Víctor Hernández-Santamaría

Kévin Le Balc'h †
email: kevin.le-balc-h@inria.fr

Liliana Peralta

Global null-controllability for stochastic semilinear parabolic equations

Keywords: Global null-controllability, Carleman estimates, forward and backward semilinear stochastic parabolic equations, Banach fixed point method

In this paper, we prove the small-time global null-controllability of forward (respectively backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and the diffusion term (respectively in the drift term). In particular, we solve the open question posed by S. Tang and X. Zhang, in 2009. We propose a new twist on a classical strategy for controlling linear stochastic systems. By employing a new refined Carleman estimate, we obtain a controllability result in a weighted space for a linear system with source terms. The main novelty here is that the Carleman parameters are made explicit and are then used in a Banach fixed point method. This allows to circumvent the well-known problem of the lack of compactness embeddings for the solutions spaces arising in the study of controllability problems for stochastic PDEs.

Introduction

Let T > 0 be a positive time, D be a bounded, connected, open subset of R N , N ∈ N * , with a C 4 boundary Γ := ∂D. Let D 0 be a nonempty open subset of D. As usual, we introduce the notation χ D 0 to refer to the characteristic function of the set D 0 . To abridge the notation, hereinafter we write Q T := (0, T ) × D and Σ T := (0, T ) × Γ.

Let (Ω, F, {F t } t≥0 , P) be a complete filtered probability space on which a one-dimensional standard Brownian motion {W (t)} t≥0 is defined such that {F t } t≥0 is the natural filtration generated by W (•) augmented by all the P-null sets in F. Hereinafter, we denote {F t } t≥0 by F unless we want to emphasize what (F t ) t≥0 is.

Let X be a real Banach space, for every p ∈ [1, +∞], we introduce the functional space L p F (0, T ; X) := {φ : φ is an X-valued F-adapted process on [0, T ] and φ ∈ L p ([0, T ] × Ω; X)}, endowed with the canonical norm and we denote by L 2 F (Ω; C([0, T ]; X)) the Banach space consisting on all X-valued F-adapted process φ(•) such that E φ(•) 2

C([0,T ];X) < ∞, also equipped with the canonical norm.

Let us consider the stochastic forward semilinear equation

  
  dy = (∆y + f (ω, t, x, y) + χ D 0 h)dt + (g(ω, t, x, y) + H)dW (t) in Q T , y = 0 on Σ T , y(0) = y 0 in D.

(1)

In the controlled system (1), y ∈ L 2 F (Ω; C([0, T ]; L 2 (D))) is the state variable, the couple (h, H) ∈ L 2 F (0, T ; L 2 (D 0 )) × L 2 F (0, T ; L 2 (D)) is the control and y 0 ∈ L 2 (Ω, F 0 ; L 2 (D)) is the initial datum.

We assume for the moment that f and g verify the following conditions:

f (•, •, •, z), g(•,
•, •, z) are F-adapted, L 2 -valued stochastic processes for each z ∈ L 2 (D), (2) ∃L > 0, ∀(ω, t, x, s 1 , s 2 ) ∈ Ω × [0, T ] × D × R 2 , |f (ω, t, x, s 1 ) -f (ω, t, x, s 2 )| ≤ L|s 1 -s 2 |, (3) ∃K > 0, ∀(ω, t, x, s 1 , s 2 ) ∈ Ω × [0, T ] × D × R 2 , |g(ω, t, x, s 1 ) -g(ω, t, x, s 2 )| ≤ K|s 1 -s 2 |, (4)

∀(ω, t, x) ∈ Ω × [0, T ] × D, f (ω, t, x, 0) = 0. (5)
Under these conditions, by taking y 0 ∈ L 2 (Ω, F 0 ; L 2 (D)) and (h, H) ∈ L 2 F (0, T ; L 2 (D 0 )) × L 2 F (0, T ; L 2 (D)), it is known (see [START_REF] Lü | A mini-course on stochastic control[END_REF]Theorem 2.7] or [LZ21, Theorem 1.56]) that system (1) is globally defined in [0, T ]. More precisely, we can establish the existence and uniqueness of the solutions to (1) in the class

y ∈ W T := L 2 F (Ω; C([0, T ]; L 2 (D))) ∩ L 2 F (0, T ; H 1 0 (D)). (6)
One of the key questions in control theory of parabolic equations is to determine whether a system enjoys the so-called null-controllability property. System (1) is said to be globally null-controllable if for any initial datum y 0 ∈ L 2 (Ω, F 0 ; L 2 (D)), there exists a control (h, H) ∈ L 2 F (0, T ; D 0 ) × L 2 F (0, T ; L 2 (D)) such that the corresponding solution satisfies y(T, •) = 0 in D, a.s.

Observe that the regularity (6) justifies the definition we have introduced.

In this paper, we are interested in studying this controllability notion for system (1). Before introducing our main results we give a brief panorama of previous results available in the literature and emphasize the main novelty of this work.

Known results

The controllability of parabolic partial differential equations (PDEs) has been studied by many authors and the results available in the literature are very rich. In the following paragraphs, we focus on (small-time) global null-controllability results for scalar parabolic equations.

Deterministic setting. In the case where g ≡ H ≡ 0 and f and y 0 are deterministic functions, system (1) has been studied by several authors. In the mid 90's, Fabre, Puel & Zuazua in [START_REF] Fabre | Approximate controllability of the semilinear heat equation[END_REF] studied the so-called global approximate null-controllability in the case where f is a globally Lipschitz nonlinearity and condition (7) is replaced by the weaker constraint y(T ) L 2 (D) ≤ . Later, Imanuvilov [START_REF] Yu | Boundary controllability of parabolic equations[END_REF] and Fursikov & Imanuvilov [START_REF] Fursikov | Controllability of evolution equations[END_REF] improved this result and proved that the global null-controllability holds, see also [START_REF] Lebeau | Contrôle exacte de l'équation de la chaleur[END_REF] for the case of the (linear) heat equation, i.e. f ≡ 0. After these seminal works, Fernández-Cara [FC97], Fernández-Cara & Zuazua [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] have considered slightly superlinear functions f leading to blow-up without control, see also [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF] and the more recent work [START_REF] Le Balc'h | Global null-controllability and nonnegative-controllability of slightly superlinear heat equations[END_REF] by the second author. Results for nonlinearities including ∇y and depending on Robin boundary conditions have also been studied for instance in [START_REF] Doubova | On the controllability of parabolic systems with a nonlinear term involving the state and the gradient[END_REF][START_REF] Fernández-Cara | Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: the semilinear case[END_REF].

One common feature among these results is that the authors study the controllability problem by using the following general strategy, due to Zuazua in the context of the wave equation, see [START_REF] Zuazua | Exact controllability for semilinear wave equations in one space dimension[END_REF] or [Cor07, Chapter 4.3]; first, linearize the system and study the controllability of the system (1) replacing f (t, x, y) by a(t, x)y(t, x) where a ∈ L ∞ (Q T ), then, use a suitable fixed point method (commonly Schauder or Kakutani) for addressing the controllability of the nonlinear system. At this point, the important property of compactness is needed. In fact, compact embeddings relying on Aubin-Lions lemma like W (0,

T ) := {y ∈ L 2 (0, T ; H 1 0 (D)), y t ∈ L 2 (0, T ; H -1 (D))} → L 2 (0, T ; L 2 (D)) are systematically used.
Stochastic setting. In the case where f (y) = αy and g(y) = βy, α, β ∈ R, the controllability results for (1) were initiated by Barbu, Răşcanu & Tessitore in [START_REF] Viorel Barbu | Carleman estimates and controllability of linear stochastic heat equations[END_REF]. Under some restrictive conditions and without introducing the control H on the diffusion, they established a controllability result for linear forward stochastic PDEs. Later, Tang & Zhang in [START_REF] Tang | Null controllability for forward and backward stochastic parabolic equations[END_REF] improved this result and considered more general coefficients α and β (depending on t, x and ω). The main novelty in that work was to introduce the additional control H and prove fine Carleman estimates for stochastic parabolic operators. The same methodology has been used to study other cases like the ones of Neumann and Fourier boundary conditions ([Yan18]), degenerate equations ( [START_REF] Liu | Carleman estimates of some stochastic degenerate parabolic equations and application[END_REF]) and fourth-order parabolic equations ( [START_REF] Gao | Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equations[END_REF]). As a side note, we shall mention the work by Lü in [L 11] who, by using the classical Lebeau-Robbiano strategy ([LR95]), noticed that the action of the control H can be omitted at the prize of considering random coefficients α and β only depending on the time variable t.

In the framework proposed in this paper, as far the author's knowledge, there are not any results available in the literature. Compared to the deterministic setting, while establishing controllability properties for stochastic PDEs, many new difficulties arise. For instance, the solution of stochastic PDEs are usually not differentiable with respect to the variable with noise (i.e., the time variable). Also the diffusion term introduces additional difficulties while analyzing the problem. But most importantly, as remarked in [TZ09, Remark 2.5], the compactness property, which is one of the key tools in the deterministic setting, is known to be false for the functional spaces related to stochastic PDEs. This is the main obstruction for employing some classical methodologies like in [START_REF] Fursikov | Controllability of evolution equations[END_REF], [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF] for establishing null-controllability of semilinear problems at the stochastic level.

Statement of the main results and presentation of the methodology

Our first main result reads as follows.

Theorem 1.1. Under assumptions (2), (3) and (5), system (1) is small-time globally nullcontrollable, i.e. for every T > 0 and for every y 0 ∈ L 2 (Ω, F 0 ; L 2 (D)), there exist contols h ∈ L 2 F (0, T ; L 2 (D 0 )) and H ∈ L 2 F (0, T ; L 2 (D)) such that the unique solution y of (1) satisfies y(T, •) = 0 in D, a.s.

We first present the main arguments to obtain Theorem 1.1. Remove the semilinearity in the diffusion term. First, we highlight the fact that the null-controllability of the equation (1) can be reduced to the null-controllability of

     dy = (∆y + f (ω, t, x, y) + χ D 0 h)dt + HdW (t) in Q T , y = 0 on Σ T , y(0) = y 0 in D. (8)
Indeed, assume that one can construct a solution y ∈ L 2 8) such that y(T, •) = 0 in D, a.s. Noting that the control H is distributed in the whole domain D, we remark that y satisfies (1) with controls h = h and

F (Ω; C([0, T ]; L 2 (D))) associated to controls h ∈ L 2 F (0, T ; L 2 (D 0 )) and H ∈ L 2 F (0, T ; L 2 (D)) for (
H = H -g(•, •, •, y) ∈ L 2 F (0, T ; L 2 (D))
, which is well-defined by (2). Moreover, we still have y(T, •) = 0 in D a.s. This is why we can drop the Lipschitz condition on g, i.e. (4) in Theorem 1.1.

Controllability of a linear system despite a source term and a Banach fixed-point argument. To overcome the lack of compactness mentioned in the last section, we propose a new tweak on an old strategy for controlling parabolic systems. We use a classical methodology for controlling a linear system with a source term

F ∈ L 2 F (0, T ; L 2 (D)) of the form      dy = (∆y + F + χ D 0 h)dt + HdW (t) in Q T , y = 0 on Σ T , y(0) = y 0 in D, (9)
in a suitable weighted space. Note that this strategy has been widely used in the literature and it has been revisited in [START_REF] Liu | Single input controllability of a simplified fluid-structure interaction model[END_REF] to obtain local results. In turn, such weighted space is naturally defined through the weights arising in the Carleman estimates needed for studying the observability of the corresponding linear adjoint system to (9) (see Theorem 2.1), which in this case is a backward parabolic equation. Previous to this work, such Carleman estimate was not available in the literature (see [BEG16, Theorem 2.5] for a similar estimate in the deterministic case).

The methodology employed to prove the result is the weighted identity method introduced in the stochastic framework in [START_REF] Tang | Null controllability for forward and backward stochastic parabolic equations[END_REF]. But, unlike many other works out there, we make precise the dependency on the parameters involved in the construction of the Carleman weights and use them in a second stage to prove that the nonlinear map N (F ) → f (t, x, ω, y), with y solution of (9), is well-defined and is strictly contractive in a suitable functional space. In this way, the controllability of the system (8) is ensured through a Banach fixed point method which does not rely on any compactness argument. As compared to some results in the deterministic framework, on the one hand notice that here we are not considering any differentiability condition on the nonlinearities, that is, f is merely a C 0 -function. On the other hand, our method does not permit to establish global controllability result for slightly superlinearities as considered in [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF], see Section 4.2 below for a more detailed discussion.

As classical in the stochastic setting, for completeness, using the same strategy described above, it is possible to establish a controllability result for semilinear backward parabolic equations. More precisely, consider

     dy = (-∆y + f (ω, t, x, y, Y ) + χ D 0 h) dt + Y dW (t) in Q T , y = 0 on Σ T , y(T ) = y T in D, (10)
where f satisfies the following assumptions:

f (•, •, •, z, Z) is F-adapted, L 2 (D)-valued for each z, Z ∈ L 2 (D), (11)
∃L > 0, ∀(ω, t, x, s 1 , s 1 , s 2 , s 2 ) ∈ Ω × [0, T ] × D × R 4 , |f (ω, t, x, s 1 , s 1 ) -f (ω, t, x, s 2 , s 2 )| ≤ L (|s 1 -s 2 | + |s 1 -s 2 |) , (12)
∀(ω, t, x) ∈ Ω × [0, T ] × D, f (ω, t, x, 0, 0) = 0. ( 13
)
Under these conditions, by taking y T ∈ L 2 (Ω, F T ; L 2 (D)) and h ∈ L 2 F (0, T ; L 2 (D 0 )), it is known (see [LZ19, Theorem 2.12] or [LZ21, Theorem 1.62]) that system (10) is also globally well-defined in [0, T ]. In this case, we can establish the existence and uniqueness of the solutions to (10) in the class

(y, Y ) ∈ W T × L 2 F (0, T ; L 2 (D)). (14)
Our second main result is as follows.

Theorem 1.2. Under assumptions (11)-(13), system (10) is small-time globally null-controllable, i.e. for every T > 0 and for every y T ∈ L 2 (Ω, F T ; L 2 (D)), there exists h ∈ L 2 F (0, T ; L 2 (D 0 )) such that the unique solution y of (10) satisfies y(0, •) = 0 in D, a.s.

Theorem 1.2 extends to the nonlinear setting the previous results in [BRT03, Corollary 3.4] and [TZ09, Theorem 2.2] for the backward equation.

The strategy to prove Theorem 1.2 is very close to the one of Theorem 1.1 but one major difference can be spotted. For this case, it is not necessary to prove a Carleman estimate for forward stochastic parabolic equations. Actually, it suffices to use the deterministic Carleman inequality of [BEG16, Thm. 2.5] and employ the duality method introduced by Liu in [START_REF] Liu | Global Carleman estimate for stochastic parabolic equations, and its application[END_REF].

Outline of the paper

The rest of the paper is organized as follows. In Section 2, we present the proof of Theorem 1.1. In particular, Section 2.1 is devoted to prove the new Carleman estimate for the adjoint system of (9). Section 3 is devoted to prove Theorem 1.2. Finally in Section 4 we present some concluding remarks.

2 Controllability of a semilinear forward stochastic parabolic equation

A new global Carleman estimate for a backward stochastic parabolic equation

This section is devoted to prove a new Carleman estimate for a backward stochastic parabolic equation. The main novelty here is that the weight does not degenerate as t → 0 + (compared with the classical work [START_REF] Fursikov | Controllability of evolution equations[END_REF]). This estimate has been proved in the deterministic case in [BEG16, Theorem 2.5] in a slightly more general framework. Here, we use many of the ideas presented there and adapt them to the stochastic setting.

To make a precise statement of our result, let D be a nonempty subset of D such that

D ⊂⊂ D 0 . Let us introduce β ∈ C 4 (D) such that      0 < β(x) ≤ 1, ∀x ∈ D, β(x) = 0, ∀x ∈ ∂D, inf D\D {|∇β|} ≥ α > 0. (15)
The existence of such a function is guaranteed by [FI96, Lemma 1.1].

Without loss of generality, in what follows we assume that 0 < T < 1. For some constants m ≥ 1 and σ ≥ 2 we define the following weight function depending on the time variable

               γ(t) = 1 + (1 -4t T ) σ , t ∈ (0, T /4], γ(t) = 1, t ∈ [T /4, T /2], γ is increasing on [T /2, 3T /4], γ(t) = 1 (T -t) m , t ∈ [3T /4, T ], γ ∈ C 2 ([0, T )). ( 16
)
We take the following weight functions ϕ = ϕ(t, x) and ξ = ξ(t, x) ϕ(t, x) := γ(t) e µ(β(x)+6m) -µe 6µ(m+1) , ξ(t, x) := γ(t)e µ(β(x)+6m) ,

where µ is a positive parameter with µ ≥ 1 and σ is chosen as

σ = λµ 2 e µ(6m-4) , (18)
for some parameter λ ≥ 1. Observe that with these elections on µ and λ, the parameter σ is always greater than 2 and this also ensures that γ(t) ∈ C 2 ([0, T )). We finally set the weight θ = θ(t, x) as θ := e where (t, x) := λϕ(t, x).

Using these notations, we state the main result of this section which is a Carleman estimate for backward stochastic parabolic equations.

Theorem 2.1. For all m ≥ 1, there exist constants C > 0, λ 0 ≥ 1 and µ 0 ≥ 1 such that, for any

z T ∈ L 2 (Ω, F T ; L 2 (D)) and any Ξ ∈ L 2 F (0, T ; L 2 (D)), the solution (z, z) ∈ W T ×L 2 F (0, T ; L 2 (D)) to      dz = (-∆z + Ξ)dt + zdW (t) in Q T , z = 0 on Σ T , z(T ) = z T in D, (20)
satisfies

E D λ 2 µ 3 e 2µ(6m+1) θ 2 (0)|z(0)| 2 dx + E Q T λµ 2 ξθ 2 |∇z| 2 dxdt + E Q T λ 3 µ 4 ξ 3 θ 2 |z| 2 dxdt ≤ CE T 0 D 0 λ 3 µ 4 ξ 3 θ 2 |z| 2 dxdt + Q T θ 2 |Ξ| 2 dxdt + Q T λ 2 µ 2 ξ 3 θ 2 |z| 2 dxdt (21)
for all µ ≥ µ 0 and λ ≥ λ 0 .

Before giving the proof of Theorem 2.1, we make the following remark.

Remark 2.2. The following comments are in order.

• The proof of this result is rather classical except for the definition of the weight function γ(t) which does not blow-up as t → 0 + and thus preventing θ to vanish at t = 0. This change introduces some additional difficulties to the classical proof of the Carleman estimate for backward stochastic parabolic equations shown in [TZ09, Theorem 6.1], but which can be handled just as in the deterministic case (see [BEG16, Appendix A.1]).

• Different to the Carleman estimate in [TZ09, Eq. (6.2)], the power of ξ in the last term of ( 21) is 3 rather than 2. This is due to the definition of the weight γ which modifies a little bit the estimate of ϕ t in [0, T /4] as compared, for instance, to [START_REF] Tang | Null controllability for forward and backward stochastic parabolic equations[END_REF]. This does not represent any problem for proving our main controllability result.

• Just as in [TZ09, Remark 6.1], we can estimate the last term in (21) by weighted integrals of Ξ and z, more precisely

E Q T λ 2 µ 2 ξ 3 θ 2 |z| 2 dxdt ≤ CE Q T λ 4 µ 4 ξ 6 θ 2 |z| 2 dxdt + Q T θ 2 |Ξ| 2 dxdt .
Nevertheless, the new term of z cannot be controlled by its counterpart in the left-hand side of (21) and this does not improve our result.

Proof of Theorem 2.1. As we have mentioned before, the proof of this result is close to other proofs for Carleman estimates in the stochastic setting (see, e.g. [START_REF] Tang | Null controllability for forward and backward stochastic parabolic equations[END_REF][START_REF] Yan | Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications[END_REF] or [FLZ19, Ch. 3]). Some of the estimates presented in such works are valid in our case but others need to be adapted. For readability, we have divided the proof in several steps and we will emphasize the main changes with respect to previous works.

Step 1. A point-wise identity for a stochastic parabolic operator. We set θ = e where we recall that = λϕ with ϕ is defined in (17). Then, we write ψ = θz and for the operator dz + ∆zdt we have the following identity

θ(dz + ∆zdt) = I 1 + Idt, (22)
where

     I 1 = dψ -2 i i ψ i dt + Ψψdt, I = Aψ + i ψ ii , A = -t + i ( 2 i -ii ) -Ψ, (23)
where Ψ = Ψ(x, t) is a function to be chosen later. Hereinafter, to abridge the notation, we simply write ρ i = ∂ x i ρ and ρ t = ∂ t ρ and we denote i and i,j to refer to N i=1 and N i=1 N j=1 , respectively.

From Itô's formula, we have that

d(Aψ 2 ) = Aψdψ + ψd(Aψ) + d(Aψ)dψ = 2Aψdψ + A t ψ 2 dt + 2ψdAdψ + A(dψ) 2 + (dA)(dψ) 2
and

ψ ii dψ = (ψ i dψ) i -ψ i dψ i = (ψ i dψ) i - 1 2 d(ψ 2 i ) + 1 2 (dψ i ) 2 . Therefore Idψ = Aψ + i ψ ii dψ = i (ψ i dψ) i - 1 2 d i ψ 2 i + 1 2 i (dψ i ) 2 + 1 2 d(Aψ 2 ) - 1 2 A t ψ 2 dt - 1 2 A(dψ) 2 . ( 24
)
On the other hand, a direct computation gives

-2 i i ψ i I = - i (A i ψ 2 ) i + i (A i ) i ψ 2 - i j (2 j ψ i ψ j -i ψ j ψ j ) i + i,j k,h [2δ ih δ kj kh -δ ij δ kh kh ] ψ i ψ j , (25)
where δ ij = 1 if i = j and 0 otherwise. Multiplying both sides of ( 22) by I and taking into account identities ( 24)-( 25), we get the following point-wise identity

θI(dz + ∆zdt) =   I 2 + i,j c ij ψ i ψ j + F ψ 2 + IΨψ + ∇ • V   dt + i (ψ i dψ) i + 1 2 i (dψ i ) 2 - 1 2 A(dψ) 2 - 1 2 d i ψ 2 i -Aψ 2 , ( 26
)
where

           V = V 1 , V 2 , . . . , V N , V i = -j (2 j ψ i ψ j -i ψ j ψ j ) -A i ψ 2 , i = 1, . . . , N, c ij = k,h [2δ ih δ kj kh -δ ij δ kh kh ] , F = -1 2 A t ψ 2 + i (A i ) i ψ 2 .
Step 2. Some old and new estimates. The main goal of this step is to start building our Carleman estimate taking as a basis the point-wise identity (26). Integrating with respect to time in both sides of (26), we get

T 0 θI(dz + ∆zdt) (27) = - 1 2 i ψ 2 i -Aψ 2 T 0 (28)
+ T 0 F ψ 2 + IΨψ dt (29) + T 0 I 2 dt + T 0 i,j c ij ψ i ψ j dt + T 0 ∇ • V dt + T 0 i (ψ i dψ) i (30)
+ T 0 1 2 i (dψ i ) 2 - T 0 1 2 A(dψ) 2 , ( 31
)
for a.e. x ∈ R N and a.s. ω ∈ Ω. We will pay special attention to terms (28) and (29) which yield positive terms that are not present in other Carleman estimates using the classical weight vanishing both at t = 0 and t = T . At this point, we shall choose the function Ψ = Ψ(x, t) as

Ψ := -2 i ii (32)
and, for convenience, we give some identities that will be useful in the remainder of the proof.

From the definition of in (19), we have

i = λγµβ i e µ(β+6m) , ii = λγµ 2 β 2 i e µ(β+6m) + λγµβ ii e µ(β+6m) . (33)
For shortness, we have dropped the explicit dependence of x and t on the expressions above.

-Positivity of the term (28). From the definitions of ψ and , we readily see that lim t→T -(t, •) = -∞ and thus the term at t = T vanishes. Therefore, (28) simplifies to

- 1 2 i ψ 2 i -Aψ 2 T 0 = 1 2 i ψ 2 i (0) + A(0)ψ 2 (0) . ( 34
)
It is clear that the first term in the right-hand side of (34) is positive. For the second one, we will generate a positive term by using the explicit expression of the function γ. Using definition (16), we obtain that

γ (t) = - 4σ T 1 - 4t T σ-1 , ∀t ∈ [0, T /4],
whence, from (18) and the above expression, we get

t (0, •) = - 4λ 2 µ 2 e µ(6m-4) T e µ(β(•)+6m) -µe 6µ(m+1) ≥ cλ 2 µ 3 e µ(12m+2) (35)
for all µ ≥ 1 and some constant c > 0 uniform with respect to T . On the other hand, from the derivatives (33) and using the facts that γ(0) = 2 and β ∈ C 4 (D), we get

| 2 i (0, •) + ii (0, •)| ≤ Cλ 2 µ 2 e 2µ(6m+1) . (36)
In this way, by using (34), the definition of A in (23) and estimates (35)-(36), there exists µ 1 > 0, such that for all µ ≥ µ 1 ≥ 1, we get

- 1 2 i ψ 2 i -Aψ 2 T 0 ≥ 1 2 |∇ψ(0)| 2 + cλ 2 µ 3 e 2µ(6m+1) ψ 2 (0) -Cλ 2 µ 2 e 2µ(6m+1) ψ 2 (0) ≥ c 1 |∇ψ(0)| 2 + c 1 λ 2 µ 3 e 2µ(6m+1) ψ 2 (0), (37)
for some constant c 1 > 0 only depending on D and D .

-Estimate of the term (29). This term is the most cumbersome one since the combination of some terms of F ψ 2 and IΨψ will yield a positive term that does not appear in the classical Carleman estimate with weight vanishing at t = 0 and t = T .

Recalling the definition of A, we see that that the first term in (29) can be written as

T 0 F ψ 2 dt = T 0 (F 1 + F 2 + F 3 ) ψ 2 dt, (38)
where

F 1 = 1 2 tt , F 2 = - 1 2 i ( 2 i + ii ) t , F 3 = i (A i i + A ii ). (39)
For the first term of (38) we argue as follows. For t ∈ (0, T /4), using the definition of γ(t), it is not difficult to see that |γ tt | ≤ Cλ 2 µ 4 e 2µ(6m-4) thus

| tt | ≤ Cλ 3 µ 5 e 2µ(6m-4) e 6µ(m+1) ≤ Cλ 3 µ 2 ξ 3 (40)
where we recall that ξ = ξ(t, x) is defined in (17). Here, we also have used that µ 3 e -2µ < 1/2 for all µ > 1.

For t ∈ (T /2, T ), using once again the definition of γ we have |γ tt | ≤ Cγ 3 . Noting that ϕ tt = γtt γ ϕ and using the estimate |ϕγ| ≤ µξ 2 we get

| tt | = λ γ tt γ ϕ ≤ Cλµξ 3 . ( 41
)
Since obviously tt vanishes for t ∈ (T /4, T /2), we can put together estimates ( 40) and (41) to deduce that

T 0 F 1 ψ 2 dt ≥ -Cλ 3 µ 2 T 0 ξ 3 ψ 2 dt. ( 42
)
We move now to the second and third terms of (38). To abridge the notation, in what follows, we set α(x) := e µ(β(x)+6m) -µe 6µ(m+1) .

Notice that α(x) < 0 for all x ∈ D. From (33), a direct computation yields

( 2 i + ii ) t = 2λ 2 µ 2 β 2 i e 2µ(β+6m) γγ t + λµ 2 β 2 i e µ(β+6m) γ t + λµβ ii e µ(β+6m) γ t , =: M i (43)
On the other hand, after a long but straightforward computation, we get from (33) that

A i i + A ii = P (1) i + P (2) i (44)
where

P (1) i := -λ 2 αγ t γµ 2 β 2 i e µ(β+6m) -λ 2 αγ t γµβ ii e µ(β+6m) -λ 2 γ t γµ 2 β 2 i e 2µ(β+6m) , (45)
P (2) i := k 3λ 3 µ 4 ξ 3 β 2 k β 2 i + 2λ 2 µ 4 ξ 2 β 2 k β 2 i + λ 2 µ 3 ξ 2 β kk β 2 i + λ 3 µ 3 ξ 3 β k β i β ki +2λ 2 µ 3 ξ 2 β k β ki β i + λ 3 µ 3 ξ 3 β 2 k β ii + λ 2 µ 3 ξ 2 β 2 k β ii + λ 2 µ 2 ξ 2 β kk β ii . (46)
In the term P

(2) i , we have further simplified the notation by recalling that ξ = e µ(β+6m) γ. Also observe that we have deliberately put together all the terms containing γ t in the above expression.

We will use now the term IΨψ in (29) to collect other terms containing γ t . Indeed, from the definition of Ψ (see eq. ( 32)), we see that this term can be rewritten as

T 0 IΨψdt = T 0 AΨψ 2 dt -2 T 0   i,k ψ ii kk   ψdt = 2 T 0 i P (3) i ψ 2 dt -2 T 0   i,k ( 2 i + ii ) kk   ψ 2 dt -2 T 0   i,k ψ ii kk   ψdt, (47) where P
(3) i := λ 2 αγγ t µ 2 β 2 i e µ(β+6m) + λ 2 αγγ t µβ ii e µ(β+6m) . Hence, from (39), ( 43), (44), and (47), we get

T 0 (F 2 + F 3 )ψ 2 dt + T 0 IΨψdt = T 0 i - 1 2 M i -P (1) i + P (3) i ψ 2 dt =:Q 1 + T 0 i P (2) i -2 k ( 2 i + ii ) kk ψ 2 dt =:Q 2 -2 T 0   i,k ψ ii kk   ψdt =:Q 3 . ( 48
)
We shall focus on the term Q 1 . From the definition of M i , (45) and using that ξ = e µ(β+6m) γ and ϕ = αγ, we see that

- M i 2 -P (1) i + P (3) i = - γ t γ 2λ 2 µ 2 ξ 2 β 2 i + 1 2 λµ 2 ξβ 2 i + 1 2 λµξβ ii - γ t γ λ 2 (-ϕ)ξµ 2 β 2 i + λ 2 (-ϕ)ξµβ ii . (49)
From the definition of γ, it is clear that the above expression vanishes on (T /4, T /2). On (T /2, T ), we use the fact that there exists C > 0 such that |γ t | ≤ Cγ 2 . Hence, for all (t, x) ∈ (T /2, T ) × D, there exists a constant C > 0 only depending on D and D such that

i - M i 2 -P (1) i + P (3) i ≤ Cλ 2 µ 2 ξ 2 + ξϕ ≤ Cλ 2 µ 3 ξ 3 , (50)
where we have used that |ϕγ| ≤ µξ 2 . On (0, T /4), we are going to use the fact that γ t ≤ 0, ϕ < 0, and γ ∈ [1, 2] to deduce that Q 1 has the good sign outside D . Indeed, from (15), we can find

µ 2 = µ 2 (α, ∆ψ ∞ ) such that for all µ ≥ µ 2 ≥ µ 1 ≥ 0 i 2λ 2 µ 2 ξ 3 β 2 i + 1 2 λµ 2 ξβ 2 i + 1 2 λµ ξ β ii + i λ 2 (-ϕ)ξµ 2 β 2 i + λ 2 (-ϕ)ξµβ ii ≥ cλ 2 µ 2 |ϕ|ξ, x ∈ D \ D . (51)
In this way, in a subsequent step, by ( 49), (51), we will obtain from Q 1 a positive term in (0, T ) × D and a localized term at D in the right-hand side of the inequality. The conclusion of this sub-step is quite classical. For the term Q 2 in (48), we can readily see that the leading term in (46) is positive. Hence, from (33) and straightforward computation, we have

Q 2 ≥ T 0 λ 3 µ 4 ξ 3 |∇β| 4 ψ 2 dt -C T 0 λ 2 µ 4 ξ 2 + λ 2 µ 3 ξ 3 + λ 3 µ 3 ξ 3 ψ 2 dt (52) for some constant C = C( ∇β ∞ , D 2 β ∞ ) > 0.
As in the previous case, using (15) will yield a positive term, a localized term on the right-hand side. The terms with lower powers of µ and λ will be absorbed later.

Finally, for analyzing Q 3 , we will use that -ψ ii kk ψ = -(ψ i kk ψ) i + ψ i kki ψ + ψ 2 i kk . Thus,

Q 3 = 2 T 0 i,k ψ 2 i kk dt + 2 T 0 i,k ψ i ψ kki dt -2 T 0 i,k (ψ i kk ψ) i dt. ( 53
)
We will leave this term as it is. In the next sub-step we will use it for producing a positive term depending on |∇ψ|.

-Estimates on the gradient of ψ. The last positive term we shall obtain in this step, comes from the second term in (30) and the first term in (53). Using (33) it can be readily seen that

T 0 i,j 2ψ 2 i jj + c ij ψ i ψ j dt ≥ T 0 λµ 2 ξ|∇β| 2 |∇ψ| 2 -C T 0 λµξ|∇ψ| 2 , ( 54
)
for some C > 0 only depending on D and D . From here, using the properties of β we will obtain a positive term and a localized term in D .

From the second term in (53) and the fact that

kki = 2λξµ 2 β k β ki + λξµ 3 β 2 k β i + λξµβ kki + λξµ 2 β kk β i ,
we can use Cauchy-Schwarz and Young inequalities to deduce

T 0 i,k ψ i ψ kki dt ≥ -C T 0 µ 2 |∇ψ| 2 dt -C T 0 λ 2 µ 4 ξ 2 |ψ| 2 dt. ( 55
)
Notice that the term containing ∇ψ does not have any power for λ so it can be absorbed later.

The last term in ( 53) is left as it is, since by the divergence theorem we will see later that this term is actually 0.

Step 3. Towards the Carleman estimate. We begin by integrating ( 27)-(31) in D and take expectation in both sides of the identity. Taking into account the estimates obtained in the previous step, i.e., (37), (42), (48), and (50)-(55), we get

c 1 E D |∇ψ(0)| 2 dx + D λ 2 µ 3 e 2µ(6m+1) |ψ(0)| 2 dx + cE T /4 0 D\D λ 2 µ 2 ξ|ϕ||γ t ||ψ| 2 dxdt + E Q T λ 3 µ 4 ξ 3 |∇β| 4 |ψ| 2 dxdt + Q T λµ 2 ξ|∇β| 2 |∇ψ| 2 dxdt + E Q T I 2 dxdt + 1 2 E Q T i (dψ i ) 2 dx ≤ E Q T θI(dz + ∆zdt) dx + 1 2 E Q T A(dψ) 2 dx + BT + R, (56)
where

BT := 2E   Q T i,k (ψ i k ψ) i dxdt   -E Q T i (ψ i dψ) i -E Q T ∇ • V dxdt , (57)
R := CE Q T λ 2 µ 3 ξ 3 + λ 2 µ 4 ξ 2 + λ 3 µ 3 ξ 3 |ψ| 2 dxdt + Q T µ 2 + λµξ |∇ψ| 2 dxdt . ( 58
)
We remark that the positive constants c 1 and C in (56)-(58) only depend on D and D , while c > 0 depends only on D, D and α (see ( 15)).

We proceed to estimate the rest of the terms. We begin with those gathered on BT , defined in (57). It is clear that z = 0 on Σ T implies ψ = 0 on Σ T . Moreover, ψ i = ∂ψ ∂ν ν i , with ν = (ν 1 , . . . , ν N ) being the unit outward normal vector of D at x ∈ ∂D. Also, by the construction of the weight β, we have

i = λµξψ i = λµξ ∂ψ ∂ν ν i and ∂ψ ∂ν < 0, on Σ T .
Hence, it is not difficult to see that using divergence theorem we have

2E Q T (ψ i k ψ) i dxdt = 2E   Σ T i,k ψ i k ψν i dxdt   = 0, -E Q T i (ψ i dψ) i = -E Σ T i ψ i ν i dψ dx = 0,
and

-E Q T ∇ • V dxdt = E   Σ T i,j (2 i ψ i ψ j -i ψ j ψ j ) + A i ψ 2 ν j dxdt   = E   Σ T λµξ ∂β ∂ν ∂z ∂ν 2 i,j ν i ν j 2 dxdt   ≤ 0.
Thus, we get BT ≤ 0.

For the following three terms, we will use the change of variables ψ = θz and the fact that z solves system (20). First, we see that

E Q T i (dψ i ) 2 dx = E Q T θ 2 i (z i + i z) 2 dxdt ≥ 0. ( 60
)
In the same spirit, using the equation verified by z and Cauchy-Schwarz and Young inequalities, we get

E Q T θI(dz + ∆zdt) dx ≤ 1 2 E Q T I 2 dxdt + 1 2 E Q T θ 2 |Ξ| 2 dxdt . (61)
Lastly, from (33) and the fact that |ϕ t | ≤ Cλµξ 3 for (t, x) ∈ (0, T ) × D, a direct computation shows that

E Q T A(dψ) 2 dx = E Q T θ 2 A|z| 2 dxdt ≤ C Q T θ 2 λ 2 µ 2 ξ 3 |z| 2 dxdt . ( 62
)
Using that inf x∈D\D |∇β| ≥ α > 0, we can combine estimate (56) with ( 59)-(62) to deduce

E D |∇ψ(0)| 2 dx + D λ 2 µ 3 e 2µ(6m+1) |ψ(0)| 2 dx + E T /4 0 D λ 2 µ 2 ξ|ϕ||γ t ||ψ| 2 dxdt + E Q T λ 3 µ 4 ξ 3 |ψ| 2 dxdt + Q T λµ 2 ξ|∇ψ| 2 dxdt + 1 2 E Q T I 2 dxdt ≤ CE T /4 0 D λ 2 µ 2 ξ|ϕ||γ t ||ψ| 2 dxdt + T 0 D λ 3 µ 4 ξ 3 |ψ| 2 dxdt + T 0 D λµ 2 ξ|∇ψ| 2 dxdt + CR + CE Q T θ 2 |Ξ| 2 dxdt + Q T θ 2 λ 2 µ 2 ξ 3 |z| 2 dxdt ,
for some C > 0 only depending on D, D and α. We observe that, unlike the traditional Carleman estimate with weight vanishing at t = 0 and t = T , we have three local integrals, one of those being only for t ∈ (0, T /4). We will handle this in the following step. Also notice that all of the terms in R have lower powers of λ and µ, thus, we immediately see that there exists some µ 3 ≥ µ 2 and λ 1 ≥ C such that, for all µ ≥ µ 3 and λ ≥ λ 1

E D |∇ψ(0)| 2 dx + D λ 2 µ 3 e 2µ(6m+1) |ψ(0)| 2 dx + E T /4 0 D λ 2 µ 2 ξ|ϕ||γ t ||ψ| 2 dxdt + E Q T λ 3 µ 4 ξ 3 |ψ| 2 dxdt + Q T λµ 2 ξ|∇ψ| 2 dxdt ≤ CE T /4 0 D λ 2 µ 2 ξ|ϕ||γ t ||ψ| 2 dxdt + T 0 D λ 3 µ 4 ξ 3 |ψ| 2 dxdt + T 0 D λµ 2 ξ|∇ψ| 2 dxdt + CE Q T θ 2 |Ξ| 2 dxdt + Q T θ 2 λ 2 µ 2 ξ 3 |z| 2 dxdt . ( 63
)
Step 4. Last arrangements and conclusion. As usual, the last steps in Carleman strategies consist in removing the local term containing the gradient of the solution and coming back to the original variable. We will see that the original strategy also helps to remove the local term in (0, T /4).

First, using that

z i = θ -1 (ψ i -i ψ), it is not difficult to see that θ 2 |∇z| 2 ≤ 2|∇ψ| 2 + 2Cλ 2 µ 2 ξ 2 |ψ| 2
for some C > 0 only depending on D and D , hence from (63) we have

E D θ 2 (0)|∇z(0)| 2 dx + D λ 2 µ 3 e 2µ(6m+1) θ 2 (0)|z(0)| 2 dx + E T /4 0 D θ 2 λ 2 µ 2 ξ|ϕ||γ t ||z| 2 dxdt + E Q T θ 2 λ 3 µ 4 ξ 3 |z| 2 dxdt + Q T θ 2 λµ 2 ξ|∇z| 2 dxdt ≤ CE T /4 0 D θ 2 λ 2 µ 2 ξ|ϕ||γ t ||z| 2 dxdt + T 0 D θ 2 λ 3 µ 4 ξ 3 |z| 2 dxdt + T 0 D θ 2 λµ 2 ξ|∇z| 2 dxdt + CE Q T θ 2 |Ξ| 2 dxdt + Q T θ 2 λ 2 µ 2 ξ 3 |z| 2 dxdt , (64)
for all λ ≥ λ 1 and µ ≥ µ 3 . We choose a cut-off function

η ∈ C ∞ c (D) such that 0 ≤ η ≤ 1, η ≡ 1 in D , η ≡ 0 in D \ D 0 ( 65
)
with the additional characteristic that

∇η η 1/2 ∈ L ∞ (D) N . ( 66
)
This condition can be obtained by taking some η 0 ∈ C ∞ c (D) satistying (65) and defining η = η 4 0 . Then η will satisfy both (65) and (66).

Using Itô's formula, we compute d θ 2 ξz 2 = (θ 2 ξ) t z 2 + 2θ 2 ξzdz + θ 2 ξ(dz) 2 and thus, using the equation verified by z, we get

E D 0 θ 2 (0)ξ(0)|z(0)| 2 η dx + 2E T 0 D 0 θθ t ξ|z| 2 η dxdt + 2E T 0 D 0 θ 2 ξ|∇z| 2 η dxdt + E T 0 D 0 θ 2 ξ|z| 2 η dxdt = -E T 0 D 0 θ 2 ξ t |z| 2 η dxdt -2E T 0 D 0 θ 2 ξzΞη dxdt -2E T 0 D 0 θ 2 ξ∇η • ∇zz dxdt -2E T 0 D 0 ∇(θ 2 ξ) • ∇zzη dxdt . ( 67
)
We readily see that the first and last terms in the left-hand side of (67) are positive, so they can be dropped. Also, notice that using the properties of η, the third term gives (up to the constants µ and λ) the local term containing |∇z|.

We shall focus on the second term on the left-hand side of (67). Similar to

Step 2 above, we analyze it on different time intervals. Obviously, for t ∈ (T /4, T /2) this term vanishes since γ t = 0. For t ∈ (0, T /4), we notice that θθ t = θ 2 λϕ γt γ ξ and since γ t ≤ 0, ϕ < 0 and γ ∈ [1, 2], this yields a positive term. Lastly, in the interval (T /2, T ), we use that |ϕ t | ≤ Cλµξ 3 to obtain the bound |θ t | ≤ Cθλ 2 µξ 3 . Summarizing, we have

2E T 0 D 0 θθ t ξ|z| 2 η dxdt ≥ E T /4 0 D 0 θ 2 λξ|γ t ||ϕ||z| 2 η dxdt -CE T T /2 D 0 θ 2 λ 2 µξ 3 |z| 2 η dxdt . (68)
Let us estimate each term on the right-hand side of (67). For the first one, using that |ξ t | ≤ Cλµξ 3 for all (t, x) ∈ (0, T ) × D, we get

E T 0 D 0 θ 2 ξ t |z| 2 η dxdt ≤ CE T 0 D 0 θ 2 λµξ 3 |z| 2 η dxdt . ( 69
)
For the second one, using Cauchy-Schwarz and Young inequalities yields

E T 0 D 0 θ 2 ξzΞη dxdt ≤ 1 2 E T 0 D 0 θ 2 λ -1 µ -2 |Ξ| 2 η dxdt + 1 2 E T 0 D 0 θ 2 λµ 2 ξ 2 |z| 2 η dxdt . (70)
For the third one, we will use property (66) and Cauchy-Schwarz and Young inequalities to deduce that

E T 0 D 0 θ 2 ξ∇η • ∇zz dxdt ≤ E T 0 D 0 θ 2 ξ|∇z| 2 η dxdt + C( )E T 0 D 0 θ 2 ξ|z| 2 dxdt ( 71
)
for any > 0. For the last term, using that |∇(θ 2 ξ)| ≤ Cθ 2 λµξ 2 and arguing as above, we get

E T 0 D 0 ∇(θ 2 ξ) • ∇zzη dxdt ≤ E T 0 D 0 θ 2 ξ|∇z| 2 η dxdt + C( )E T 0 D 0 θ 2 ξ 3 µ 2 λ 2 |z| 2 η dxdt . (72)
Therefore, taking = 1 2 and using estimates (68)-(72) together with the properties of the cut-off η, we get

E T /4 0 D θ 2 λξ|γ t ||ϕ||z| 2 dxdt + E T 0 D θ 2 ξ|∇z| 2 dxdt ≤ CE T 0 D 0 θ 2 (λ 2 µξ 3 + λµ 2 ξ 2 + λ 2 µ 2 ξ 3 )|z| 2 dxdt + CE Q T θ 2 λ -1 µ -2 |Ξ| 2 dxdt . (73)
As usual, we have paid the price of estimating locally the gradient by enlarging a little bit the observation domain. Notice that this procedure gives us the local estimate in (0, T /4) by using the properties of the weight function ϕ and γ t . Finally, the desired estimate follows by multiplying both sides of (73) by λµ 2 and using the result to bound in the right-hand side of (64). We conclude the proof by setting µ 0 = µ 3 and λ 0 = λ 1 .

A controllability result for a linear forward stochastic heat equation with one source term and two controls

In this section, we will prove a controllability result for a linear forward equation. More precisely, recall the equation defined in (9)

     dy = (∆y + F + χ D 0 h)dt + HdW (t) in Q T , y = 0 on Σ T , y(0) = y 0 in D. (74)
In ( 74), (h,

H) ∈ L 2 F (0, T ; L 2 (D 0 )) × L 2 F (0, T ; L 2 (D)
) is a pair of controls and F is a given source term in L 2 F (0, T ; L 2 (D)). Observe that given y 0 ∈ L 2 (Ω, F 0 ; L 2 (D)) and the aforementioned regularity on the controls and source term, system (74) admits a unique solution y ∈ W T , see [LZ19, Theorem 2.7].

Under the notation of Section 2.1, let us set the parameters λ and µ to a fixed value sufficiently large, such that inequality (21) holds true. We define the space

S λ,µ = F ∈ L 2 F (0, T ; L 2 (D)) : E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt 1/2 < +∞ , (75)
endowed with the canonical norm.

Our linear controllability result reads as follows.

Theorem 2.3. For any initial datum y 0 ∈ L 2 (Ω, F 0 ; L 2 (D)) and any source term F ∈ S λ,µ , there exists a pair of controls (h,

H) ∈ L 2 F (0, T ; L 2 (D 0 )) × L 2 F (0, T ; L 2 (D)
) such that the associated solution y ∈ W T to system (74) satisfies y(T ) = 0 in D, a.s. Moreover, the following estimate holds

E Q T θ -2 |y| 2 dxdt + E T 0 D 0 θ -2 λ -3 µ -4 ξ -3 |h| 2 dxdt + E Q T θ -2 λ -2 µ -2 ξ -3 |H| 2 dxdt ≤ C 1 E y 0 2 L 2 (D) + C F 2 S λ,µ , (76)
where C 1 > 0 is a constant depending on D, D 0 , λ, µ and C > 0 only depends on D and D 0 .

Remark 2.4. Using classical arguments, see for instance [LTT13, Proposition 2.9], from Theorem 2.3 one can construct a linear continuous mapping that associates every initial datum y 0 ∈ L 2 (Ω, F 0 ; L 2 (D)) and every source terms F ∈ S λ,µ , to a trajectory (y, h, H) such that y(T ) = 0 in D, a.s. and (76) holds.

The proof of Theorem 2.3 is based on a classical duality method, called penalized Hilbert Uniqueness Method, which ideas can be traced back to the seminal work [START_REF] Glowinski | Exact and approximate controllability for distributed parameter systems[END_REF]. The general strategy consists in three steps:

-Step 1. Construct a family of optimal approximate-null control problems for system (74).

-Step 2. Obtain a uniform estimate for the approximate solutions in terms of the data of the problem, i.e., the initial datum y 0 and the source terms F and G.

-Step 3. A limit process to derive the desired null-controllability result.

We shall mention that in the stochastic setting, similar strategies have been used for deducing controllability results and Carleman estimates for forward and backward equations, see e.g. [START_REF] Liu | Global Carleman estimate for stochastic parabolic equations, and its application[END_REF][START_REF] Yan | Carleman estimates for stochastic parabolic equations with Neumann boundary conditions and applications[END_REF][START_REF] Liu | Carleman estimates of some stochastic degenerate parabolic equations and application[END_REF].

In what follows, C will denote a generic positive constant possibly depending on D, D 0 , but never on the parameters λ and µ.

Proof of Theorem 2.3. We follow the steps described above.

Step 1. For any > 0, let us consider the weight function γ (t) given by

           γ (t) = 1 + (1 + 4t T ) σ , t ∈ [0, T /4], γ (t) = 1, t ∈ [T /4, T /2 + ], γ (t) = γ(t -), t ∈ [T /2 + , T ],
σ as in (18). Defined in this way, it is not difficult to see that γ does not blow up as t → T -and that γ (t) ≤ γ(t) for t ∈ [0, T ]. With this new function, we set the weight ϕ as in (17) by replacing the function γ by γ . In the same manner, we write θ = e λϕ .

With this notation, we introduce the functional

J (h, H) := 1 2 E Q T θ -2 |y| 2 dxdt + 1 2 E T 0 D 0 θ -2 λ -3 µ -4 ξ -3 |h| 2 dxdt + 1 2 E Q T θ -2 λ -2 µ -2 ξ -3 |H| 2 dxdt + 1 2 E D |y(T )| 2 dx (77)
and consider the minimization problem

min (h,H)∈H J (h, H), subject to equation (74), ( 78
)
where

H = (h, H) ∈ L 2 F (0, T ; L 2 (D)) : E T 0 D 0 θ -2 λ -3 µ -4 ξ -3 |h| 2 dxdt < +∞, E Q T θ -2 λ -2 µ -2 ξ -3 |H| 2 dxdt < +∞ .
It can be readily seen that the functional J is continuous, strictly convex and coercive. Therefore, the minimization problem (78) admits a unique optimal pair solution that we denote by (h , H ). From classical arguments: the Euler-Lagrange equation for (77) at the minimum (h , H ) and a duality argument (see, for instance, [START_REF] Lions | Optimal control of systems governed by partial differential equations. Translated from the French[END_REF]), the pair (h , H ) can be characterized as

h = -χ D 0 θ 2 λ 3 µ 4 ξ 3 z , H = -θ 2 λ 2 µ 2 ξ 3 Z in Q, a.s., (79)
where the pair (z , Z ) verifies the backward stochastic equation

     dz = (-∆z -θ -2 y )dt + Z dW (t) in Q T , z = 0 on Σ T , z (T ) = 1 y (T ) in D, (80)
and where (y , y (T )) can be extracted from y the solution to (74) with controls h = h and

H = H . Observe that since y ∈ L 2 F (Ω; C([0, T ]; L 2 (D)
)) the evaluation of y at t = T is meaningful and ( 80) is well-posed for any > 0.

Step 2. Using Itô's formula, we can compute d(y z ) and deduce

E D y (T )z (T ) dx = E D y (0)z (0) dx + E Q T (∆y + F + χ D 0 h )z dxdt + E Q T (-∆z -θ -2 y )y dxdt + E Q T H Z dxdt
whence, replacing the initial data of systems ( 74), (80) and using identity (79), we get

E T 0 D 0 θ 2 λ 3 µ 4 ξ 3 |z | 2 dxdt + E Q T θ 2 λ 2 µ 2 ξ 3 |Z | 2 dxdt + E Q T θ -2 |y | 2 dxdt + 1 E D |y (T )| 2 dx = E D y 0 z (0) dx + E Q T F z dxdt . ( 81
)
Now, we will use the Carleman estimate in Theorem 2.1. We will apply it to equation (80) with Ξ = -θ -2 y and z = Z . Then, after removing some unnecessary terms, we get for any λ and µ large enough

E D λ 2 µ 3 θ 2 (0)|z (0)| 2 dx + E Q T λ 3 µ 4 ξ 3 θ 2 |z | 2 dxdt + E Q T λ 2 µ 2 ξ 3 θ 2 |Z | 2 dxdt ≤ CE T 0 D 0 λ 3 µ 4 ξ 3 θ 2 |z | 2 dxdt + Q T θ 2 |θ -2 y | 2 dxdt + Q T λ 2 µ 2 ξ 3 θ 2 |Z | 2 dxdt . (82)
Notice that we have added an integral of Z on the left-hand side of the inequality. This increases a little bit the constant C on the right-hand side but it is still uniform with respect to λ and µ.

In view of (82), we use Cauchy-Schwarz and Young inequalities in the right-hand side of (81)

to obtain

E T 0 D 0 θ 2 λ 3 µ 4 ξ 3 |z | 2 dxdt + E Q T θ 2 λ 3 µ 4 ξ 3 |z | 2 dxdt + E Q T θ -2 |y | 2 dxdt + 1 E D |y (T )| 2 dx ≤ δ E D θ 2 (0)λ 2 µ 3 |z (0)| 2 dx + E Q T θ 2 λ 3 µ 4 ξ 3 |z | 2 dxdt + Q T θ 2 λ 2 µ 2 ξ 3 |Z | 2 dxdt + C δ E D θ -2 (0)λ -2 µ -3 |y 0 | 2 dx + E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt ( 83
)
for any δ > 0. Using inequality (82) to estimate in the right-hand side of (83) and the fact that θ 2 θ -2 ≤ 1 for all (t, x) ∈ Q T , we obtain, after taking δ > 0 small enough, that

E T 0 D 0 θ 2 λ 3 µ 4 ξ 3 |z | 2 dxdt + E Q T θ 2 λ 2 µ 2 ξ 3 |Z | 2 dxdt + E Q T θ -2 |y | 2 dxdt + 1 E D |y (T )| 2 dx ≤ C E D θ -2 (0)λ -2 µ -3 |y 0 | 2 dx + E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt .
Recalling the characterization of the optimal control h in (79) we obtain

E T 0 D 0 θ -2 λ -3 µ -4 ξ -3 |h | 2 dxdt + E Q T θ -2 λ -2 µ -2 ξ -3 |H | 2 dxdt + E Q T θ -2 |y | 2 dxdt + 1 E D |y (T )| 2 dx ≤ C E D θ -2 (0)λ -2 µ -3 |y 0 | 2 dx + E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt . ( 84
)
Observe that the right-hand side of (84) is well-defined and finite since θ -2 (0) < +∞ and the source term F belongs to S λ,µ , defined in (75).

Step 3. Since the right-hand side of ( 84) is uniform with respect to , we readily deduce that there exists ( h, y, Y ) such that

     h h weakly in L 2 (Ω × (0, T ); L 2 (D 0 )), H H weakly in L 2 (Ω × (0, T ); L 2 (D)), y y weakly in L 2 (Ω × (0, T ); L 2 (D)). ( 85
)
We claim that y is the solution to (74) associated to ( h, H). To show this, let us denote by ỹ the unique solution in L 2 F (0, T ; C([0, T ]; L 2 (D))) ∩ L 2 F (0, T ; H 1 0 (D)) to (74) with controls ( h, H). For any m ∈ L 2 F (0, T ; L 2 (D)), we consider (z, Z) the unique solution to the backward equation

     dz = (-∆z -m)dt + ZdW (t) in Q T , z = 0 on Σ T , z(T ) = 0 in D. (86)
Then, using Itô's formula, we compute the duality between ( 86) and (74) associated to (h, H) = (h , H ) and (h, H) = ( h, H), respectively. We have

-E D y 0 z(0) dx = -E Q T my dxdt + E Q T F z dxdt + E T 0 D 0 h z dxdt + E Q T H z dxdt (87)
and

-E D y 0 z(0) dx = -E Q T mỹ dxdt + E Q T F z dxdt + E T 0 D 0 hz dxdt + E Q T Hz dxdt . ( 88
)
Then, using (85) in (87) to pass to the limit → 0 and subtracting the result from (88), we get ỹ = y in Q T , a.s.

To conclude, we notice from (84) that y(T ) = 0 in D, a.s. Also, from the weak convergence (85), Fatou's lemma and the uniform estimate (84) we deduce (76). This ends the proof.

Proof the nonlinear result for the forward equation

Now, we are in position to prove Theorem 1.1. To this end, let us fix the parameters λ and µ in Theorem 2.3 to a fixed value sufficiently large. Recall that in turn, these parameters come from Theorem 2.1 and should be selected as λ ≥ λ 0 and µ ≥ µ 0 for some λ 0 ≥ 1 and µ 0 ≥ 1, so there is no contradiction.

Note that at this point, we have preserved explicitly the parameters λ and µ in the controllability result of Theorem 2.3. This was possible due to the selection of the weight θ in the Carleman estimate (21), which allows to have a term depending on z(0) in the left-hand side.

Proof of Theorem 1.1. Let us consider a nonlinearity f fulfilling (2), (3), and (5). We define the nonlinear map

N : F ∈ S λ,µ → f (ω, t, x, y) ∈ S λ,µ ,
where y is the trajectory of (74) associated to the data y 0 , F , see Theorem 2.3 and Remark 2.4. In what follows, to abridge the notation, we simply write f (y).

We will check the following facts for the nonlinear mapping N . The mapping N is well-defined. To this end, we need to show that for any F ∈ S λ,µ , N (F ) ∈ S λ,µ . We have from (2), (3), and (5), that

N (F ) 2 S λ,µ = E Q T θ -2 λ -3 µ -4 ξ -3 |f (y)| 2 dxdt ≤ λ -3 µ -4 L 2 E Q T θ -2 ξ -3 |y| 2 dxdt .
Using (76) and ξ -1 ∞ ≤ 1 for all (t, x) ∈ Q T , we get

N (F ) 2 S λ,µ ≤ L 2 λ -3 µ -4 C 1 E y 0 2 L 2 (D) + C F 2 S λ,µ < +∞.
This proves that N is well-defined. The mapping N is strictly contractive. Let us consider source terms F i ∈ S λ,µ , i = 1, 2. We denote the solutions of the corresponding equations by y 1 and y 2 , respectively. Using the fact that the nonlinearity f is globally Lipschitz, i.e. (3), we have

N (F 1 ) -N (F 2 ) 2 S λ,µ = E Q T θ -2 λ -3 µ -4 ξ -3 |f (y 1 ) -f (y 2 )| 2 dxdt ≤ L 2 λ -3 µ -4 E Q T θ -2 |y 1 -y 2 | 2 dxdt .
where we have used again that ξ -1 ∞ ≤ 1 for all (t, x) ∈ Q T . Then applying Theorem 2.3 and Remark 2.4, and using estimate (76) for the equation associated to F = F 1 -F 2 , y 0 = 0, we deduce from the above inequality that

N (F 1 ) -N (F 2 ) 2 S λ,µ ≤ CL 2 λ -3 µ -4 E Q T θ -2 λ -3 µ -4 ξ -3 |F 1 -F 2 | 2 dxdt = CL 2 λ -3 µ -4 F 1 -F 2 2 S λ,µ , (89)
where C = C(D, D 0 ) > 0 comes from Theorem 2.3. Observe that all the constants in the righthand side of (89) are uniform with respect to λ and µ thus, if necessary, we can increase their value so CL 2 λ -3 µ -4 < 1. This yields that the mapping N is strictly contractive. Once we have verified these two conditions, by the Banach fixed point theorem, it follows that N has a unique fixed point F in S λ,µ . By setting y the trajectory associated to this F , we observe that y is the solution to

     dy = (∆y + f (ω, t, x, y) + χ D 0 h)dt + HdW (t) in Q T , y = 0 on Σ T , y(0) = y 0 in D, (90)
and verifies that y(T, •) = 0 in D, a.s. This concludes the proof of Theorem 1.1.

Controllability of a semilinear backward stochastic parabolic equation

As for the forward equation, the main ingredient to prove Theorem 1.2 is a controllability result for a linear system with a source term. In this case, we shall focus on studying the controllability of

     dy = (-∆y + χ D 0 h + F )dt + Y dW (t) in Q T , y = 0 on Σ T , y(T ) = y T in D,
where F ∈ L 2 F (0, T ; L 2 (D)) and y T ∈ L 2 (Ω, F T ; L 2 (D)) are given. Unlike the previous section, we shall not devote to prove a Carleman estimate for the corresponding adjoint system (i.e. a forward equation). Although this is possible, we will see later that we can greatly simplify the problem by studying a random parabolic equation, for which a deterministic Carleman estimate will suffice.

A deterministic Carleman estimate and its consequence

As we mentioned in Section 2.1, in [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF] the authors have proved a Carleman estimate for the (backward) heat equation with weights that do not vanish as t → 0 + (see ( 16) and ( 17)). Following their approach it is possible to prove the analogous result for a forward equation. For this, we need to introduce some new weight functions which are actually the mirrored version of ( 16) and (17).

In more detail, let us consider the function β as in (15) and let 0 < T < 1. We define the function γ(t) as

                 γ(t) = 1 t m , t ∈ (0, T /4], γ is decreasing on [T /4, T /2], γ(t) = 1, t ∈ [T /2, 3T /4], γ(t) = 1 + 1 -4(T -t) T σ , t ∈ [3T /4, T ], γ ∈ C 2 ([0, T ]), (91)
where m ≥ 1 and σ ≥ 2 is defined on (18). Observe that γ(t) is the mirrored version of γ(t) in ( 16) with respect to T /2. Analogous to the properties of γ, the function γ preserves one important property which is that for the interval [3T /4, T ] the derivative of γ has a prescribed sign, i.e., γ t ≥ 0.

With this new function, we define the weights ϕ = ϕ(t, x) and ξ = ξ(t, x) as

ϕ(t, x) := γ(t) e µ(β(x)+6m) -µe 6µ(m+1) , ξ(t, x) := γ(t)e µ(β(x)+6m) , (92)
where µ ≥ 1 is some parameter. In the same spirit, we set the weight θ = θ(t, x) as

θ := e where (t, x) = λ ϕ(t, x) for a parameter λ ≥ 1.
In what follows, to keep the notation as light as possible and emphasizing that there is no possibility for confusion since the notation is specific for this section, we simply write γ = γ, θ = θ, and so on.

We have the following Carleman estimate for the heat equation with source term

     ∂ t q -∆q = g(t, x) in Q T , q = 0 on Σ T , q(0) = q 0 (x) in D. (93)
Theorem 3.1. For all m ≥ 1, there exist constants C > 0, λ 0 ≥ 1 and µ 0 ≥ 1 such that for any q 0 ∈ L 2 (D) and any g ∈ L 2 (Q T ), the weak solution to (93) satisfies

Q T θ 2 λµ 2 ξ|∇q| 2 dxdt + Q T θ 2 λ 3 µ 4 ξ 3 |q| 2 dxdt + D λ 2 µ 3 e 2µ(6m+1) θ 2 (T )|q(T )| 2 dx ≤ C Q T θ 2 |g| 2 dxdt + D 0 ×(0,T ) θ 2 λ 3 µ 4 ξ 3 |q| 2 dxdt ,
for all µ ≥ µ 0 and λ ≥ λ 0 .

The proof of this result is a straightforward adaptation of [BEG16, Theorem 2.5], just by taking into account that in this case the weight γ verifies γ t ≥ 0 in [3T /4, T ], contrasting with the fact that γ t ≤ 0 in [0, T /4] as in [START_REF] Badra | Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations[END_REF] or as we have used in the proof of Theorem 2.1.

Let us consider the forward parabolic equation given by

     dq = (∆q + G 1 )dt + G 2 dW (t) in Q T , q = 0 on Σ T , q(0, x) = q 0 (x) in D, (94)
where

G i ∈ L 2 F (0, T ; L 2 (D)), i = 1, 2, and q 0 ∈ L 2 (Ω, F 0 ; L 2 (D)
). An immediate consequence of Theorem 3.1 is a Carleman estimate for a random parabolic equation. More precisely, we have the following. Lemma 3.2. Assume that G 2 ≡ 0. For all m ≥ 1, there exists constants C > 0, λ 0 ≥ 1 and µ 0 ≥ 1 such that for any q 0 ∈ L 2 (Ω, F 0 ; L 2 (D)) and any G 1 ∈ L 2 F (0, T ; L 2 (D)), the corresponding solution to (94) with G 2 = 0 satisfies

E Q T θ 2 λµ 2 ξ|∇q| 2 dxdt + E Q T θ 2 λ 3 µ 4 ξ 3 |q| 2 dxdt + E D θ 2 (T )|∇q(T )| 2 dx + E D λ 2 µ 3 e 2µ(6m+1) θ 2 (T )|q(T )| 2 dx ≤ CE Q T θ 2 |g| 2 dxdt + D 0 ×(0,T ) θ 2 λ 3 µ 4 ξ 3 |q| 2 dxdt , (95)
for all µ ≥ µ 0 and λ ≥ λ 0 .

A controllability result for a linear backward stochastic heat equation with source term and one control

Inspired by the duality technique presented in [Liu14, Prop. 2.2], we present a controllability result for a linear backward stochastic heat equation with a source term. To this end, consider the linear control system given by

     dy = (-∆y + χ D 0 h + F )dt + Y dW (t) in Q T , y = 0 on Σ T , y(T ) = y T in D, (96)
where F ∈ L 2 F (0, T ; L 2 (D)) is a given fixed source term and h ∈ L 2 F (0, T ; L 2 (D 0 )) is a control. In what follows, we consider constants µ and λ large enough such that (95) holds. We define the space S λ,µ := F ∈ L 2 F (0, T ; L 2 (D)) : E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt < +∞ , endowed with the canonical norm. We have the following global null-controllability result for system (96).

Theorem 3.3. For any initial datum y T ∈ L 2 (Ω, F T ; L 2 (D)) and any F ∈ S λ,µ , there exists a control h ∈ L 2 (0, T ; L 2 (D 0 )) such that the associated solution (y,

Y ) ∈ [L 2 F (Ω; C[0, T ]; L 2 (D)) ∩ L 2 F (0, T ; H 1 0 (D))] × L 2 F (0, T ; L 2 (D))
to system (96) satisfies y(0) = 0 in D, a.s. Moreover, the following estimate holds

E Q T θ -2 |y| 2 dxdt + E Q T θ -2 λ -2 µ -2 ξ -2 |Y | 2 dxdt + E Q T θ -2 λ -3 µ -4 ξ -3 |h| 2 dxdt ≤ C 1 E y T 2 L 2 (D) + CE Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt , (97)
where C 1 > 0 is a constant depending on D, D 0 , µ, λ and C > 0 only depends on D and D 0 .

Remark 3.4. As before, from classical arguments, see e.g. [LTT13, Proposition 2.9], from Theorem 3.3 we can construct a linear continuous mapping that associates every initial datum y T ∈ L 2 (Ω, F T ; L 2 (D)) and every source term F ∈ S λ,µ , to a trajectory ( y, h) such that y(0) = 0 in D, a.s. and (97) holds.

Proof. The proof is very similar to the one of Theorem 2.3 and requires only some adaptations. We emphasize their main differences.

Step 1. For any > 0, let us consider the weight function γ (t) given by

           γ (t) = γ(t + ), t ∈ [0, T /2 -], γ (t) = 1, t ∈ [T /2 -, 3T /4] γ (t) = 1 + (1 + 4(T -t) T ) σ , t ∈ [3T /4, T ], σ as in (18).
In this way, γ (t) ≤ γ(t) for t ∈ [0, T ]. We set the corresponding weight ϕ as in (92) by replacing the function γ by γ . Also, we write θ = e λϕ .

We introduce the cost functional

I (h) := 1 2 E Q T θ -2 |y| 2 dxdt + 1 2 E T 0 D 0 θ -2 λ -3 µ -4 ξ -3 |h| 2 dxdt + 1 2 E D |y(0)| 2 dx
and consider the minimization problem

min h∈H I (h), subject to equation (96), (98)
where

H = h ∈ L 2 F (0, T ; L 2 (D 0 )) : E T 0 D 0 θ -2 λ -3 µ -4 ξ -3 |h| 2 dxdt < +∞ .
It can be readily seen that the functional I is continuous, strictly convex and coercive. Therefore, the minimization problem (98) admits a unique optimal solution that we denote by h . As in the proof of Theorem 2.3 the minimizer h can be characterized as

h = χ D 0 λ 3 µ 4 ξ 3 θ 2 q in Q, a.s., (99)
where q verifies the random forward equation

     dq = (∆q + θ -2 y )dt in Q T , q = 0 on Σ T , q (0) = 1 y (0) in D, (100)
and where (y , y (0)) can be extracted from (y , Y ) the solution to (96) with control h = h . Observe that since y ∈ L 2 F (Ω; C([0, T ]; L 2 (D))) the evaluation of y at t = 0 is meaningful and (100) is well-posed for any > 0. Also notice that there is no term containing W (t) so (100) is regarded as a random equation. This greatly simplifies our task, since we only need to use the Carleman estimate of Lemma 3.2 to deduce the uniform estimate for the solutions to (y , Y ) in the next step.

Step 2. Using Itô's formula, we can compute d(y q ) and deduce

E D y (T )q (T ) dx = E D y (0)q (0) dx + E Q (-∆y + F + χ D 0 h )q dxdt + E Q (∆q + θ -2 y )y dxdt
whence, using equations (96), (100), and identity (99), we get

E T 0 D 0 λ 3 µ 4 ξ 3 θ 2 |q | 2 dxdt + E Q T θ -2 |y | 2 dxdt + 1 E D |y (0)| 2 dx = E D y T q (T ) dx -E Q T F q dxdt . (101)
In view of (95), we use Cauchy-Schwarz and Young inequalities in the right-hand side of (101) to introduce the weight function as follows

E T 0 D 0 θ 2 λ 3 µ 4 ξ 3 |q | 2 dxdt + E Q T θ -2 |y | 2 dxdt + 1 E D |y (0)| 2 dx ≤ δ E D λ 2 µ 3 θ 2 (T )|q (T, x)| 2 dx + E Q T θ 2 λ 3 µ 4 ξ 3 |q | 2 dxdt + C δ E D λ -2 µ -3 θ -2 (T )|y T | 2 dx + E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt ( 102
)
with δ > 0. Applying inequality (95) to (100) and using it to estimate in the right-hand side of (102), we obtain, after taking δ > 0 small enough, that

E T 0 D 0 θ 2 λ 3 µ 4 ξ 3 |q | 2 dxdt + E Q T θ -2 |y | 2 dxdt + 1 E D |y (0)| 2 dx ≤ C E D λ -2 µ -3 θ -2 (T )|y T | 2 dx + E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt
for some constant C > 0 only depending on D and D 0 . At this point, we have used the fact that

θ 2 θ -2 ≤ 1 for all (t, x) ∈ Q T .
Recalling the characterization of the optimal control h in (99) we obtain

E T 0 D 0 θ -2 λ -3 µ -4 ξ -3 |h | 2 dxdt + E Q T |y | 2 dxdt + 1 E D |y (0)| 2 dx ≤ C E D θ -2 λ -2 µ -3 |y T | 2 dx + E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 s -3 ξ -3 dxdt . (103)
Now, our task is to add a weighted integral of the process Y on the left-hand side of the above inequality. To do that, using Itô's formula and equation (96

) with h = h yield d(θ -2 λ -2 ξ -2 y 2 ) =(θ -2 λ -2 ξ -2 ) t y 2 dt + θ -2 λ -2 ξ -2 Y 2 + 2θ -2 λ -2 ξ -2 y [(-∆y + χ D 0 h + F )dt + Y dW (t)]
and after some integration by parts and substituting the initial datum, we get

E Q T θ -2 λ -2 ξ -2 |Y | 2 dxdt + 2E Q T θ -2 λ -2 ξ -2 |∇y | 2 dxdt + E Q T (θ -2 λ -2 ξ -2 ) t |y | 2 dxdt = E D θ -2 (T )λ -2 ξ -2 (T )|y T | 2 dx -2E Q T ∇(θ -2 λ -2 ξ -2 ) • ∇y y dxdt -2E Q T θ -2 λ -2 ξ -2 y F dxdt -2E T 0 D 0 θ -2 λ -2 ξ -2 y h dxdt (104)
Observe that the term containing y T is well defined since, by construction, the weight θ -1 does not blow up at t = T . Also, notice that there is no term of y (0, x) since ξ -1 (0) = 0 and the weight θ -1 does not blow up at t = 0.

Let us analyze the term containing (θ -2 λ -2 ξ -2 ) t in the left-hand side of the above identity. We split the integral as

E Q T (θ -2 λ -2 ξ -2 ) t |y | 2 dxdt = E 3T /4 0 D (θ -2 λ -2 ξ -2 ) t |y | 2 dxdt + E T 3T /4 D (θ -2 λ -2 ξ -2 ) t |y | 2 dxdt . (105)
We note that for t ∈ [3T /4, T ], γ (t) = γ(t), so we can drop the dependence of . Also notice that on this time interval γ t ≥ 0 and 1 ≤ γ ≤ 2. Thus, computing explicitly, we have

(θ -2 λ -2 ξ -2 ) t = -2θ -2 λ -1 γ t γ ϕξ -2 -2θ -2 λ -2 γ t γ ξ -2 . ( 106
) Recall that ϕ < 0, thus (θ -2 λ -2 ξ -2 ) t ≥ cθ -2 λ -1 γ t |ϕ|ξ -2 . ( 107
)
for all t ∈ [3T /4, T ], where c > 0 only depends on D and D 0 . Therefore,

E T 3T /4 D (θ -2 λ -2 ξ -2 ) t |y | 2 dxdt ≥ 0 (108)
and this term can be dropped. For t ∈ [0, 3T /4], we can use expression (106) (replacing everywhere the weights depending on ) and the fact that

|∂ t γ | ≤ Cγ 2 to obtain (θ -2 λ -2 ξ -2 ) t ≤ Cθ -2 λ -1 µ,
where the constant C > 0 is uniform with respect to λ and µ. Therefore,

E 3T /4 0 D (θ -2 λ -2 ξ -2 ) t |y | 2 dxdt ≤ CE 3T /4 0 D θ -2 λ -1 µ|y | 2 dxdt . (109)
Thus, using formulas (105) and ( 108)-(109) we deduce from (104) that

E Q T θ -2 λ -2 ξ -2 |Y | 2 dxdt + 2E Q T θ -2 λ -2 ξ -2 |∇y | 2 dxdt ≤ CE D θ -2 (T )λ -2 |y T | 2 dx + CE 3T /4 0 D θ -2 λ -1 µ|y | 2 dxdt + 2 E Q T ∇(θ -2 λ -2 ξ -2 ) • ∇y y dxdt + 2 E Q T θ -2 λ -2 ξ -2 y F dxdt + 2 E T 0 D 0 θ -2 λ -2 ξ -2 y h dxdt . ( 110
)
For the first term in the right-hand side, we have used that θ -2 (T ) = θ -2 (T ) and ξ -1 (T ) ≤ C for some C > 0 only depending on D and D 0 .

Employing Cauchy-Schwarz and Young inequalities, we estimate the last three terms of the above inequality. For the first one, we have

0 D 0 θ -2 λ -3 µ -4 ξ -3 |h | 2 dxdt . (114)
At this point, we have adjusted the powers of λ and ξ in the last term by using the fact that λ -1 ξ -1 ≤ C for some constant only depending on D, D 0 .

Finally, combining (114) and (103) we get

E T 0 D 0 θ -2 λ -3 µ -4 ξ -3 |h | 2 dxdt + E Q T θ -2 (|y | 2 + λ -2 µ -2 ξ -2 |Y | 2 ) dxdt + E Q T θ -2 λ -2 µ -2 ξ -2 |∇y | 2 dxdt + 1 E D |y (0)| 2 dx ≤ C E D θ -2 (T )λ -2 µ -2 |y T | 2 dx + E Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt , (115)
for some positive constant C only depending on D, D 0 .

Step 3. The last step is essentially the same as in the proof of Theorem 2.3. Since the righthand side of (115) is uniform with respect to , we readily deduce that there exists

( h, y, Y ) such that      h h weakly in L 2 (Ω × (0, T ); L 2 (D 0 )), y y weakly in L 2 (Ω × (0, T ); H 1 0 (D)), Y Y weakly in L 2 (Ω × (0, T ); L 2 (D)). (116)
To check that ( y, Y ) is the solution to (96) associated to h can be done exactly as in Theorem 2.3, so we omit it.

To conclude, we notice from (103) that y(0) = 0 in D, a.s. Also, from the weak convergence (116), Fatou's lemma and the uniform estimate (103) we deduce (97). This ends the proof of Theorem 3.3.

Proof the nonlinear result for the backward equation

Now, we are in position to prove Theorem 1.2. The proof is very similar to the one of Theorem 1.1 but for the sake of completeness, we give it.

Let us fix the parameters λ and µ in Theorem 3.3 to a fixed value sufficiently large. Recall that in turn, this parameter comes from the Theorem 3.1 and should be selected as λ ≥ λ 0 and µ ≥ µ 0 for some λ 0 , µ 0 ≥ 1, so there is no contradiction.

Let us consider a nonlinearity f fulfilling (12) and ( 13) and define

N : F ∈ S λ,µ → f (ω, t, x, y, Y ) ∈ S λ,µ ,
where (y, Y ) is the trajectory of (96) associated to the data y T and F , defined by Theorem 3.3 and Remark 3.4. In what follows, to abridge the notation, we simply write f (y, Y ).

We will check the following facts for the nonlinear mapping N . The mapping N is well-defined. To this end, we need to show that for any F ∈ S λ,µ , N (F ) ∈ S λ,µ . We have from ( 12) and ( 13)

N (F ) 2 S λ,µ = E Q T θ -2 λ -3 µ -4 ξ -3 |f (y, Y )| 2 dxdt ≤ 2L 2 E Q T θ -2 λ -3 µ -4 ξ -3 |y| 2 + |Y | 2 dxdt ≤ 2L 2 λ -1 µ -2 E Q T θ -2 λ -2 µ -2 ξ -2 |Y | 2 dxdt + E Q T θ -2 |y| 2 dxdt ≤ 2L 2 λ -1 µ -2 C 1 E y T 2 L 2 (D) + CE Q T θ -2 λ -3 µ -4 ξ -3 |F | 2 dxdt < +∞,
where we have used (97) and that ξ -1 ∞ ≤ 1. This proves that N is well-defined. The mapping N is a strictly contraction mapping. Let us consider F i ∈ S λ,µ , i = 1, 2. From the properties of the nonlinearity f , we have

N (F 1 ) -N (F 2 ) 2 S λ,µ = E Q T θ -2 λ -3 µ -4 ξ -3 |f (y 1 , Y 1 ) -f (y 2 , Y 2 )| 2 dxdt ≤ 2L 2 λ -1 µ -2 E Q T θ -2 λ -2 µ -2 ξ -2 |Y 1 -Y 2 | 2 dxdt + Q T θ -2 |y 1 -y 2 | 2 dxdt .
Then applying Theorem 3.3 to the equation associated to F = F 1 -F 2 , y T = 0, and using the corresponding estimate (97), we deduce from the above inequality that

N (F 1 ) -N (F 2 ) 2 S λ,µ ≤ 2CL 2 λ -1 µ -2 E Q T θ -2 λ -3 µ -4 ξ -3 |F 1 -F 2 | 2 dxdt = 2CL 2 λ -1 µ -2 F 1 -F 2 2 S λ,µ , (117)
where C = C(D, D 0 ) > 0 comes from Theorem 3.3. Observe that all the constants in the righthand side of (117) are uniform with respect to λ and µ thus, if necessary, we can increase the value of λ and µ so CL 2 λ -1 µ -2 < 1. This yields that the mapping is strictly contractive. Once we have verified these two conditions, it follows that N has a unique fixed point F in S λ,µ . By setting (y, Y ) the trajectory associated to this F , we observe that (y, Y ) is the solution to (10) and verifies y(0, •) = 0 in D, a.s. This concludes the proof of Theorem 1.2. The controllability result provided by Theorem 3.3 yields as a byproduct the obtention of a new global Carleman estimate for forward stochastic parabolic equations with a weight that do not vanish as t → T -. In fact, under the construction of weights shown in ( 91) and (92) (where again we drop the tilde notation for simplicity), we are able to prove the following result.

Further results and remarks

Proposition 4.1. For all m ≥ 1, there exist constants C > 0, λ 0 ≥ 1 and µ 0 ≥ 1 such that for any q 0 ∈ L 2 (Ω, F 0 ; L 2 (D)) and G i ∈ L 2 F (0, T ; L 2 (D)), i = 1, 2, the solution y ∈ W T to (94) satisfies

E Q T θ 2 λµ 2 ξ|∇q| 2 dxdt + E Q T θ 2 λ 3 µ 4 ξ 3 |q| 2 dxdt + E Ω θ 2 (T )λ 2 µ 2 |q(T )| 2 dx ≤ CE Q T θ 2 |G 1 | 2 dxdt + Q T θ 2 λ 2 µ 2 ξ 2 |G 2 | 2 dxdt + D 0 ×(0,T )
θ 2 λ 3 µ 4 ξ 3 |q| 2 dxdt , for all µ ≥ µ 0 and λ ≥ λ 0 .

The proof of Proposition 4.1 can be achieved by following the proof of [Liu14, Theorem 1.1] with a few straightforward adaptations. For completeness, we give a brief sketch below.

The starting point is to use Theorem 3.3 with F = θ 2 λ 3 µ 4 ξ 3 q and y T = -s 2 µ 2 θ 2 (T )q(T ), where q is the solution to (94) with given G 1 and G 2 . Observe that the weight functions in these data are well defined and bounded. We also remark that since the solution q belongs to W T , we have that y T = -λ 2 µ 2 θ 2 (T )q(T ) ∈ L 2 (Ω, F T ; L 2 (D)) and thus system (96) with these given data is well-posed.

Thus, from Theorem 3.3, we get that there exists a control h ∈ L 2 F (0, T ; L 2 (D)) such that the solution y to

     d y = (-∆ y + χ D 0 h + θ 2 λ 3 µ 4 ξ 3 q)dt + Y dW (t) in Q T , y = 0 on Σ T , y(T ) = -θ 2 λ 2 µ 2 q(T ) in D, (118)
satisfies y(0) = 0 in D, a.s. Moreover, the following estimate holds

E Q T θ -2 | y| 2 dxdt + E Q T θ -2 λ -2 µ -2 ξ -2 | Y | 2 dxdt + E Q T θ -2 λ -3 µ -4 ξ -3 | h| 2 dxdt ≤ CE D λ 2 µ 2 θ 2 (T )|q(T )| 2 dx + Q T e -2sϕ s 3 ξ 3 |q| 2 dxdt , (119)
for some constant C > 0 only depending on D, D 0 .

From (94), (118) and Itô's formula, we get

E D θ 2 (T )λ 2 µ 2 |q(T )| 2 dx + E Q T e -2sϕ s 3 ξ 3 |q| 2 dxdt = -E Q T yG 1 dxdt -E Q T Y G 2 dxdt -E T 0 D 0 hq dxdt .
Using Cauchy-Schwarz and Young inequalities, together with (119), it can be obtained from the above identity that

E D θ 2 (T )λ 2 µ 2 |q(T )| 2 dx + E Q T θ 2 λ 3 µ 4 ξ 3 |q| 2 dxdt ≤ CE Q T θ 2 |G 1 | 2 dxdt + Q T θ 2 λ 2 µ 2 ξ 2 |G 2 | 2 dxdt + T 0 D 0 θ 2 λ 3 µ 4 ξ 3 |q| 2 dxdt .
To add the integral containing ∇q, it is enough to compute d(e -2sϕ λξq 2 ) and argue as in Step 2 of the proof of Theorem 3.3. For brevity, we omit the details.

Other type of nonlinearities

where f and g are two globally Lipschitz nonlinear functions. We may wonder if (120) is smalltime globally null-controllable. A good starting point seems to obtain a Carleman estimate for the backward equation (20), with a source term Ξ ∈ L 2 F (0, T ; H -1 (D)). This seems to be possible, according to [START_REF] Liu | Global Carleman estimate for stochastic parabolic equations, and its application[END_REF]Remark 1.4]. By a duality argument, this would lead to a null-controllability result for the system (74) similar to Theorem 2.3, with an estimate of ρ y in L 2 F (0, T ; H 1 0 (D)), where ρ is some suitable weight function. Details remain to be written. Another open question is whether Theorem 1.1 can be extended to slightly superlinear nonlinearities in the spirit of [START_REF] Fernández | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]. In the recent paper [START_REF] Sylvain Ervedoza | Exact controllability of semilinear heat equations through a constructive approach[END_REF], the authors revisit the nullcontrollability of semilinear heat equations in the deterministic setting through a constructive approach based on a Banach fixed point argument similar to the one performed here. It would be interesting to see if their method can be adapted to the stochastic setting.

Extension of the method to other equations

The method introduced in this article could probably be applied to other nonlinear equations for which there is a lack of compactness embeddings for the solutions spaces and for which we are able to derive Carleman estimates in the spirit of [BEG16, Theorem 2.5]. For instance, for the Schrödinger equation, it is a well known fact that there is no regularizing effect so there is a lack of compactness. Up to our knowledge, the following question is still open. Let f : C → C be a globally Lipschitz nonlinearity and (T, D, D 0 ) be such that the so-called Geometric Control Condition holds. Is the system      i∂ t y = ∆y + f (y) + χ D 0 h in Q T , y = 0 on Σ T , y(0) = y 0 in D, globally null-controllable? See [START_REF] Zuazua | Remarks on the controllability of the Schrödinger equation[END_REF] or [START_REF] Laurent | Internal control of the Schrödinger equation[END_REF] for an introduction to this problem. We also refer to [L 13] for results in the stochastic setting.

  Known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Statement of the main results and presentation of the methodology . . . . . . . . . . . . . . . . . 1.3 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Controllability of a semilinear forward stochastic parabolic equation 2.1 A new global Carleman estimate for a backward stochastic parabolic equation . . . . . . . . . . . 2.2 A controllability result for a linear forward stochastic heat equation with one source term and two controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Proof the nonlinear result for the forward equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Controllability of a semilinear backward stochastic parabolic equation 3.1 A deterministic Carleman estimate and its consequence . . . . . . . . . . . . . . . . . . . . . . . 3.2 A controllability result for a linear backward stochastic heat equation with source term and one control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof the nonlinear result for the backward equation . . . . . . . . . . . . . . . . . . . . . . . . . 4 Further results and remarks 4.1 A new Carleman estimate for forward equation as a consequence of Theorem 3.3 . . . . . . . . . 4.2 Other type of nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Extension of the method to other equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. 1

 1 A new Carleman estimate for forward equation as a consequence of Theorem 3.3

  It should be interesting to extend Theorem 1.1 to the case where the semilinearities f and g depend on the gradient of the state. More precisely, let us consider ∆y + f (ω, t, x, y, ∇y)+ χ D 0 h)dt + (g(ω, t, x, y, ∇y) + H)dW (t) in Q T , y = 0 on Σ T , y(0) = y 0 in D,

Acknowledgements

We are deeply grateful to the anonymous referees for the very careful reading and valuable comments and suggestions that have led to an improvement of the results and the presentation of this work.

The work of the first author was supported by the programme "Estancias posdoctorales por México" of CONACyT, Mexico. The second author has benefited from financial support of the French State, managed by the French National Research Agency (ANR) in the frame of the "Investments for the future" Programme IdEx Bordeaux -SysNum (ANR-10-IDEX-03-02). The second author has also been supported by the grant "Numerical simulation and optimal control in view of temperature regulation in smart buildings" of the Nouvelle Aquitaine Region. The second author is partially supported by the Project TRECOS ANR-20-CE40-0009 funded by the ANR (2021-2024).