A p ] : B ∈ g k-1 , p = 0, . . . , m} (k ≥ 1) g = span k≥0
g k (4.12) h = Lie algebra generated by A 0 , . . . , A m .

Here [A, B] = AB -BA is the usual matrix commutator. Note that g is a Lie sub-algebra of h, and that h = g+RA 0 . In fact, g is even an ideal in h, meaning that [g, h] ⊆ g. Let G denote the connected Lie subgroup of SO(d) generated by e B , B ∈ g, and H the connected Lie group similarly generated by h. Then g (respectively h) is the Lie algebra of G (respectively H).

Finally, define Gx 0 = {Qx 0 : Q ∈ G}, the orbit of x 0 under G, and similarly Hx 0 . These are smooth submanifolds of S d-1 , and their tangent spaces are given by

T x (Gx 0 ) = gx and T x (Hx 0 ) = hx.
In view of the identities

Q -1 gQ = g and Q -1 hQ = h for all Q ∈ G, it follows that T Qx (Gx 0 ) = Q T x (Gx 0 ) and T Qx (Hx 0 ) = Q T x (Hx 0 ), Q ∈ G. (4.13)
All these properties are well-known, and simply reflect the fact that Gx 0 and Hx 0 are homogeneous spaces for G and H, respectively. The natural state space for solutions to (4.7) starting from x 0 ∈ S d-1 is Hx 0 . However, by slightly adjusting such a solution one obtains a process which remains in Gx 0 . Specifically, one has the following lemma.

Lemma 4.8. Let X be the solution to (4.7) with X 0 = x 0 , and define a process Z by

Z t = e -tA 0 X t . Then Z t ∈ Gx 0 for all t ≥ 0.
Proof. Since g is an ideal in h, we have for each p ∈ {1, . . . , m}, e -tA 0 A p e tA 0 z ∈ gz = T z (Gx 0 ) for all z ∈ Gx 0 , t ≥ 0.

(4.14)

To see this, write f (t) = e -tA 0 A p e tA 0 . Then

f (k+1) (0) = [f (k) (0), A 0 ] for all k ∈ N. Since g is an ideal in h one has f ′ (0) = [A p , A 0 ] ∈ g, whence f (k) (0) ∈ g for all k ∈ N by induction.
Thus e -tA 0 A p e tA 0 = f (t) ∈ g since f is analytic with infinite radius of convergence. Hence (4.14) follows. Now, a brief calculation shows that Z satisfies the time-inhomogeneous SDE

dZ t = e -tA 0 A 1 e tA 0 Z t • dW 1 t + • • • + e -tA 0 A m e tA 0 Z t • dW m t ,
which admits a unique strong solution. In view of (4.14), the vector fields on the right-hand side are tangent to Gx 0 , which by Hsu (2002, Theorem 1.2.9) implies that Z takes values there.

One now has the following fairly precise result regarding existence of smooth densities in the case where BQ(x, y) admits a sum of squares representation.

Theorem 4.9. Let X be the solution to (4.7) with X 0 = x 0 ∈ S d-1 ; in particular, BQ(x, y) is a sum of squares of polynomials. Then X takes values in Hx 0 . Moreover, the following conditions are equivalent:

(i) X t has a smooth density with respect to surface area measure on Hx 0 for all t > 0.

(ii) Hx 0 = Gx 0 .

(iii) A 0 x 0 ∈ gx 0 .

Proof. It follows from Lemma 4.8 and the definition of H that X takes values in Hx 0 . We now prove that the stated conditions are equivalent. To this end, first note that since Gx 0 ⊆ Hx 0 , and Hx 0 is connected due to the connectedness of H, one has Gx 0 = Hx 0 if and only if T x (Gx 0 ) = T x (Hx 0 ) for all x ∈ Gx 0 . Thanks to (4.13) this yields

Gx 0 = Hx 0 ⇐⇒ T x (Gx 0 ) = T x (Hx 0 ) for some x ∈ Gx 0 . (4.15) ⇐⇒ T x 0 (Gx 0 ) = T x 0 (Hx 0 ).
The latter condition is equivalent to gx 0 = hx 0 . Since h = g + RA 0 , this holds if and only if A 0 x 0 ∈ gx 0 . This proves (ii) ⇐⇒ (iii).

To prove (i) =⇒ (ii), let σ denote surface area measure on Hx 0 and note that for any measurable subset U ⊆ Hx 0 , one has σ(U ) > 0 if and only if σ(e tA 0 U ) > 0, where t ∈ R is arbitrary. Fix some t > 0. Since Lemma 4.8 yields X t ∈ e tA 0 Gx 0 , it follows that σ(e tA 0 Gx 0 ) > 0 and hence σ(Gx 0 ) > 0. This in turn implies that there exists some x ∈ Gx 0 for which the tangent space T x (Gx 0 ) cannot be a proper subspace of T x (Hx 0 ); that is, T x (Gx 0 ) = T x (Hx 0 ). Thus (ii) follows by (4.15)

It remains to prove (ii) =⇒ (i). Consider the vector fields V p given by V p (x) = A p x, appearing in the expression (4.8) for the generator G . One has [V p , V q ](x) = [A p , A q ]x, where [A p , A q ] = A p A q -A q A p . Hörmander's parabolic condition (see [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF], or Theorem 1.3 in [START_REF] Hairer | On Malliavins proof of Hörmanders theorem[END_REF] for a formulation suited to our current setting) is therefore equivalent to requiring T x (Hx 0 ) = gx for all x ∈ Hx 0 . This holds thanks to the assumption Hx 0 = Gx 0 . Applying Hörmander's theorem in each smooth coordinate chart of Hx 0 now gives the existence of a smooth density.

Corollary 4.10. Let X be the solution to (4.7) with X 0 = x 0 ∈ S d-1 . Then X t has a smooth density with respect to surface area measure on S d-1 for all t > 0 if and only if g = Skew(d).

Example 4.11. As a simple illustration of the distinction between G and H, let d = 4 and consider the skew-symmetric matrices

A 0 =     0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0     , A 1 =     0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0     .
Then we have A 0 A 1 = A 1 A 0 = 0, so that g = span{A 1 } and h = span{A 0 , A 1 }. Moreover, e -tA 0 A 1 e tA 0 = A 1 , whence the process Z in Lemma 4.8 satisfies

dZ t = A 1 Z t •dW 1 t . It follows that (Z 3 t ) 2 + (Z 4 t ) 2 is constant,
showing that Z is confined to a circle embedded in R 4 , namely Gx 0 = e RA 1 x 0 . On the other hand, the process X given by X t = e tA 0 Z t is not restricted to any fixed subset of Hx 0 , although for a fixed time t ≥ 0 it necessarily lies in the transformed circle e tA 0 Gx 0 (we suppose that A 0 x 0 = 0 to avoid a trivial situation). This is a nullset in Hx 0 , so a density cannot exist.

By means of a projection argument, Corollary 4.10 can be used to derive conditions that guarantee the existence of smooth densities for a particular class of polynomial diffusions taking values in the ball B d . Corollary 4.12. Let X be the solution to (4.2) with X 0 = x 0 ∈ S d-1 and B = -

1 2 α. Write α = a 1 a ⊤ 1 +• • •+a d a ⊤
d for some vectors a 1 , . . . , a d ∈ R d , and define matrices A p ∈ Skew(d+1) by A p = A p 0 0 0 for p = 0, . . . , m, and

A p = 0 a i -a ⊤ i 0 for p = m + i, i = 1, . . . , d.
Let g be defined as in (4.12) with A p replaced by A p . If g = Skew(d + 1), then for all t > 0, X t has a density that is smooth on the interior of B d .

Proof. Let z 0 = (x 0 , 1x 0 2 ) ∈ S d , and define the S d -valued process Z as the unique solution to the SDE

dZ t = A 0 dt + (• d W 1 t ) A 1 + • • • (• d W m+d t ) A m+d Z t , Z 0 = z 0 , where W = ( W 1 , . . . , W m+d ) is an (m + d)-dimensional Brownian motion. Let π : S d → R d , (z 1 , . . . , z d , z d+1 ) → (z 1 , . . . , z d )
be the canonical projection of S d onto R d , and define X ′ = π(Z). Then X ′ satisfies

dX ′ t = A 0 X ′ t dt + Z d+1,t d i=1 a i • d W m+i t + m p=1 A p • d W p t X ′ t .
Moreover, since

dZ d+1,t = - d i=1 a ⊤ i X ′ t • d W m+i t , one has Z d+1 , W m+i t = -a ⊤ i X ′ t dt.
Thus, writing the middle term in the above expression for dX ′ t in Itô form, one obtains

dX ′ t = A 0 - 1 2 α X ′ t dt + Z d+1,t d i=1 a i d W m+i t + m p=1 A p • d W p t X ′ t .
Note that Z d+1,t = 1 -X ′ t 2 sign(Z d+1,t ). Thus X ′ satisfies an SDE of the form (1.2), and its generator coincides with that of X. Thus X and X ′ have the same law by Lemma 2.4.

Next, let σ be surface area measure on S d . By Corollary 4.10, Z t has a smooth density p Z (t; z) with respect to σ for all t > 0. We claim that this implies that X ′ t has a density, smooth on the interior of B d , for every t > 0. Since X and X ′ have the same law, this will complete the proof of the corollary. be the pushforward under π of the restriction of σ to S d + , and recall that one has

µ(dx) = (1 -x 2 ) -1/2 dx, x ∈ B d .
For any measurable subset A ⊆ B d we compute:

P(X ′ t ∈ A) = P(Z t ∈ π -1 (A)) = π -1 (A) p Z (t; z)σ(dz) = S d + ∩π -1 (A) p Z (t; z 1 , . . . , z d , z d+1 ) + p Z (t; z 1 , . . . , z d , -z d+1 ) σ(dz) = A p Z (t; x, 1 -x 2 ) + p Z (t; x, -1 -x 2 ) µ(dx).
Thus X ′ t , hence X t , has a density p X (t; x) given by

p X (t; x) = p Z (t; x, 1 -x 2 ) + p Z (t; x, -1 -x 2 ) (1 -x 2 ) -1/2 ,
which is smooth on the interior of B d .

Proofs for Section 3

In this section we collect the proofs of the main results in Section 3. In particular, the proof of Theorem 3.5 relies on results regarding the space K defined in (3.3); we develop these results in Section 5.2.

Proof of Theorem 3.3

Recall that we set

(iii) dim D -dim E = 2 d 4 + 3 d 3 + d 2 .
This follows from a direct calculation.

(iv) T is surjective. To see this, fix any i, j, k, l ∈ {1, . . . , d}. We need to find c ∈ D such that c(x)x = x i x j x k e l . One such c is given by c(x) = x i (x j E kl + x k E jlx l E jk )/2, where we define E st = e s e ⊤ t + e t e ⊤ s .

Since the collection {D p xx ⊤ D ⊤ q + D q xx ⊤ D ⊤ p : 1 ≤ p < q ≤ m} is a subset of ker T , the ranknullity theorem together with (iii) and (iv) implies (3.4) once we prove that this collection contains 2 d 4 + 3 d 3 + d 2 linearly independent elements. To this end, recall the elementary skew-symmetric matrices S ij = e i e ⊤ je j e ⊤ i appearing in (1.5), and consider the sub-collection

B = {S ij xx ⊤ S ⊤ kl + S kl xx ⊤ S ⊤ ij : 1 ≤ i < j < k < l ≤ d} ∪{S ik xx ⊤ S ⊤ jl + S jl xx ⊤ S ⊤ ik : 1 ≤ i < j < k < l ≤ d} ∪{S ij xx ⊤ S ⊤ ik + S ik xx ⊤ S ⊤ ij : 1 ≤ i < j < k ≤ d} ∪{S ij xx ⊤ S ⊤ jk + S jk xx ⊤ S ⊤ ij : 1 ≤ i < j < k ≤ d} ∪{S ik xx ⊤ S ⊤ jk + S jk xx ⊤ S ⊤ ik : 1 ≤ i < j < k ≤ d} ∪{2S ij xx ⊤ S ⊤ ij : 1 ≤ i < j ≤ d}.
It contains 2 d 4 + 3 d 3 + d 2 elements, which as we now show are linearly independent. For any i < j, the matrix 2S ij xx ⊤ S ⊤ ij is the only matrix in B with x 2 i appearing in position (j, j). Moreover, for any i

< j < k, S ij xx ⊤ S ⊤ jk + S jk xx ⊤ S ⊤ ij is the only matrix in B with x 2 j in position (i, k); S ik xx ⊤ S ⊤ jk + S jk xx ⊤ S ⊤
ik is the only matrix with x 2 k in position (i, j); and S ij xx ⊤ S ⊤ ik + S ik xx ⊤ S ⊤ ij is the only matrix with x 2 i in position (j, k). Finally, for any i < j < k < l, S ij xx ⊤ S ⊤ kl +S kl xx ⊤ S ⊤ ij is the only matrix in B with x i x l in position (j, k); and S ik xx ⊤ S ⊤ jl + S jl xx ⊤ S ⊤ ik is the only matrix with x k x l in position (i, j). These observations imply that B is a linearly independent set, as claimed. Hence (3.4) is proved.

It remains to show that dim K = d 4 . To see this, observe that K is the kernel of the linear map

S m → C , H → c H .
By (3.4), this map is surjective, and the dimension of its kernel is

dim S m -dim C = m + 1 2 -2 d 4 -3 d 3 - d 2 = d 4 ,
where the last equality follows from a direct calculation.

The space K and the Plücker relations

The goal is now to develop a better understanding of the space K in (3.3). The main outcome is Lemma 5.3 below, which is crucial for the proof of Theorem 3.5. The lemma shows in particular that the set K can be identified with the Plücker relations from algebraic geometry.

3 2 -1. Applying Lemma 5.5 with r = 1 then yields a rank-one matrix zz ⊤ , z ∈ R m \ {0}, in the intersection A ∩ S m + . In particular, z ⊤ Kz = 0 for all K ∈ K . Identifying z with the skew-symmetric matrix A whose entries are given by a ij = z π(i,j) for i < j, this says that A, K[A] = 0 for all K ∈ K . Thus rank A = 2 by Lemma 5.3, whence A = xy ⊤yx ⊤ for some x, y ∈ R d by Lemma 5.1. Consequently, (5.1) yields y ⊤ c H (x)y = A, H[A] /2 = z ⊤ Hz = zz ⊤ , H = -1, as required. Theorem 3.5(i) is proved.

Remark 5.6. A key step of the proof is to apply Lemma 5.5 to obtain a rank one element of A ∩ S m + . This is where the proof breaks down for d = 5. Indeed, Lemma 5.5 then only provides a rank three element, which is not enough to continue the proof.

Proof of Theorem 3.5(ii)

We will use the construction given in [START_REF] László | On sum of squares decomposition for a biquadratic matrix function[END_REF] to obtain an example for d = 6. In order to make the presentation self-contained, we summarize the construction here. Note that it suffices to consider the case d = 6. Indeed, suppose f (x 1 , . . . , x 6 , y 1 , . . . , y 6 ) is a nonnegative biquadratic form in 6 + 6 variables vanishing on the diagonal that is not a sum of squares. For d > 6, define g(x, y) = f (x 1 , . . . , x 6 , y 1 , . . . , y 6 ).

The left-hand side is then a nonnegative biquadratic form in d + d variables vanishing on the diagonal that is not a sum of squares.

From now on we thus take d = 6 and hence m = 15. In view of Lemma 3.4 we must find H ∈ S 15 such that y ⊤ c H (x)y ≥ 0 for all x, y, but at the same time (H + K ) ∩ S 15 + = ∅;

(5.5) in other words, we must find a counterexample to (5.4). A candidate H is obtained from the inequality 2( X 2 Y 2 -Tr(X ⊤ Y ) 2 ) -XY -Y X 2 ≥ 0, (5.6) which is valid for any d × d matrices X and Y ; see for instance Böttcher and Wenzel (2008, Theorem 2.2). Here X = Tr(X ⊤ X) is the Frobenius norm. Note also that the left-hand side defines a biquadratic form vanishing on the diagonal. Consider now the special case where for x, y ∈ R 6 , X = where the matrix H ∈ S 15 can be found to be

H = 1 2                           
2 0 0 0 0 0 0 0 -2 -1 0 0 0 0 0 0 2 0 0 0 0 -1 0 0 0 -2 0 0 0 0 0 0 4 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 4 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 2 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 4 0 -1 0 0 0 0 0 -2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 4 0 0 0 0 0 0 -2 0 0 0 0 -1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4

                          
.

The characteristic polynomial of H is given by det(s Id -H) = 2 -5 (s -2) 7 (2s 2 -2s -1) (4s 3 -16s 2 + 14s + 1) 2 .

Here 2s 2 -2s -1 has one single negative root λ = (1 -√ 3)/2. Moreover, the polynomial φ(s) = 4s 3 -16s 2 + 14s + 1 has one single negative root µ, since φ(0) = 1 > 0, φ(2) = -3 < 0, lim s→∞ φ(s) = ∞, and lim s→-∞ φ(s) = -∞. We now define the three vectors where e i denotes the ith canonical basis vector in R 15 . A calculation shows that v 1 is an eigenvector of H corresponding to the eigenvalue λ, and v 2 , v 3 are linearly independent eigenvectors of H corresponding to the eigenvalue µ. We now define the positive semidefinite matrix

B = δv 1 v ⊤ 1 + v 2 v ⊤ 2 + v 3 v ⊤ 3 ,
where δ = µ(µ -2)(2µ -1)/λ > 0. This matrix satisfies H, B = Tr(HB) = λδ v 1 2 + µ v 2 2 + µ v 3 2 < 0.

(5.7)

Theorem 2. 1 .

 1 Assume a and b satisfy (1.1) and the state space is E = B d . The following conditions are equivalent: (i) There exists a continuous map σ : R d → R d×d with σσ ⊤ = a on B d such that (1.2) admits a B d -valued weak solution for any initial condition in B d .

  Theorem 3.5. (i) If d ≤ 4, then any nonnegative biquadratic form in d + d variables vanishing on the diagonal is a sum of squares. Equivalently, any c ∈ C + is of the form c = c H for some H ∈ S m + .

  and Y is a Skew(d)-valued correlated Brownian motion with drift, independent of W . Here • dY t denotes Stratonovich differential.

  Let S d + = {z ∈ S d : z d+1 > 0} denote the upper hemisphere of S d . Let µ = σ| S d + • π -1

  d k = 0 for k > d. Consider the linear spaces D = c : R d → S d : c ij ∈ Hom 2 for all i, j E = f : R d → R d : f i ∈ Hom 3 for all i as well as the linear map T : D → E defined by (T c)(x) = c(x)x. Then ker T = C , so we need to prove that ker T = {c H : H ∈ S d }. We make the following observations. (i) dim D = d+1 2 2 , since each c ∈ D is specified by d+1 2 independently chosen elements of Hom 2 , each of which is specified by dim Hom 2 = d+1 2 independent parameters. (ii) dim E = d d+2 3 , since each f ∈ E is specified by d independently chosen elements of Hom 3 , each of which is specified by dim Hom 3 = d+2 3 independent parameters.

  As in Section 5.2 we use the notationA = xy ⊤yx ⊤ .One then easily verifies the identities 2( X 2 Y 2 -Tr 2 (X ⊤ Y )) = A 2 and XY -Y X 2 = (a 12 + a 26 ) 2 + (a 45a 46 ) 2 + (a 13 + a 24 + a 35 ) 2 . Furthermore, Lemma 5.2(i) yields P 1234 (A) = P 2345 (A) = 0. In view of (5.6) it follows that the functionf (x, y) = A 2 -(a 12 + a 26 ) 2 -(a 45a 46 ) 2 -(a 13 + a 24 + a 35 ) 2 -P 1234 (A) -P 2345 (A)is a nonnegative biquadratic form vanishing on the diagonal. The correspondence established in (5.1) then yields f (x, y) = y ⊤ c H (x)y,

  µ(µ -2) e 10 ,

  ). Thus (4.2) has the same drift and diffusion coefficients as (1.2), which by Lemma 2.4 implies that the law of any solution to (4.2) with initial condition x is indeed P x , as desired. Finally, the existence of a B d -valued solution to (4.2) follows, as in the proof of Theorem 2.1, from Example 2.5 and Theorem 2.2 in Da Prato and Frankowska (2007).

anonymous referees for their useful comments. Martin Larsson gratefully acknowledges support from SNF Grant 205121 163425. The research of Sergio Pulido benefited from the support of the "Chair Markets in Transition" under the aegis of Louis Bachelier laboratory, a joint initiative of École Polytechnique, Université d' Évry Val d'Essonne and Fédération Bancaire Française, and the project ANR 11-LABX-0019. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 307465-POLYTE.

Let m = d 2 = dim Skew(d). Each matrix H = (h pq ) ∈ S m can be viewed as a symmetric linear map on Skew(d), acting on matrices A = (a ij ) ∈ Skew(d) by the formula

with the remaining components of H [A] given by skew-symmetry. Here the map π lists strictly upper-triangular matrix elements in lexicographic order, (i, j) (1, 2) (1, 3) (1, 4)

The symmetry property A, H[B] = H[A], B is immediate from the symmetry of H as a matrix. Using this notation, the biquadratic form corresponding to an element c H ∈ C , where H ∈ S m , can be expressed as

where A = xy ⊤yx ⊤ .

(5.1)

This follows from (3.2) using that x ⊤ D π(i,j) y = x i y jx j y i . Together with Lemma 5.1 below this immediately yields

In particular, this shows that C ∼ = S m /K can be identified with the set of restrictions to Skew(2, d) of quadratic forms on Skew(d).

Lemma 5.1. Let A ∈ Skew(d). Then rank A = 2 if and only if A = xy ⊤yx ⊤ for some linearly independent x, y ∈ R d .

Proof. Only the necessity needs proof, so pick A ∈ Skew(2, d). Any rank-two matrix A can be written A = xx ⊤ A + yy ⊤ A, where {x, y} is an orthonormal basis for the range of A.

By skew-symmetry of A, we have A ⊤ x = γy and A ⊤ y = -γx, where γ = x ⊤ Ay. Thus A = (γx)y ⊤y(γx) ⊤ , which is of the stated form.

For each ordered 4-tuple i < j < k < l of indices from {1, . . . , d}, define a polynomial P ijkl in the entries of A ∈ Skew(d) by

These polynomials generate the variety of rank-two skew-symmetric matrices: all P ijkl vanish at a nonzero A ∈ Skew(d) if and only if rank A = 2. Moreover, they constitute a basis for the homogeneous ideal generated by this variety: any homogeneous polynomial vanishing on Skew(2, d) is a polynomial linear combination of the P ijkl . The classical use of this fact is to give the Grassmannian manifold of two-dimensional subspaces of R d the structure of a projective variety. This is achieved via the Plücker embedding, which maps a subspace V = span{x, y} of R d to A = xy ⊤yx ⊤ ∈ Skew(d). Note that, up to scaling, A only depends on V , not on the choice of spanning vectors x and y. Conversely, any A ∈ Skew(2, d) corresponds to a unique two-dimensional subspace V ⊂ R d due to Lemma 5.1. The entries a ij of A are called the Plücker (or Grassmann) coordinates of V , and the quadratic equations P ijkl (A) = 0 that they satisfy are called the Plücker relations. For us, the relevant result is the following:

Lemma 5.2. The polynomials P ijkl have the following properties:

(ii) Let P (A) be a homogeneous polynomial in the entries of A ∈ Skew(d). Then P (A) = 0 for all A ∈ Skew(2, d) if and only if it is of the form

for some polynomials Q ijkl .

Proof. Keeping in mind the identification of Skew(2, d) with the set of two-dimensional subspaces as above, part (i) follows from Theorem II on page 312 in [START_REF] Hodge | Methods of algebraic geometry[END_REF], and part (ii) follows from Theorem I on page 315 in the same reference.

The following lemma is the primary reason for studying K in such detail; it is a crucial ingredient in the proof of Theorem 3.5. In particular, it shows that the set K can be identified with the Plücker relations.

Lemma 5.3. The set K can be identified with the linear span of the polynomials P ijkl , i < j < k < l. Moreover, one has

(5.3)

Proof. Lemma 5.2(ii) together with (5.2) implies that K ∈ K holds if and only if the polynomial

is a linear combination of the P ijkl (A). The latter polynomials can therefore be identified with a basis for K . It follows that the right-hand side of (5.3) is equal to {A ∈ Skew(d) \ {0} : P ijkl (A) = 0 for all i < j < k < l}. This, in turn, is equal to Skew(2, d) by Lemma 5.2(i).

Remark 5.4. The identification of K with the linear span of the polynomials P ijkl leads to an explicit description of a basis for K . Indeed, one has

where K (i,j,k,l) is the linear map on Skew(d) characterized by

The collection of all such K (i,j,k,l) for i < j < k < l thus constitutes a basis for K . Note that there are d 4 ways of choosing the indices i, j, k, l, which is in agreement with the dimension of K as given in Theorem 3.3.

Proof of Theorem 3.5(i)

A preliminary result is required for the proof of Theorem 3.5(i). Recall that an affine subspace of a vector space is a subset

A is an affine subspace and x ∈ A , then Ax is a linear subspace that does not depend on x. The dimension of A is the dimension of Ax. The following result is Theorem 1.1 in [START_REF] Barvinok | A remark on the rank of positive semidefinite matrices subject to affine constraints[END_REF]; see Theorem 1.3 and its reformulation (2.2) in [START_REF] Barvinok | Problems of distance geometry and convex properties of quadratic maps[END_REF] for a proof.

Lemma 5.5. Let A ⊂ S m be an affine subspace such that the intersection A ∩S m + is nonempty and codim A ≤ r+2 2 -1 for some r ∈ N. Then there exists some B ∈ A ∩ S m + with rank B ≤ r.

We now proceed with the proof of Theorem 3.5(i). We need to show that, for any H ∈ S m ,

(5.4) Indeed, then any c H ∈ C + is equal to c H+K for some K ∈ K such that H + K ∈ S m + . We prove the contrapositive. Thus, consider any H ∈ S m such that (H + K ) ∩ S m + = ∅. We need to prove that y ⊤ c H (x)y < 0 for some (x, y) ∈ R d × R d . The separating hyperplane theorem gives some B ∈ S m such that C, B ≥ 0 for all C ∈ S m + and H + K, B < 0 for all K ∈ K . The former inequality yields B ∈ S m + by self-duality of S m + . The latter inequality yields K, B = 0 for all K ∈ K , since otherwise H + tK, B would become positive for some K ∈ K and some sufficiently large t. Thus, after scaling B if necessary, we have

+ and H, B = -1. Hence A ∩ S m + = ∅ for the affine subspace

Note that A -B = {C ∈ S m : C ∈ K ⊥ and C, H = 0}. Our initial assumption excludes H ∈ K , since otherwise we would have 0 ∈ (H + K ) ∩ S m + . Thus codim A = 1 + dim K , so that codim A = 2 for d = 4, and codim A = 1 for d ∈ {2, 3}. In either case, codim A ≤ In addition, by considering each of the 6 4 = 15 basis elements K (i,j,k,l) of K described in Remark 5.4, one can check by direct computation that K, B = 0 for all K ∈ K .

(5.8)

Since B is nonzero and positive semidefinite, (5.7) and (5.8) imply (5.5). This completes the proof of Theorem 3.5(ii).

Remark 5.7. The condition (5.5) and its relation to sum of squares representations can also be expressed via the following primal-dual pair of semidefinite programs:

{λ : H -λId + K 0} .

(5.10)

Note that v(D) ≥ 0 if and only if (H + K ) ∩ S 15 + = ∅. Moreover, weak duality v(D) ≤ v(P ) always holds. Indeed, for any primal feasible solution B and dual feasible solution (λ, K) one has 0 ≤ B, H -λId + K = H, Bλ. Thus, for a given candidate c = c H , a primal feasible solution B with H, B < 0 provides a certificate that no sum of squares representation can exist. Furthermore, (5.7) and (5.8) imply that the matrix B/ Tr(B) is thus a feasible solution to the primal problem (5.9) with negative value, and hence certifies that no sum of squares representation can exist.

Remark 5.8. One can write down the component functions of the resulting map c = c H :

Remark 5.9. A natural question is whether c = c H is non-degenerate on the sphere, that is, if all eigenvalues are strictly positive except the one corresponding to the eigenvector x. This turns out not to be the case. With x 2 = x 4 = x 6 = 0 and under the restriction x = 1, the eigenvalues of c(x) are 0, (x 1 + x 5 ) 2 , 2x 2 1 , 2x 2 5 , 2, 2. Thus c(x) is degenerate along the curve S d-1 ∩ {x 2 = x 4 = x 6 = 0} ∩ {x 1 + x 5 = 0}. In particular, we do not know whether the corresponding diffusion on the sphere can be realized as the solution to an SDE for which pathwise uniqueness holds.


Guillermo Gomez

Carole Plasson

Frédéric Elisei

Frédéric Noël

Gérard Bailly

Qualitative assessment of an immersive teleoperation environment for collaborative professional activities in a "beaming" experiment

Keywords: (according to ACM CCS): I, 2, 9 [Computer Graphics]: Robotics/Operator interfaces-H, 1, 2 [ Information Systems ]: User/Machine Systems/Human factors -H, 5, 3 [Information Systems]: Group and Organization Interfaces-

The current study assesses an immersive teleoperation platform for professional collaborative activities. This platform uses an iCub robot and virtual-reality hardware (VR headset and motion capture) to elicit an embodiment situation, where the pilot collaborates with remote actors through the robot ("beaming platform"): through the robot, the pilot can act and perceive using his natural affordances (head moving, vision and hearing) that are mapped to/from the robot effectors/sensors. One challenge is to measure how the robot expression capacity level can modulate the efficiency of the participation of the remote pilot with his collaborators. An experience where the pilot trains two subjects to assemble a mechanical system was organised: we observed the capacity of the two subjects to realize the task they discover and to collaborate with the pilot, with a more or less expressive robot. This experience clearly highlighted the importance of embodiment and the opportunities opened by beaming platforms, allowing the enhancement of the physical space by agents:i.e. social robots or conversational agents.

Introduction

The ever-growing technologies in the fields of telerobotics have led both the scientific and industrial community wonder when and how they can start to adapt and deploy these technologies in their respective fields of work. A particular application has turn out to be particularly interesting: gifting a person with the capacity to transport to a distant location, thanks to a humanoid robot that serves as a platform between the individual and another group of people on a determined physical location, for acting there as well as perceiving from there : it is a beaming [STB * 12] process where the individual is replicated by the robot, and common affordances are used (for example, moving one's head or hand to move the robot's head or hand) and sense the environment accordingly.

Beaming refers to technologies digitally transporting a representation of yourself to a distant place, where you can interact with the people there as if you were there (acting, perceiving). The possibilities for such a technology are tremendous, and this research focuses on the industrial engineering application field, which is often a collaborative work.

Although humanoid robots are not expected to replace humans, they could be very helpful in some particular tasks. However there is still plenty of work to design robots that can be socially and industrially competent, in order to have a satisfying and successful interaction with humans in diverse environments and domains.

It is important to study how humans react when interacting with a robot instead of a real person, and the impact on their behaviour. Humans are very sensitive to verbal and non-verbal communication, and humanoid robots are still being developed in terms of their "human-likeness" and "sociability". The evaluation of a beaming platform implies many different features to study in order to provide a deep analysis of the strengths and weaknesses.

The current study uses a system of interaction in a multisite collaborative work, allowing collaborators to be "telerepresented" by a humanoid robot with an active and physical presence. An experience was organized to perform a remote assembly training scenario with the confrontation of different objects. It is expected to check the condition for an actor to participate intuitively from a remote location. The same experience is run while degrading the functions of the environment to identify the most important functions which are associated here with the robot degrees of freedom allowing more or less expression transmission.

The next section introduces the technical state of the art about remote teleoperation. Then a short state of the art presents the domain of application we are concerned with (collaborative design activities). At this step the experience built for the current study will be described and the final section will introduce the first level of the results from this experience.

Remote teleoperation issues

Beaming platforms

Telerobotics [She89, She92] is the core technology of the assessed environment. It allows to remotely drive a robotic system. Robots are an extremely good alternative to humans for multiple applications (medicine, military, nuclear plant maintenance, construction) and also for collaborative distant work.

The "Wizard of Oz" (WoZ) refers to a person remotely operating a robot [START_REF] Riek | Wizard of oz studies in hri: a systematic review and new reporting guidelines[END_REF], controlling its movement, navigation, speech, gestures, for example by triggering predefined actions by pressing keys or teleoperating movements of some body segments from a mouse, a joystick or any other motion capture device. WoZ controls a robot from fully autonomous to fully tele-operated. The controlled robot serves more as a proxy for humans than as an independent entity. A valid WoZ simulation expects [Rie12, GPL * 12] "to simulate the future system, given human limitations", "to specify the future system's behaviour" and "to make the simulation convincing".

Part of telerobotics, the beaming concept (Being in Augmented Multimodal Naturally Networked Gatherings) was introduced [STB * 12]. Beaming is the name for the process that allows someone -here the pilot -to instantaneously transfer himself from one physical place to any given distant location, thanks to the monitoring of the pilot's actions, physiological states and emotions which can be streamed over the internet while simultaneously streaming visual, audio, spatial, and context back from the distant destination. This flow of data is synchronized and the result is an unified virtual environment representing the physical space of the destination in real time. The destination is thus inhabited by both local and virtual users.

Beaming allows to be embodied in a virtual interface. The destination/visitor interface is non-symmetrical, but beaming is defined to support symmetric social interaction between the visitor and locals. Thus an intensive attention is given to the quality of the visual immersion for the pilot, copying the physical remote destination. But indeed visualisation is just a part of a successful beaming platform, since all human senses should participate to the embodiment experience up to the "self-perception" of the avatar in the remote location.

With a virtual filter, human senses may be tricked introducing a infinite range of applications where perception may be faked [NSVW * 12] and where awareness becomes a real ethical issue. Gathering immersive virtual reality environment with tele-operated systems opens applications which otherwise would be impossible to achieve. For example, [BEFS15] uses a humanoid robot, tele-operated by a human pilot to artificially provide the robot with social skills. Such a WoZ platform was used to experience effects of culture in human robot interaction and social robotics [BV13] highlighting the role of head nods for social interaction. A basic analysis grid is used to assess a robot performance through various subjective criteria: likeability, engaging, satisfaction, useful, human-likeness, efficiency, credibility, human-presence.

[GPL * 12] focused on the perception of gaze and head movements through an iCub robot. It allows the interaction to get closer to humans because it feels more "human-like", with the help of another sensor for capturing gaze and head motion.

Human robot interaction

With a telerobotic beaming platform, a human robot interaction is taking place at the shared physical place. Many key issues arise then [PDGA14]: mutual interaction learning, human presence undertaking, integrating social rules and protocols, as well as legible motion. There are also more specific issues like: robustness and efficiency, comfort and intentionality to specify robot behaviour, the capacity of the robot to adapt and plan according to its environment just like a human would do it, among many others.

[BCBJ * 09, SLMF06, CKH * 15] deeply analysed the effects of head nodding that can be simulated or reproduced by a robot. They analysed how people use visual feedback while conversing with robots. Humans apparently nod at the robot during conversation.

[CKH * 15] debates over social cues of interaction as well. They assume that human robot interaction comes only when the capacities of perception of the environment as much as movement of the robot, conjugate harmoniously to attain the individual expectations. Likewise, signal and gestural responses from the robot such as head nodding along with other non-verbal behaviour, help individuals to know the intentions of their interlocutor, and so react and adapt their behaviour accordingly.

Multiparty dialog and situated interaction

The robot behaviour must integrate qualities that favour the communication between multiple parties. Gaze perception plays a crucial role in human-human interaction [Lan00, MEB12]. It has been proved that it affects several aspects of communication and dialogue. An embodied conversational avatar must integrate an accurate gaze model. Gaze remains one of the strongest nonverbal cues in human interaction and this result should be assessed for human-robot interaction.

Interaction with humans expects to consider also human factors. Primary causes of malfunctions in processes are due to human factors [Nar99]. Among all the human factors to consider that might have an impact on repairing, problem solving and effectiveness, we can cite a few: attitude, fatigue, workload, schedule, etc. For [HC98] having eyes and hands busy, drive decision about the usage or not of voice recognizing systems instead of classic systems like a keyboard. In addition, speech is a strongly personal activity which contains more information than just the text produced: then voice acts also as an important human factor.

Collaboration within design and manufacturing activities

The current study applies a beaming platform to support collaborative professional activity, specially in a design and/or manufacturing context. Design and manufacturing engineering is a cognitive activity [Cho09, GK04]. There are several design modes: the prescriptive approach, which describes the process of design as a succession of stages. This approach masters processes, acknowledge the creative part, and is teachable. Design is also viewed [Sim96] as a "problem solving" process where it becomes a non-linear and nondeterministic process of resolution, and has multiple solutions.

[Sch83] presents design as a process of reflexive conversation: designers work in virtual worlds, and their talent lies on the capacity to build and manipulate these worlds.

Then design is an art in the professional activity, it is a form of intelligence that is not teachable but can be learned, where identifying a problem and improvising is a necessary "art" for designers.

To collaborate designers use intermediary objects [START_REF] Blanco | Intermediary objects as a means to foster co-operation in engineering design[END_REF]. Objects are a second essential component in design and manufacturing processes. Design is often supported by visual reasoning with object representations. [Kua06] built experiences to analyse the links between cognitive micro-activities during design and demonstrated the separation between design reasoning, intention of design, intention of representation, and modeling actions.

[OOCS92] observed how design often takes place in a context of a collaborative face-to-face meeting. They try to analyse how designers use their time, what kind of activities they develop, how they organise these activities and if there exist any similar "models" between different projects and teams of people. They conclude that people expose their problems, alternatives and criteria to evaluate them, and they share their expertise. Results show that 40% of the time is spent on design discussions and briefings account for 30% of the time, which implies that reunions apparently have a role of coordination, generating a discussion in order to clarify the best way to proceed, and moreover that meetings produce an important amount of clarifications, since participants make and answer questions on diverse subjects.

Design and manufacturing activities provide a good context to assess the usability and utility of a beaming platform.

An experience for collaborative remote training

Research question

This study expects to assess the conditions for creating a beaming context to provide an efficient support for design and manufacturing professional activities. A basic beaming platform is controlled and its functions depreciated to check their importance. Especially the following research questions are submitted:

1. What is the impact of the different grades of movement and expression replicated by a humanoid robot in the interaction with individuals on a collaborative work environment? 2. How to assess this collaboration between individuals and the robot?

Increasing the freedom and expressiveness of the robot's movements should raise the fluency of the avatar representing the distant person, and thus improve the collaboration with a team of individuals, allowing a more meaningful interaction and a greater implication of individuals in the task. Such an outcome comes with the interest of building a platform for interaction of people in a collaborative setting, which has applications in the industrial sector within a realistic use-case.

A beaming platform

The beaming platform used for the experience integrates an iCub robot, which is 1 m tall. This robot has 53 degrees of freedom, plus 5 extra DoF for the jaw and mouth [PRM * 15]. In the used beaming platform version, only a few are mapped from a real person : it can move the head using the three degrees of freedom, and it can move the jaw and lips thanks to the generation of a model adapted to track pilot's lips. To create the model for the lips movement, the pilot had to glue a few motion infra-red reflectors strategically placed around the mouth, and run a calibration before each experiment.

The robot eyes are instrumented with two cameras allowing a stereoscopic capture of the robot vision. Microphones were inserted at robot ears location, allowing the capture of the sound environment around the robot. A loudspeaker was also inserted in the robot mouth, creating audio source for verbalization reproduction.

A Sony c Head Mounted Device (HMD), like to the one from Figure 1, was used to immerse the pilot in the robot space that transmit the views from the two mounted cameras (Figure 2). The HMD earphones receive the sounds captured by the iCub ear microphones. Pilot's speech is captured by a close microphone and played back via the robot's mouth loudspeaker. The robot acts as the distant avatar of the pilot. A motion tracking system was used for capturing both global movements of the pilot and mouth articulations.

Use case selection about mechanism assembly

A scenario that could be played by actors in a reasonable amount of time was expected to make experiences acceptable by subjects. The proposed scenario should be of a bearable difficulty, considering that something too hard to achieve is not useful here. To seek with utility within design and manufacturing applications, a jigsaw system was selected. This system provides a realistic context; on the inside it is a traditional mechanical system, with simple components such as a rotor, a gear, a switch, a rod, etc. and it is a real system for real engineering context whether it is for design, maintenance, production, etc. Additionally, it is easy enough for people to quickly identify its main parts and even if they have never seen such a system before, to be able to handle it without requiring training or whatsoever. This system can be mounted in a reasonable time. Involved subjects were not familiar with it. The disassembled system is presented to two subjects. The two subjects are expected to collaborate to re-assemble the system (see Figure 3) under the supervision of the pilot. The role of the pilot is limited to supervision (our beaming platform does not have an exoskeleton to support arm control yet). The scenario can be viewed as a training situation where two persons discover and learn a professional task supported by a remote expert. This context is closed to a context of project review where the pilot plays the role of manager and moderator of a discussion between two experts.

The selection of the jigsaw system is a compromise between complexity of task to assemble it, realism of the system and level of details: indeed the pilot guides the robot with a vision replication which has a low resolution and quality; it is due to the resolution of camera installed in the robot eyes (two 640x480 cameras with Bayer filter) and to the HMD which reproduce vision with a better resolution than the camera but creates new sources of uncomfort: weight, vergence for eye accommodation, etc.

Experimental protocol

The pilot is driving tasks within a predefined sequence of actions. The pilot is expert of the system, knows every details about the mechanism and must have enough possible situations in mind to react promptly as soon as there is a deviation from the predefined sequence of tasks to ensure to make possible comparisons between tests. The sub-tasks are thus organised into a series of short collaborations or parallel actions of the two subjects to ensure that the predefined process is followed.

The position of parts when the subjects enter the room is pre-defined as organised in Figure 4. When entering the room the jigsaw is masked. Three questionnaires have been prepared which cannot be fully reported here but we provide some question example to fix ideas : Memory Questionnaire: Q 1 : 10 basic questions to test what the person remind by himself even if no special attention was drawn to these details during the assembly process:

Q 1.1
What is the color of the main switch of the sigsaw ? black red yellow green Q 1.2 How many screw sizes belongs to the system ? 1 2 3 4 Q 1.i etc.

Individual perception of collaboration: Q 2 : 10 new questions were devoted to measure the perception of a more or less living robot:

Q 2.1 How will you qualify the robot from machine to alife ? 1 2 3 4 5 Q 2.2 How will you qualify pilot/robot from stupid to clever ? 1 2 3 4 5 Q 2.i etc.

Individual comparison of configurations: Q 3 : 5 questions to compare configurations 'A' or 'B'.

Q 3.1 Mark from 1 to 3 the importance of the following modalities ?

• voice transmission : 1 2 3 • head movement : 1 2 3

• lip movement : 1 2 3

Q 3.2
In which configuration the pilot specifications were more clear ? A or B or None Q 3.3 In which configuration the pilot was the more attentive ? A or B or None Q 3.4 In which configuration did you feel more easy when interacting with the pilot ? A or B or None Q 3.5 What configuration did you prefer ? A or B or None

After filling a participation agreement form, both the avatar and pilot are presented to subjects. The wizard of Oz context is clearly highlighted: they know that the robot is driven by a person. They are informed that they will have to follow the directives of the pilot and that they will assemble twice the same system following different sequences. Each sequence will not overpass 20 minutes: the task is stopped after this delay.

There are two assembly sequences, S 1 and S 2 . For each S i , assembly sub-tasks are executed in different orders and the roles of the two subjects are swapped. Sub-tasks are either individual or collaborative, and can be executed in parallel or sequentially.

For statistics issues configurations A and B are associated to sequence S 1 and S 2 leading to four conditions that are applied alternatively to the tuples of subjects.

Then, the mask over the mechanism is removed and the pilot drives the first assembly sequence. At the end of this sequence the questionnaire Q 1 is filled individually by the two subjects. Then they also answer to the questionnaire Q 2 to collect their subjective feedback about the collaboration with the pilot through the robot. While filling the questionnaire the system is disassembled and prepared again for the second sequence. The second sequence is played with the remaining configuration. After the second sequence, the questionnaire Q 2 is answered again but in reference to the new configuration. At last, the two subjects answer to the questionnaire Q 3 . This protocol was repeated for 9 couples of subjects and indeed 18 different persons were involved. The distribution of participants could be presented as follow: 10 females and 8 males, with 3 teams of women, 2 teams of men and 4 mixed teams. The average age of the participants in the experiments was 25 years old, and the majority had an engineering background.

Analysis of collaboration between subjects and a remote drive

4.1. Analysis of questionnaire Q 1

The overall objective of the questionnaire Q 1 is to check if subjects have a better performance in configuration B than in configuration A. The average percentage of correct answers was medium to low in both cases: 45% of success rate for the actors that answered after experiencing the configuration A, and 62% for actors that answered after configuration B. It must be reminded that in both cases they answer after the first sequence of assembly. These values seem to indicate that actors in configuration B are more engaged in the task because they have a better communication whenever the robot presents a more human-like expression and record more information.

4.2. Analysis of questionnaire Q 2 The second questionnaire deals with the subject individual impression about the robot. The average mark of each question for all participants are representative numbers to analyse. For every question, the average mark is higher for configuration B than for configuration A. Figure 5 details results for every question and the configuration B is clearly appreciated but questions can be separated into two sets depending on a more or less difference between the two configurations. Then it seems that:

• Configuration B provides a real impact to understand who (between the two subjects) is concerned when the pilot interacts. This configuration also enhances the frequency of interaction and the demand for interaction with the pilot. • Configuration B seems to be also better for interaction quality and when assessing the overall pleasure to interact through the avatar but the difference is less obvious.

The average and the standard deviation on the overall questions (Figure 6) demonstrate the tendency and the clear advantage of configuration B which is here much more obvious than for questionnaire Q 1 . Indeed enhancing the expressiveness of the robot is a major asset in human-robot interaction, as well as a useful tool for communication between humans. The third questionnaire serves as a summary to confirm the analysis of the first questionnaires. The first question (Figure 7) provides a quite obvious result but which is clearly measured. The voice is the preferred interaction modality but the gaze identified by head movements is a major interaction modality. The mouth articulation are not unnecessary (indeed, they might play an important role for speech inteligibility), but from the conscious self-report task, they are perceived as less important. The other questions of Q 3 were direct comparisons of the two configurations. The results described by Figure 8 are incontrovertible. The configuration B is clearly preferred even if 1 or 2 subjects reported that they did not care at all about the robot, which explains a few answers with no preference: video shows that these persons were concentrated on the task while listening the pilot but without watching the robot.

Conclusion and perspectives

The current study clearly demonstrates again the importance of gaze and head movements within social interaction. It also demonstrates the capacity of a beaming system to telepresent a remote human coach as soon as the beaming system reproduces the good interaction modalities. It demonstrates also a beaming process within a realistic professional collaborative task. Usually, research about user interfaces focuses on usability demonstration. With the current realistic process, a first step towards utility assessment is passed.

Obviously the statistics sample should be extended but the first results seem incontrovertible. We need to explore which parts of the dialog benefit from gazing at head or lips movements, and how these gaze patterns evolve with the duration of the interaction. Another direction of assessment will be exploited soon to get objective measures: Indeed all the scenarios have been recorded (video, audio, motion capture of the pilot, as well as robot's sensors streams) providing a rich set of corpus which can be deeply analysed (video labeling of the eye contacts, of the number of head movements from the human partners...) Another perspective is also to go ahead with this experience using other type of avatars, to check how presence filling may be produced on a distant location. A video of the pilot face may be streamed at the remote location but it can be expected to face the Mona-Lisa cue: a 2D image cannot propagate the gaze direction of the pilot. An image avatar could provide an immersive environment where the importance of gaze could be checked with respect to the quality of the avatar. The image could provide really realistic human face. To avoid the Mona-Lisa cue, stereoscopic displays or holographic displays could create an interesting 3D avatar; for a single remote subject, stereoscopy may be used while for multiple remote subjects holography will be expected. It will be worth exploring the respective properties of the streaming of 3D videos vs. the sensorimotor monitoring of virtual/robotic avatars in terms of users' experience and interaction efficiency.

Figure 1 :

 1 Figure 1: The beaming platform: pilot side.

Figure 2 :

 2 Figure 2: The beaming platform: avatar side.

Figure 3 :

 3 Figure 3: Jigsaw system on the inside (left) and completely mounted (right).

Figure 4 :

 4 Figure 4: Jigsaw presentation when initializing the process.

Figure 5 :

 5 Figure 5: Average results for questionnaire 2.

Figure 6 :

 6 Figure 6: Average results for questionnaire 2.

Figure 7 :

 7 Figure 7: Results for question Q 3.1

Figure 8 :

 8 Figure 8: Results for questionnaire Q 3 , question 2 to 5.

Aknowledgement•

  This paper was written within the scope of the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).• This work has also been partially supported by the VI-SIONAIR infrastructure project funded by the European Commission under grant agreement 262044. • The experimental robotic platform has been supported by the ANR (Robotex ANR-10-EQPX-44-0 and SOM-BRERO ANR-14-CE27-0014) and the UJF (EMSOC 2012084RECF991).