Guillaume Barraquand

Alexandre Krajenbrink

Pierre Le Doussal

Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case

We study the Kardar-Parisi-Zhang equation on the half-line x 0 with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter u of the dynamics, and the drift -v of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time t, it is natural to rescale parameters as (u, v) = t -1/3 (a, b) to study the critical region. In the special case a + b = 0, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase (a < 0) but not in the unbound phase. For instance, starting from the flat or droplet initial condition, the height field near the boundary converges to the stationary process with a > 0 and b = 0, which is not Brownian. For a + b 0, we determine exactly the large time distribution F stat a,b of the height function h(0, t). As an application, we obtain the exact covariance of the height field in a half-line at two times 1 t 1 t 2 starting from stationary initial condition, as well as estimates, when starting from droplet initial condition, in the limit t 1 /t 2 → 1.

Introduction and main results

Introduction

The Kardar-Parisi-Zhang (KPZ) equation [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF] in one dimension describes the time evolution of the height field h(x, t) of an interface which undergoes a local growth process driven by white noise. It is a paradigmatic element of a large universality class of one dimensional models with identical universal behavior at large scale, the so-called KPZ class. Since the interface is growing, with h(x, t) ∼ v ∞ t at large time, it is intrisically an out of equilibrium problem. One can nevertheless ask whether a stationary state can be reached at large time.

While the height at one point grows linearly in time with non trivial t 1/3 fluctuations, the height difference between any two points, h(x, t)-h(y, t), reaches a stationary distribution. Even when this distribution is known, it says nothing about the increments of the height field in time, say h(0, t) -h(0, 0). In this paper, we are interested in computing these temporal increments and their asymptotics at stationarity (i.e. starting from a stationary initial condition), and apply it to compute the two-time covariance of the height field.

Let us first briefly review what is known about stationary distributions for the KPZ equation. They depend on whether one considers the equation on the full-line, or in a restricted geometry such as a half-line or an interval. On the full-line, it has been predicted for a long time [START_REF] Forster | Large-distance and long-time properties of a randomly stirred fluid[END_REF][START_REF] Parisi | On the replica approach to random directed polymers in two dimensions[END_REF] that the KPZ equation admits the Brownian motion (BM) with an arbitrary drift as a stationary measure. This was proved rigorously in [START_REF] Bertini | Stochastic burgers and KPZ equations from particle systems[END_REF], and in [START_REF] Hairer | The strong Feller property for singular stochastic PDEs[END_REF] for periodic boundary conditions. Interestingly, in the cases of the half-line and the interval, the generic situation is more complicated (not translation invariant, not Gaussian, see below). One typically imposes Neumann type boundary conditions (that is, one fixes the derivative of the height field at the boundary) so that stationary measures depend on boundary parameters. For the interval it depends on the two boundary parameters, while for the half-line it depends on one boundary parameter and on the drift at infinity. In the special case when boundary and drift parameters are such that the slope imposed at the origin has the same value as the drift at infinity, the BM (with the same drift) is again stationary, as was shown in the case of the half-line in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] (this specific half-line stationary measure was studied in the equivalent directed polymer context in [START_REF] Barraquand | Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall[END_REF]). But this Brownian stationary measure is not unique, and for arbitrary values of the drift parameter, the stationary measures have been found only recently.

For the KPZ equation on an interval [0, L], an explicit formula for the Laplace transform (LT) of the stationary height distribution was obtained in [START_REF] Corwin | Stationary measure for the open KPZ equation[END_REF] (for L = 1, and for some range of parameters). Explicit Laplace inversion was performed shortly after in [START_REF] Bryc | Markov processes related to the stationary measure for the open KPZ equation[END_REF] and [START_REF] Barraquand | Steady state of the KPZ equation on an interval and Liouville quantum mechanics[END_REF] (see also [START_REF] Bryc | Markov limits of steady states of the KPZ equation on an interval[END_REF]). In [START_REF] Barraquand | Steady state of the KPZ equation on an interval and Liouville quantum mechanics[END_REF] a rather simple and explicit characterization of the process h(x, t) -h(x, 0) was obtained by two of the present authors. This characterization allowed in particular to predict the stationary measures in the limit of the half-line, letting the size of the interval to infinity. This large interval limit is actually quite non-trivial, but surprisingly, the limit of stationary measures on [0, L] had been already studied in the mathematics literature [START_REF] Hariya | Limiting distributions associated with moments of exponential Brownian functionals[END_REF], with completely different motivations. This is why we refer below to the resulting processes on the half-line (i.e. the non-Brownian stationary distribution for KPZ in a half-space) as Hariya-Yor processes.

Let us now describe in more details the question we address in this paper. While the steady-state distribution of h(x, t) -h(0, t) for the KPZ equation on the half-line x 0 is now clear, it remains to understand the global height, that is the height at one point, for instance h(0, t). Apart from situations where the boundary is very attractive, already studied in [START_REF] Barraquand | Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall[END_REF], the height grows at large time as h(0, t) v ∞ t + t 1/3 χ, where χ is a random variable whose distribution depends on some details of the initial condition h(x, 0). For the KPZ equation on the full-line with a stationary, i.e. BM initial condition, χ follows the Baik-Rains distribution [START_REF] Baik | Limiting distributions for a polynuclear growth model with external sources[END_REF][START_REF] Imamura | Stationary correlations for the 1D KPZ equation[END_REF][START_REF] Imamura | Exact solution for the stationary Kardar-Parisi-Zhang equation[END_REF][START_REF] Borodin | Height fluctuations for the stationary KPZ equation[END_REF]. It is universal over the KPZ class and, remarkably, was measured in recent experiments on liquid crystals [START_REF] Iwatsuka | Direct evidence for universal statistics of stationary Kardar-Parisi-Zhang interfaces[END_REF]. For the half-line, the corresponding question was addressed only recently, but until now only for the special case mentioned above where the BM (with drift) is still the stationary measure. The analog of the Baik-Rains distribution then depends on one boundary parameter, we denote it F Brownian a , and it was obtained in [START_REF] Betea | Stationary half-space last passage percolation[END_REF] (in the context of last-passage-percolation, which is equivalent by universality) and in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] directly for the KPZ equation for a = 0.

The aim of this paper is to obtain the analog of the Baik-Rains distribution for the KPZ equation on the half-line starting from a stationary initial condition in the generic case, that is for the Hariya-Yor initial conditions. We show in this paper that these Hariya-Yor processes are integrable, in the sense that one can write down simple exact formulas for the mixed exponential moments, using the framework of half-space Macdonald processes [START_REF] Barraquand | Half-space Macdonald processes[END_REF]. Through the usual Hopf-Cole mapping h(x, t) = log Z(x, t), the process Z(x, t) solves the multiplicative noise stochastic heat equation (that is the partition function of a continuous Brownian directed polymer in a random potential) and we compute explicitly all moments of Z(x, t) via a Bethe ansatz type approach. Following a similar line as in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] (and previous works including [START_REF] Borodin | Directed random polymers via nested contour integrals[END_REF][START_REF] Krajenbrink | Replica bethe ansatz solution to the Kardar-Parisi-Zhang equation on the half-line[END_REF]), we express the Laplace transform of Z(x, t) as Fredholm Pfaffians and determinants, which we analyze asymptotically to obtain the limiting distribution of χ.

It is important to note that although our results are obtained from the large-scale analysis of the KPZ equation, we expect that they hold universally for all half-space models in the KPZ universality class. As an application, following a method introduced in [START_REF] Ferrari | Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process[END_REF] for full-space models, we have obtained the two-time covariance of the KPZ height in a half-line geometry. These results can also be translated in terms of the free energy of a directed polymer in the presence of a wall.

Outline. In the following sections, we first review the stationary measures for the KPZ equation on the half-line (Section 1.2) and discuss the various distributions that arise for the large time fluctuations of the height field (Section 1.3). Our main new results are presented in Section 1.3.3. We then present an important application of the main results to the computation of two-time covariance in half-space KPZ growth in Section 1.4.

The remaining sections are devoted to details of the derivations. In Section 2, we obtain the moments and Laplace transform formulas characterizing the distribution of h(0, t). We analyze the formulas asymptotically in Section 3 and obtain explicit formulas for the cumulative distribution function (CDF) of the limiting distributions. We provide the details of computations of two-time covariance in half-space KPZ growth in Section 4. Finally, in Appendix A, we extend the results of our previous work [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] to provide an explicit formula for the CDF F Brownian a when the boundary parameter a = 0. In Appendix B, we explain how to compute the distribution of h(x, t) instead of h(0, t), when x is scaled of order t 2/3 , and the initial condition is a Hariya-Yor processes.

Half-space KPZ stationary measures

Let Z(x, t) denote the solution to the half-space stochastic heat equation

∂ t Z(x, t) = ∂ xx Z(x, t) + √ 2η(x, t)Z(x, t), (x 0, t 0), (1)
with standard white noise η, initial condition Z 0 (x) and boundary parameter A ∈ R, corresponding formally to

∂ x Z(x, t) x=0 = AZ(0, t).
We will often denote the boundary parameter rather by the letter u where u =1 2 + A. Then, we will say that h u (x, t) = log Z(x, t) solves the Kardar-Parisi-Zhang equation on

R + ∂ t h u = ∂ xx h u + (∂ x h u ) 2 + √ 2η (2)
with boundary parameter u.

It was noticed in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] that if the initial condition x → → h u (x, 0) is distributed as

h u (x, 0) = B(x) + ux (3)
that is a Brownian motion with drift u, then for all t 0, the process x → h u (x, t)-h u (x, 0) has the same distribution, that is

h u (x, t) -h u (0, t) (d) = B(x) + ux. ( 4
)
In other terms, the Brownian motion with drift u is a stationary distribution for the KPZ equation in a half-space with boundary parameter u. In infinite volume (that is for dynamics on functions of R or R + ), there is no reason to expect that the stationary process is unique. Indeed, there exist other, more complicated, stationary measures for the half-space KPZ equation, recently described in [START_REF] Barraquand | Steady state of the KPZ equation on an interval and Liouville quantum mechanics[END_REF], based on results for the stationary measure on an interval from [START_REF] Corwin | Stationary measure for the open KPZ equation[END_REF] (see also [START_REF] Bryc | Markov processes related to the stationary measure for the open KPZ equation[END_REF][START_REF] Bryc | Markov limits of steady states of the KPZ equation on an interval[END_REF] for an equivalent description of stationary measures on an interval). These additional stationary measures depend on a parameter v, where -v is the drift of the process at infinity, and they arise only when u v, v 0. It is convenient to represent them on the diagram of Fig. 1, which explains which stationary processes arise in the large time limit, depending on the boundary parameter u and the drift of the initial condition. They are defined in terms of a process that we call the Hariya-Yor process, defined below and denoted HY u,v , For u > v, v 0 with u + v > 0, we define the Hariya-Yor process, denoted

1 HY u,v (x), by exp HY u,v (x) := w 1 x 0 dte B 1 (t)+B 2 (x)-B 2 (t) + w 1 w 2 e B 2 (x) , ( 5
)
where B 1 , B 2 are independent standard Brownian motions with drifts -v and v respectively, and w 1 , w 2 are independent inverse gamma random variables w 1 ∼ Gamma -1 (u+v)

R 1 = {u 0, v 0}, R 2 = {u 0, u v} and R 3 = {u v, v 0}.
Along the line u + v = 0, the stationary measure is a Brownian motion with drift u = -v. and w 2 ∼ Gamma -1 (u -v). We say that a random variable X follows the Gamma -1 (θ) distribution if 1/X is a Gamma random variable with scale parameter 1 and shape parameter θ. In other terms, X is a positive random variable with density 1 Γ(θ) 1 x>0 x -θ e -1/x dx x . In order to give a unified description of the stationary measures, is is convenient to define, for any u, v ∈ R, a stationary initial condition h stat u,v , where

h stat u,v (x) =          HY u,0 (x) -HY u,0 (0), for u > 0, v 0, that is (u, v) ∈ R 1 , B(x) + ux for u 0, v u, that is (u, v) ∈ R 2 , HY u,v (x) -HY u,v (0), for u > v, v 0, that is (u, v) ∈ R 3 . ( 6
)
Let us stress that while in the phase u > v, v 0 the process h stat u,v (x) describes exactly the spatial increments of HY u,v (x), it is defined differently in the other phases and this is why we needed a new notation. We will use alternatively h stat u,v (x) and HY u,v (x) for the following reason: h stat u,v (x) is the natural way to describe the stationary height field, in particular it is normalized so that h stat u,v (0) = 0. The process HY u,v (x), however, is not normalized but contains the appropriate random shift that makes exact computations possible.

Remark 1.1. The Hariya-Yor process ( 5) is defined only when u > v, v 0 with u+v > 0. It depends on a random variable w 1 ∼ Gamma -1 (u + v), which explains the condition u + v > 0. However,

h stat u,v (x) = HY u,v (x) -HY u,v (0) = HY u,v (x) -log w 1 -log w 2
depends only on w 2 ∼ Gamma -1 (u -v), and on two Brownian motions B 1 , B 2 , hence it is well-defined for any u > v, v 0, without imposing the condition u + v > 0.

Remark 1.2. For any u > 0,

HY u,-u (x) -HY u,-u (0) = B(x) + ux (7)
where B(x) is a standard Brownian motion. This is a nontrivial result, discovered by Hariya and Yor in [START_REF] Hariya | Limiting distributions associated with moments of exponential Brownian functionals[END_REF].

Remark 1.3. As v → u, it is easy to check that HY u,v (x) -HY u,v (0) ==⇒ v→u B(x) + ux. ( 8
)
Remark 1.4. Our definition of HY u,v (x) may appear different from the process defined in [10, Eq. ( 35)] but it is equal up to a shift by log(w 1 w 2 ). Indeed, the process is defined in [START_REF] Barraquand | Steady state of the KPZ equation on an interval and Liouville quantum mechanics[END_REF] as

H(x) = B (1) (x) + B (2) (x) + log 1 + γ u-v x 0 e -2B (2) (z) dz , ( 9
)
where B (1) (x), B (2) (x) are independent Brownian motions with variance 1/2 and drifts 0 and v respectively. Using the change of variables

B (1) = B 1 + B 2 2 , B (2) = B 2 -B 1 2 ,
B 1 , B 2 are two independent standard Brownian motions of drifts -v and v respectively, and we have that, if we identify γ u-v in (9) with 1/w 2 in (5), then

e H(x) = e HY u,v (x) w 1 w 2 , ( 10
) so that H(x) = HY u,v (x) -log(w 1 ) -log(w 2 ). (11)
Remark 1.5. The integral (5) is similar to the partition function of the semi-discrete O'Connell-Yor polymer [START_REF] O'connell | Brownian analogues of Burke's theorem[END_REF][START_REF] O'connell | A representation for non-colliding random walks[END_REF] with two rows. Other O'Connell-Yor partition functions decorated by inverse Gamma variables appeared in the literature, [START_REF] Barraquand | Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall[END_REF][START_REF] Borodin | Height fluctuations for the stationary KPZ equation[END_REF][START_REF] Talyigás | Borodin-Péché fluctuations of the free energy in directed random polymer models[END_REF], though the partition function in (5) is different. In the limit u → ∞, we recover exactly the O'Connell-Yor partition function.

1.3 Limiting distribution for the KPZ height on a half-line 1.3.1 Droplet initial condition. For the droplet initial condition, i.e. e hu(x,t) → t→0 δ(x), and boundary parameter u, a phase transition occurs based on the sign of u. We have

lim t→∞ P h u (0, t) + t 12 t 1/3 s =          F GSE (s) for u > 0, F GOE (s) for u = 0, 0 for u < 0. ( 12
)
In the phase u < 0, the scaling and statistics are different: h u (0, t) t -1 12 + u 2 [START_REF] Kardar | Depinning by quenched randomness[END_REF] and statistics are Gaussian on the t 1/2 scale (see [START_REF] Barraquand | Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall[END_REF][START_REF] De Nardis | Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions[END_REF]). If we scale u close to the critical point as u = at -1/3 , we have that

lim t→∞ P h at -1/3 (0, t) + t 12 t 1/3 s = F droplet a (s). ( 13
)
The existence of a transition was anticipated in [START_REF] Kardar | Depinning by quenched randomness[END_REF]. In the equivalent directed polymer problem, it corresponds to a transition to polymers bound to the wall at x = 0 when u < 0 to unbound polymers when u 0. The statistics occurring around the phase transition [START_REF] Hariya | Limiting distributions associated with moments of exponential Brownian functionals[END_REF] and the exact formula for F droplet a were first discovered in [START_REF] Baik | The asymptotics of monotone subsequences of involutions[END_REF][START_REF] Baik | Random matrix models and their applications[END_REF] in the context of asymptotic fluctuations of symmetrized last passage percolation models. For the KPZ equation, these asymptotics were obtained in [START_REF] Gueudré | Directed polymer near a hard wall and KPZ equation in the half-space[END_REF] for u = +∞, in [START_REF] Borodin | Directed random polymers via nested contour integrals[END_REF] for u = 1/2, in [START_REF] Barraquand | Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process[END_REF] for a = 0 (i.e. the critical case, u = 0), and in [START_REF] Krajenbrink | Replica bethe ansatz solution to the Kardar-Parisi-Zhang equation on the half-line[END_REF][START_REF] De Nardis | Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions[END_REF] in the general case. We have in particular F droplet 0 = F GOE .

Brownian initial condition.

For a Brownian initial condition h u (x, 0) = B(x)+ ux, where B(x) is a standard Brownian motion, which is stationary for any u ∈ R, we have for u = at -1/3

lim t→∞ P h at -1/3 (0, t) + t 12 t 1/3 s = F Brownian a (s). ( 14
)
This statement and an exact formula for F Brownian a was obtained in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] in the special case a = 0. In the context of last-passage percolation, the analogous statement for any a ∈ R was shown earlier in [START_REF] Betea | Stationary half-space last passage percolation[END_REF]. We also provide in Appendix A an alternative formula for F Brownian a , based on an extension of our earlier result for a = 0 in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] (The formula for F Brownian a (s) appears in (133)). It remains to be shown that the exact formulas from [START_REF] Betea | The half-space Airy stat process[END_REF] and our formulas in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] and Section A are equivalent. Note that in [START_REF] Betea | Stationary half-space last passage percolation[END_REF][START_REF] Betea | The half-space Airy stat process[END_REF], formulas were also obtained for the height distribution and multipoint correlations at points away from the boundary.

According to the phase diagram of Fig. 1, in the bound phase, i.e. when u < 0, for any initial condition with drift at infinity not exceeding -u, the spatial process h u (x, t) -h u (0, t) should converge, as t goes to infinity, to a Brownian with drift u, hence the importance of this case. Consequences about the geometry of the polymer and the distribution of the endpoint were investigated in [START_REF] Barraquand | Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall[END_REF].

In the unbound phase u 0 however, the stationary process obtained at large time is the Hariya-Yor process HY u,v , when the drift of the initial condition is positive and equals -v > 0, and HY u,0 when the drift is negative. Brownian stationary measures arise only when u + v = 0, see Remark 1.2, which is a very special case. For droplet or flat initial condition for instance, the stationary process observed at large times is the Hariya-Yor process HY u,0 . This makes the study of fluctuations starting from the Hariya-Yor initial condition particularly important in the unbound phase.

Main new result: Hariya-Yor initial condition.

We now focus on the regions R 1 and R 3 . We assume that the initial condition h u (x, t) = log Z(x, t) is given by h u (x, 0) = HY u,v (x), for u > v, v 0 as defined in [START_REF] Hairer | The strong Feller property for singular stochastic PDEs[END_REF], under the additional technical assumption u + v > 0. We will show that, for

u = at -1/3 , v = bt -1/3 , with a + b > 0, b 0 lim t→∞ P h at -1/3 (0, t) + t 12 t 1/3 s = G HY a,b (s) ( 15
)
where the CDF G HY a,b (s) will be explicitly determined. An explicit formula for G HY a,b (s) is given in (78). When a > 0 and b = 0, that is in the maximal current phase, the formula simplifies and we obtain

G HY a,0 (s) = ∂ s Det(I + Ãs ) - 2 a + s + 2R + , ( 16
)
where Ãs is an integral operator acting on L 2 (0, +∞) with kernel Ãs (x, y) = Ã(s + x + y) where Ã(x) = dz 2iπ

a + z a -z e -xz+ z 3 3 , ( 17
)
where the contour is a vertical line with real part between 0 and a, and R + is a scalar product defined using quantum mechanical notations as (see Section 3.2 for details)

R + = 1| Ãs 1 + Ãs |1 . ( 18
)
Remark 1.6. As a → +∞, we notice that G HY a,0 (s) → F Brownian 0 given in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF]Eq. (7.24)]. This limit is obvious from the formula but the reason is non-trivial, it can be seen as a consequence of a surprising identity in distribution obtained in [6, The distribution given by G HY a,b is not centered, unlike the Baik-Rains distribution or the distribution F Brownian a , but this is an artefact due to our definition of the Hariya-Yor process. Indeed, under the scaling u = at -1/3 , v = bt -1/3 , lim t→∞ t -1/3 HY u,v (0)

(d) = E a+b + E a-b
, where E a+b and E a-b are independent exponential random variables with parameters a ± b. We can construct a centered variable as follows: we may write that

lim t→∞ P h at -1/3 (0, t) -h at -1/3 (0, 0) + t 12 t 1/3 s = F HY a,b (s) (19)
where

F HY a,b (s) is such that E F HY a,b (s -E a+b -E a-b ) = G HY a,b (s). ( 20
)
More explicitly, this means that

G HY a,b (s) = (a 2 -b 2 ) +∞ 0 du e -ua sinh(bu) b F HY a,b (s -u). ( 21
)
This can be inverted as

F HY a,b (s) = G HY a,b (s) + ∂ s G HY a,b (s) 2a a 2 -b 2 + ∂ 2 s G HY a,b (s) 1 a 2 -b 2 . ( 22
)
Equivalently, if we denote by χ a,b a random variable with distribution G HY a,b and ξ a,b a random variable with distribution F HY a,b , we have that (recall that we have assumed in this section that b 0, a + b > 0)

ξ a,b + E a+b + E a-b = χ a,b . ( 23
)
where ξ a,b , E a+b and E a-b are independent. By stationarity, we can argue that E ξ a,b = 0. Indeed,

E ξ a,b = lim t→∞ t -1/3 E h at -1/3 (0, t) -h at -1/3 (0, 0) + t 12 = lim t→∞ t -1/3 t 0 ∂ s E h at -1/3 (0, s) -h at -1/3 (0, 0) + 1 12 ds
The quantity inside the integral is independent of s, by stationarity, so it is a constant. This constant is necessarily 0, otherwise we would have that lim t→∞

h at -1/3 (0,t) t = -1 12 .
F HY a,b (s) ----→ b→-a F Brownian a (s), (24)
although this is not immediately apparent from our formulas.

Summary.

The results presented above for the different phases of the diagram in Fig. 1 can be summarized in a unified manner. We expect that for any a, b ∈ R, scaling u = at -1/3 , v = bt -1/3 , when one considers the initial condition h(x, 0) = h stat u,v (x) defined in (6), we have that

lim t→∞ P h at -1/3 (0, t) + t 12 t 1/3 s = F stat a,b (s), (25)
for some CDF F stat a,b . We have obtained [START_REF] Talyigás | Borodin-Péché fluctuations of the free energy in directed random polymer models[END_REF] and determined a formula for F stat a,b in a number of cases (see Fig. 2), in particular:

F stat a,b (s) =              F HY a,0 (s) for a > 0, b 0, F Brownian a for a 0, b a, F HY a,b (s) for a > b, b 0, a + b > 0, F Brownian a for a 0, a + b = 0, ( 26
)
where F Brownian a is defined in Section 1.3.2 and F HY a,b is defined in Section 1.3.3. In the next Sections, we will denote by ξ a,b a random variable with distribution F stat a,b .

Away from the boundary.

While we understand now the distribution and asymptotics of h u (0, t) and we know as well the distribution of stationary spatial increments h u (x, t) -h u (0, t), we cannot immediately deduce the asymptotics of h u (x, t) for

x > 0. This is due to the fact that the random variables h u (0, t) and h u (x, t) -h u (0, t) are correlated in a very nontrivial way. However, the distribution of h u (x, t) can be computed explicitly, at least in the large time limit, for the same initial condition as in the previous sections. When x = xt 2/3 , we obtain

lim t→∞ P   h at -1/3 (t 2/3 x, t) + t 12 t 1/3 y   = G HY a,b (y, x). ( 27
)
The CDF is computed in Appendix B and given in (191).

Two-time covariance

An interesting application of our results concerns the two-time covariance. For general initial conditions, one can study the correlation of the KPZ height field at two different times, i.e. the correlation of h(0, t 1 ) and h(0, t) (here we focus on the same space point). In the limit where both times are large 1 t 1 < t with a fixed ratio τ = t 1 /t, this covariance becomes a universal function C(τ ) of τ ∈ [0, 1], whose specific form depends only on the class of initial condition (stationary, droplet, flat, etc. . . ). This covariance, as well as the full two-time height distribution, were studied for the full-space geometry [START_REF] Ferrari | On time correlations for KPZ growth in one dimension[END_REF][START_REF] De Nardis | Memory and universality in interface growth[END_REF][START_REF] Nardis | Tail of the two-time height distribution for KPZ growth in one dimension[END_REF][START_REF] Doussal | Maximum of an Airy process plus Brownian motion and memory in Kardar-Parisi-Zhang growth[END_REF][START_REF] Johansson | The two-time distribution in geometric last-passage percolation[END_REF][START_REF] De Nardis | Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica[END_REF][START_REF] Johansson | Long and short time asymptotics of the two-time distribution in local random growth[END_REF][START_REF] Liao | Multi-point distribution of discrete time periodic TASEP[END_REF][START_REF] Ferrari | Time-time covariance for last passage percolation with generic initial profile[END_REF][START_REF] Corwin | KPZ equation correlations in time[END_REF] and measured in experiments [START_REF] Takeuchi | Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence[END_REF]. One defines for a general initial condition,

C(τ ) = lim t→+∞ 1 t 2/3 Cov(h u (0, tτ ), h u (0, t)) (28)
The results of this paper allow to obtain some predictions for C(τ ) in the half-space geometry. In particular we obtain C(τ ) for the stationary initial conditions for all 0 < τ < 1, and for droplet initial conditions for close times, i.e. 1 -τ 1. Of particular interest are the dimensionless ratios

(τ ) = lim t→+∞ Cov(h u (0, tτ ), h u (0, t)) Varh u (0, tτ )Varh u (0, t) = C(τ ) τ 1/3 C(1) , ( 29
)
R(τ ) = lim t→+∞ Cov(h u (0, tτ ), h u (0, t)) Varh u (0, tτ ) = C(τ ) τ 2/3 C(1) , ( 30
)
which measure the overlap of the two polymer configurations (of lengths τ t and t respectively). In particular a finite value of R(0), i.e. in the limit of infinitely separated times is a measure of memory or ergodicity breaking [START_REF] De Nardis | Memory and universality in interface growth[END_REF][START_REF] Doussal | Maximum of an Airy process plus Brownian motion and memory in Kardar-Parisi-Zhang growth[END_REF][START_REF] Takeuchi | Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence[END_REF].

Let us first consider the droplet initial condition, and start by recalling the result for the full-space problem. For close times, i.e. in the limit τ → 1, one expects that the height profile reaches local stationarity at the intermediate time, which allows to compute the two-time covariance. Indeed it was shown that [START_REF] Ferrari | On time correlations for KPZ growth in one dimension[END_REF]

C droplet (τ ) = Var[ξ GUE ] - 1 2 Var[ξ BR ](1 -τ ) 2/3 + O(1 -τ ), (31)
where ξ GUE follows the GUE Tracy-Widom distribution and ξ BR denotes a random variable following the Baik-Rains distribution [START_REF] Baik | Limiting distributions for a polynuclear growth model with external sources[END_REF]. We have Var[ξ BR ] ≈ 1.1504. In this paper we extend the arguments of [START_REF] Ferrari | On time correlations for KPZ growth in one dimension[END_REF] to the half-line geometry. The results depend on the value of the boundary parameter u. In the critical region (u close to zero) it is natural to scale the boundary parameter u = at -1/3 , and, as shown below, we obtain for the droplet initial condition (centered at x = 0), as τ → 1,

C droplet (τ ) = Var ξ droplet a - 1 2 Var ξ 0,0 (1 -τ ) 2/3 + O(1 -τ ), (32)
where ξ droplet a denotes a random variable with distribution F droplet a defined in [START_REF] Hariya | Limiting distributions associated with moments of exponential Brownian functionals[END_REF], and ξ 0,0 denotes a random variable with distribution F stat 0,0 = F Brownian 0 . The variance was computed in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF], we have Var[ξ 0,0 ] ≈ 1.649. In the particular case a = 0 the variable ξ droplet 0 has the GOE Tracy Widom distribution and its variance is Var[ξ droplet 0 ] ≈ 1.6078. In the limit a → +∞ one enters the unbound phase and ξ droplet ∞ has the GSE Tracy-Widom distribution [START_REF] Tracy | On orthogonal and symplectic matrix ensembles[END_REF] with variance is Var[ξ droplet ∞ ] ≈ 0.5177.

Consider now stationary initial conditions. In that case one can compute C(τ ) for arbitrary τ ∈ [0, 1]. Let us recall the result in full-space, obtained in [START_REF] Ferrari | On time correlations for KPZ growth in one dimension[END_REF]. If the initial condition is given by a standard Brownian motion, one obtains

C(τ ) = 1 2 1 + τ 2/3 -(1 -τ ) 2/3 Var[ξ BR ]. ( 33
)
Note that it leads to the dimensionless ratio R(τ

) = C(τ ) τ 2/3 → τ →0 1
2 which shows a memory effect at very separated times. We have extended the arguments of [START_REF] Ferrari | On time correlations for KPZ growth in one dimension[END_REF] to the half-line geometry. The results depend on the value of the boundary and drift parameters u, v. Let us start with the discussion of the critical region with u = at -1/3 , v = bt -1/3 . We find, for any a, b, starting from the initial condition

h stat at -1/3 ,bt -1/3 , C(τ ) = 1 2 Var(ξ a,b ) + τ 2/3 Var(ξ aτ 1/3 ,bτ 1/3 ) -(1 -τ ) 2/3 Var(ξ a(1-τ ) 1/3 ,b(1-τ ) 1/3 ) . ( 34
)
where ξ a,b a random variable with distribution F stat a,b defined in [START_REF] Kardar | Depinning by quenched randomness[END_REF].

Remark 1.8. For τ → 0 one finds R(0

+ ) = 1 2 Var(ξ 0,0 ) Var(ξ a,b )
Remark 1.9. Consider the case a = 0 and b 0. For b 0, ξ a,b = ξ a,0 by definition, and for a = 0, ξ a,0 = ξ 0,0 = ξ Brownian 0 , so that the dimensionless ratio defined in [START_REF] Baik | Random matrix models and their applications[END_REF] is asymptotically

(τ ) = 1 2τ 1/3 1 + τ 2/3 -(1 -τ ) 2/3 , ( 35
)
that is exactly the same as for the KPZ equation in full-space [START_REF] Ferrari | On time correlations for KPZ growth in one dimension[END_REF]. For general a, b this is not the case however. Using Remark 1.6, the same formula is true in the case a = +∞, b 0 (since ξ +∞,0 = ξ Brownian 0

).

Let us now discuss what happens for fixed u, v. There are three phases which are depicted in Fig. 3 where the main results are summarized. The first phase is defined by u, v > 0 corresponding by to taking the limit a, b → +∞ in the previous discussion. We find that the two-time covariance does not depend on u, v and is given by (35) (indeed, as explained in Remark 1.9, ξ +∞,+∞ = ξ Brownian 0 ). If u < 0 or v < 0, however, the scalings will be different with height fluctuations of order t 1/2 instead of t 1/3 . We define a variant of the coefficient C(τ ) by

C(τ ) = lim t→+∞ 1 t Cov(h u (0, tτ ), h u (0, t)). ( 36
)
To discuss the phase u < 0 (with u < v), we consider the equivalent polymer picture. In that phase, the polymer is bound to the wall (with Gaussian free energy fluctuations) so that Cov(h u (0, tτ ), h u (0, t)) Var(h u (0, tτ )) at large time, which is known [START_REF] Barraquand | Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall[END_REF][START_REF] De Nardis | Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions[END_REF] to be asymptotically equivalent to -2utτ . This implies that C(τ ) = -2uτ . The R ratio being now equal to R(τ ) = C(τ ) τ C [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF] one finds that it is equal to R(τ ) = 1 in that phase. In the phase v < 0 (with v < u), the covariance will be determined by the initial condition, which can be approximated at large scale by a Brownian motion with drift -v. The free energy equals

R(τ ) = 1 C(τ ) = -2uτ R(τ ) = 1 C(τ ) = -2vτ R(τ ) = 1 2τ 2/3 1 + τ 2/3 -(1 -τ ) 2/3 C(τ ) = Var[ξ Brownian 0 ]τ 2/3 R(τ )
h u (0, t) = log Z(0, t) = log +∞ 0 dxZ(x, 0|0, t)e hu(x,0) ≈ max x log Z(x, 0|0, t) + h u (x, 0) . ( 37
)
The optimal x = x max can be approximated by arg max x 2 4t + vt so that x max -2vt and the fluctuations of the free energy log Z(x, 0|0, t) are subdominant compared to fluctuations of h(x, 0). Thus, we again have that Cov(h u (0, tτ ), h u (0, t))

Var(h u (0, tτ )) which is asymptotically equivalent to the variance of the initial condition Var(h u (x, 0)) at the point x = -tτ v. Since the initial condition can be approximated at large scale by a Brownian motion with drift -v, we find that Cov(h u (0, tτ ), h(0, t)) = -2vtτ . This implies that C(τ ) = -2vτ , and again the R ratio is again equal to R(τ ) = C(τ ) τ C [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF] one finds that it is equal to R(τ ) = 1 in that phase.

Mathematical aspects

Let us stress a few points that deserve further consideration from a mathematical perspective. First of all, outside of the Brownian phase (a 0, a b), the fact that the Hariya-Yor processes defined in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] are stationary for the half-space KPZ equation was discovered in [START_REF] Barraquand | Steady state of the KPZ equation on an interval and Liouville quantum mechanics[END_REF]. These processes arose as L → +∞ limits of stationary processes for the KPZ equation on [0, L], for which formulas had been found in [START_REF] Corwin | Stationary measure for the open KPZ equation[END_REF]. That the L → +∞ limit of stationary processes on [0, L] are stationary for the dynamics on R + is a very reasonable hypothesis, but it still needs to be formally proven mathematically.

The computation of limiting distributions obtained in this article rely on a combination of physics and mathematics methods, but we focus in this article on physics results and do not attempt to prove the results according to the standards of writing in the mathematics literature. We refer to [6, Section 2.4] where we had already discussed the interplay between these physics and mathematics arguments, as well as the main challenges that would arise to turn these results into mathematical proofs. The results of [START_REF] Imamura | Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials[END_REF]Section 10] would likely be useful in order to prove rigorously the Pfaffian formula for the generating series [START_REF] Quastel | Airy processes and variational problems[END_REF].

The results of Section 1.4 also rely on a mixture of established facts, and assumptions based on analogies with the full-space setting, about the universal processes describing the large time fluctuations of h(x, t), starting from various initial conditions. It would be interesting to confirm those assumptions by rigorous proofs.

Acknowledgements

Moment formula

Nested contour moment formula

In order to study this initial condition ( 5), we will first study formulas from the more general initial condition

Z u,v 1 ,v 2 (x) = w 1 x 0 dte B 1 (t)+B 2 (x)-B 2 (t) + w 1 w 2 e B 2 (x) , ( 38
)
where B 1 , B 2 are independent standard Brownian motions with drifts -v 1 and -v 2 respectively, and w 1 , w 2 are independent inverse gamma random variables w 1 ∼ Gamma -1 (u + v 1 ) and w 2 ∼ Gamma -1 (u + v 2 ). Equation [START_REF] De Nardis | Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica[END_REF] can be interpreted as a two-row O'Connell-Yor semi-discrete polymer partition function with inverse gamma weights in the beginning of each row. Eventually, we will let

v 2 = -v 1 = v, so that log(Z u,v 1 ,v 2 (x)) = HY u,v (x).
Remark 2.1. Similar partition functions of O'Connell-Yor type polymers with inverse Gamma decorations have been considered earlier in [START_REF] Borodin | Height fluctuations for the stationary KPZ equation[END_REF][START_REF] Talyigás | Borodin-Péché fluctuations of the free energy in directed random polymer models[END_REF] in the context of full-space KPZ growth and in [START_REF] Barraquand | Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall[END_REF] in the context of KPZ equation in a half-space.

For x 1 x 2 • • • x k 0, define f (t, x) = E Z(x 1 , t) . . . Z(x k , t)
where we assume the initial condition Z(x, 0) = Z u,v 1 ,v 2 (x). The function f (t, x) satisfies the following conditions [START_REF] Borodin | Directed random polymers via nested contour integrals[END_REF] (see also [START_REF] Kardar | Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities[END_REF][START_REF] Gaudin | La fonction d'onde de Bethe[END_REF]). It satisfies the heat equation

∂ t f ( x, t) = k i=1 ∂ 2 x i f (t, x), ( 39
)
on the sector x 1 x 2 • • • x k 0, subject to the two-body boundary condition

∂ x i+1 -∂ x i -1 f x i =x i+1 = 0, (40)
with a boundary condition at 0 given by

∂ x k -(u - 1 2
)

f x k =0 = 0. ( 41
)
0 r1 + iR r2 + iR r3 + iR ... ... r k + iR v1 -1 2 v2 -1 2 1 2 -u > 1
Figure 4: The contours used in [START_REF] Takeuchi | Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence[END_REF].

The function f (t, x) must also satisfy the initial condition

f ( x, t = 0) = E   k i=1 Z(x i , 0)   . ( 42
)
Fix k 1 and assume that u,

v 1 , v 2 ∈ R 0 are such that v 1 -1 2 > k -1, v 2 -1 2 > k -1 (
note that this hypothesis is necessary and it was missing in [6, Claim 4.7]) and u+v 1 , u+v 2 > k. The function

f ( x, t|u, v 1 , v 2 ) = 2 k Γ(v 1 + v 2 ) Γ(v 1 + v 2 -k) r 1 +iR dz 1 2iπ • • • r k +iR dz k 2iπ 1 a<b k z a -z b z a -z b -1 z a + z b z a + z b -1 × k i=1 z i z i + u -1 2 1 (v 1 -1/2) 2 -z 2 i 1 (v 2 -1/2) 2 -z 2 i e tz 2 i -x i z i , ( 43
)
where the contours (see Fig. 4) are chosen so that

min v 1 - 1 2 , v 2 - 1 2 > r 1 > r 2 + 1 > . . . , > r k + k -1 > max{k -1 -u + 1 2 , k -1},
satisfies the equations ( 39), ( 40), [START_REF] Ferrari | Time-time covariance for last passage percolation with generic initial profile[END_REF] and the initial condition [START_REF] Corwin | KPZ equation correlations in time[END_REF] with

Z 0 (x) = Z u,v 1 ,v 2 (x)
defined in [START_REF] De Nardis | Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica[END_REF]. Assuming that there is at most one solution to these equations, we obtain that for

x 1 x 2 • • • x k 0, u + v 1 > k, u + v 2 > k, v 1 -1/2 > k -1 and v 2 -1/2 > k -1, E Z(x 1 , t) . . . Z(x k , t) = f ( x, t|u, v 1 , v 2 ) ( 44
)
where f ( x, t|u, v 1 , v 2 ) is defined in [START_REF] Takeuchi | Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence[END_REF].

Remark 2.2. When v 1 → +∞ one has that Z 0 (x) → w 1 w 2 e B 2 (x) and thus one recovers the result (4.19) in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF], taking into account that x) and the same works.

w k 1 = Γ(v 1 +u-k) Γ(v 1 +u) and w k 2 = Γ(v 2 +u-k) Γ(v 2 +u) . When v 2 → +∞ one has v 2 Z 0 (x) → w 1 e B 1 (
Let us now explain why f ( x, t|u, v 1 , v 2 ) satisfies each of the equations ( 39), ( 40), [START_REF] Ferrari | Time-time covariance for last passage percolation with generic initial profile[END_REF] and [START_REF] Corwin | KPZ equation correlations in time[END_REF] one by one (except for [START_REF] Corwin | KPZ equation correlations in time[END_REF], the arguments are based on [START_REF] Borodin | Directed random polymers via nested contour integrals[END_REF] and [START_REF] Borodin | Macdonald processes[END_REF]).

The function k i=1 e tz 2 i -x i z i is a solution of (39) for any z, so that by linearity, the function f ( x, t|u, v 1 , v 2 ) is also a solution.

Let us apply the operator ∂ x i+1 -∂ x i -1 to f ( x, t|u, v) and let us assume that x i = x i+1 for some i. The application of the operator brings a factor (-z i+1 + z i -1) inside the integrand in [START_REF] Takeuchi | Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence[END_REF]. This extra factor cancels with the denominator z a -z b -1 for a = i, b = i + 1, so that there is no pole anymore at z i+1 = z i -1. Hence, we may deform the contour for z i+1 to be the same as the one for z i . Because of the factor z a -z b for a = i, b = i + 1, the integrand is now antisymmetric with respect to exchanging z i and z i+1 , and since both variables are integrated along the same contour, their integral is zero. We conclude that f ( x, t|u, v) satisfies [START_REF] Liao | Multi-point distribution of discrete time periodic TASEP[END_REF].

Let as apply the operator

∂ x k -u + 1 2 to f ( x, t|u, v).
This brings an extra factor -z k -u + 1 2 to the integrand, which cancels the denominator z k + u -1 2 already present in the formula. After this cancellation, and if one assumes that x k = 0, then the integrand is antisymmetric with respect to changing z k into -z k . Furthermore, the integrand has no pole anymore at z k = -(u - 1 2 ) so that the contour of z k can be freely deformed to the left, regardless of the value of u. Since v -1 2 > k -1, it is then always possible to shift the z k contour to the left so that v-

1 2 > r 1 > r 2 +1 > • • • > r k-1 +k-2 > max{k-1, k-2-(u-1 2
)} and r k = 0. Now, the integration contour for z k is symmetric with respect to changing z k into -z k , and thus the integral is zero. We conclude that f ( x, t|A, B) satisfies [START_REF] Ferrari | Time-time covariance for last passage percolation with generic initial profile[END_REF].

Let us assume that t = 0 and x 1 > x 2 > . . . x k > 0. The formula agrees with moments of Z u,v 1 ,v 2 (x) using a scaling limit of the half-space log-gamma polymer moment formula from [START_REF] Barraquand | Half-space Macdonald processes[END_REF] (see [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF]Proposition 4.2]). More precisely, we take α

• = u, α 1 = v 1 , α 2 = v 2 and for i 3, α i = 1 2 +
√ n and consider the log-gamma partition function Z( √ nx/2, 2). The scaling limit is explained in [6, Section 4.3], see also [START_REF] Barraquand | Kardar-Parisi-Zhang equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall[END_REF]Appendix C.3].

Pfaffian formula

Let us define the two functions

G(z) = z 3 3 - z 2 2 + z 6 , Ω(z) = Γ(u -z) Γ(u + z) Γ(v 1 -z) Γ(v 1 + z) Γ(v 2 -z) Γ(v 2 + z) Γ(2z) (45)
Then, for u -1 2 , v 1 -1 2 , v 2 -1 2 > k -1, using the moment formula [START_REF] Tracy | On orthogonal and symplectic matrix ensembles[END_REF] and similar manipulations as in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF]Sections 4.4 and 4.7], based on [20, Conjecture 5.2], we obtain the expression for the moment of Z(0, t) as

E[Z(0, t) k ] = 4 k k! Γ(v 1 + v 2 ) Γ(v 1 + v 2 -k) λ k λ=1 m 1 2 m 2 ... (-1) (λ) m 1 !m 2 ! . . . iR dw 1 2iπ • • • iR dw (λ) 2iπ × Pf u i -u j u i + u j 2 (λ) i,j=1 (λ) j=1
e tG(w j +λ j ) e tG(w j )

(w j + 1/2) λ j -1 4(w j ) λ j Γ(-w j + 1)Γ(w j + λ j ) Γ(-w j -λ j + 1)Γ(w j ) × Γ(u + 1 2 -w j -λ j )Γ(v 1 + 1 2 -w j -λ j )Γ(v 2 + 1 2 -w j -λ j ) Γ(u -1 2 + w j + λ j )Γ(v 1 -1 2 + w j + λ j )Γ(v 2 -1 2 + w j + λ j ) × Γ(u -1 2 + w j )Γ(v 1 -1 2 + w j )Γ(v 2 -1 2 + w j ) Γ(u + 1 2 -w j )Γ(v 1 + 1 2 -w j )Γ(v 2 + 1 2 -w j ) , ( 46
)
where (u 1 , . . . , u 2 (λ) ) = (-

w 1 + 1 2 , w 1 -1 2 + λ 1 , . . . , -w (λ) + 1 2 , w (λ) -1 2 + λ (λ)
) and the sum runs over integer partitions λ of k. This can be rewritten as

E[Z(0, t) k ] = 4 k k! Γ(v 1 + v 2 ) Γ(v 1 + v 2 -k) λ k λ=1 m 1 2 m 2 ... (-1) (λ) m 1 !m 2 ! . . . iR dw 1 2iπ • • • iR dw (λ) 2iπ × Pf u i -u j u i + u j 2 (λ) i,j=1 (λ) j=1
e tG(w j +λ j ) e tG(w j ) (w j + 1/2) λ j -1 4(w j ) λ j Γ(-w j + 1)Γ(w j + λ j ) Γ(-w j -λ j + 1)Γ(w j )

× Ω(w j -1 2 + λ j )Ω(-

w j + 1 2 ) Γ(2w j -1 + 2λ j )Γ(1 -2w j ) , (47)
We recognize the same formula as [6, Claim 4.11, Eq. (4.35)] with the only difference that the ratio Ω(z) here should be replaced by

G(z) = Γ(A + 1/2 -z) Γ(A + 1/2 + z) Γ(B + 1/2 -z) Γ(B + 1/2 + z) Γ(2z). ( 48
)

Moment series in terms of a Fredholm Pfaffian

We will now write the moment generating function of Z(0, t). Let W be a inverse gamma random variable with parameter v 1 + v 2 , independent from the initial condition and from the noise ξ. We define, for ς > 0,

g(ς) = E exp(-ςe t 12 W Z(0, t)) . ( 49
)
Ignoring the fact that the summation over k cannot be exchanged with the expectation due to the divergence of moments, we will consider the following formal power series

1 + ∞ k=1 (-ςe t 12 ) k k! E W k Z(0, t) k , ( 50
)
that we will again denote by g(ς). This generating series was computed in [6, Section 5], and leads to the Fredholm Pfaffian formula [6, (5.16)] with the kernel [6, (2.12)]. Since our moment formula has exactly the same form with a different choice of Ω, we apply the same manipulations (it suffices to replace Ω(z) in the present paper by G(z) in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF]) and obtain the following.

g(ς) = 1 + ∞ =1 (-1) ! p=1 R dr p ς ς + e -rp Pf K(r i , r j ) 2 (λ) i,j=1 (51)
This series is a Fredholm Pfaffian,

g(ς) = E exp(-ςe t 12 W Z(0, t)) = Pf(J -σ ς K) L 2 (R) . ( 52
)
The kernel K is matrix valued and represented by a 2 × 2 block matrix with elements

K 11 (r, r ) = C 2 dw 2iπ dz 2iπ
w -z w + z Ω(w)Ω(z) cos(πw) cos(πz)e -rw-r z+t w 3 +z 3 (

) 53
where the dependence in parameters u, v 1 , v 2 only appears in the function Ω(z) which was defined in [START_REF] Imamura | Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials[END_REF], and the contour C is an upwardly oriented vertical line parallel to the imaginary axis with real part between 0 and min{u, v 1 , v 2 , 1}. The function σ ς is given by σ ς (r) = ς ς+e -r and the 2 × 2 symplectic kernel J is given by J(r, r ) = 0 1 -1 0 1 r=r .

As in [6, Section 2.2.2] we may also rewrite g(ς) as the square root of a Fredholm determinant with a scalar kernel. We obtain

g(ς) = E exp(-ςe t 12 W Z(0, t)) = Pf(J -σ ς K) L 2 (R) = Det(I -Kt,ς ) L 2 (R + ) . ( 54
)
where the kernel Kt,ς is defined by In the sequel, we will only be interested in the large time limit, in which case, asymptotics can be directly extracted from the asymptotics of g(ς).

Kt,ς (x, y) = 2∂ x C 2 dwdz (2iπ) 2 Ω(z)Ω(w) sin(π(z -w)) sin(π(z + w)) ς w+z e -xz-
3 Large time height distribution at x = 0

Limiting distribution in terms of a Fredholm determinant

As we have found in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF], at large times, g(e -t 1/3 s ) P h u (0, t) + log W + t 12 t 1/3 s .

(56)

We will also rescale parameters as

u = at -1/3 , v 1 = bt -1/3 , v 2 = ct -1/3 . ( 57
)
Under this scaling, log(W )/t 1/3 converges to an exponential distribution independent from h at -1/3 (0, t), so that [22, Eq. (4.3)]

lim t→+∞ P h at -1/3 (0, t) + t 12 t 1/3 s = 1 + ∂ s b + c lim t→∞ g(e -t 1/3 s ). ( 58
)
To compute the limit of g(e -t 1/3 s ), we use (55) with the scalings (w, z) → t -1/3 (w, z). The kernel t 1/3 Kt,e -t 1/3 s (xt 1/3 , yt 1/3 ) converges to K(a,b,c) (x + s, y + s) where

K(a,b,c) (x, y) = 1 2 Γ 2 A dwdz (2iπ) 2 ω(w)ω(z) w -z w + z 1 w e -xz-yw+ w 3 +z 3 3 , ( 59
)
where the contour Γ A is an upwardly oriented vertical line with real part between 0 and min{a, b, c} as previously, and

ω(z) = a + z a -z b + z b -z c + z c -z . ( 60
)
Thus, for any a, b, c > 0,

G (a,b,c) (s) := lim t→+∞ P h(0, t) + t 12 t 1/3 s (61) = 1 + ∂ s b + c Det(I -K(a,b,c) ) L 2 (s,+∞) , ( 62
)
where K(a,b,c) is defined in (59). Using the decomposition 1 2 w-z

(w+z)w = 1 w+z -1 2w , we obtain that K(a,b,c) (x, y) = +∞ 0 dλA (a,b,c) (x + λ)A (a,b,c) (λ + y) - 1 2 A (a,b,c) (x) +∞ 0 A (a,b,c) (y + λ) dλ, (63)
where the function A (a,b,c) (x) is defined by

A (a,b,c) (x) = Γ A dz 2iπ ω(z)e -xz+ z 3 3 , ( 64
)
where the contour Γ A is a vertical line with real part between 0 and min{a, b, c}. Note that the function A (a,b,c) has exponential decay at +∞, that is for any d ∈ (0, min{a, b, c}), there exist C ∈ R such that A (a,b,c) (x) Ce -dx . Let us introduce an operator Âs acting on L 2 (0, +∞) with kernel where all operators act on L 2 (0, +∞). Hence one has

A s (x, y) = A (a,b,c) (x + y + s), (65)
G (a,b,c) (s) = 1 2 1 + ∂ s b + c Det(I -Âs ) + Det(I + Âs ) (67)
In order to compute G (a,b,c) (s), we had assumed that a, b, c > 0. However, if we fix some a > 0, we expect that the RHS of (61) is analytic in b and c in the region {(b, c) ∈ R 2 ; b+c > 0, a + b > 0, a + c > 0} (recall that when a + b 0 or a + c 0, the initial condition would not even be defined, and when b + c 0 we cannot compute the moments). In the sequel, we implicitly extend the definition of G (a,b,c) (s) by analyticity. To be more precise, G (a,b,c) (s) is defined in (67) in terms of the operator Âs with kernel A (a,b,c) (x+y +s) where the function A (a,b,c) is defined in (64). This function can be readily extended analytically to b < 0 (for example). Indeed, the value when b < 0 can be expressed by first moving the integration contour to the right of b, taking into account the associated residue, and finally setting b to the desired negative value.

Stationary limit c → -b

Assume that a > b, c, 0. In this section we set c = -b + and let go to 0 to obtain the stationary limit. Let us rewrite

A (a,b,c) (x) = Ã(a,b,c) (x) + 2 b + c b -c h c (x) -h b (x) , ( 68
)
where

Ã(a,b,c) (x) = Γ Ã dz 2iπ ω(z)e -xz+ z 3 3 , ( 69
)
the contour being now a vertical line with real part between max{0, b, c} and a, and

h b (x) = b a + b a -b e b 3 3 -bx . ( 70
)
From now on, we use quantum mechanical notations with kets and bra. For functions f, g ∈ L 2 (0, +∞), and an operator O on L 2 (0, +∞) acting with a kernel O(x, y), we denote by

f | O |g the integral +∞ 0 dx +∞ 0 dyf (x)O(x, y)g(y)
and we denote by |f g| the operator acting on L 2 (R + ) with kernel f (x)g(y). In particular, 1| , |1 below corresponds to the constant functions f (x) = 1, g(y) = 1.

We rewrite the kernel as

A s (x, y) = Ã(a,b,c) s (x + y) + 2 b + c b -c |f c (x) g c (y)| -|f b (x) g b (y)| , ( 71
)
where

f α (x) = α a + α a -α e α 3 3 -(x+s)α , g β (x) = e -xβ . ( 72
  1 ± 2 b + c b -c I cc 1 ∓ 2 b + c b -c I bb + 4 b + c b -c 2 I bc I cb   (73)
where

I α,β = f α |g β ∓ f α | Ã(a,b,c) s I ± Ã(a,b,c) s |g β (74)
Explicitly, we have

I α,β = e α 3 3 -sα a + α a -α α α + β ∓ αR ± a,b,c (α, β) , ( 75
)
where

R ± a,b,c (α, β) = e -xα | Ã(a,b,c) s 1 ± Ã(a,b,c) s |e -xβ . ( 76
)
where the braket notation denotes the two-sided Laplace transform and has the following definition: for any operator O acting on

L 2 (R + ) with kernel (u, v) → O(u, v) we have e -xα | O |e -xβ = R 2 + dudv e -αu O(u, v)e -βv (77)
In the limit when c → -b, we find, after simplifications using Mathematica, that lim b→-c

F (a,b,c) (s) = G HY a,b (s) := 1 2 ∂ s Det(1 + Ãs )Q + (a, b, s) + Det(1 -Ãs )Q -(a, b, s) , ( 78
)
where Ãs is now an operator acting with kernel Ãs (x, y) = Ã(s + x + y) with

Ã(x) = dz 2iπ a + z a -z e -xz+ z 3 3 , ( 79
)
where the contour is a vertical line with real part between 0 and a. We define

Q ± (a, b, s) = S ± (a, b, s) + S ± (a, -b, s),
where

S + (a, b, s) = 1 2 s -b 2 + 2R + (b, -b) + (a + b) 2 2b(a 2 -b 2 ) e b 3 /3-bs (2bR + (b, b) -1) - a a 2 -b 2 , (80) S -(a, b, s) = 1 2 s -b 2 -2R -(b, -b) + (a + b) 2 2b(a 2 -b 2 ) e b 3 /3-bs (2bR -(b, b) + 1) - a a 2 -b 2 . ( 81
)
and

R ± (α, β) = e -xα | Ãs 1 ± Ãs |e -xβ . ( 82
) 3.3 Case b = 0, a > 0
This case corresponds to the maximal current phase. Only the + term remains, as in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF].

In the special case b = 0, we have the simplifications

Q + (a, 0, s) = 2 a -2 + as + 2aR + (0, 0) , ( 83
)
Q -(a, 0, s) = 0, ( 84
)
so that

G HY a,0 (s) = ∂ s Det(I + Ãs ) - 2 a + s + 2R + (0, 0) . ( 85
)
Remark 3.1. As a → +∞, we recover F Brownian a studied in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] (denoted simply F in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF]). This is due to a symmetry between boundary and initial condition parameters [6, Section 4.6] (see also [START_REF] Barraquand | Half-space Macdonald processes[END_REF][START_REF] Parekh | The KPZ limit of ASEP with boundary[END_REF]). The law of h(0, t) for the half-line KPZ equation with Dirichlet boundary condition (that is a = +∞) and with initial condition given by a O'Connell-Yor polymer partition function, is the same as the law of h(0, t) for the half-line KPZ equation with Robin boundary condition and Brownian initial condition, for appropriately chosen parameters, see details in 4 Two-time covariance

Full-space variational formulas

In full space one has, for the droplet initial condition e h(x,0) = δ 0 (x), at large time t, h(x, t)

+ t 12 t 1/3 (A 2 (x) -x2 ), x = x 2t 2/3 (86)
For a Brownian IC one has

h(x, t) + t 12 t 1/3 A stat (x) (87)
where the process A stat was introduced in [START_REF] Baik | Limit process of stationary TASEP near the characteristic line[END_REF], and can be characterized by the following formula [START_REF] Quastel | Airy processes and variational problems[END_REF]: for any fixed x,

A stat (x) = max ŷ∈R ( √ 2B(ŷ) + A 2 (x -ŷ) -(x -ŷ) 2 ). ( 88
)
Since the Brownian motion is stationary for the KPZ equation, (87) implies that, as processes in x,

A stat (x) -A stat (0) (d) = √ 2B(x). ( 89
)
where B is a two-sided Brownian motion.

Half-space universal processes

In half-space, for droplet IC and boundary parameter u = at -1/3 one has, for large t and fixed a, the solution h u (x, t) behaves as

h u=at -1/3 (x, t) + t 12 t 1/3 (A a (x) -x2 ), x = x 2t 2/3 0, ( 90
)
where A a (x) is a half-space variant of the Airy 2 process, having explicit finite-dimensional marginal distributions computed in [START_REF] Baik | Pfaffian Schur processes and last passage percolation in a half-quadrant[END_REF] (this limiting process was obtained as a limit of a model of last passage percolation in a half-space, but by universality, the same should arise as a limit of the KPZ equation). More generally, for an initial condition h 0 (x) = h(x, 0), such that the rescaled process

h rescaled 0 (x) = lim y→∞ 1 √ y h 0 (2xy) (91)
exists, we expect that the solution h u (x, t) of the half-space KPZ equation with boundary parameter u behaves asymptotically as

h u=at -1/3 (0, t) + t 12 t 1/3 max ŷ 0 h rescaled 0 (ŷ) + A a (ŷ) -ŷ2 . ( 92
)
For the stationary initial condition h(x, 0) = h stat u,v (x), we similarly expect that for u = at -1/3 and v = bt -1/3 , there exists a process A stat a,b (x) such that at large t

h at -1/3 (x, t) + t 12 t 1/3 A stat a,b (x), x = x 2t 2/3 0. ( 93
)
In particular, we have that

A stat a,b (0) (d) = ξ a,b . ( 94
)
In the case a + b = 0 and a 0, a b, that is when the initial condition is Brownian this process is defined and studied in [START_REF] Betea | The half-space Airy stat process[END_REF]. We expect that

A stat a,b (0) = max ŷ 0 h rescaled a,b (ŷ) + A a (ŷ) -ŷ2 . ( 95
)
where the process h rescaled a,b is defined as

h rescaled a,b (x) = lim r→∞ 1 √ r h stat ar -1/2 ,br -1/2 (2rx). ( 96
)
We may describe this process very explicitly using [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF], even in cases where it is not Brownian. For a b, b 0, we have that

h rescaled a,b (x) = max B 2 (2x), -E a-b + max t∈[0,x] B 1 (2t) + B 2 (2x) -B 2 (2t) , (97)
where B 1 , B 2 are independent standard Brownian motions with drifts -b and b respectively, and E a-b is an independent exponential random variables with parameter a -b.

For a 0, b 0,

h rescaled a,b (x) = max B 2 (2x), -E a + max t∈[0,x] B 1 (2t) + B 2 (2x) -B 2 (2t) , ( 98
)
where B 1 , B 2 are independent standard Brownian motions (without drift). In the limit a → +∞ and b = 0, this process has the same law as the maximum of two non intersecting Brownians [START_REF] O'connell | A representation for non-colliding random walks[END_REF]. For a b, a 0,

h rescaled a,b (x) = B(x) + ax, ( 99
)
where B is a standard Brownian motion.

By stationarity, we have the equality in distribution of processes in the variable x:

A stat a,b (x) -A stat a,b (0) (d)
= h rescaled a,b (x). (100)

Computation of two-time covariances

In this Section we compute two-time covariances starting from various initial conditions.

To this purpose we adapt the argument from [START_REF] Ferrari | On time correlations for KPZ growth in one dimension[END_REF] to the half-space geometry. We also use similar notations as in that paper. Let us use the notation, where t is the late time and τ t the earlier time, 0 < τ < 1,

C(τ ) = lim t→∞ Cov X t (τ ), X t (1) , X t (τ ) = h u (0, tτ ) + tτ 12 t 1/3 . ( 101
)
We will use the formula

Cov X t (τ ), X t (1) = 1 2 Var X t (1) + 1 2 Var X t (τ ) - 1 2 Var X t (1) -X t (τ ) . ( 102
)

Stationary Hariya-Yor initial condition.

Here we relate the two-time covariance of the scaled KPZ height field with stationary initial condition to the variance of the random variable ξ a,b studied in this paper of CDF denoted F stat a,b , defined in [START_REF] Kardar | Depinning by quenched randomness[END_REF]. For the moment we focus on the regions in the regions R 1 and R 3 , that is u > v, v 0. Assume that we start from the initial condition h(x, 0) = h stat u,v (x). Recall that by stationarity, E X t (τ ) = 0, so that we have

C(τ ) = lim t→∞ 1 2 VarX t (1) + VarX t (τ ) -E (X t (1) -X t (τ )) 2 . ( 103
)
Scaling u, v as u = at -1/3 , v = bt -1/3 , we have, by replacing t → τ t and a → aτ 1/3 , b → bτ 1/3 in (93) that the height field at the earlier time satisfies

h at -1/3 (x, τ t) + τ t 12 (τ t) 1/3 A stat aτ 1/3 ,bτ 1/3 x 2(τ t) 2/3 (104)
This field can then be used as an initial condition for the evolution from time τ t to time t. Using formula (92) with t → (1 -τ )t, a → a(1 -τ ) 1/3 and ŷ =

x 2(t(1-τ )) 2/3 it leads to the variational formula, where we denote τ = τ 1-τ

X t (1) (1 -τ ) 1/3 max ŷ>0 τ 1/3 A stat aτ 1/3 ,bτ 1/3 τ -2/3 ŷ + A a(1-τ ) 1/3 (ŷ) -ŷ2 , ( 105
)
where the processes A stat aτ 1/3 ,bτ 1/3 and A a(1-τ ) 1/3 are independent, because they describe the growth over two disjoint time intervals. Thus, using (104) with x = 0, X t (1) -X t (τ ) has asymptotically the same distribution as

(1 -τ ) 1/3 max ŷ>0 τ 1/3 A stat aτ 1/3 ,bτ 1/3 τ -2/3 ŷ -A stat aτ 1/3 ,bτ 1/3 (0) + A a(1-τ ) 1/3 (ŷ) -ŷ2 .
(106) Using ( 100) and (95), we obtain that (106) has the same distribution as

(1 -τ ) 1/3 max ŷ>0 τ 1/3 h rescaled aτ 1/3 ,bτ 1/3 (τ -2/3 ŷ) + A a(1-τ ) 1/3 (ŷ) -ŷ2 . ( 107
)
Now, we use the fact that the process h rescaled satisfies the scaling property

1 √ r h rescaled ar -1/2 ,br -1/2 (rx) (d) = h rescaled a,b (x), (108)
so that using (92),

X t (1) -X t (τ ) (d) = (1 -τ ) 1/3 max x>0 h rescaled a(1-τ ) 1/3 ,b(1-τ ) 1/3 (x) + A a(1-τ ) 1/3 (x) -x 2 , (d) = (1 -τ ) 1/3 ξ a(1-τ ) 1/3 ,b(1-τ ) 1/3 .
where in the last formula we used (95) and (94). Finally, we obtain that using (93) and ( 103)

C(τ ) = 1 2 Var(ξ a,b ) + τ 2/3 Var(ξ aτ 1/3 ,bτ 1/3 ) -(1 -τ ) 2/3 Var(ξ a(1-τ ) 1/3 ,b(1-τ ) 1/3 ) . (109)
4.3.2 Droplet initial condition. Now we consider the solution h u of the half-space KPZ equation with boundary parameter u and droplet initial condition at the origin. At large time, the height field should converge locally to one of the invariant distributions.

The limiting distribution depends on the boundary parameter, u, and the drift of the initial condition, according to the diagram in [7, Fig. 2]. In the case of the droplet initial condition, the drift parameter v = +∞, so the height field converges to the invariant process HY u,0 , see [10, Fig. 2]. More precisely, for x in a domain of order 1, and u > 0,

lim t→∞ h u (x, t) -h u (0, t) (d) = HY u,0 (x) -HY u,0 (0). ( 110
) When u < 0, lim t→∞ h u (x, t) -h u (0, t) (d) = B(x) + ux. ( 111
)
Hence, for any u,

lim t→∞ h u (x, t) -h u (0, t) (d) = h stat u,0 (x). (112)
On the other hand, we also know by (90) that on the scale t 2/3 , if u = at -1/3 the height field converges to A a , that is

lim t→∞ h u (2xt 2/3 , t) -h u (0, t) (d) = t 1/3 A a (x) -A a (0) -x2 . ( 113
)
We expect that (112) and (113) match when x = 2xt 2/3 goes to infinity and x goes to zero. This implies that, for x going to zero,

lim t→∞ t -1/3 h stat at -1/3 (2xt 2/3 ) (d) = A a (x) -A a (x). (114)
In other terms, for x going to zero and any fixed a ∈ R,

h rescaled a,0 (x) A a (x) -A a (0). ( 115
)
Scaling u as u = at -1/3 , we obtain, by replacing t → τ t and a → aτ 1/3 in (90), that the height field at the earlier time satisfies for large t

h at -1/3 (x, τ t) + τ t 12 (τ t) 1/3   A aτ 1/3 x 2(τ t) 2/3 - x 2 4(τ t) 4/3   (116)
This field can then be used as an initial condition for the evolution from time τ t to time t. Using formula (92

) with t → (1 -τ )t, a → a(1 -τ ) 1/3 and ŷ = x 2(t(1-τ )) 2/3 it leads to the variational formula, where we denote τ = τ 1-τ X t (1) (1 -τ ) 1/3 max ŷ>0 τ 1/3 A aτ 1/3 τ -2/3 ŷ -τ -1 ŷ2 + Ãa(1-τ) 1/3 (ŷ) -ŷ2 , (117)
where the processes A aτ 1/3 and Ãa(1-τ) 1/3 are independent, because they describe the growth over two disjoint time intervals. Thus, using (116) with x = 0, X t (1) -X t (τ ) has asymptotically the same distribution as

(1 -τ ) 1/3 max ŷ>0 τ 1/3 A aτ 1/3 τ -2/3 ŷ -A aτ 1/3 (0) -τ -4/3 ŷ2 + Ãa(1-τ) 1/3 (ŷ) -ŷ2 .
(118) In the limit 1 -τ 1, the argument τ -2/3 ŷ is small and one can use (115) and the scale invariance property (108) leading to

X t (1) -X t (τ ) (1 -τ ) 1/3 max ŷ>0 h rescaled a(1-τ ) 1/3 ,0 (ŷ) + Ãa(1-τ) 1/3 (ŷ) -ŷ2 , (1 -τ ) 1/3 ξ a(1-τ ) 1/3 ,0
where in the last formula we used (95) and (94) . So that, the formula (102) yields, for any a ∈ R and τ → 1,

C(τ ) = 1 2 Var A a (0) + 1 2 τ 2/3 Var A aτ 1/3 (0) - 1 2 (1 -τ ) 2/3 Var ξ a(1-τ ) 1/3 ,0 + O(1 -τ ).
(119) Since this formula is valid only in the limit τ → 1, we may simplify it using Var[ξ a(1-τ ) 1/3 ,0 ] = Var[ξ 0,0 ] + O((1 -τ ) 1/3 ) and Var A aτ 1/3 (0) = Var A a (0) + O(1 -τ ), so that

C(τ ) = Var A a (0) - 1 2 (1 -τ ) 2/3 Var ξ 0,0 + O(1 -τ ), (120)
thus we obtain (32) as announced.

Appendix A Brownian case

In [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF], we computed only the function F Brownian a (s) when a = 0. We show in this Section that very similar arguments as those already developed in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] also yields an expression for F Brownian a (s) for any a ∈ R. We start from

F Brownian a (s) = lim b→-a F (a,b) (s) (121)
where F (a,b) (s) was given in [6, (7.11)] as

F (a,b) (s) = 1 2 1 + ∂ s a + b Det(I -Âs ) + Det(I + Âs ) (122)
We will follow the same notations as in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] (up to minor changes). The operator A s acts

L 2 (0, +∞) with kernel Âs (x, y) = A (a,b) (x + y + s), ( 123
)
where the function A (a,b) (x) is defined by

A (a,b) (x) = dz 2iπ a + z a -z b + z b -z e -xz+ z 3 3 , ( 124
)
and the contour is a vertical line with real part between 0 and min{a, b}. Moving the contour to the right, we obtained in [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF] that Âs is a rank-2 perturbation of Ãs , in the sense that Âs (x, y) = Ãs (x, y)

A (a,b) (x) = Ã(a,b) (x) + 2 a + b a -b (h b (x) -h a (x)), (125)
+ 2 a + b a -b (b |f b (x) f b (y)| -a |f a (x) f a (y)|), (127)
I α,β = f α |f β ± f α | Ãs I ∓ Ãs |f β . ( 129
)
The scalar products are evaluated as is of order . Hence, dividing by and using ( 121) and ( 122),

f α |f β = 1 α + β e
F Brownian a (s) = 1 2 ∂ s Det(I -Ai s )S - a + Det(I + Ai s )S + a ( 133
)
where Ai s denotes the operator with kernel Ai(x + y + s),

S - a = e as-a 3 3 R - -a,-a + e a 3 3 -as R - a,a -2R - a,-a + sinh 1 3 a a 2 -3s a -a 2 + s, ( 134
)
S + a = e as-a 3 3 R + -a,-a + e a 3 3 -as R + a,a + 2R + a,-a - sinh 1 3 a a 2 -3s a -a 2 + s (135)
and

R ∓ α,β = e -αx | Ai s I ∓ Ai s |e -βx . ( 136
)
We may check that for a = 0, S - 0 = 0 and S + 0 = 4R + 0 + 2s, so that we recover exactly the result from [START_REF] Barraquand | Half-space stationary Kardar-Parisi-Zhang equation[END_REF]. Using the Sherman-Morrison formula, we have that

R ∓ α,β = ∓ Det(I ∓ Ai s ∓ |Ai s e -βy e -αx |) Det(I ∓ Ai s ) ± 1. ( 137
)
so that F Brownian a can be written in terms of Fredholm determinants and simple functions, and could be evaluated numerically (to compute the Fredholm determinants of rank one perturbation of Ai s , one may need to conjugate the kernel so that all kernels involved are decaying at infinity).

B Limiting one-point distribution away from the wall at x > 0

In this Appendix we study the distribution of the height h(x, t) at x > 0.

B.1 Moment formula

We start from the moment formula [START_REF] Tracy | On orthogonal and symplectic matrix ensembles[END_REF], that is

E[Z(x, t) k ] = 2 k Γ(v 1 + v 2 ) Γ(v 1 + v 2 -k) r 1 +iR dz 1 2iπ • • • r k +iR dz k 2iπ 1 a<b k z a -z b z a -z b -1 F ( z) F ( z) = 1 a<b k z a + z b z a + z b -1 k i=1 z i z i + u -1/2 1 (v 1 -1/2) 2 -z 2 i 1 (v 2 -1/2) 2 -z 2 i e tz 2 i -z i x , (138)
We use [START_REF] Borodin | Directed random polymers via nested contour integrals[END_REF]Proposition 5.1] specializing to a function F ( z) which is symmetric in its arguments

r 1 +iR dz 1 2iπ • • • r k +iR dz k 2iπ 1 a<b k z a -z b z a -z b -1 F ( z) = k! λ k 1 m 1 (λ)!m 2 (λ)! . . . aw+iR dw 1 2iπ • • • aw+iR dw (λ) 2iπ Det 1 w i + λ i -w j (λ) i,j=1
× F (w 1 , w 1 + 1, . . . , w 1 + λ 1 -1, . . . , w (λ), w (λ) + 1, . . . , w (λ) + λ (λ) -1) (139)

For the last equation to be valid, the function F need to be holomorphic in each variable in the region spanned by the contour deformation, that is, it needs to be holomorphic in the whole region between r 1 + iR and r k + iR. This will be the case if we choose u -1 2 > r 1 and r k > max{0, -v 1 + 1 2 , -v 2 + 1 2 } on the left hand side, so that we may take the contour in the right hand side such that v 1 -k + 1 > a w > max{1/2 -u, 0} (one can simply take a w = r k ).

B.2 Laplace transform formula

We can evaluate the function F into strings using the same manipulations as in Section (2.2), and using the idenity

λ 1 -1 a=0 λ 2 -1 b=0 w 1 + w 2 + a + b w 1 + w 2 + a + b -1 = Γ (w 1 + w 2 -1) Γ (w 1 + w 2 + λ 1 + λ 2 -1) Γ (w 1 + w 2 + λ 1 -1) Γ (w 1 + w 2 + λ 2 -1) (140)
and

0 a<b λ-1 2w + a + b 2w + a + b -1 = 2 -λ Γ(w)Γ 2(w + λ) -1 Γ(w + λ)Γ(2w + λ -1) . ( 141
)
We obtain

E[Z(x, t) k ] = 2 k Γ(v 1 + v 2 ) Γ(v 1 + v 2 -k) k! λ k 1 m 1 (λ)!m 2 (λ)! . . . aw+iR dw 1 2iπ • • • aw+iR dw (λ) 2iπ Det 1 w i + λ i -w j (λ) i,j=1 (λ) j=1 e tG(w j +λ j )-x 2 (w j +λ j ) 2 + x 2 (λ j +w j ) e tG(w j )-x 2 w 2 j + x 2 w j × (λ) j=1 2 -λ j Γ 2(w j + λ j ) -1 Γ(2w j + λ j -1) η(w j + λ j -1/2) η(w j -1/2) × 1 i<j (λ) Γ w i + w j -1 Γ w i + w j + λ i + λ j -1 Γ w i + w j + λ i -1 Γ w i + w j + λ j -1 . ( 142
)
where

η(z) = Γ(v 1 -z) Γ(v 1 + z) Γ(v 2 -z) Γ(v 2 + z) 1 Γ(u + z) . ( 143
)
The contour for the variables w i has to be chosen so that

max{1/2 -u, 1/2} < a w < v -1/2 -λ i + 1 (144)
for any λ i and v = v 1 , v 2 , and the moment formula was valid for u+v > k and v-1 2 > k-1, which implies that u+v > λ i , v -1 2 > λ i -1 so that one can always find a w satisfying (144). Summing to obtain the generating function, and using Mellin-Barnes the Mellin-Barnes integral representation, we obtain

0 1/2 v 1 + 1/2 1/2 -u a s w w + 1 w + 2 a w + iR C as [w]
E exp -ςe t 12 W Z(x, t) = +∞ =0 (-1) ! aw+iR dw 1 2iπ • • • aw+iR dw 2iπ Ca s [w 1 ] ds 1 2iπ • • • Ca s [w ] ds 2iπ Det 1 s i -w j i,j=1 × j=1 e tG(s j )-x 2 s 2 j + x 2 (s j -w j ) e tG(w j )-x 2 w 2 j π sin(π(s j -w j )) (ςe t 12 ) s j -w j η(s j -1/2) η(w j -1/2) × j=1 Γ 2s j -1 Γ(w j + s j -1) 1 i<j Γ w i + w j -1 Γ s i + s j -1 Γ s i + w j -1 Γ w i + s j -1 . (145)
The contour C as [w] (depicted on Fig. 5) is formed by two semi-infinite rays going to ∞ in the direction ±π/3, starting from the horizontal axis at the point a s and the union of negatively oriented circles around the poles at w + 1, w + 2, . . . when these lie to the left of the semi-infinite rays. The infinite part of the contour is oriented from bottom to top.

The real numbers a s and a w has to be chosen so that (recall that after the analytic continuation,

[λ] = [s -w] = a s -a w ) a w < a s < a w +1, max{1/2-u, 1/2, 1/2-v} < a w , a s < v 1 +1/2, u+v 1 > a s -a w -1.
(146) Note that the condition a s < a w + 1 is not really needed since we have added the small circles to the contour for s i . Additionally, we need to chose a s > 1/2, so that Γ(2s -1) has no poles on the right of contours. One also needs to discuss the convergence of the w integral. A sufficient condition for the integrals over w i to be convergent is that a w -1/2 < x/2.

B.3 Pfaffian formula in the large time limit

We now consider the large time limit, and the critical region, hence we rescale

u = at -1/3 , v 1 = bt -1/3 , v 2 = ct -1/3 , ( 147
)
w i → 1 2 + t -1/3 w i , s i → 1 2 + t -1/3 s i , ( 148
) x = t 2/3 x, ς = exp(-yt 1/3 ). ( 149
)
Defining the rational function

η(z) = (a + z) (b + z)(c + z) (b -z)(c -z) , ( 150
)
the right hand side becomes

E exp -ςe t 12 W Z(x, t) = +∞ =0 (-1) ! aw+iR dw 1 2iπ • • • aw+iR dw 2iπ Ca s ds 1 2iπ • • • Ca s ds 2iπ Det 1 s i -w j i,j=1 × j=1 exp s 3 j 3 - w 3 j 3 - x 2 (s 2 j -w 2 j ) -y(s j -w j ) s j + w j s j -w j η(s j ) η(w j ) 1 2s j × 1 i<j (s i + w j )(s j + w i ) (w i + w j )(s i + s j ) (151)
The contour C as is now formed by two semi-infinite rays going to ∞ in the direction ±π/3, starting from the horizontal axis at the point a s (without additional small circles). The values of a w , a s now satisfy max{-a, -b, -c, 0} < a w < a s < min{b, c}, (

which can be happen only when b, c > 0 and a + b, a + c > 0. Note that the condition that a w , a s are positive is important because of the denominators (w i + w j ) and (s i + s j ), and also the pole at s = 0. Now, the condition for the convergence of the integrals over w i is that a w < x 2 . Observe that we may write j=1 (s j + w j )

1 i<j (s i + w j )(s j + w i ) (w i + w j )(s i + s j ) Det 1 s i -w j i,j=1 = i=1 s i + w i s i -w i i<j (s i -s j )(-w i + w j )(s i + w j )(-w i -s j ) (-w i -w j )(s i + s j )(s i -w j )(-w i + s j ) = i<j u i -u j u i + u j , (153)
where u = (s 1 , -w 1 , s 2 , -w 2 , . . . , s , -w ). We recognize Schur's Pfaffian formula, so that we obtain

E exp -ςe t 12 W Z(x, t) = +∞ =0 (-1) ! aw+iR dw 1 2iπ • • • aw+iR dw 2iπ Ca s ds 1 2iπ • • • Ca s ds 2iπ Pf u i -u j u i + u j 2 i,j=1 × j=1 exp s 3 j 3 - w 3 j 3 -x 2 (s 2 j -w 2 j ) -y(s j -w j ) (s j -w j )(2s j ) η(s j ) η(w j ) (154)
This corresponds to

E exp -ςe t 12 W Z(x, t) = Pf[J -K] L 2 (0,+∞) (155) = +∞ =0 (-1) ! R + dr 1 • • • R + dr k Pf K(r i , r j ) i,j=1 , (156)
where, after a change of variables w → -w in the integrals, we may define the kernel by (157c)

K 11 (r, r ) = Ca s dz 2iπ Ca s dw 2iπ z -w z + w e z 3 3 + w 3 3 -x 2 (z 2 +w 2 )-(r+y)z-(r +y)w η(z)η(w) 4zw , ( 157a
) K 12 (r, r ) = Ca s dz 2iπ -aw+iR dw 2iπ z -w z + w e z 3 3 + w 3 3 -x 2 (z 2 -w 2 )-(r+y)z-(r +y)w η(z) 2zη(-w) , (157b)
We now shift the contour for -a w +iR to the right of 0. In K 12 we can deform the contours without crossing any pole. In K 22 , we do cross a pole and have to take into account the residue.

K 11 (r, r ) = Ca s dz 2iπ Ca s dw 2iπ z -w z + w e z 3 3 + w 3 3 -x 2 (z 2 +w 2 )-(r+y)z-(r +y)w η(z)η(w) 4zw , (158a) K 12 (r, r ) = Ca s dz 2iπ Ca s dw 2iπ z -w z + w e z 3 3 + w 3 3 -x 2 (z 2 -w 2 )-(r+y)z-(r +y)w η(z) 2zη(-w) , (158b) K 22 (r, r ) = Ca s dz 2iπ Ca s dw 2iπ z -w z + w e z 3 3 + w 3 3 -x 2 (-z 2 -w 2 )-(r+y)z-(r +y)w 1 η(-z)η(-w) (158c) -2 Ca s dz 2iπ e xz 2 +(r -r)z z a 2 -z 2 . ( 158d
)
The kernel has a particuliar structure. Define an operator D such that for a function f written as

f (r) = Ca s dz 2iπ f (z)e -rz , (159)
then

Df (r) = Ca s dz 2iπ D(z) f (z)e -rz (160)
where

D(z) = -2 ze xz 2 a 2 -z 2 = -e xz 2 1 a -z - 1 a + z . ( 161
)
We have that

K = K 11 K 12 K 21 K 22 = K 11 -K 11 D -DK 11 DK 11 D + Dε ( 162
)
where ε is an operator with kernel ε(r, r ) = δ(r -r ). In other terms, ε is the identity operator, i.e. ε = I. The operator D acts on the left, i.e. f | D |g = ∞ 0 drDf (r)g(r).

B.4 From Fredholm Pfaffians to Fredholm determinants with scalar kernels

Consider operators B, D, ε :

L 2 (R + ) → L 2 (R + ).
The operator D acts by multiplication in Fourier space (as in (160)) and the operator ε is such that Dε has an anti-symmetric kernel. Recall the expression of the symplectic matrix

J = 0 1 -1 0 . ( 163
)
We can then perform the manipulations (similarly to [START_REF] Krajenbrink | From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies[END_REF])

Pf   J - B -BD -DB DBD + Dε   2 = Pf   J - 1 0 0 D B -BD -B BD + ε   2 = Det   I + B -BD -B BD + ε J 1 0 0 D   = Det   I + BD BD -BD -ε -BD   .
where the contours pass between 0 and a. We can see that L 0 can be bounded, for any M > 0, as |L 0 (r, r )| Ce -M r-ar for some constant C. Let us also define

f 0 ± (r) = lim →0 f ± (r) = (a ∓ b) ±1 exp - b 3 3 ∓ x 2 b 2 + (r + y)b . ( 185
)
We compute now the inner products in (177). Since the functions g + and g -decay exponentially fast at infinity, we simply have

b + c b -c g + | 1 1 + L |g -= 2b g + | 1 1 + L 0 |g -+ o( ). ( 186
)
To compute the remaining inner products, we use the decomposition

1 1 + L = 1 - L 1 + L .
where the contour passes to the right of b and to the left of a.

Hence our final result is that for the initial condition h(x, 0) = HY u,v (x), where u = at -1/3 , v = bt

3 +2yb + 1 2 f 0 -| L 0 1 + L 0 |f 0 + , ( 194
)
M 21 = 1 2b g + | 1 1 + L 0 |g -, ( 195
)
M 22 = bx + y + 1 2b -b 2 - 1 a + b + 1 2 f 0 -| L 0 1 + L 0 |g - (196)

Figure 1 :

 1 Figure 1: Phase diagram of stationary measures for the KPZ equation in the half-space R + with boundary parameter u. The diagram means that if the initial condition h(x, 0) has drift -v at infinity, the height field should converge at large time under mild assumptions to the stationary measure indicated in one of the three regions of the (u, v) plane, namelyR 1 = {u 0, v 0}, R 2 = {u 0, u v} and R 3 = {u v, v 0}.Along the line u + v = 0, the stationary measure is a Brownian motion with drift u = -v.

Figure 2 :Remark 1 . 7 .

 217 Figure 2: Definition of F stat a,b as in (26) for various values of a, b. In a small portion of the phase diagram below the line a + b = 0, we are not able to characterize the distribution of large time fluctuations.

Figure 3 :

 3 Figure 3: Phase diagram of two time covariances, for stationary initial condition h stat u,v with fixed u, v.

.

  This article is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the three authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2021 semester. G.B. was partially supported by the ANR grant COR-TIPOM. AK acknowledges support from ERC under Consolidator grant number 771536 (NEMO). PLD acknowledges support from ANR grant ANR-17-CE30-0027-01 RaMaTraF.

3 ,K 22 (-r z+t w 3 +z 3 3 ,K 3 ,K

 32233 12 (r, r ) = C 2 dw 2iπ dz 2iπ w -z w + z Ω(w)Ω(z) cos(πw) sin(πz) π e -rw-r z+t w 3 +z 3 21 (r, r ) = -K 12 (r , r).

  and an operator K(a,b,c) s acting on L 2 (0, +∞) with kernel K(a,b,c) s (x, y) := K(a,b,c) (x+s, y + s). For any s ∈ R, and a, b, c > 0, we have [6, Claim 7

  = Ã(a,b,c) (x + s), and we will also denote by the same symbol Ã(a,b,c) s the operator with kernel Ã(a,b,c) s (x + y). Then we have Det I ± Â(a,b,c) s = Det I ± Ã(a,b,c)

[ 6 ,

 6 Sections 4.5 and 4.6]. More generally, using the results of [6, Sections 4.6] we obtain that lim a→∞ G HY a,b = F Brownian -b.

  with h b (x) = be -xb+b 3 /3 . Letting b = -a + , we haveÃ(a,b) (x) = dz 2iπ a + z a -z b + z b -z e -xz+z 3 /3 = Ai(x) + 2 +∞ x dλ cosh(ay)Ai(λ + y) + O( 2 ),(126)(I corrected that formula) where in the integral over z, the contour passes to the right of a, b. We also introduce the operator Ãs acting on L 2 (0, +∞) with kernel Ãs (x, y) = Ã(a,b) (s + x + y).

  with f b (x) = e b 3 /6-bs/2-bx . Using the matrix determinant lemma, we have Det(I∓ Âs ) = Det(I∓ Ãs )

|f β = e α 3 6 + β 3 6

 33 e -α+β 2 s e -αx | Ãs I ∓ Ãs |e -βx . (131) Putting all this into Mathematica, we find that Det(I -Âs ) + Det(I + Âs ) (132)

Figure 5 :

 5 Figure 5: The contour C as [w] used in (145).

K 22 (

 22 z 2 -w 2 )-(r+y)z-(r +y)w 1 η(-z)η(-w) .

= 3 3 3 ∓ x 2 c 2 - 2 --w 2 w 2 -yw+ z 3 3 -x 2 z 2

 3322222 line to the second one and subtracting the second column to the first one, Det I + B(D -D) + BDε = Det I + (D -D)B + DεB . (165)To go from the first line with a matrix-valued kernel to the second line with a scalar kernel, we used a Schur's complement formula. to go from the second line to the third one we used Det(I + M N ) = Det(I + N M ).For our specific kernel K in (162), we choose B = K 11 and ε = I, so that we obtain the simple formulaPf(J -K) = Det (I + D K 11 ) = Det (I + K 11 D ),(166)where now K 11 D is a scalar kernel, acting on L 2 (0, +∞). Explicitly, we have(K 11 D )(r, r ) = -Ca s dz 2iπ Ca s dw 2iπ z -w z + w e z + w 3 3 -x 2 (z 2 -w 2 )-(r+y)z-(r +y)w η(z) 2zη(-w) (167)with the rational factorsη(z) = (a + z) (b + z)(c + z) (b -z)(c -z) Det (I + K 11 D ). (169)in order to obtain the cumulative distribution of the height at large time for a stationary initial condition with parameters a, b. For now a s in (167) satisfies 0 < a s < a, b, c. To be able to perform the limit c → -b we move c across the contours for w, z. It gives two additional residue terms and no double residue. Definef ± (r) = (a ± c) ±1 exp c 3 (r + y)c . (170) g + (r) = -(r + y)w) (b + w) (b -w)(a -w) Res z=c (K 11 D ) = b + c b -c f + (r)g + (r ), -Res w=c (K 11 D ) = c b + c b -c f -(r )g -(r). (173)Hence we may write(K 11 D )(r, r ) = L(r, r ) + b + c b -c f + (r)g + (r ) + cf -(r )g -(r) , (174)or in the operator formalism(K 11 D ) = L + b + c b -c |f + g + | + c |g -f -| , 2 -w 2 )-(r+y)z-(r +y)w η(z) 2zη(-w) (176)with both contours going between {0, c} and {a, b}. Thus, we haveDet(I +K 11 D ) = Det(I +L)Det 1 + b+c b-c g + | 1 1+L |f + b+c b-c c f -| 1 1+L |f + b+c b-c g + | 1 1+L |g - 1 + b+c b-c c f -| 1 1+L |g -.(177)Letting = b + c, we will write upon expanding in(177) (Det(I + L 0 + X)(C 0 + C 1 + C 2 2 ) (178)and show that C 0 = C 1 = 0, so thatG HY a,b (y, x) = lim →0 1 + ∂ y b + c Det(I + K 11 D ) = ∂ y Det(I + L 0 )C 2 . (179)Let us calculate some inner products explicitly and expand in each term in (177) is defined only for c > 0 but we will consider the analytic continuation to -b < c < 0. z)(a + z) (b -z)(z + c) = -1 + bx + y z)(a + z) (z -b) 2(181) where we have moved the contour to z > b and taken the residue, which for c = -b has a pole. The remaining integral has no singularity at c = -b. • One has b + c b -c g + |f + = -b + c b -c (a + c)e -yw (b + w) (b -w)(a -w)(w + c) = -1 + y --yz (b + w)(b + z)(a + z) (b -w)(a -w)(b -z) 2 -w 2 )-(r+y)z-(r +y)w a + z 2z(a -w) ,

z 2 1 +w 2

 212 -yz (b + z)(a + z) z(z -b) L 0 |g -+ o( ) (188)where the contour passes to the right of b.-yw b + w (a -w)(w -b) 2 -2b g + | L 0 1 + L 0 |f 0 + +o( ),

12 t 1

 121 -1/3 , with a + b > 0, b 0 lim t→∞ P   h at -1/3 (t 2/3 x, t) + t the solution with Hariya-Yor initial condition is G HY a,b (y, x) = ∂ y Det(I + L 0 )Det(M ) (191)where M is the 2 by 2 matrixM = M 11 M 12 M 21 M 22 ,

z 2

 2 -yz (b + z)(a + z) z(z -b) 2 .

our notations are different from[START_REF] Barraquand | Steady state of the KPZ equation on an interval and Liouville quantum mechanics[END_REF], where a slightly different process was denoted HY -v u (x))

We recall that L 0 is defined in (184), f 0 ± in (185), g ± in (171), (172). Although this result is fully explicit, it is quite involved and it remains to be studied how the various known limits can be obtained from it.


Year

Data volume (PB) [START_REF] Soille | A versatile data-intensive computing platform for information retrieval from big geospatial data[END_REF] Annual data volume Spectro-temporal kernel Spatio-spectro-temporal kernel

x = {x φ , x λt } 1. λt-GP: k l (x, x ) = k λt (x λt , x λt ) 2. φλt-GPSC: k l (x, x ) = α 2 φ ×k φ (x φ , x φ )+α 2 λt ×k λt (x λt , x λt ) 3. φλt-GPPC: k l (x, x ) = k φ (x φ , x φ ) × k λt (x λt , x λt )

Experimental setup

  Process Classi cation Multi-output Stochastic Variational Sparse Gaussian Process Classi cation [3]: To overcome large scale issues [4], Gaussian Process f ∼ GP(µ, k) can be approximated using inducing points Z = {z i } m i=1 with m n (i.e. n number of training inputs X = {x i } n i=1 ). The complexity is reduced from O(n 3 ) to O(nm 2 ). (µ and k are modeled by some parametric functions with hyperparameters θ) These inducing points Z can be learned with the hyperparameters θ using the ELBO ε (stochastic optimization): ε = data driven term + regularization term (Kullback-Leibler divergence) c ŷ g l ∼ GP(µ l , k l ) Sparse GP Stochastic Variational SGP Multi-output GP cation Spectro-temporal and spatial covariates:

  Study area: SITS and reference data • 27 Sentinel-2 tiles • acquisitions of level-2A from January to December 2018 linearly resampled (interval of 10 days) (total of 37 virtual dates) • 13 spectral features λt: 10 spectral-bands (10m ground sampling distance) + 3 spectral indices (NDVI, NDWI, brightness) • 2 spatial features φ: geographic coordinates (latitude, longitude) in meters in Lambert 93 projection • 23 land cover classes ranging from arti cial areas to vegetation and water bodies • ∼ 5 TB Color Name Area (km 2 )

Acknowledgements

This work is supported by the Natural Intelligence Toulouse Institute (ANITI) from Universite Federale Toulouse Midi-Pyrenees under grant agreement ANITI ANR-19-PI3A-0004 (this PhD is co-founded by CS-Group and by the Centre National d'Etudes Spatiales (CNES)). Special thanks to Benjamin Tardy for the support.

Maximum number of pixels for each class

Region X Global Dataset A (train-val) 4 000-1 000 32 000 -8 000 Dataset B (train-val) 16 000-4 000 128 000-32 000 Dataset A&B (test) 10 000 80 000

Classi cation con guration:

Global con guration: Qualitative results: (with strati cation con guration using iota2 chain with the model λt-GP)

Perspectives • Dimensionality reduction

Optimization of the inducing points Z = {z i } m i=1 with z i ∈ R φ+λt can be di cult for large dimension: feature reduction.

• Spatial constraints in boundary zones:

Join optimization between models of 2 regions [START_REF] Park | Patchwork Kriging for Large-scale Gaussian Process Regression[END_REF]:

-L 1 : ELBO region 1 -L 2 : ELBO function region 2

f 1 = f 2 in boundary zones