Aurélien Deya
email: deya@iecn.u-nancy.fr

Samy Tindel
email: tindel@iecn.u-nancy.fr

Malliavin calculus for fractional heat equation

Keywords: November 3, 2013. 2000 Mathematics Subject Classification. Primary 60H35; Secondary 60H07, 60H10, 65C30 fractional Brownian motion, heat equation, Malliavin calculus

Dedicated to David Nualart on occasion of his 60th birthday

Introduction

The definition and resolution of evolution type PDEs driven by general Hölder continuous signals have experienced tremendous progresses during the last past years. When the Hölder regularity of the driving noise is larger than 1/2, this has been achieved thanks to Young integrals [START_REF] Gubinelli | Young integrals and SPDEs[END_REF] or fractional integration [START_REF] Maslowski | Evolution equations driven by a fractional Brownian motion[END_REF] techniques. The more delicate issue of a Hölder exponent smaller than 1/2 has to be handled thanks to rough paths techniques, either by smart transformations allowing to use limiting arguments [START_REF] Caruana | Partial differential equations driven by rough paths[END_REF][START_REF] Caruana | A (rough) pathwise approach to a class of non-linear stochastic partial differential equations[END_REF][START_REF] Friz | Rough path stability of SPDEs arising in non-linear filtering[END_REF] or by an adaptation of the rough paths formalism to evolution equations [START_REF] Deya | A discrete approach to Rough Parabolic Equations[END_REF][START_REF] Deya | Non-linear rough heat equations[END_REF][START_REF] Gubinelli | Rough evolution equations[END_REF]. Altogether, those contributions yield a reasonable definition of rough parabolic PDEs, driven at least by a finite dimensional signal.

With those first results in hand, a natural concern is to get a better understanding of the processes obtained as solutions to stochastic PDEs driven by rough signals. This important program includes convergence of numerical schemes (see [START_REF] Deya | Numerical schemes for rough parabolic equations[END_REF] for a result in this direction), ergodic properties and a thorough study of the law of those processes. The current article makes a first step towards the last of these items. Indeed, we shall consider here a simple case of rough evolution PDE and see what kind of result might be obtained as far as densities of the solution are concerned. More specifically, we focus on the following mild heat equation on (0, 1)

Y t = S t ϕ + t 0 S t-u (F i (Y ) u ) dB i u , t ∈ [0, T ], (1)
where T > 0 is a finite horizon, S t stands for the heat semi-group associated with Dirichlet boundary conditions, ϕ is a smooth enough initial condition, F i : L 2 (0, 1) → L 2 (0, 1) and B : [0, T ] → R d is a d-dimensional fractional Brownian motion with Hurst parameter H > 1/2. For this equation, we obtain the following results:

(1) Existence of a density for the random variable Y t (ξ) for any t ∈ (0, T ] and ξ ∈ (0, 1), when the F i 's are rather general Nemytskii operators F i (ϕ)(ξ) := f i (ϕ(ξ)). See Theorem 3.8 for a precise statement. (2) When the F i 's are defined through some regularizing kernel (see Hypothesis 2), we obtain that the density of Y t (ξ) is smooth. This will be the content of Theorem 4.8. To the best of our knowledge, these are the first density results for solutions to nonlinear PDEs driven by fractional Brownian motion. Let us point out that we could have obtained the same kind of results for a more general class of equations (operator under divergence form, general domain D ⊂ R n , drift term, Gaussian process as driving noise). We prefer however to stick to the simple case of the fBm-driven stochastic heat equation for the sake of readability and conciseness.

Our main results will obviously be based on a combination of pathwise estimates for integrals driven by rough signals and Malliavin calculus tools. In particular, a major part of our effort will be dedicated to the differentiation of the solution to equation [START_REF] Adams | Sobolev spaces[END_REF] with respect to the driving noise B and to a proper estimate of the derivative. Since the equations for derivatives are always of linear type they lead to exponential type estimates, which are always a delicate issue. This is where we shall consider some regularizing vector fields F i in [START_REF] Adams | Sobolev spaces[END_REF], and proceed to a careful estimation procedure (see Section 4.1). It should also be noticed at this point that the basis of our stochastic analysis tools is contained in the celebrated book [START_REF] Nualart | The Malliavin calculus and related topics[END_REF] by D. Nualart, plus the classical reference [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF] as far as equations driven by fBm are concerned.

Here is how our article is structured: Section 2 is devoted to recall basic facts on both pathwise noisy evolution equations and Malliavin calculus for fractional Brownian motion. We differentiate the solution to (1) and obtain the existence of the density at Section 3. Finally, further estimates on the Malliavin derivative and smoothness of the density are derived at Section 4.

Notation. Throughout the paper, we will use the generic notation c to refer to the constants that only depend on non-significant parameters. The constants which are to play a more specific role in our reasoning will be labelled c 1 , c 2 ,...

For any k ∈ N, we will denote by C k,b (R; R) the space of functions on R which are k-times differentiable with bounded derivatives. For any γ ∈ (0, 1), C γ = C γ ([0, T ]; R d ) will stand for the set of (d-dimensional) γ-Hölder paths on [0, T ].

Setting

One of the technical advantages of dealing with the simple case of a stochastic heat equation on (0, 1) is a simplification in the functional analysis setting, based on rather elementary Fourier series considerations (notice in particular that the L p considerations of [START_REF] Deya | Non-linear rough heat equations[END_REF] can be avoided). We shall first detail this setting, and then recall some basic facts on equations driven by noisy signals and fractional Brownian motion. Throughout the section, we assume that a (finite) horizon T has been fixed for the equation. We shall denote by (y n ) n the (Fourier) decomposition of any function y ∈ B on this orthonormal basis. Sobolev spaces based on B are then easily characterized by means of Fourier coefficients. We label their definition for further use: Definition 2.1. For any α ≥ 0, we denote by B α the fractional Sobolev space of order α based on B, defined by

B α := y ∈ L 2 (0, 1) : ∞ n=1 λ 2α n (y n ) 2 < ∞ . ( 2
)
This space is equipped with its natural norm

y 2 Bα := ∆ α y 2 B = ∞ n=1 λ 2α n (y n ) 2 . We also set B ∞ = C(0, 1).
The above-defined fractional Sobolev spaces enjoy the following classical properties (see [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF]): Proposition 2.2. Let B α , B ∞ be the Sobolev spaces introduced at Definition 2.1. Then the following hold true: • Sobolev inclusions: If α > 1/4, then we have the continuous embedding

B α ⊂ B ∞ . (3)
• Algebra: If α > 1/4, then B α is a Banach algebra with respect to pointwise multiplication, or in other words

ϕ • ψ Bα ≤ ϕ Bα ψ Bα . (4)
• Composition:

If 0 ≤ α < 1/2, ϕ ∈ B α and f : R → R belongs to C 1,b , then f (ϕ) ∈ B α and f (ϕ) Bα ≤ c f {1 + ϕ Bα } . (5)
Here, f (ϕ) is naturally understood as f (ϕ)(ξ) := f (ϕ(ξ)).

Let now S t be the heat semigroup associated with ∆, and notice that if an element y ∈ L 2 (0, 1) can be decomposed as y = n≥1 y n e n , then S t y = n≥1 e -λnt y n e n . The general theory of fractional powers of operators provides us with sharp estimates for the semigroup S t (see for instance [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]):

Proposition 2.3. The heat semigroup S t satisfies the following properties:

• Contraction: For all t ≥ 0, α ≥ 0, S t is a contraction operator on B α .

• Regularization: For all t ∈ (0, T ], α ≥ 0, S t sends B on B α and

S t ϕ Bα ≤ c α,T t -α ϕ B . (6)
• Hölder regularity. For all t ∈ (0, T ], ϕ ∈ B α ,

S t ϕ -ϕ B ≤ c α,T t α ϕ Bα . (7)
2.2. Young convolutional integrals. The stochastic integrals involved in equation (1) will all be understood in the Young sense. In order to define them properly, let us first introduce some notation concerning Hölder type spaces in time. To begin with, for any α ≥ 0 and any subinterval I ⊂ [0, T ], set C 0 (I; B α ) for the space of continuous B α -valued functions on I, equipped with the supremum norm. Then Hölder spaces of B α -valued functions can be defined as follows: for κ ∈ (0, 1), set

C κ (I; B α ) := y ∈ C 0 (I; B α ) : sup s<t∈I y t -y s Bα |t -s| κ < ∞ .
Observe now that the definition of our stochastic integrals weighted by the heat semigroup will require the introduction of a small variant of those Hölder spaces (see [START_REF] Gubinelli | Rough evolution equations[END_REF][START_REF] Deya | Non-linear rough heat equations[END_REF] for further details): we define Ĉκ (I; B α ) as

Ĉκ (I; B α ) := y ∈ C 0 (I; B α ) : sup s<t∈I y t -S t-s y s Bα |t -s| κ < ∞ .
In order to avoid confusion, the natural norms on the spaces C κ (I; B α ), Ĉκ (I; B α ) are respectively denoted by N [•; C κ (I; B α )], N [•; Ĉκ (I; B α )], etc. For the sake of conciseness, we shall often write

C κ (B α ) (resp. Ĉκ (B α )) instead of C κ ([0, T ]; B α ) (resp. Ĉκ ([0, T ]; B α )).
We also need to introduce a family of spaces Ĉ0,κ (I; B κ ) in the following way:

More generally, for every λ ≥ κ,

N [y; C κ (I; B)] ≤ N [y; Ĉκ (I; B λ )] + c λ |I| λ-κ N [y; C 0 (I; B λ )]. (9)
Proof. Indeed, owing to [START_REF] Deya | Non-linear rough heat equations[END_REF], one has, for every s < t ∈ I,

y t -y s B ≤ y t -S t-s y s B + (S t-s -Id)y s B ≤ y t -S t-s y s B λ + c λ |t -s| λ y s B λ .
With those definitions in hand, the following proposition (borrowed from [START_REF] Deya | Non-linear rough heat equations[END_REF]) will be invoked in the sequel in order to give a meaning to our stochastic integrals weighted by the heat semigroup: Proposition 2.5. Consider a γ-Hölder real-valued function x defined on [0, T ]. Let I = [ℓ 1 , ℓ 2 ] be a subinterval of [0, T ] and fix κ ∈ [0, γ] such that γ + κ > 1. Suppose that z ∈ C 0 (I; B λ ) ∩ C κ (I; B λ-α ) for some parameters λ ≥ 0, 0 ≤ α ≤ min(κ, λ). Then, for every s < t ∈ I, the convolutional Riemann sum

t k ∈Π S t-t k+1 z t k+1 x t k+1 -x t k
converges in B λ as the mesh of the partition Π := {s = t 0 < t 1 < . . . < t n = t} tends to 0, and we denote the limit by

t s S t-u z u dx u .
Moreover, for every ϕ ∈ B λ , there exists a unique path y ∈ Ĉγ (I; B λ ) such that y ℓ 1 = ϕ and y t -S t-s y s = t s S t-u (z u ) dx u if s < t ∈ I. For this function, the following estimate holds:

N [y; Ĉγ (I; B λ )] ≤ c x γ N [z; C 0 (I; B λ )] + |I| κ-α N [z; C κ (I; B λ-α )] , (10)
for some constant c that only depends on (γ, κ, λ, α).

Malliavin calculus techniques.

This subsection is devoted to present the Malliavin calculus setting which we shall work in, having in mind the differentiability properties of the solution to (1).

2.3.1.

Wiener space associated to fBm. Let us first be more specific about the probabilistic setting in which we will work. For some fixed H ∈ (1/2, 1), we consider (Ω, F , P) the canonical probability space associated with the fractional Brownian motion with Hurst parameter H. That is, Ω = C 0 ([0, T ]; R d ) is the Banach space of continuous functions vanishing at 0 equipped with the supremum norm, F is the Borel sigma-algebra and P is the unique probability measure on Ω such that the canonical process B = {B t , t ∈ [0, T ]} is a d-dimensional fractional Brownian motion with Hurst parameter H, with covariance function

E B i t B j s = 1 2 t 2H + s 2H -|t -s| 2H 1 (i=j) , s, t ∈ [0, T ]. (11)
In particular, the paths of B are almost surely γ-Hölder continuous for all γ ∈ (0, H).

Consider then a fixed parameter H > 1/2, and let us start by briefly describing the abstract Wiener space introduced for Malliavin calculus purposes (for a more general and complete description, we refer the reader to [13, Section 3]).

Let (e 1 , . . . , e d ) be the canonical basis of R d , E be the set of R d -valued step functions on [0, T ] and H the completion of E with respect to the semi-inner product

1 [0,t] e i , 1 [0,s] e j H := R H (s, t) 1 (i=j) , s, t ∈ [0, T ].
Then, one constructs an isometry

K * H : H → L 2 ([0, T ]; R d ) such that K * H (1 [0,t] e i ) = 1 [0,t] K H (t, •) e i , where the kernel K = K H is given by K(t, s) = c H s 1 2 -H t s (u -s) H-3 2 u H-1 2 du and verifies that E[B i s B i t ] = s∧t 0 K(t, r)K(s, r
) dr, for some constant c H . Moreover, let us observe that K * H can be represented in the following form:

[K * H ϕ] t = T t ϕ r ∂ r K(r, t) dr = d H t H-1/2 I H-1/2 T - u H-1/2 ϕ t , (12)
where I α Tstands for the fractional integral of order α. The fractional Cameron-Martin space can be introduced in the following way: let

K H : L 2 ([0, T ]; R d ) → H H := K H (L 2 ([0, T ]; R d )) be the operator defined by [K H h](t) := t 0 K(t, s) h(s) ds, h ∈ L 2 ([0, T ]; R d ).
Then, H H is the Reproducing Kernel Hilbert space associated with the fractional Brownian motion B. Observe that, in the case of the classical Brownian motion, one has

that K(t, s) = 1 [0,t] (s), K * is the identity operator in L 2 ([0, T ]; R d
) and H H is the usual Cameron-Martin space.

In order to deduce that (Ω, H, P) defines an abstract Wiener space, we remark that H is continuously and densely embedded in Ω. In fact, one proves that the operator R H : H → H H given by

R H ψ := • 0 K(•, s)[K * ψ](s) ds
defines a dense and continuous embedding from H into Ω; this is due to the fact that R H ψ is H-Hölder continuous (for details, see [13, p. 400]).

Let us also recall that there exists a d-dimensional Wiener process W defined on (Ω, H, P) such that B can be expressed as

B t = t 0 K(t, r) dW r , t ∈ [0, T ]. ( 13
)
This formula will be referred to as Volterra's representation of fBm.

Malliavin calculus for B.

Let us introduce now the Malliavin derivative operator on the Wiener space (Ω, H, P). Namely, we first let S be the family of smooth functionals F of the form

F = f (B(h 1 ), . . . , B(h n )),
where h 1 , . . . , h n ∈ H, n ≥ 1, and f is a smooth function having polynomial growth together with all its partial derivatives. Then, the Malliavin derivative of such a functional F is the H-valued random variable defined by

DF = n i=1 ∂f ∂x i (B(h 1 ), . . . , B(h n ))h i .
For all p > 1, it is known that the operator D is closable from L p (Ω) into L p (Ω; H) (see e.g. [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]Chapter 1]). We will still denote by D the closure of this operator, whose domain is usually denoted by D 1,p and is defined as the completion of S with respect to the norm

F 1,p := (E(|F | p ) + E( DF p H )) 1
p . Sobolev spaces D k,p for any k ∈ N and p ≥ 1 can be defined in the same way, and we denote by D k,p loc the set of random variables F for which there exists a sequence

(Ω n , F n ) n≥1 ⊂ F × D k,p such that Ω n ↑ Ω a.s. and F = F n a.s. on Ω n . We also set D ∞ = ∩ k,p D k,p .
Remark 2.6. For F ∈ D 1,2 , one can write DF = d j=1 D j F e j , where D j F denotes the Malliavin derivative with respect to the j th component of B.

Since we deal with pathwise equations, we shall also be able to differentiate them in a pathwise manner. The relation between almost sure and Malliavin derivatives has been established by Kusuoka, and we quote it according to [

= DF (B)(R H h), h ∈ H, (14)
where we recall that D stands for the Malliavin derivative and D for the pathwise differentiation operator.

Stochastic analysis techniques are widely used in order study laws of random variables defined on a Wiener space. Let us recall the main criterions we shall use in this direction: Proposition 2.8. Let F be a real-valued random variable defined on (Ω, F , P). Then (i) If F ∈ D 1,p loc for p > 1 and DF H > 0 almost surely, then the law of F admits a density p with respect to Lebesgue measure.

(ii) If F ∈ D ∞ and E[ DF -p
H ] is finite for all p ≥ 1, then the density p of F is infinitely differentiable.

Existence of the density in the case of Nemytskii-type vector fields

In this section, we first consider a general equation of the form

y t = S t ϕ + t 0 S t-u (F i (y) u ) dx i u , ϕ ∈ L 2 (0, 1) , t ∈ [0, T ], (15)
driven by a d-dimensional noise x = (x 1 , . . . , x d ) considered as a C γ function with γ ∈ (1/2, 1). We shall be able to handle the general case of a perturbation involving Nemytskii operators, that is

F i (ϕ)(ξ) := f i (ϕ(ξ)) , ϕ ∈ B , ξ ∈ (0, 1).
for smooth enough functions f i : R → R, i = 1, . . . , d. Thus, Equation ( 15) can here be written as

y t = S t ϕ + t 0 S t-u (f i (y u )) dx i u , ϕ ∈ B , t ∈ [0, T ], (16)
or equivalently, in a multiparameter setting,

y(t, ξ) = 1 0 G t (ξ, η)ϕ(η) dη + 1 0 t 0 G t-u (ξ, η)f i (y(u, η)) dx i u dη , t ∈ [0, T ], ξ ∈ (0, 1),
where G t stands for the heat kernel on (0, 1) associated with Dirichlet boundary conditions.

It is readily checked that if each f i belongs to C 1,b (R; R) and y ∈ Ĉ0,κ (B κ ) for some κ ∈ (max(1γ, 1/4), 1/2), then the integral in the right-hand-side of ( 16) can be interpreted with Proposition 2.5. Indeed, owing to (5), we know that f (y) ∈ C 0 (B κ ), while, due to the embedding [START_REF] Friz | Rough path stability of SPDEs arising in non-linear filtering[END_REF],

N [f (y); C κ (B)] ≤ f ′ ∞ N [y; C κ (B)] ≤ c f ′ ∞ N [y; Ĉ0,κ (B κ )] < ∞.
For the remainder of the section, we shall rely on the following regularity assumptions.

Hypothesis 1. We consider κ ∈ (max(1γ, 1/4), 1/2) and an initial condition ϕ ∈ B κ . The family of functions {f 1 , . . . ,

f d } is such that f i is an element of C 3,b (R; R) for i = 1, . . . , d.
In this context, the following existence and uniqueness result has been proven in [START_REF] Deya | Non-linear rough heat equations[END_REF]Theorem 3.10]: Proposition 3.1. Under Hypothesis 1, Equation ( 16) interpreted with Proposition 2.5 admits a unique solution y ∈ Ĉ0,κ (B κ ), where we recall that the space Ĉ0,κ (B κ ) has been defined at Lemma 2.4.

As a preliminary step towards Malliavin differentiability of the solution to (1), we shall study the dependence on x of the deterministic equation (16).

3.1. Differentiability with respect to driving noise. For equation ( 16), consider the application Φ :

C γ → Ĉ0,κ (B κ ), x → y, (17)
for a given initial condition ϕ. We shall elaborate on the strategy designed in [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF] in order to differentiate Φ. Let us start with a lemma on linear equations: Lemma 3.2. Suppose that (x, y) ∈ C γ × Ĉ0,κ (B κ ) and fix t 0 ∈ [0, T ]. Then for every w ∈ Ĉ0,κ ([t 0 , T ]; B κ ), the equation

v t = w t + t t 0 S t-u (f ′ i (y u ) • v u ) dx i u , t ∈ [t 0 , T ], (18)
admits a unique solution v ∈ Ĉ0,κ ([t 0 , T ]; B κ ), and one has

N [v; Ĉ0,κ ([t 0 , T ]; B κ )] ≤ C x,y,T • N [w; Ĉ0,κ ([t 0 , T ]; B κ )], (19)
where C x,y,T := C( x γ , N [y; Ĉ0,κ (B κ )], T ) for some function C : (R + ) 3 → R + growing with its arguments.

Proof. The existence and uniqueness of the solution stem from the same fixed-point argument as in the proof of Proposition 3.1 (see [START_REF] Deya | Non-linear rough heat equations[END_REF]Theorem 3.10]), and we only focus on the proof of (19). Let I = [ℓ 1 , ℓ 2 ] be a subinterval of [t 0 , T ]. One has, according to Proposition 2.5,

N [v; Ĉκ (I; B κ )] ≤ N [w; Ĉκ (B κ )] + c |I| γ-κ x γ N [f ′ i (y) • v; C 0 (I; B κ )] + N [f ′ i (y) • v; C κ (I; B)] . (20
) Now, by using successively (4) and ( 5), we get

N [f ′ i (y) • v; C 0 (I; B κ )] ≤ cN [f ′ i (y); C 0 (B κ )]N [v; C 0 (I; B κ )] ≤ c 1 + N [y; C 0 (B κ )] N [v; C 0 (I; B κ )],
while, owing to (8) and (3),

N [f ′ i (y) • v; C κ (I; B)] ≤ N [f ′ i (y); C κ (I; B)]N [v; C 0 (I; B ∞ )] + N [f ′ i (y); C 0 (I; B ∞ )]N [v; C κ (I; B)] ≤ c 1 + N [y; Ĉ0,κ (B κ )] N [v; Ĉ0,κ (I; B κ )].
Going back to (20), these estimates lead to

N [v; Ĉκ (I; B κ )] ≤ N [w; Ĉκ (B κ )] + c x,y |I| γ-κ N [v; Ĉ0,κ (I; B κ )],
and hence

N [v; Ĉ0,κ (I; B κ )] ≤ v ℓ 1 Bκ + cN [w; Ĉκ (B κ )] + c x,y |I| γ-κ N [v; Ĉ0,κ (I; B κ )].
Control ( 19) is now easily deduced with a standard patching argument.

The following lemma on flow-type linear equations will also be technically important for our computations below. Lemma 3.3. Fix (x, y) ∈ C γ × Ĉ0,κ (B κ ) and for every u ∈ [0, T ], consider the system of equations

Ψ i t,u = S t-u (f i (y u )) + t u S t-w (f ′ j (y w ) • Ψ i w,u ) dx j w , t ∈ [u, T ] , i ∈ {1, . . . , m}. (21)
Then, for every i ∈ {1, . . . , m} and t ∈ [0, T ], the mapping u → Ψ i t,u is continuous from [0, t] to B κ . In particular, for every ξ ∈ (0, 1), u

→ Ψ i t,u (ξ) is a continuous function on [0, t]. Proof. Let us fix i ∈ {1, . . . , m}, t ∈ [0, T ]. For any 0 ≤ u < v ≤ t, set Γ i v,u (s) := Ψ i s,v -Ψ i s,u , s ∈ [v, T ]. It is easy to check that Γ i v,u is solution of the equation on [v, T ] Γ i v,u (s) = S s-v (Ψ i v,v -Ψ i v,u ) + s v S s-w (f ′ j (y w ) • Γ i v,u (w)) dx j w .
Therefore, according to the estimate (19),

Ψ i t,v -Ψ i t,u Bκ = Γ i v,u (t) Bκ ≤ N [Γ i v,u ; C 0 (|v, T ]; B κ )] ≤ c x,y,T Ψ i v,v -Ψ i v,u Bκ . Now, observe that Ψ i v,v -Ψ i v,u = f i (y v ) -S v-u (f i (y u )) + v u S v-w (f ′ j (y w ) • Ψ i w,u ) dx j w ,
and since y ∈ Ĉ0,κ (B κ ), it becomes clear that Ψ

i v,v -Ψ i v,u Bκ v→u -→ 0.
We now show how to differentiate a function which is closely related to equation (16).

Lemma 3.4. The application F :

C γ × Ĉ0,κ (B κ ) → Ĉ0,κ (B κ ) defined by F (x, y) t := y t -S t ϕ - t 0 S t-u (f i (y u )) dx i u ,
is differentiable in the Fréchet sense and denoting by D 1 F (resp. D 2 F ) the derivative of F with respect to x (resp. y), we obtain

D 1 F (x, y)(h) t = - t 0 S t-u (f i (y u )) dh i u , (22)
D 2 F (x, y)(v) t = v t - t 0 S t-u (f ′ i (y u ) • v u ) dx i u . (23)
Besides, for any x ∈ C γ , the mapping D 2 F (x, Φ(x)) is a homeomorphism of Ĉ0,κ (B κ ).

Proof. One has, for every h ∈ C γ , v ∈ Ĉ0,κ (B κ ),

F (x + h, y + v) t -F (x, y) t = v t - t 0 S t-u (f ′ i (y u ) • v u ) dx i u - t 0 S t-u (f i (y u )) dh i u -R 1 t (v) + R 2 t (h, v) , (24)
with

R 1 t (v) := t 0 S t-s z i s dx i s , z i s := 1 0 dr 1 0 dr ′ r f ′′ i (y s + rr ′ v s ) • v 2 s , R 2 t (v, h) := t 0 S t-s zi s dh i s , zi s := 1 0 dr f ′ i (y s + rv s ) • v s ,
and we now have to show that

N [R 1 . (v) + R 2 . (h, v); Ĉ0,κ (B κ )] = o h 2 γ + N [v; Ĉ0,κ (B κ )] 2 1/2 . Observe first that N [R 1 . (v); Ĉ0,κ (B κ )] ≤ cN [R 1 . (v); Ĉκ (B κ )].
Thanks to (4) and ( 5), we get

z s Bκ ≤ c [0,1] 2 drdr ′ f ′′ i (y s + rr ′ v s ) Bκ v s 2 Bκ ≤ c {1 + y s Bκ + v s Bκ } v s 2 Bκ .
Besides, owing to ( 3) and ( 8) and setting

M ts (r, r ′ ) = f ′′ i (y t + rr ′ v t ) -f ′′ i (y s + rr ′ v s ) we end up with z t -z s B ≤ [0,1] 2 drdr ′ M ts (r, r ′ ) B v s 2 B∞ + c v t -v s B { v t B∞ + v s B∞ } ≤ c { y t -y s B + v t -v s B } v s 2 Bκ + c v t -v s B { v t Bκ + v s Bκ } . ≤ c |t -s| κ N [y; Ĉ0,κ (B κ )] + N [v; Ĉ0,κ (B κ )] N [v; C 0 (B κ )] 2 + N [v; Ĉ0,κ (B κ )]N [v; C 0 (B κ )] .
The estimate [START_REF] Gubinelli | Rough evolution equations[END_REF] for the Young convolutional integral now provides us with the expected control

N [R 1 . (v); Ĉ0,κ (B κ )] = O(N [v; Ĉ0,κ (B κ )] 2
). In the same way, one can show that

N [R 2 . (h, v); Ĉ0,κ (B κ )] = O( h γ • N [v; Ĉ0,κ (B κ )]
), and the differentiability of F is thus proved. Of course, the two expressions ( 22) and ( 23) for the partial derivatives are now easy to derive from (24). As for the bijectivity of D 2 F (x, Φ(x)), it is a consequence of Lemma 3.2.

We are now ready to differentiate the application Φ defined by (17): Proposition 3.5. The map Φ : C γ → Ĉ0,κ (B κ ) is differentiable in the Fréchet sense. Moreover, for every x ∈ C γ and h ∈ C ∞ , the following representation holds: if t ∈ [0, T ], ξ ∈ (0, 1),

DΦ(x)(h) t (ξ) = t 0 Ψ i t,u (ξ) dh i u , (25)
where Ψ i t,. ∈ C([0, t]; B κ ) is defined through the equation

Ψ i t,u = S t-u (f i (Φ(x) u )) + t u S t-w (f ′ j (Φ(x) w ) • Ψ i w,u ) dx j w . (26)
Proof. Thanks to Lemma 3.4, the differentiability of Φ is a consequence of the implicit function theorem, which gives in addition

DΦ(x) = -D 2 F (x, Φ(x)) -1 • D 1 F (x, Φ(x)), x ∈ C γ .
In particular, for every x, h ∈ C γ , z := DΦ(x)(h) is the (unique) solution of the equation

z t = t 0 S t-u (f i (Φ(x) u )) dh i u + t 0 S t-u (f ′ i (Φ(x) u ) • z u ) dx i u , t ∈ [0, T ]. ( 27
)
If x ∈ C γ and h ∈ C ∞ , an application of the Fubini theorem shows (as in the proof of [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF]Proposition 4]) that the path zt := t 0 Ψ i t,u dh i u (which is well-defined thanks to Lemma 3.3) is also solution of ( 27), and this provides us with the identification (25).

As the reader might expect, one can obtain derivatives of any order for the solution when the coefficients of equation ( 16) are smooth: Proposition 3.6. Suppose that f i ∈ C ∞,b (R; R) for every i ∈ {1, . . . , m}. Then the function Φ : C γ → Ĉ0,κ (B κ ) defined by (17) is infinitely differentiable in the Fréchet sense. Moreover, for every n ∈ N * and every x, h 1 , . . . , h n ∈ C γ , the path z t := D n Φ(x)(h 1 , . . . , h n ) satisfies a linear equation of the form

z t = w t + t 0 S t-u (f ′ i (Φ(x) u ) • z u ) dx i u , t ∈ [0, T ], ( 28
)
where w ∈ Ĉ0,κ (B κ ) only depends on x, h 1 , . . . , h n .

Proof. The details of this proof are omitted for the sake of conciseness, since they simply mimic the formulae contained in the proof of [START_REF] Nualart | Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion[END_REF]Proposition 5]. As an example, let us just observe that for x, h, k ∈ C γ , the path z t := D 2 Φ(x)(h, k) t is the unique solution of (28) with

w t := t 0 S t-u (f ′ i (Φ(x) u ) • DΦ(x)(h) u ) dk i u + t 0 S t-u (f ′ i (Φ(x) u ) • DΦ(x)(k) u ) dh i u + t 0 S t-u (f ′′ i (Φ(x) u ) • DΦ(x)(h) u • DΦ(x)(k) u ) dx i u .
3.2. Existence of the density. We will now apply the results of the previous section to an evolution equation driven by a fractional Brownian motion B = (B 1 , . . . , B d ) with Hurst parameter H > 1/2. Namely, we fix κ ∈ (max(1/4, 1γ), 1/2) and an initial condition ϕ ∈ B κ . We also assume that f i ∈ C 3,b (R; R) for i = 1, . . . , m. We denote by Y = Φ(B) the solution of

Y t = S t ϕ + t 0 S t-u (f i (Y u )) dB i u , t ∈ [0, T ]. (29)
Notice that since H > 1/2 the paths of B are almost surely γ-Hölder continuous with Hölder exponent greater than 1/2. Thus, equation (29) can be solved by a direct application of Proposition 3.1. Moreover, one can invoke Proposition 3.5 in order to obtain the Malliavin differentiability of Y t (ξ):

Lemma 3.7. For every t ∈ [0, T ], ξ ∈ (0, 1), Y t (ξ) ∈ D 1,2 loc and one has, for any

h ∈ H, D(Y t (ξ)), h H = DΦ(B)(R H h) t (ξ). ( 30
)
Proof. According to [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], we have that

D(Y t (ξ)), h H = D(Y t (ξ))(R H h) = d dε|ε=0 Φ(B + εR H h) t (ξ).
Furthermore, Proposition 3.5 asserts that Φ :

C γ → Ĉ0,κ (B κ ) is differentiable. Therefore 1 ε [Φ(x + εR H h) t (ξ) -Φ(x) t (ξ)] = DΦ(x)(R H h) t (ξ) + 1 ε R(εR H h) t (ξ), with |R(εR H h) t (ξ)| ≤ N [R(εR H h); C 0 (B ∞ )] ≤ cN [R(εR H h); C 0 (B κ )] = o(ε),
and hence d dε |ε=0 Φ(B + εR H h) t (ξ) = DΦ(x)(R H h) t (ξ), which trivially yields both the inclusion Y t (ξ) ∈ D 1,2
loc and expression (30).

With this differentiation result in hand plus some non degeneracy assumptions, we now obtain the existence of a density for the random variable Y t (ξ): Theorem 3.8. Suppose that for all λ ∈ R, there exists i ∈ {1, . . . , d} such that f i (λ) = 0. Then for all t ∈ (0, 1] and ξ ∈ (0, 1), the law of Y t (ξ) is absolutely continuous with respect to Lebesgue measure.

Proof. We apply Proposition 2.8 part (i), and we will thus prove that D(Y t (ξ)) H > 0 almost surely. Assume then that D(Y t (ξ)) H = 0. In this case, owing to (30), we have DΦ(B)(R H h) t (ξ) = 0 for every h ∈ H. In particular, due to (25), one has

t 0 Ψ i t,u (ξ) dh i u = 0 for every h ∈ C ∞ . As u → Ψ i t,u (ξ)
is known to be continuous, it is easily deduced that Ψ i t,u (ξ) = 0 for every u ∈ [0, t] and every i ∈ {1, . . . , d}, and so 0 = Ψ i t,t (ξ) = f i (Y t (ξ)) for every i ∈ {1, . . . d}, which contradicts our non-vanishing hypothesis.

Smoothness of the density in the case of regularizing vector fields

Up to now, we have been able to differentiate the solution to (16) when the coefficients are fairly general Nemytskii operators. However, we have only obtained the inclusion Y t (ξ) ∈ D 1,2 loc . Additional problems arise when one tries to prove Y t (ξ) ∈ D 1,2 , due to bad behavior of linear equations driven by rough signals in terms of moment estimates. This is why we shall change our setting here, and consider an equation of the following type

y t = S t ϕ + t 0 S t-u (L(f i (y u ))) dx i u , t ∈ [0, T ] , ϕ ∈ B, (31)
where x ∈ C γ ([0, T ]; R d ) with γ > 1/2, each f i : R → R (i ∈ {1, . . . , d}) is seen as a Nemytskii operator (see the beginning of Section 3), and L stands for a regularizing linear operator of B. Let us be more specific about the assumptions in this section:

Hypothesis 2. We assume that for every i ∈ {1, . . . , d}, f i is infinitely differentiable with bounded derivatives. Moreover, the operator L : B → B is taken of the form

L(φ)(ξ) := 1 0 dη U(ξ, η)φ(η),
for some positive kernel U such that: (i) U is regularizing, i.e., L is continuous from B to B λ for every λ ≥ 0, and (ii) one has c U := min ξ∈(0,1)

1 0 dη U(ξ, η) > 0.
In other words, we are now concerned with the following equation on [0, T ] × (0, 1):

y(t, ξ) = 1 0 G t (ξ, η)ϕ(η) dη + 1 0 1 0 t 0 G t-u (ξ, η)U(η, µ)f i (y(u, µ)) dx i u dµdη,
with U satisfying the above conditions (i)-(ii). This setting covers for instance the case of an (additional) heat kernel U = G ε on (0, 1) for any fixed ε > 0. The following existence and uniqueness result then holds true: For the sake of clarity, we henceforth assume that T = 1. The generalization to any (fixed) horizon T > 0 easily follows from slight modifications of our estimates.

Moreover, for some technical reasons that will arise in the proofs of Propositions 4.2 and 4.3, we will focus on the case λ = 2 + γ in the statement of Proposition 4.1. In other words, from now on, we fix the initial condition ϕ in the space B 2+γ . 4.1. Estimates on the solution. Under our new setting, let us find an appropriate polynomial control on the solution to (31) in terms of x. Proposition 4.2. Suppose that y is the solution of (31) in Ĉγ (B 2+γ ) with initial condition ϕ. Then there exists a constant C γ,f,L such that

N [y; Ĉγ ([0, 1]; B 2+γ )] ≤ C γ,f,L (1 + x γ ) max x 1/γ γ , ϕ 1/2 B 2+γ 1-γ . ( 33
)
Proof. For any N ∈ N * , let us introduce the two sequences

ε k = ε N,k := 1 N + k , ℓ 0 := 0 , ℓ k+1 = ℓ N k+1 := ℓ N k + ε N,k .
The first step of the proof consists in showing that we can pick N such that for every k,

ε 2 k y ℓ k B 2+γ ≤ 1. ( 34
)
For the latter control to hold at time 0 (i.e., for k = 0), we must first assume that N ≥ ϕ 1/2 B 2+γ . Now, observe that for any k, one has, owing to [START_REF] Gubinelli | Rough evolution equations[END_REF],

N [y; Ĉγ ([ℓ k , ℓ k+1 ]; B 2+γ )] ≤ c x γ N [L(f (y)); C 0 ([ℓ k , ℓ k+1 ]; B m 2+γ )] + ε γ k N [L(f (y)); C γ ([ℓ k , ℓ k+1 ]; B m 2+γ )] (35) ≤ c x γ L L(B,B 2+γ ) {1 + ε γ k N [y; C γ ([ℓ k , ℓ k+1 ]; B)]} ≤ c x γ 1 + ε γ k N [y; Ĉγ ([ℓ k , ℓ k+1 ]; B 2+γ )] + ε γ+2 k N [y; C 0 ([ℓ k , ℓ k+1 ]; B 2+γ )] ≤ c 1 x γ 1 + ε γ k N [y; Ĉγ ([ℓ k , ℓ k+1 ]; B 2+γ )] + ε γ+2 k y ℓ k B 2+γ ,
where we have used [START_REF] Gubinelli | Young integrals and SPDEs[END_REF] to get the third inequality. Consequently, if we take N such that 2c 1 N -γ x γ ≤ 1 (i.e. N ≥ (2c 1 x γ ) 1/γ ), we retrieve

N [y; Ĉγ ([ℓ k , ℓ k+1 ]; B 2+γ )] ≤ 2c 1 x γ + ε 2 k y ℓ k B 2+γ (36)
and hence

y ℓ k+1 B 2+γ ≤ 1 + (1 + ε 2+γ k ) y ℓ k B 2+γ . From this estimate, if we assume that ε 2 k y ℓ k B 2+γ ≤ 1, then ε 2 k+1 y ℓ k+1 B 2+γ ≤ ε 2 k+1 + ε 2 k+1 y ℓ k B 2+γ + ε γ k ≤ 2ε 2 k+1 + ε 2 k+1 ε 2 k = 2 + (N + k) 2 (N + k + 1) 2 ≤ 1
and (34) is thus proved by induction. Going back to (36), we get, for every k,

N [y; Ĉγ ([ℓ k , ℓ k+1 ]; B 2+γ )] ≤ 2c 1 x γ + 1. (37)
By a standard patching argument, this estimate yields

N [y; Ĉγ ([0, 1]; B 2+γ )] ≤ {2c 1 x γ + 1} K 1-γ ,
where K stands for the smallest integer such that K

k=0 ε k ≥ 1. Finally, observe that 2 ≥ K k=0 ε k = N +K k=N 1
k , and thus one can check that K ≤ (e 2 -1)N ≤ 7N. To achieve the proof of (33), it now suffices to notice that N can be picked proportional to max( x

1/γ γ , ϕ 1/2 B 2+γ ).
We now consider a linear equation, which is equivalent to (18) in our regularized context:

z t = w t + t 0 S t-u (L(f ′ i (y u ) • z u )) dx i u , t ∈ [0, 1], (38)
where w ∈ Ĉγ (B 2+γ ) and y stands for the solution of (31) with initial condition ϕ ∈ B 2+γ . The existence and uniqueness of a solution for (38) can be proved along the same lines as Proposition 4.1, that is to say via a fixed-point argument. We shall get a suitable exponential control for this solution.

Proof. The absolute continuity of the law of Y 1 (ξ) can be obtained by following the lines of Section 3, which gives Y 1 (ξ) ∈ D ∞ loc as well. Then, like in Proposition 3.6, observe that n-th (Fréchet) derivatives Z n of the flow associated with (43) satisfy a linear equation of the form

Z n t = W n t + t 0 S t-u (L(f ′ i (Y u ) • Z n u )) dB i u , t ∈ [0, 1].
The explicit expression for W n (n ≥ 1) can be derived from the formulae contained in [13, Proposition 5], and it is easy to realize that due to (33), one has N [W n ; Ĉ0,γ (B 2+γ )] ∈ L p (Ω) for any n and any p. Then, thanks to (39), we deduce that N [Z n ; C 0 (B 2+γ )] is a square-integrable random variable, which allows us to conclude that Y 1 (ξ) ∈ D ∞ (see [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]Lemma 4.1.2]).

The following proposition, which can be seen as an improvement of Lemma 3.3 (in this regularized situation), provides us with the key-estimate to prove the smoothness of the density: Proposition 4.6. For every s ∈ [0, 1], consider the system of equations

Ψ i t,s = S t-s (L(f i (Y s ))) + t s S t-u (L(f ′ j (Y u ) • Ψ i u,s )) dB j u , t ∈ [s, 1] , i ∈ {1, . . . , m}.
(44) Then, for every i ∈ {1, . . . , m} and every t ∈ [0, 1], Ψ i t,. ∈ C γ ([0, t]; B 2+γ ). In particular, for any ξ ∈ (0, 1), Ψ i t,. (ξ) ∈ C γ ([0, t]). Moreover, one has the following estimate

N [Ψ i t,. ; C γ ([0, t]; B 2+γ )] ≤ Q( ϕ B 2+γ , B γ ) • exp c max ϕ 1/2 B 2+γ , B 1/γ γ , (45)
for some polynomial expression Q.

Proof. As in the proof of Lemma 3.3, we introduce the path

Γ i v,u (s) := Ψ i s,v -Ψ i s,u , s ∈ [v, 1] , 0 ≤ u < v ≤ t,
and it is readily checked that Γ i v,u solves the equation on [v, 1]

Γ i v,u (s) = S s-v (Ψ i v,v -Ψ i v,u ) + s v S s-w (L(f ′ j (Y w ) • Γ i v,u (w))) dB j w .
Therefore, thanks to the estimate (39), we get

Ψ i t,v -Ψ i t,u B 2+γ = Γ i v,u (t) B 2+γ ≤ N [Γ i v,u ; C 0 ([v, 1]; B 2+γ )] ≤ c Ψ i v,v -Ψ i v,u B 2+γ exp c max ϕ 1/2 B 2+γ , B 1/γ γ . (46)
Then, by writing

Ψ i v,v -Ψ i v,u = L(f i (Y v )-f i (Y u ))-[S v-u -Id] (L(f i (Y u )))- v u S v-w (L(f ′ j (Y w )•Ψ i w,u )) dB j w ,
we deduce that Proposition 4.6 implies in particular that the Young integral t 0 Ψ i t,u (ξ) dh i u is welldefined for every h ∈ C γ , t ∈ [0, 1] and ξ ∈ (0, 1). We are thus in a position to apply the Fubini-type argument of [13, Propositions 4 and 7] so as to retrieve the following convenient expression for the Malliavin derivative: Corollary 4.7. For every ξ ∈ (0, 1), the Malliavin derivative of Y 1 (ξ) is given by

Ψ i v,v -Ψ i v,
D i s (Y 1 (ξ)) = Ψ i 1,s (ξ) , s ∈ [0, 1] , i ∈ {1, . . . , m}, (47)
where Ψ i .,s stands for the solution of (44) on [s, 1]. Theorem 4.8. Suppose that there exists λ 0 > 0 such that for every i ∈ {1, . . . , m} and every η ∈ R, f i (η) ≥ λ 0 . Then, for every ξ ∈ (0, 1), the density of Y 1 (ξ) with respect to the Lebesgue measure is infinitely differentiable.

Proof. We shall apply here the criterion stated at Proposition 2.8 item (ii). Notice that we already know that Y 1 (ξ) ∈ D ∞ , so it remains to show that for every p ≥ 2, there exists ε 0 (p) > 0 such that if ε < ε 0 (p), then P ( D . (Y 1 (ξ)) H < ε) ≤ ε p .

To this end, we resort to the following practical estimate, borrowed from [2, Corollary 4.5]: for every β > H -1/2, there exist α > 0 such that

P ( D . (Y 1 (ξ)) H < ε) ≤ P ( D . (Y 1 (ξ)) ∞ < ε α ) + P D . (Y 1 (ξ)) β > ε -α . ( 48
)
The first term in the right-hand-side of (48) is easy to handle. Indeed, owing to the expression (47) for the Malliavin derivative of Y 1 (ξ), one has (Y 1 (ξ)) β ∈ L q (Ω) for every q ≥ 1.

D . (Y 1 (ξ)) ∞ ≥ inf

2. 1 .

 1 Fractional Sobolev spaces. As mentioned above, we are working here with the heat equation in the Hilbert space B := L 2 (0, 1) with Dirichlet boundary conditions.The Laplace operator ∆ on B can be diagonalized in the orthonormal basis e n (ξ) := √ 2 sin(πnξ) (n ∈ N * ), with eigenvalues λ n := π 2 n 2 .

Lemma 2 . 4 .

 24 For any κ ∈ (0, 1) and any subinterval I ⊂ [0, T ], let Ĉ0,κ (I; B κ ) be the space associated with the norm N [•; Ĉ0,κ (I; B κ )] := N [•; C 0 (I; B κ )] + N [•; Ĉκ (I; B κ )]. Then the following continuous embedding holds true: Ĉ0,κ (I; B κ ) ⊂ C κ (I; B).

Proposition 4 . 1 .

 41 Under Hypothesis 2, for any λ ≥ γ and any initial condition ϕ ∈ B λ , Equation (31) interpreted with Proposition 2.5 admits a unique solution in Ĉγ (B λ ).Proof. As in the proof of Proposition 3.1, the result can be obtained with a fixed-point argument. Observe indeed that if y ∈ Ĉγ (I;B λ ) (I := [ℓ 1 , ℓ 2 ] ⊂ [0, 1]) and z is the path defined by z ℓ 1 = y ℓ 1 , z t -S t-s z s = t s S t-u (L(f i (y u ))) dx i u (s < t ∈ I),then, according to Proposition 2.5, z ∈ Ĉγ (I; B λ ) and one has N [z; Ĉγ (I; B λ )] ≤ c x γ N [L(f (y)); C 0 (I; B m λ )] + |I| γ N [L(f (y)); C γ (I; B m λ )] . (32) Now, owing to the regularizing effect of L, it follows that N [L(f (y)); C 0 (I; Bm λ )] ≤ L L(B,B λ ) f ∞ and N [L(f (y)); C γ (I; B m λ )] ≤ L L(B,B λ ) f ′ ∞ N [y; C γ (I; B)] ≤ cN [y; Ĉ0,γ (I; B λ )],which, together with (32), allows to settle the fixed-point argument.

  u B 2+γ ≤ c |v -u| γ L L(B,B 2+γ ) N [Y ; C γ (B)] + L L(B,B 2+2γ ) + L L(B,B 2+γ ) B γ N [f ′ (Y ) • Ψ i .,u ; C 0 (B m )] + N [f ′ (Y ) • Ψ i .,u ; C γ (B m )] ≤ c |v -u| γ {1 + B γ } {1 + N [Y ; C γ (B)]} 1 + N [Ψ i .,u ; C 0 (B 2+γ )] + N [Ψ i .,u ; C γ (B)]. Going back to (46), the result now easily follows from the embedding Ĉ0,γ (B 2+γ ) ⊂ C γ (B) and the three controls (33), (39) and (40).

  ξ, η)f i (Y 1 (η)) ≥ c U λ 0 > 0 (remember that U and c U have been defined in Hypothesis 2), so thatP( D . (Y 1 (ξ)) ∞ < ε α ) = 0 for ε small enough.Then, in order to cope with P ( D . (Y 1 (ξ)) β > ε -α ), one can simply rely on the Markov inequality, since, according to (45),D . (Y 1 (ξ)) β = Ψ 1,. (ξ) β ≤ c sup i∈{1,...,m} N [Ψ i 1,. ; C γ ([0, 1]; B 2+γ )] ≤ c Q ϕ B 2+γ , B γ • exp c max ϕ B 2+γ , B 1/γ γ ,which proves that D .

  12, Proposition 4.1.3]. Proposition 2.7. A random variable F is said to be H-differentiable if for almost all ω ∈ Ω and for any h ∈ H, the map ν → F (ω + νR H h) is differentiable. Those random variables belong to the space D 1,p loc , for any p > 1. Moreover, the following relation holds true: DF, h H

is partially supported by the (French) ANR grant ECRU.

Proposition 4.3. There exists constants C 1 , C 2 which only depend on f , L and γ such that

Moreover, if w t = S t ψ for some function ψ ∈ B 2+γ , there exists an additional constant C 3 which only depend on f , L and γ such that

Proof. We go back to the notation ε N,k , ℓ N k of the proof of Proposition 4.2, and set, for every c ≥ 0,

We have seen in the proof of Proposition 4.2 that there exists a constant c 1 such that for every N ≥ N(c 1 ) and every k, one has simultaneously

Suppose that N ≥ N(c 1 ) and set N w := N [w; Ĉγ (B 2+γ )]. One has, similarly to (35),

where we have used (3) and ( 9) to derive the last inequality. Therefore, if we choose N 2 ≥ max N(c 1 ), (2c 2 x γ ) 1/γ , one has, for any N ≥ N 2 and any k, 9) and (41), we know that

As a consequence, there exists c 3 such that for any

Pick now an integer N 3 ≥ N 2 such that

and we get, for any k,

, where K(N 3 ) stands for the smallest integer such that

As in the proof of Proposition 4.2, one can check that K(N 3 ) ≤ cN 3 . In order to get (39), it suffices to observe that there exists a constant c 4 such that any integer

Suppose now that w t = S t ψ. In particular, N w = 0. Then we go back to (42) to obtain, thanks to (39),

is thus proved.

Remark 4.4. For any t 0 ∈ [0, 1], the proof of Proposition 4.3 can be easily adapted to the equation starting at time t 0

and both estimates (39) and (40) remain of course true in this situation.

Smoothness of the density.

Let us now go back to the fractional Brownian situation

where γ ∈ ( 1 2 , H) is a fixed parameter. We suppose, for the rest of the section, that the initial condition ϕ is fixed in B 2+γ and that Hypothesis 2 is satisfied. We denote by Y the solution of (43) in Ĉγ (B 2+γ ) given by Proposition 4.1.

As in Subsection 3.2, we wish to study the law of Y t (ξ) for t ∈ [0, 1] and ξ ∈ (0, 1). Without loss of generality, we focus more exactly on the law of Y 1 (ξ), for ξ ∈ (0, 1).

The first thing to notice here is that the whole reasoning of Section 3 can be transposed without any difficulty to Equation (43), which is more easy to handle due to the regularizing effect of L. Together with the estimates (33) and (39), this observation leads us to the following statement: Proposition 4.5. For every ξ ∈ (0, 1), Y 1 (ξ) ∈ D ∞ and the law of Y 1 (ξ) is absolutely continuous with respect to the Lebesgue measure.


Enrique Wulff Barreiro

Emilia Curras

Enrique Wulff

Spaniards in U.S. Laboratories in the early 80s'

Mutation Carners : led by

INTRODUCTION

In a first approach, the integrated interaction of the viral and cellular genomes is due to L.A. Zilber (1894-1966),1 the theory of oncogenes was originated by R. Huebner (1914-1998) and G. Todaro (1937 -),2 H. TellÚn (1934-1994)3 elaborated the theory of the provirus, D. Baltimore (1938-) isolated the enzyme transcriptase inverse, and M. Barbacid (1949-) activated the first oncogene. [START_REF]Studies in scíenlometrics, part 2 , The reta/ion between source aUlhor and cíted author populations[END_REF] The most accurate reconstruction from the tme ancestral eaneer immunologist Zilber, may take into account of the importance of the use of cell lines free of sarcoma virus in mutagenic experiments.

In 1969, was segregated the third line 3T3, from the miee embryos in the Swiss/NIH lineo This line was to be called NIH 3T3 (after the name of the Natíonal lnstÍtute of Health where they proliferated), One of the authors was Todaro. The cells gave 20-25 steps to be cloned, until achieving the clone (clone 6) with a particularly high efficieney and low saturation density.5 Later the NIH 3T3 was used mainly to study the viral transformation in tissue-and cell-cultures. The selection of this line for transfection experiments is conditioned by a high efficiency in the integration and fixation of the exogenous DNA in the genome of the NIH 3T3 cells.

The considerable interest that had the discovery that an ample variety of tumor cells contained activated ras genes, was detectable Lev Zilber (1894-1966) contribution, L. Silber, 'Ucber das Wesen der Weil Felixschen Reaktion', Centra/bl.f Bah erc. l. Abr. Originale. (1923). 89, 250-9, where is proved the serological modification experienced by the bacteria Proteus vulgaris, hereditary transforrncd iuto a variant that is agglutinated by the serum of the guinea pig infected by louse-borne typhus virus. lt was a precedent of the experiments by Griffith on the transformation of pneumococci by way of transfer from a virulent culture, see F. Griffith, 'The Significance of Pneumococcal Types', Journal of Hygiene (1928), 27,113. 2 RJ. Huebner, OJ. Todaro, 'Oncogenes of RNA tumor viruses as deterrninants of cancer', Proc Natf Acad Sci U S A. (1969), 64, 1087-94. 3 Max Herzberg, Michel Revel, Atlas de biologie moléculaire. Hermann, 1972. • SJ. Garte, A.E. Hochwalt, 'Oncogene activation of experimental carcinogenesis: the role of carcinogen and tissue specificity', Environmental Ifealth Perspecrives (1989), 1,29-31. J.L. Jainchill, S,A. Aaronson, GJ. Todaro, 'Murine sarcoma and leukemia viruses: assay using clonallines of contact-inhibited mouse cells', Journal ofVirology (1969), 4,549-553,550. through gene transfer in NIH 3T3 cells, and displayed the work by Barbacid in August 1983. 6 It reports on the c-haslbas human proto oncogene spontaneous activation along the transfection of NIH/3T3 cells, a fact that occurred without the partícipation of known carcinogenic agents. The explanatíon of this event was obstaculized the limited knowledge of the biochemical properties of the protein and terciary structures. This protein, codified by the c has/ras gene, spontaneously activated, will incorporate aspartic acid as a 12 th amino acid residue. Even more, the adequate coupling of the alpha-helicoidal structure at the ami no terminal domain of the ras proteins, is altered by a conformational change after the replacement of the 12 th amino acid residue (Gly) by any other amino acid residue (except proline). This mechanism of malign activation of the ras oncogene, will be further supplemented by the discovery, al80 Barbacid and also by studying the morphological transformation of the NIH 3T3 cells, of the amplification mechanics of the ras gene as which the ras gene can narticinate in the This work was presented to the US Natl Acad Sci on April 1983 by the 1953 Nobel Prize, Severo Ochoa, who also selected to present there on July 1985 a development on human adenocarcinoma cells of previous NIH 3T3 results, associated to the detection of point mutations authored by M. Perucho.H 9 The point mutation malignant acquisition responsibility discovery was the why after which Barbacid was awarded in 1988,10 but a carcinogenic induction ol' ras oncogene actívation was also obtained under the resp0nl'ility of A. Pellicer, being I. Guerrero his . , • Ángel Pellicer, professor in the NYU Dpt ofPathology from 1980, worked with animal model systems of carcinogenesis, by using ras genes because of his great clinical relevance. Constantly in contact with Spain through the training of postdocs, he admitted Isabel Guerrero as a Fellow of the Spanish High Research Council in 1983. [START_REF] Pellieer Havc Bcen | Malignant transformation in vitro of cells from C57BL mouse normal pulmonary tissue[END_REF] The discovery was that a significant proportion of the tumors scored positive in the NIH 3T3 focus forming assay, and the genes responsible belonged to the ras family, After the initial observation, the two main murine genes responsible for the phenotype, N-ras and K-ras were quickly identified, The following year Pellicer sequenced the complete region of N-ras, identified the activating mutation and indícated that in sorne tumors there was los8 of the • normal allele, suggesting that the wild type copy (proto-oncogene)

was exerting some restraining effect on the action of the activated version (oncogene). Further studies continued providing evidence for the spectrum of activating mutations in different mouse strains and by different carcinogenic agents.

The Spanish team with which M. Barbacid sol ved the question that served to establish the molecular basis of human neoplasia was l'ormed by E. Santos, D. Martín Zanca, and V. Notario. The researchers group where worked A. Pellicer involved V. Corces, A. Pellicer, P. Calzada, A. Villasante and I. Guerrero. M. Perucho with a ras-proto-oncogene creates the event that initiates tumor formation", see E.P. Reddy, R.K. Reynolds, E. Santos, M. Barbacid, 'A point mutation is responsible for the aequisition of transforming properties by the T24 human bladder carcinoma', Nature (1982), 300, 149-52. For a metieulous figure depicting the importance of this last paper see M. Morange, 'From the regulatory vision of caneer to the oncogene paradigm, 1975-1985'.J. Hist. Bio/. (1997), 30, 1-29, 21. Fig. 1.

working wifu a team that included as Spanish collaborators C. Almoguera, C. Lama and 1. Jordano afforded with the characterization of human and murid cellular oncogenes and provided a method to detect mutations in transcripted genes. A previous paper reported on the constitutional history of these three groupS.13 But all these experiments arrived to function just because of the fortunated circonstance, that the NIH3T3 cells were not normal at aH, but fuat they already have accumulated a great number of genetic alterations subjacent to the gradual process of neoplastic transformation. They need one last push, provided by the dominant oncogene.

METHODOLOGY

Despite the phylogenetic anal ysis creates the text tradition that results in the production of critical editions of major written works, a trifling difficulty arises when what is wanted is to see ifit works to 14 retrace the history of a scientific program. A cladistic approach has been useful to give a quick indication of the likelihood relationships of Ángel Pellicer in fue race between the three concurrent Spanish teams.

Because the construction and characterization of a bibliographic library for mapping a given fíeld, illustrates the tendency of action to develop where there is action already. And that its cohesiveness mainly depends on how old it is. A graphic presentabon of the time series displayed by the papers whieh may be relevant to a topic is a preliminary analysis to determine what to make in terros of chronology. As the important paper for 3T3 cells was the choice of the preeursors G.J, Todaro and H. Green 15 , a highly cited eollection of Bouree record s making referenee to this original contribution was fírst considered. , 51 An analysis of fue 'Science Citatíon Index' has determined us to consider the evolution of the number of citations of the main experiments involved. Owing to that, a reference analysis was made to induce the Price index (percentage references dated in last five years) of a selected set of A. Pellicer. 3T3 CELLS, A TOOL POR THE JOB Testing for transforming activity by isolating viral DNA from cells that have been infected wifu a tumor pathogen is a function of the sort of cells used. Not many cells express, and absorb foreign DNA. But sueh are the NIH 3T3 cells, an all-purpose line of skin like cells, from the US National Institutes of Health. It could be a marked eontribution to the present-day objectives methodological approaches for analysis of infections by screening for signs for active virus, if it • was not a bright support to put foreign genes into cells by eircumventing aH the eomplexities of pathogens, The differenee is that it is the calcium messenger system that ushers in fue alíen gene, not a víruS. 16 The outcome of exploring the 3T3 cell type, comes to discuss on the questionable views related to their origins. '7 And as it has been successfully applied to deteet mutated cellular KRAS oncogenes in human tumors, and single mutations in the human and in the mouse genes, 18 it can be deseribed as a tool for the job of mutatíon deteetion,19 This very active mutation and selection proeess at the celIular level, has something truly Darwinian in character. 20 This most prolific cellline for virus oncogenic essays, 3T3, was created to shape ID This replies the (according to Marx extremely fruilful for Ibe history of science) notion of action as desclibing the situational 1 ¡mil tbat grounds the most general N; fferentiating characteristics between natural and technologícal sciences. R,S. Barkan, N.N, Nikol'skiI, 'Minimally transformed 3T3 celllines as an object for research on the rnechanisms ofproliferation', Tsitologiia (1985), 27, 5-27. 1& C. López-Galíndez. J.A, López, JA, Melero, L. de la Fuente, C. Martínez. J. Ortín, M. Perucbo, 'Analysis of genetic variability and mapping oí' point mutations in influenza virus by the RNase A mismatch cleavage method', froc. Nall. Acall. Sci.

(2004),85,3522-26. R.G. Cotton, 'Current methods of mutation detection', Mutar Res. (1993), 285, 125 44, lO W. Anderson, M. Jackson, B.G, Rosenkrantz, 'Towards an unnatural history oí' immunology', Journal of the Hístor)' ofBiology (1994), 27, 575-94, mouse embryo cells in 1963. 21 Popular in these days were other tissue culture systems, like the important for tumorigenic conversion lines of hamster kidney cells (designated BHK21).22 Or the C57Bl strain of mouse, that was introduced by fue French G. Barski (lnstitut Pasteud 3 , the man who will participate his anticipations on oncogenic properties as early as 1968. [START_REF] Barski | Evidence for oncogenic properties o[ heteroploid celllines-discussion[END_REF] Barski could be claimed príority in this early of today widely used lab mouse strains as its first reviewer, in socialist Czechoslovakia journal Folia biologíca. 25 He was also involved in the early essays of transfection,26 because of his reported formation of chromosomal hybrids by joint culti valÍon of two celIlines with morphological chromosomal markers. But unfortunately these in vitro and in vivo transformation experiments were not reproducible.

A given method in molecular genetics, the use for experimental results of 3T3 cells is found to be in correspondence one-to-one with transfection essays. The work of the scientists who introduced their study is known only to others working in the field. The assumed concordance between diligence and viability that lies behind varíes inordinately from one case to another. [START_REF] Even | mere is a report suggesting alleasl two NIH 3T3 celllines of different origins circulalinl! in the worfd. What makes depend on the cell lines tesled, failurcs of of specific seIs of genes in NIH 3T3 cells[END_REF] As welI as the amount of scientific work that they represent. In its real use, 3T3 cells are an uncertain form of the gradual process of neoplastic transformatíon in which a good number of genetics alterations are subjacent. It is what draws attention to the emphasis on these instruments. As such they need be ungovernable; the tdal and error used to develop them is not the same as expedmentation! ..

The outline of the analysis of this methodical testing device become a candidate to transcript the dialogues between those with the magnetic materialism to own it and those on which other priorities makes their use remarkably distinct. An interesting sidelight on this is that for a brief while the substantially active science is domínated by a lot of pieces of apparatus that could be connected with the use of these cell lines, and by the test procedures with which all manner of experiments could be done. 28 This relationship between scientists and those who develop the instruments is a revealing example of the dialectical interaction of science and technology. As historically the arrow of causalitv is largely from the technology to the science. The Spanish immunological tradition, on the base of technical applications of these cells, started with J. Salas 2Y the doctoral advisor of M. Perucho, on DNA binding proteins from hamster fibroblasts, normal or transfonned by oncogenic retroviruses. Salas was involved during his post-doctoral fellowship at NYU, in the early seventies, with H Green, the creator with GJ Todaro of this cellline. 30

In the middle eighties, Ángel Pellicer research agenda is represented in Fig. 1. 31 The general topology of this picture differentiates the upper zone with the methods used along the researches, fmm the down zone with aims and results obtained. As observed by the use of ir is precisely the express ion 'research front' that me work of scientlsts, eXPlonanon of technology by identificanon of it with ,9 See R, Pollack, J. Salas, R, Wang, T. Kusano, H. Green, 'Human-mouse hybrid cell Hnes and susceptibility to species-specific viruses', Jm4mal of Cellular Physiology (1977),77,117-9: and]' Salas, H. Green, 'Proteíns binding to DNA and their relation to growth in cultured mammalian cells', Nature: New mology (1971).229.165-9. 30 On the labora!ory of Howard Green as where Ilnes, ste G,W, Teebor, 'History of the Deparlment Pathology of University School of Medicine', s.a. " A maximum Iikelihood (ML) analysis was carríed out with Treecloud, applying semantic proxímity algorithms, see p, Gambette, J, Véronis, 'Visualising a text with a Tree Cloud'. IFCS'09: ¡nterna/lonal Federa/ion of Classification Socielies Cmiference.2009.

Todaro develooed the 3T3 intensity in bold type, all the articles were the result of researeh supported by governmental programs .. Criteria of gene purification and cell-transformation observation on animals, like miee, served to experiment with mutations in oneogenes. Less highest intensity visualizatíon of terms, as the corpus is associated with dates refleets superimposed information that refleets less disseminated researehes. The strueture of the tree is hidden in the center but the elades obtained at the two tips of the main branch substantialIy shows the chemics vs. radiation eareinogenic proeedures. With Ángel Pellicer 32 it must be said that "until 1977, the methods of gene transfer were successful only when DNA of low complexity was used", and that "subsequent improvements made the transfer feasible using total vertebrate and mammalian". Pellicer's historieal three viral positive timidine kinase colonies were obtained by using the mouse Ltk-eell line, one of the cell lines at this date that pos ses sed exogene DNA ineorporation and subsequent stabilization efficieney enough high to permit the gene transfer by using total 32 See A. Pellicer, 'Gene purification by transfectioll methods', in R, Kucherlapaty (ed.). Gene transfer, New York: Plenum press, 1986, p. 264 (rer. in vid, nI. 35). cellular DNA, with the Ciciencies reached by Michel Wigler modifieations of the original method. The other line was NlH 3T3.

The experiment

Pellieer conclusiveJy attained the theoretical efficieneies of the mínimum value in need to transfer a single copy gene; in eontrast with the first suceesses of the sixties this experiment was reproducible. And as sueh it was published in Cell, July 1978 issue. 33 A result on gene transl'er by using total mammalian DNA, that was obtained by Pellicer in the Columbia Univcrsity in 1978, This genetic transformatíon of somatic cells attained its historical model in the exploration of the idea of cotransformation, ie the co-oceurrenee in transfer with rclative hi~h frequency of unselectable genes, together with a selectable one. 3 That specifies in virtue of what property purified genes can be introduced in any kind of cultured ceIls. 35

The next year, 1979, Perucho went to the Cold Spring Harbor Laboratory, from the Max Planck in Berlín. In the heat of the argurnent developed by Szybalski 36 and through his involvement in the integrabon by eotransformation ol' all the donor cell genome into recipient chromosomes, the question of whether or not spontaneous neoplastic transformation of 3T3 cells were responsible for unstability by transfection led to further analysis. Created in the eontext of 3T3 eells and at the same time,37 as a reference scenario for how experiments can "go",38 3T6 eells (a ¡ine tuetb dettct e,ati\' c.inl1 the problem requiring much more authority than they have assigned, continual competitive behaviour linked their productions.

On the question of how much affects individual scientists in this demography, or its constituent parts (groups of seientists), the faet is that only a woman (Guerrero) is allocated to treat tii..e programo This knowledge must be described as an insularity that reql.res a distortion of the original structure before its narrative can be fitted neatly into place. 45 Figure 2 provides a sketch of this social power process. 46 ONCOGENIC ORIGIN OF CANCER

The potentialIy greater adversity of epigenetic alterations than nucleotide mutations (since their effeets on regional chromatin structure can spread, thereby affecting multiple genetic loci) will gain the support of M. Heidelberger for Ángel Pellicer contributíons to the 47 NAS. By associating his highly cited documents with the contexts ( where these citations occurred, is revealed the role of Pellicer as the transcriber, for both his friends (Barbacid and Perucho), of the "ownership" ofthe transfection experiments in the NIH 3T3 celllines. 48 In fact as the mean Jife of a scientific publication is shorter as far as the literature growth is fast, the studies on oncogenes had grown up so much by 1993, that new researchers were disengaging from the 45 DJ. de S. Price, "loe development and structure of the biomedical Iíterature', in K.S. Warren (ed) Coping with the bio1nedical literature : II primer for the sdentist and the e/inician, Praeger. 1981, 3-16. Theories of research, Seasidc, California, lntersystcms 1984, 164-169, 'Rcfcrence frames'. P. Calzada, A. Mayer, A. Pel1icer, 'A molecular approach to leukemogenesis: lllouse Iymphomas contain an activated c-ras oneogene', Proc. Nat!. A cad, Sci. USA (1984) The 1984 Pellicer contribution "Isolation, characterizatíon, and chromosome assignment of mouse N-ras gene from carcinogen induced thyrnic Iymphoma" was clue in his work, as it suggested a role for the N-ras in mammals hematologíc malignancies. Pellicer ascertained with this paper that the transmission of the modified N-r~s gene followed Mendelian rules, and concluded that the absence of a ras gene might result in reduced leveIs of malignancy . 51 Both the 1983 Barbacid paper "Spontaneous activation of a human proto-oncogene"S2 and the 1985 Perucho article HA method to S3 detect and characterize point mutations in transcribed genes .. were presented by the discoverer of RNA enzymatic synthesis (S.ochoa) to the US NAS, They were signaled results from the labos of Barbacid at Bethesda and Perucho at Stony Brook, after weIl reported previous successes,54 and both used NIH 3T3 celIs to be driven to malignancy. Although Todaro found that Nlli 3T3 line is already heteroploid in chromosome constitution,55 as beginning heteroploidy usually accompanies the initial event in lhe acquisítion of transforming properties, NIH 3T3 clones transformed by the mutant oncogenes on the human lung adenocarcinomas maintained in mice were homozygous for the mutant allele advanced Perucho. And Barbacid reported on an exception in an NIH 3T3 clone transformed by multiple copies of the c-haslbas proto-oncogenes, that successfully induced foci when tested in transfection assays. As a real system of published research can be viewed like a sort of arboreal process, 56 the relative autonomy of a team where the idiosyncrasy of a singular person (A. Pellicer) is prominent, holds with the assumption that his publications will show a special timewise progression. For a field that assemblages 9 contributions (see Table when it is considered the felicitous choice of the ratio to the last five years in the references lhat they use (Price's Index),57 a guess can be made that this proportíon range up to 75 percent in 8 of the cases. ss As 57 By Pricc's reasoning, science as a whole looks to a very recent past of about 5 years for roughly half of all citations , see B.e. Gríffith, 'Oerek Príce's puzzles : numerical m .. t.nhors for the operatíon of scicnce', Seienee. Tec/mology. & Human Values 13,351-60. Price hazarded that for a "subject that was al! research front and no general archive" a Príce's Index range up lo 75 percent, see DJ. de S. Price, 'Citation measures of hard sciencc. soft science, lechnology, and nonscíence', in: E.N, Carnol. O. PoIlack (eds.) 61 the rate of growth of the literature is lhe vehicle for a field to estimate how is its activity, an estimatíon is made that the Pellicer contributions was about 88% all research fronl. Separately, a paper developed the comprehensiveness proper of the reviews (upper left comer, PI 73.52), as in fact it í8 no. 4 in a series entitled "Genetic mechanisms in tumour initiation and progression".

In Fig, 4, fros;¡ the two more active papers (right side), lhere 18 a single best scored contribution (PI 923) to be ranged between the biochemistry discussions on the presence of point mutations in lhe ras genes; it acknowledged Ariel Ruiz í Altaba, from the Dpt. Cell Biology, NYU. The second position (lower right comer) comes from a contribution (PI 89.47) that could be close to an E'( Cathedra59 a"Pproach to the ras mutations irnrnunogenetics. It was written by Angel Pellicer once I. As such, the appearance of I. Guerrero in tbis race keeps the characteristics of a research front worker. In the period, Pellicer two highly cited papers (scores 176, 140, see Table 1) were two studies dated in 1984. The first (PI 77.77), that supported the normative criteria of the US Natl Acad, was between lhe last presentations to the

62

NAS by M Heidelberger (1888-1991), the Ochoa colleague at the Society.60 So enhancing its value for the of cancer. The interpretation must take in charge that this of work in an way makes the choice of María Luisa Duran-Re~ls between its sources. This Spanish bom medical doctor provitréd excellent models for the genetic and epigenetic factors in Iymphoid tumors working at Albert Einstein Medical School and at the facilities that had her husband, F. Duran-Reynals, at Yale. 61 Toblo 1: Historiclll approach 'o!bt "",sr """""¡»ST (5 Y"m) This unifying concept of genetic cell abnormalities as a central determinant of cancer emerged from a spedal cell lineo To be true in terms of exogenous and endogenous factors which do trigger cellular oncogenes the new carcinogen theory worked with a cell line (NIH 3T3) with a good number of genetic alteratíons subjacent to neoplastic transformation. In spite of the bias introduced in the analvsis bv the . method of detection, derived from the hazardous amount infection behind NIH 3T3 cells, this se1ective controversy made them tool perceived as a frontier assocíated to in vivo. Under the of Pellicer, an answer was to the question on if it were the cultivated cells that activated their gene after transfection while a different oncogen was operative in primary tumor cells. By suggesting a unique role for oncogene N-ras in marnmals hematologic malignancies.

Together with his postdoc Guerrero, Pellicer debated whether or not oncogenc mutations occur as the primary event in the development of the tumor. By providing addiúonal evidences in favor of ras gene activation in early stages, Pellicer supplied robust support to the hypothesis that ras mutation is the primary event in cardnogenic models. He also reported on thymic lymphomas in favor of n~""rrl..".. link,np the mutated version of ras genes with tumor claims that ras genes are activated by mutation.

working with brain mouse tumors, he provided definite evidence in favor of y-radiation as a source of the first event that triggers the carcinogenic process. By way of consequence he crucially intervened in the debate on mutation specificity for a carcinogen and advanced the correlation between the activation of a particular l"" I L 65 oncogene and its expression in the target organ considering a preferred ras gene target for each experimental tumor analyzed.

The inestimable position of Spanish biologists, in touch with NIH 3T3 cells from its inception, made them technically robust in the search for ras oncogenes. A medical doctor, Ángel Pellicer activated and characterized the genes involved in point mutations by using viral transforming genes obtained through transfer methods deve10ped by him, first in Columbia University and later in the Dpt of Pathology, NYU. The responsibility for the initiation or progression of tumors, and the issues of mutation specificity have been his successful path to solve the dialectic between the chemist and the biolo!l:ist in the fírst steps of molecular onc.olC\pv

  IJ E. Currás, E. Wulff Barreiro, 'Integration in Europe of human genetics results obtained by Spaniards in the USA: a historical perspective', Scienrometrics (2008), 75, 473-93. 14 M. Spencer, E.A. Davídson, A,e. Barbrook, c.J. Howe, 'Phylogenetics of artificial manuscrípts', Journal ofTheoretical Biology (2004),227,503-11. 15 GJ, Todaro, H. Green. 'Quantitative studíes of growth of mouse embryo cells in culture and their development into cstablished lilles', Journal of cell biology (1963), 17,299-313.

(

Figure 1 ,

 1 Figure1, Maximum likelíhood unrooted tree of the tumors sCllred positive and !he genes responsible of it. discovered by Ángel Pellicer,

  poUlf lUUlattonBarbacid, Pellicer and Perueho. This relation between individual1/ i -------l\'YU1 one careers consist of being as source authors and as cited authors aem'a.uon 1~S) NYU :-''YU (btln'U'U1) I continually approached by the dynamics of the scientific community.43 H-m.l'-1llutatmn earlvevmt Ntl A number of interacting processes can be advanced, each embedded in Beth~sda ¡ the next.

  81,202-5; L Guerrero, A. ViIlasante, V. Corees, A. Pelliccr, 'Loss of the normal N-ras allele in a mOURe thYlllic Iymphoma induced by a chemical earcinogen', Proc. Natl. Acad. Sci. USA (1985), 82, 7810-4. 4' Transfection started working at hands of Pellicer in 1977, sce M. Wigler, S. Silverstein, L.S. Lee, A. Pellicer, Y.e. Cbeng, R. Axel, 'Transfer of purified Herpes virus thymidine kinasc gene to cultured mouse ceJJs" Cell (1977), 11,223-32. Figure 3 is then constructed until 1993, with the three Spanish raeing teams citation data. 50

Figure 3 .

 3 Figure 3. Different frontal upsurge that the citation curves provide in the three racing teams.

  56 R. Menon. 'Singletons and Multiples in Scíentific Discovery : A Chapter in the Sociology of Sciencc', in: American Phí/o:iOphical Sodety, Proceedings 105. 1961, 470-486, 483.

Figure 4 .

 4 Figure 4. Scatter pIot of share of Price indexes vs. number of references per paper.

Communicatiol1

  Among Scientists and Engil1eers. Lexington. Mass: Lexington Books. O.C. Heath and Company, 1970,3-22.

  sciellce (1999), 1.87-98. " . _..... ' . _ -_.. 63 With a sorne less archival nature (PI 83.33) Pellicer contributed on oncogene mutations produced by y-radiatíon, close to the chemically induced benígn skin papillomas found activated coK-ras and N-ras. His discovery was that the alignment of the coding sequence from the isolated gene ídentified a single base change that produced an amino acid substitution. It carne after Barbacid activation of H-ras-I locus by mutation, and was conceived with the help of Perucho for of Nlli 3T3 cel!s wíth recombinant plasmid. CONCLUSION The competing oncogene theory, of Robert Huebner and George Todaro, hypothesízing a genetic rather than an infectious mechanism for cancer, and Temin's evolutionary protovirus hypothesis were within the scope of the biology performed in Spain from the early seventies when José Salas was involved during his post-doctoral • fellowship at NYU, with H Oreen, the creator with GJ Todaro of the NIH 3T3 cell lineo Ángel Pellicer, whose doctoral advísor M.L. Salas was a close colleague of J Salas, worked 1976-80 in cell proliferation & cancer at the Columbia University's Medical Center. Transfection started working at hands of Pellicer in 1977. And two years later, in 1979, Pellicer communicated the protocols for transfection experiments to Mariano Barbacid, from the Laboratory of RNA Tumor Viruses, NCI, Betllesda, and Manuel Perucho, from the Max Planck Institut fúr Molekulare Genetik (Berlin), Thus a fríendship that started in Madrid along the doctorate in the early 70's is linked to the creation of three research groups in molecular oncology, leadered by Spaniards in the USo As indeed the next year, in 1980, Ángel Pellicer obtained a post as a teacher at the NYU Dpt. of Pathology, Perucho moved from Berlin to Cold Spring Harbor Laboratory and later to the Dpt. of Biochemistry, State University of New York at Stony Brook, and Barbacid created the Laboratory of cel!ular and molecular biology, in the National Cancer Institute (Bethesda), Once they knew what they wanted from Pellicer, in 1979, how to make growth fibroblasts after successive transfer associated with properties in vivo by using NIH 3T3 cells, al! three molecular oncology research groups leadered by Spaniards decided to identífy and isolate human oncogenes through the use of transfection techniques. In 1982, both the groups of Mariano Barbacid and Eugenio Santos of the National Cancer Institute and of Michael Wigler and Manuel Perucho of the Cold Spring Harbor Laboratory found the oncogene of the T24 line of human bladder .&cinoma cells. And Pellicer decided to work in the identification and isolation of murine oncogenes also using transfection techniques, in his own labo at NYU Dept of Pathology from 1980.

  Guerrero had retumed to Europe (Oxford); • where she was working wíth Drosophila melanogaster, her doctoral dissertation subject and subsequent research field in Madrid,

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		animal-modol 'Y''''''> of carcinos"",is". Muta .. R.,. 185 (3), 293
		308.
	80	30 I Chang, R-Y.; auerrero, l; Lak•. R.; Pelli...., A.; D'Eustachio, P. I 9
		(1987) ""Mous. N-ras genes: orgaoizalÍOll of ,he functiolUll locus and
		of. truncoted cDNA-lik. ",eudo«"",," Oncc~ene Res. 1, 129-136.
		19 I Diamond, L.E.; Guerrero,!.; Pellicer, A. (1988, Mo:y) "Concomitant K-I 44

ras and N-ras gene point mutarions tu clonal mu.rine lympooDld." MoL

	....	,	I CeU. BicI. 8 (5). 2233-2236.
	Ochoa was the 1953-54 Harvey Society President, and its [952-53 Vice-Presiden!
	under the Presidency of the irnmunologist Heidelberger. Sec, The Harvey lectures:
	delivered under the auspices of lhe Harvey Society of New York 1977-1978.
	AcadellÚc Press, 1979. p. 321.
	T,E Glíck. A. Roca Rosell, 'Francesc Duran Reynals (Barcelona. 1899 -New
	Haven, USA. 1958). Virus and cancer: a controversial lheory'. Conlributions lo

nr rrf-number oíre~mce:s-; nr cit.-number oí citattoU$.

E. Santos, E.P. Reddy. S. Pulciani, R.J. Feldmann, M. Barbacid, 'Spontaneous aclÍvation of a human proto-oncogene', Proc Natl Acad Sci USA (1983),80.4679-83.

S. Pulcia¡ú, E. Santos, L.K. Long, V. Sorrentíno, M. Barbacid, 'ras Gene amplification and malignant transformation', Mol! Cel! Biol (1985).5.2836-41.

E. Winter, F. Yamamoto, C. Almoguera, M. Perucho, 'A method to detect and characterize point mutations in transeribed genes: Amplification and overexpression of (he mutanl e-IG-ras allele in human tumor eells', Proc Natl Acad Sc; USA (1985), 82,7575-9.

In 1981, Perucho provided a powerful technique for efficiently transform NIH 3T3 cells, see M. Perucho, M. Goldfarb, K. Shimizy, C. Lama, j, fogh, M, Wigler, 'Human-tumor-derived cell Iines eontain eornmon and different transforming genes', Cell (l981), 27, 467-76.

The "caneer Nobel" was attríbuted (in 1988) by the Switzerland Josef Steiner Foundation to Barbacid "for prnviding evidence that the ínteractíon (Jf a earcinogen

Op. cil. (15). 22 1. Macpherson, M. Stoker, 'Polyoma transformatíon of hamster cell clones -an investigation, of genetic factors affectíng cell competence', Virology (1962),16. 147

J.L. Marx, 'Oncogenes reach a milestone', Science (1994). 266,1942-4.

By the criticism of the number of citations that they received, we arrive to handle the main individual scientifíc papers dcscribed, so that their history becol1le operationaL

J. Leon, 1. Guerrero, A. Pellicer, 'Differential expression of the ras gene family in mice', Mol. Cel!o Biof. (1987), 1535-40. Ex~Calhedra contributions are considered as wíth less Ihan 16 references. op. cit. (58).