Pedro Gajardo
email: pedro.gajardo@usm.cl

Hector Ramirez Cabrera

Julio C Rodriguez
email: julio.rodriguez@uv.cl

Héctor Ramírez
email: hramirez@dim.uchile.cl

Julio C Rodríguez

Tools for improving feeding strategies in a SBR with several species

Keywords: Sequential batch reactors, Minimal time problem, Singular arc strategies

published or not. The documents may come

Tools for improving feeding strategies in a SBR with several species

Introduction

Sequential batch reactors (SBRs) typically consist of a tank filled with biological micro-organisms that are capable of degrading an undesirable substrate. These devices are often used in biotechnological applications, most notably in wastewater treatments (see [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF][START_REF] Irvine | Sequencing batch reactors for biological wastewater treatment[END_REF][START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF][START_REF] Smith | The Theory of the Chemostat[END_REF] for details about the fundamental role of SBRs in bioengineering). For this purpose, we use the typical sequence of cycles, which is composed of three phases:

-Phase 1: fill the reactor with water to be treated, -Phase 2: wait for the concentration of the undesirable substrate to decrease to a given (low) concentration, -Phase 3: remove the clean water from the reactor, leaving the sludge inside.

In this paper, we focus on a SBR in which several species compete for a single substrate and on the corresponding optimal control problem of feeding to minimize the time in which the SBR reaches a desired (low) concentration of the substrate (i.e., Phases 1 and 2 described above). This optimal control problem is the same as that presented in [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF][START_REF] Gajardo | On optimal strategies for feeding in minimal time a SBR with several species[END_REF], which corresponds to an extension of the one-species model introduced in [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]. This problem has been solved for certain cases; for instance, explicit optimal feeding strategies are given in [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF], in which the SBR contains only one species. Feeding strategies are also given in [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF], in which the SBR contains two species with nondecreasing growth functions. See also [START_REF] Bayen | Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima[END_REF][START_REF] Rapaport | Minimal time control of fed-batch processes with growth functions having several maxima[END_REF] for other related feeding problems involving SBRs. Nonetheless, bioprocesses generally involve a significant amount of biodiversity, particularly in wastewater treatment. Microbial communities can contain thousands of species [START_REF] Forney | Structure of Microbial Communities in Activated Sludge: Potential Implications for Assessing the Biodegradability of Chemicals[END_REF]. Thus, it is relevant to study optimal feeding strategies in the presence of several species. This multi-species setting leads to an optimization problem that is difficult to solve. However, [START_REF] Gajardo | On optimal strategies for feeding in minimal time a SBR with several species[END_REF] presented a method for characterization when a singular arc is presented in an optimal feeding strategy. Indeed, this characterization establishes that an extremal curve (eventually optimal) is an singular arc if and only if the substrate level remains constant during that period of time. Because the existence of singular arcs can lead to complex solutions that are not easily tractable from a mathematical point of view, this characterization is very valuable for verifying the optimality of various feeding strategies. We thus provide explicit conditions for which the above-mentioned characterization holds. These conditions depend on the growth functions of the species involved in the depollution process. When the appropriate conditions are fulfilled, we can propose potential optimal feeding strategies, based on the existence of a singular arc, and can compare them with other more intuitive strategies via numerical simulations. In this section, we define the optimal control problem studied in this article. This problem was introduced by Gajardo, Ramírez and Rapaport [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF], as an extension of the minimal time problem presented by Moreno in [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]. We finish this section by exhibiting some basic results concerning Pontryagin's principle, which permits us, in the next section, to understand the meaning of a singular arc strategy in this problem.

The dynamics of a SBR with several species can be described as follows (see [START_REF] Smith | The Theory of the Chemostat[END_REF]):

           ẋi = µ i (s)x i - F v x i , x i (t 0 ) = x i0 (i = 1 • • • n), ṡ = - n j=1 µ j (s)x j + F v (s in -s), s(t 0 ) = s 0 , v = F, v(t 0 ) = v 0 , (2.1)
where x i , s and v represent the concentration of the ith species, the concentration of the substrate and the current volume of water in the tank, respectively. The parameter s in > 0 is a constant that represents the substrate concentration in the input flow. The growth functions µ i (•) are non-negative smooth functions such that µ i (0) = 0, and the input flow F is a non-negative control variable.

The aim is to treat the maximum quantity of water in the SBR as quickly as possible. From a mathematical point of view, this means that, given a (desirable) substrate concentration s out ∈]0, s in [ and the volume of the reactor v max , we want to solve the following minimal time problem: inf

F (•) t -t 0 | s t0,ξ,F (t) ≤ s out , v t0,ξ,F (t) = v max , (2.2)
where s t0,ξ,F (•), v t0,ξ,F (•) denote solutions of (2.1) with the initial condition ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) at time t 0 and when a particular control strategy F (•) is used. It is worth emphasizing that our theoretical results are valid provided that the initial condition ξ is in the domain D = (IR n + \{0})×]0, s in ]×]0, v max [. Here, F (•) is allowed to be a non-negative measurable function combined with possible positive impulses; that is, dF (t) = u(t)dt + dσ, where u(•) is a measurable non-negative control that we impose to be bounded from above by some positive value u max because it corresponds to the use of a pump device, and at time t, the non-negative impulse dσ corresponds to an instantaneous addition of volume. The latter is useful for modeling practical situations in which large amounts of polluted water are added to the SBR in a short time (e.g., [START_REF] Ciappelloni | On-line supervision and control of an aerobic SBR process[END_REF][START_REF] Donoso-Bravo | Anaerobic treatment of low-strength wastewater with a high fraction of particulate matter in an unconventional two-phase ASBRs system[END_REF]).

From [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF], we know that a time parameterization τ ≥ t 0 such that dt = r(τ )dτ with r(τ ) = 1, when the pump device is used 0, when an impulse is used, allows us to replace the dynamics (2.1) by the system

                 dx i dτ = rµ i (s)x i - u v x i (i = 1 • • • n) , ds dτ = -r n j=1 µ j (s)x j + u v (s in -s) , dv dτ = u , (2.3)
where the controls u(•) and r(•) are sought among measurable functions with respect to τ , taking values in [0, u max ] and {0, 1}, respectively. In this formulation, u(•) acts as an ordinary control when r = 1 and controls the amplitude of the impulse when r = 0, with the same single constraint u ∈ [0, u max ]. Consequently, control r(•) indicates the presence of an instantaneous dilution. This conclusion can also be interpreted from system (2.3) by noting that r = 0 causes all of the terms related to biological interactions to vanish. Indeed, from a process point of view, one considers impulses when the time needed to add a given volume of water is sufficiently small, such that growth can be neglected during this time. This assumption also allows us to neglect this time in the total time of the process. Thus, we obtain the following value function for the reformulated dynamics (2.3)

V (ξ) = inf (u,r)(•) τ t0 r(θ)dθ such that s t0,ξ,u,r (τ ) ≤ s out , v t0,ξ,u,r (τ ) = v max , (2.4)
where s t0,ξ,u,r (•), v t0,ξ,u,r (•) denote solutions of (2.3) with the initial condition ξ ∈ D at time t 0 and with controls u(•) and r(•).

Remark 1 As one can always take r = 0 and u = 0 for an arbitrarily large τ -interval without modifying the total time, the minimal time problem has no unique solution. Hence, we will be only interested in controls satisfying r(τ ) = 0 or u(τ ) = 0 for all time τ .

Now, set z = (x 1 , • • • , x n , s, v) ⊤ , u 1 = r and u 2 = u.
In [START_REF] Gajardo | On optimal strategies for feeding in minimal time a SBR with several species[END_REF], the Pontryaguin principle (cf. [START_REF] Pontryagin | The mathematical theory of optimal processes, Translated from the Russian[END_REF]) applied to problem (2.4) can be stated as follows:

If

u * = (u 1 * , u 2 *
) is an optimal control and z * is its corresponding trajectory on R n+2 , then there exists a constant λ 0 ≥ 0 and an absolutely continuous function τ → λ(τ ) = (λ 1 (τ ), . . . , λ n+2 (τ )) such that for almost every τ , (λ, λ 0 ) never vanish and satisfy

ż = ∂H ∂λ (z, λ, u * ), λ = - ∂H ∂z (z, λ, u * ), (2.5)
where the Hamiltonian H is given by Here, we define the following functions

H(z, λ, u 1 , u 2 ) = u 1 φ u1 (z, λ) + u 2 φ u2 (z, λ). ( 2
       φ u1 (z, λ) := λ 0 + n i=1 µ i (s)x i (λ i -λ n+1 ), φ u2 (z, λ) := - n i=1 xi v λ i + (sin-s) v λ n+1 + λ n+2 .
(2.7)

Furthermore, the optimal control u * minimizes the Hamiltonian over the control admissible set, through the curve (λ(τ ), z(τ )).

For the sake of simplicity, we will simply use φ ui (τ ) instead of φ ui (z(τ ), λ(τ )), and we assume that λ 0 = 1. The latter condition corresponds to the normal extremals.

Characterization of singular arcs

In our context, a singular arc corresponds to an extremal curve for which there exists a nontrivial interval [τ 1 , τ 2 ] ⊂ [0, T ] where both switching functions φ ui (τ ), i = 1, 2, are identically zero. This can be understood as an extension of the standard definition of singular arcs when only one control is considered (e.g., [3, Part III, Ch. 2]).

When a singular arc is presented in the optimal strategy, minimization of the Hamiltonian H introduced in (2.6) can lead to complex feeding strategies. The next result thus provides important qualitative information about singular arcs, constraining them only to extremal curves for which the substrate level s remains constant. This constraint permits us to limit our study of possible optimal feeding strategies.

Singular strategies have been known to appear in many control problems of particular interest to chemical engineers and have already been addressed with the help of optimal control theory [START_REF] Bayen | Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima[END_REF][START_REF] D'ans | Time optimal control for a model of bacterial growth[END_REF][START_REF] Hong | Optimal substrate feeding policy for fed batch fermentation with substrate and product inhibition kinetics[END_REF][START_REF] Irvine | Sequencing batch reactors for biological wastewater treatment[END_REF][START_REF] Rapaport | Minimal time control of fed-batch processes with growth functions having several maxima[END_REF]. In addition, [START_REF] Edgar | The computation of optimal singular bang-bang control II. Nonlinear systems[END_REF] and [START_REF] Siebenthal | The Application of Pontryagins Methods to the Control of a Stirred Reactor[END_REF] have demonstrated the possible appearance of singular arcs in the start-up of bioreactors and in a variety of chemical reactor optimization problems.

The issue of singular arcs is well-known in the classical theory of optimal control; see, for instance, [START_REF] Kelley | Singular extremals, Topics in Optim[END_REF]. Currently, this type of trajectory plays an important role in optimal control theory. These trajectories are studied in high-order optimality conditions; see, for instance, [START_REF] Bonnard | Théorie des singularités de lápplication entrée/sortie et optimalité des trajectoires singuliéres dans le problemé du temps minimal[END_REF], [START_REF] Goh | Necessary conditions for singular extremals involving multiple control variables[END_REF][START_REF] Krener | The high order maximum principle and its application to singular extremals[END_REF].

The next characterization of singular arcs was reported in [START_REF] Gajardo | On optimal strategies for feeding in minimal time a SBR with several species[END_REF].

Theorem 1 Suppose that the matrix is nonsingular for any s ∈ (0, s in ). Here, µ (j) i denotes the j-th derivative of function µ i . Then, an extremal curve is a singular arc on

D(s) =           µ (1) 1 (s) • • • µ (1) n (s) µ (2) 1 (s) • • • µ (2) n (s) . . . . . . . . . µ (n) 1 (s) • • • µ (n) n (s)           (3.
(τ 1 , τ 2 ) ⊂ [0, T ] if and only if s(•) is constant on (τ 1 , τ 2 ).
Remark 2 The characterization stated above ensures that any optimal strategy is composed of bang-bang controls (which, in our case, means either to close the pump or to do one impulsion) or that the pump is used in order to maintain a constant substrate level, where the latter approach corresponds to a singular arc. However, this characterization does not prevent the presentation of more than one singular arc. For the sake of simplicity, in the next sections, we only consider optimal strategies containing at most one singular arc.

Applications

In this section, we apply Theorem 1 to study singular arcs for species with particular biological characteristics that have not been considered in previous articles [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF][START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]. This tool allows us to propose potential optimal feeding strategies, which can then be compared with other more intuitive strategies.

Arbitrary number of species following Monod growth functions

Suppose that each of the n species of microorganisms in competition for a single substrate in the SBR follows a Monod law growth function [START_REF] Monod | Recherches sur la croissance des cultures bactriennes[END_REF], that is

µ i (s) = µ max,i s K i + s , i = 1, • • • , n (4.1)
where, for the i-species, µ max,i and K i represent the maximal growth rate and the half saturation constant or Monod constant, respectively, and K i = K j , for all i = j. In this case, it is easy to verify that the hypothesis of Theorem 1 holds true (cf. Proposition 3.5 of [START_REF] Gajardo | On optimal strategies for feeding in minimal time a SBR with several species[END_REF]), and we obtain the following result:

Proposition 2 Under the hypotheses of this subsection, an extremal curve is a singular arc on (τ 1 , τ 2 ) ⊂ [0, T ] if and only if s(•) is constant on (τ 1 , τ 2 ).

Two species with a Haldane and a Monod growth function

Consider now that one of the two species has a Monod growth law:

µ M (s) = µ max,M s K M + s , (4.2)
(where, as before, µ max,M is the maximal growth rate and K M is the half saturation constant or Monod constant) and the other species follows a Haldane growth law [START_REF] Andrews | A Mathematical model for the continuous culture of microorganism utilizing inhibitory substrate[END_REF], which is usually used to represent a substrate inhibiting growth at high values, as given by where µ max,H is the maximal growth rate, K I is an inhibition constant and K S is the affinity or saturation constant.

µ H (s) = µ max,
In this situation, we can verify the following result:

Proposition 3 The hypothesis of Theorem 1 is fulfilled if one of the following conditions is satisfied.

(1) K I = K M and 4K S < K I .

(

) K I > K M and K I K S = K M (K I -K M ) . 2
(

) K I -K M = 2K S and K I K S < K M (K I -K M ) . ( 3
) K I -K M K S > max 2, K I K M . 4
Conditions ( 1)-( 4) of Proposition 3 are derived from an algebraic analysis of the determinant of the matrix D(s) given in (3.1). Consequently, we obtain Corollary 4 Under any of the conditions of Proposition 3, an extremal curve is a singular arc on (τ 1 , τ 2 ) ⊂ [0, T ] if and only if s(•) is constant on (τ 1 , τ 2 ).

Two species with a linear and a Monod growth function

Consider again two species, one following a Monod growth law, cf. (4.2), and one following a linear law given by

µ L (s) = cs (4.4)
where c is a positive constant. This type of function can model very large growth functions (for instance, either Monod or Haldane type) at the beginning of its range. In this situation, the determinant of D(s), defined in (3.1), is given by 2cµ max,M

(K M + s) 3 .
Therefore, the hypothesis of Theorem 1 is always satisfied.

Corollary 6

For the two-species case considered in this subsection, suppose that µ ′′ H (s) = 0, for all s ∈ (0, s in ). Then, an extremal curve is a singular arc on (τ 1 , τ 2 ) ⊂ [0, T ] if and only if s(•) is constant on (τ 1 , τ 2 ).

Numerical simulations

In this section, we compare two feeding strategies for the cases stated in Section 4; namely, we compare the immediate one-impulse strategy and the singular arc strategy. As discussed in previous articles [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF][START_REF] Gajardo | On optimal strategies for feeding in minimal time a SBR with several species[END_REF][START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF], these two types of strategies naturally arise as feeding strategies in different bioprocesses.

The immediate one-impulse strategy (denoted IOI for short) was introduced in [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF], and it consists of introducing, from a given initial state ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) at time t 0 , the following: 1) An impulse of volume v maxv 0 at t 0 . This can be achieved by r(τ ) = 0, u(τ ) = u max , for τ ∈ [t 0 , t 0 + (v max -v 0 )/u max ], and 2) A null control (no feeding) until the concentration s(τ ) reaches s out .

The singular arc strategy has been previously analyzed [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF][START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF] and depends on a given value s * for the substrate (indeed, it is denoted by SA(s * ) for short). Roughly speaking, this strategy consists of reaching, as quickly as possible, a given level s * in (s out , s in ) and then maintaining s constant and equal to s * until v reaches v max . Finally, we set u = 0 and r = 1 (which means the pump is closed) until s reaches s out . In all of our numerical experiments, the value of s * is chosen to minimize the cost of this type of strategy among all possible values of s * in (s out , s in ).

Recall that in the case of two species with increasing growth functions, it was shown in [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF] that these two strategies were unique possible optimal strategies for our optimal control problem.

As an example, we first consider the following three Monod growth functions (see Figure 1):

µ 1 (s) = s 1 + s , µ 2 (s) = 2s 1.5 + s , µ 3 (s) = 4s 2 + s ,
as well as the following parametric values v max = 10, s out = 0.1 and s in = 5 and the initial conditions x 10 = 1, x 20 = 0.001, s 0 = 3 and v 0 = 1. In Table 1, we compare, for different values of the initial condition x 30 , the times achieved by the IOI and SA(s * ) strategies. Thus, the results reported in Table 1 establish that SA(s * ) exhibits an improvement close to 25% with respect to IOI for some values of x 30 . We thus conclude that IOI is not optimal for this particular setting. Note that it is already known (see [START_REF] Gajardo | Minimal time sequential Batch Reactors with Bounded and Impulsive Controls for one or more species[END_REF]) that when there are only two Monod growth functions in the bioreactor and when one of them is clearly more performant than the other (i.e., its graphical image is always higher), the IOI strategy is indeed optimal. Thus, the previous results indicate that considering a third species (even a more performant one) can lead to a loss of the optimality of IOI. We believe that this fact can be important in practice. The case of an arbitrary number of species following increasing growth functions was analyzed in [START_REF] Gajardo | On optimal strategies for feeding in minimal time a SBR with several species[END_REF]. Therein, conditions that ensure the optimality of the IOI strategy were studied. For the rest of the cases, we consider two species that are in competition for a single substrate in the SBR. In all of these cases, we consider the parametric values v max = 10, s out = 0.1 and s in = 5 and the initial conditions x 10 = 1, s 0 = 3, and v 0 = 1. We then compare, for different values of x 20 , the times achieved by the IOI and SA(s * ) strategies.

In Table 2, we illustrate the case in which one species follows a Haldane growth law and the other follows a Monod growth law. These growth functions fall into some of the cases of Proposition 3 (see Figure 2).

K I -K M K S > max n 2, K I K M o : µ H (s) =
) K I -K M K S > max n 2, K I K M o
In Table 3, we illustrate a situation in which one of the species follows a Monod growth law and the other species follows a linear growth law, given by (see Figure 3-(a In Table 4, we illustrate a situation in which one of the species follows a Haldane growth law and the other species follows a linear growth law, given by (see Figure 3-(b))

µ H (s) = 6s 2 + s + s 2 9
, and µ L (s) = 1.5s. objective being to reach a given (low) substrate level in a minimal time. Our mathematical model allows for instantaneous dilution of the polluted water as well as continuous feeding from a pump. For this multi-species setting, the existence of singular arcs is known to be fully characterized as the intervals of time in which the substrate concentration remains constant. This characterization permits us to conclude that optimal feeding strategies are composed of only three types of feeding phases: dilutions, manipulations of the pump to maintain a constant substrate concentration and closing of the pump (which involves waiting for the concentration of the substrate to decrease to a given level). Of course, these three phases can be combined in an infinite number of ways. Our numerical simulations were conducted with the aim of comparing only two feeding strategies: the IOI strategy, in which the tank is filled with a single dilution followed by waiting, and the singular arc strategy, in which, a given substrate concentration is reached as quickly as posible and the pump is then used to keep this concentration constant until the tank of the reactor is completely filled, at which point the pump is then closed.

In this note, we have performed simulations for several cases in which our theoretical characterization of singular arcs is fulfilled. We have observed that it is not easy to determine which of the studied two strategies is optimal, even in cases when one could suppose that filling the reactor as quickly as possible is the best feeding strategy (for instance, in the case of Figure 1). Indeed, in many cases, this decision also depends on the initial conditions of the bioreactor. Thus, these numerical experiments can guide practitioners in choosing strategies for particular SBR settings.

It is worth noting that explicit optimal strategies cannot be directly obtained from our approach in the multi-species case. This result is a consequence of the lack of information about the number of commutations that can be employed between the above-mentioned phases (making dilutions, maintaining a constant substrate concentration and closing the pump) in an optimal strategy. This limit is the main line of research that we expect to address in future works.

3 2

 3 Tools for improving feeding strategies in a SBR with several species Mathematical model and Pontryagin's principle

9 Fig. 1

 91 Fig. 1 Graphs of the three growth functions considered in the example.

Fig. 2

 2 Fig. 2 Two species with Haldane and Monod law growth functions: (a) K I = K M , 4K S < K I , (b) K I -K M

11 Fig. 3

 113 Fig. 3 Two species with growth functions: (a) Linear-Monod (b) Linear-Haldane.

  Tools for improving feeding strategies in a SBR with several species 5

	F o
	r
	P
	e e r
	R e v
	i e w
	.6)

Table 1 .

 1

	x 30	T (IOI)	s *	T (SA(s * ))	(T(IOI)-T(SA)) T(IOI)
	10 -4	5.416174	4.226000	5.402767	0.2%
	10 -3	5.389022	3.540000	4.978126	7.6%
	10 -2	5.172141	3.442000	4.170769	19.4%
	0.05	4.669824	3.344000	3.548414	24.0%
	0.1	4.350146	3.246000	3.274169	24.7%
	0.5	3.458854	3.050000	2.620281	24.2%

Table 2 .

 2 K I = K M , 4K S < K I : µ H (s) =

				2s 1+s+ s 2 6	, and µ M (s) = 4s 6+s
	x 20	T (IOI)	s *	T (SA(s * ))	(T(IOI)-T(SA)) T(IOI)
	10 -4	3.914240	2.462000	3.716620	5.0487%
	10 -3	3.909991	2.462000	3.715882	4.9644%
	10 -2	3.870175	2.462000	3.708550	4.1762%
	0.05	3.733370	2.658000	3.673870	1.5937%
	0.1	3.612956	4.716000	3.613056	-0.0028%
	0.5	3.162882	4.716000	3.163642	-0.0240%

Table 3 .

 3

	x 20 s 10 -4 T (IOI) 1.894604 4.716000	1.897233	-0.1388%
	10 -3	1.600381	4.716000	1.603105	-0.1702%
	10 -2	1.317108	4.716000	1.319561	-0.1862%
	0.05	1.130736	4.716000	1.132728	-0.1762%
	0.1	1.052256	4.716000	1.053935	-0.1596%
	0.5	0.870140	4.716000	0.870980	-0.0965%

* T (SA(s * )) (T(IOI)-T(SA))

T(IOI)

Table 4 .

 4

	x 20	T (IOI)	s *	T (SA(s * ))	(T(IOI)-T(SA)) T(IOI)
	10 -4	1.435443	4.226000	1.434500	0.0657%
	10 -3	1.429464	4.618000	1.429450	0.0010%
	10 -2	1.384587	4.716000	1.385042	-0.0329%
	0.05	1.289891	4.716000	1.290788	-0.0695%
	0.1	1.231523	4.716000	1.232486	-0.0782%
	0.5	1.064653	4.716000	1.065375	-0.0678%
	6 Conclusions
	In this paper, we have studied the optimal control problem consisting of feeding
	in a SBR in which several species compete for a single substrate, with the

Acknowledgements The authors thank Francis Mairet for fruitful discussions on the topic of this paper. We also thank two anonymous reviewers who have substantially contributed to the improved quality of this revision.

Supported by FONDECYT under grant No. 1120239 (P. Gajardo), 1110888 (H. Ramírez) and 1130538 (J.C. Rodríguez), the BASAL Project (Centro de Modelamiento Matemático, Universidad de Chile), CONICYT Anillo ACT1106, project BIONATURE of CIRIC, INRIA-Chile, and UFAM-INCTMat Avanço Global e Integrado da Matemática Brasilera e Contribuções a Região, CAPES-Brasil.


R Le Guével

J Lévy Véhel

Hausdorff, Large Deviation and Legendre Multifractal Spectra of Lévy Multistable Processes

We compute the Hausdorff multifractal spectrum of two versions of multistable Lévy motions. These processes extend classical Lévy motion by letting the stability exponent α evolve in time. The spectra provide a decomposition of [0, 1] into an uncountable disjoint union of sets with Hausdorff dimension one. We also compute the increments-based large deviations multifractal spectrum of the independent increments multistable Lévy motion. This spectrum turns out to be concave and thus coincides with the Legendre multifractal spectrum, but it is different from the Hausdorff multifractal spectrum. The independent increments multistable Lévy motion thus provides an example where the strong multifractal formalism does not hold.

Introduction and background

Multifractal analysis gives a fairly complete description of the singularity structure of measures, functions or stochastic processes. Various versions of multifractal analysis exist, which include the determinations of the so-called Hausdorff, large deviation, and Legendre multifractal spectra [START_REF] Véhel | Multifractal Analysis of Choquet Capacities: Preliminary Results[END_REF]. Multifractal analysis has been performed for various measures [START_REF] Arbeiter | Random self-similar multifractals[END_REF][START_REF] Brown | On the multifractal analysis of measures[END_REF], functions [START_REF] Jaffard | Old friends revisited. The multifractal nature of some classical functions[END_REF], and stochastic processes [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF][START_REF] Barral | Multifractal analysis of a class of additive processes with correlated non-stationary increments[END_REF][START_REF] Durand | Singularity sets of Lévy processes[END_REF][START_REF] Durand | Multifractal analysis of Lévy fields[END_REF][START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF]. In the case of Lévy processes, substantially finer results have been obtained in [START_REF] Balança | Fine regularity of Lévy processes and linear (multi)fractional stable motion[END_REF] using 2-microlocal analysis.

This article deals with the multifractal analysis of extensions of Lévy stable motions known as multistable Lévy motions. Generally speaking, multistable processes extend the well-known stable processes (see, e.g. [START_REF] Samorodnitsky | Stable Non-Gaussian Random Process[END_REF]) by letting the stability index α evolve in "time". These processes have been introduced in [START_REF] Falconer | Multifractional, multistable, and other processes with prescribed local form[END_REF] and have been studied for instance in [START_REF] Ayache | Sharp estimates on the tail behavior of a multistable distribution[END_REF][START_REF] Biermé | Linear multifractional multistable motion: LePage series representation and modulus of continuity[END_REF][START_REF] Falconer | Multistable Processes and Localisability[END_REF][START_REF] Guével | Incremental moments and Hölder exponents of multifractional multistable processes[END_REF][START_REF] Guével | On two multistable extensions of stable Lévy motion and their semimartingale representation[END_REF][START_REF] Molchanov | Multifractional Poisson process, multistable subordinator and related limit theorems[END_REF]. They provide useful models in various applications where the data display jumps with varying intensity, such as financial records, EEG or natural terrains: indeed, multistability is one practical way to deal with (increments-) non-stationarities observed in various real-world phenomena, since a multistable process X is tangent, at each time u, to a stable process Z u in the following sense [START_REF] Falconer | Tangent fields and the local structure of random fields[END_REF][START_REF] Falconer | The local structure of random processes[END_REF]:

lim r→0 X(u + rt) -X(u) r h = Z u (t) (1)
for a suitable h (the limit (1) is taken either in finite dimensional distributions or, when X has a version with càdlàg paths, in distribution -one then speaks of strong localisability). Without loss of generality, we shall consider our processes on [0, 1]. We will need the following ingredients:

• α : [0, 1] → (1, 2) is a C 1 function.
• (Γ i ) i≥1 is a sequence of arrival times of a Poisson process with unit arrival time.

• (V i ) i≥1 is a sequence of i.i.d. random variables with uniform distribution on [0, 1].

• (γ i ) i≥1 is a sequence of i.i.d. random variables with distribution P (γ i = 1) = P (γ i = -1) = 1/2.

The three sequences (Γ i ) i≥1 , (V i ) i≥1 , and (γ i ) i≥1 are independent. We denote c = inf

u∈[0,1] α(u), d = sup u∈[0,1] α(u),
We shall consider two versions of Lévy multistable processes: the first one has independent but non stationary increments. It admits the following representation:

B(t) = +∞ i=1 γ i C 1/α(V i ) α(V i ) Γ -1/α(V i ) i 1 (V i ≤t) , (2)
while the second one has correlated non stationary increments and reads:

D(t) = C 1/α(t) α(t) ∞ i=1 γ i Γ -1/α(t) i 1 [0,t] (V i ), (3)
where C u = ∞ 0 x -u sin x dx -1 . Both processes are semi-martingales and are tangent, at each time t, to α(t)-stable Lévy motion. See [START_REF] Guével | On two multistable extensions of stable Lévy motion and their semimartingale representation[END_REF] and the references therein for more details on these processes. We shall also denote:

Y (t) = ∞ i=1 γ i Γ -1/α(t) i 1 [0,t] (V i ).

Hausdorff multifractal spectra

Let h Y (t) denote the pointwise Hölder exponent of Y at t. The Hausdorff multifractal analysis of Y consists in measuring the Hausdorff dimension (denoted dim H ) of the sets

F h = {t ∈ [0, 1] : h Y (t) = h}.
The Hausdorff multifractal spectrum is the function h → f H (h) := dim H F h . We will use the following notations: S = ∪ i {V i }, S = S N and R t = {(r n ) n∈N ∈ S : r n → t}. If (r n ) n ∈ R t , we put V φ(n) = r n . Finally, define the positive function δ for t / ∈ S,

δ(t) = inf (V φ(n) )n∈Rt lim inf i→∞ - log φ(i) log |V φ(i) -t| .

Main result

The Hausdorff multifractal spectra of both B and D are described by the following theorem:

Theorem 1. With probability one, the common Hausdorff multifractal spectrum f H of B and D satisfies:

f H (h) =                -∞ for h < 0; hd for h ∈ [0, 1 d ]; 1 for h ∈ ( 1 d , 1 c ); dim H ({t ∈ [0, 1] : α(t) = c}) for h = 1 c ; -∞ for h > 1 c . (4)
Theorem 1 follows from a series of lemmas that are proven in the next section:

Lemma 2. Almost surely, t → Y (t) is càdlàg. Lemma 3. Almost surely, ∀t ∈ [0, 1]\S, δ(t) ≤ 1. Lemma 4. Almost surely, ∀t ∈ [0, 1]\S, h Y (t) ≤ δ(t) α(t) .
Lemma 5. Let g : [0, 1] → R be a càdlàg function, and f be the function defined on [0, 1] by f (t) = t 0 g(x)dx. The pointwise Hölder exponent h f of f verifies: ∀t ∈ (0, 1),

h f (t) ≥ 1. Lemma 6. Almost surely, ∀t ∈ [0, 1]\S, h Y (t) ≥ δ(t) α(t) . Lemma 7. Almost surely, ∀h < 0, f H (h) = -∞. Lemma 8. Almost surely, f H (0) = 0. Lemma 9. Almost surely, ∀h ∈ (0, 1 d ], f H (h) = hd. Lemma 10. Almost surely, ∀h ∈ ( 1 d , 1 c ), f H (h) = 1. Lemma 11. Almost surely, f H ( 1 c ) = dim H ({t ∈ [0, 1] : α(t) = c}). Lemma 12. Almost surely, ∀h > 1 c , f H (h) = -∞.

Proofs of the lemmas

Proof of Lemma 2:

Set Y N (t) = N i=1 γ i Γ -1/α(t) i 1 [0,t] (V i ). Lemma 8 in [19] states that, amost surely, Y N converges to Y (t) uniformly on [0, 1]. Fix ε > 0 and choose N 0 ∈ N such that, ∀N ≥ N 0 , sup t∈[0,1] |Y N (t) -Y (t)| ≤ ε 3 . ( 5
)
1 st case : t ∈ S. Let i 0 ∈ N be such that t = V i 0 , and N 1 = max(i 0 , N 0 ). Then, for h ∈ R,

Y (t) -Y (t + h) = Y (t) -Y N 1 (t) + Y N 1 (t) -Y N 1 (t + h) + Y N 1 (t + h) -Y (t + h). Since lim h→0 + Y N 1 (t) -Y N 1 (t + h) = 0, there exists h 0 > 0 such that ∀h ∈ (0, h 0 ), |Y N 1 (t) -Y N 1 (t + h)| ≤ ε 3 . As a consequence, ∀h ∈ (0, h 0 ), |Y (t)-Y (t+h)| ≤ ε and thus lim h→0 + Y (t)-Y (t+h) = 0. lim h→0 -Y N 1 (t) -Y N 1 (t + h) = lim h→0 - N 1 i=1 γ i Γ -1/α(t) i 1 (t+h,t] (V i ) = γ i 0 Γ -1/α(V i 0 ) i 0 , thus lim h→0 -Y N 1 (t) -Y N 1 (t + h) -γ i 0 Γ -1/α(V i 0 ) i 0 = 0. Choose h 0 < 0 such that ∀h ∈ (h 0 , 0), |Y (t) -Y (t + h) -γ i 0 Γ -1/α(V i 0 ) i 0 | ≤ ε. Thus, lim h→0 -Y (t) -Y (t + h) = γ i 0 Γ -1/α(V i 0 ) i 0 . 2 nd case : t / ∈ S Since lim h→0 |Y N 0 (t + h) -Y N 0 (t)| = 0, there exists h 0 > 0 such that ∀|h| < h 0 , |Y N 0 (t) -Y N 0 (t + h)| ≤ ε 3 .
Using [START_REF] Barral | Multifractal analysis of a class of additive processes with correlated non-stationary increments[END_REF], one thus has, for |h| < h 0 , |Y (t)-Y (t+h)| ≤ ε, and thus lim h→0 |Y (t+h)-Y (t)| = 0 Note 1. We have shown precisely that Y is càdlàg with set of jump points exactly equal to S. The jump at point V i is of size

γ i Γ -1/α(V i ) i .
Proof of Lemma 3:

For j ≥ 1, k = 1, ..., 2 j , let E k,j denote the interval E k,j = k -1 2 j - 1 2 1+(j+1)(1-1 √ j ) , k -1 2 j + 1 2 1+(j+1)(1-1 √ j )
.

Let us show that lim inf

j→∞ 2 j k=1 2 j+1 -1 i=2 j {V i ∈ E k,j } ⊂ {∀t / ∈ S, δ (
t) ≤ 1} first, and then that

P lim inf j→∞ 2 j k=1 2 j+1 -1 i=2 j {V i ∈ E k,j } = 1. We denote a j = 1 2 1+(j+1)(1-1 √ j
) . Assume that there exists J 0 ∈ N such that for all j ≥ J 0 , and all k = 1, ..., 2 j , we can fix i(k, j) ∈ [2 j , 2 j+1 -1] with V i(k,j) ∈ E k,j . Let t / ∈ S. For all j ≥ J 0 , there exists k(j) ∈ [1, 2 j ] such that t ∈ E k(j),j , because 2a j ≥ 1 2 j .

As a consequence,

|t -V i(k(j),j) | ≤ 1 2 (j+1)(1-1 √ j ) ≤ 1 i(k(j), j) 1-1 √ j . Hence -log i(k(j),j) log |t-V i(k(j),j) | ≤ 1 1-1 √ j
. This entails lim inf j→∞ -log i(k(j),j) log |t-V i(k(j),j) | ≤ 1, and, since V i(k(j),j) tends to t, δ(t) ≤ 1.

Finally, distinguishing the cases k = 1, k = 2, . . . , 2 j -1 and k = 2 j , one estimates

P   2 j k=1 2 j+1 -1 i=2 j {V i / ∈ E k,j }   ≤ 2 j k=1 P   2 j+1 -1 i=2 j {V i / ∈ E k,j }   ≤ a j + (1 -( 2 j -1 2 j -a j )) + 2 j -1 k=2 (P ({V 1 / ∈ E k,j })) 2 j ≤ 2a j + 1 2 j + (2 j -2)(1 -2a j ) 2 j . Since +∞ j=1
(2 j -2)(1 -2a j ) 2 j < +∞, Borel-Cantelli lemma allows us to conclude.

Proof of Lemma 4:

Recall Note 1. Lemma 1 of [START_REF] Jaffard | Old friends revisited. The multifractal nature of some classical functions[END_REF] entails that, for all sequences V φ(i) ∈ R t , and all t / ∈ S,

h Y (t) ≤ lim inf i→+∞ log |Γ - 1 α(V φ(i) ) φ(i) | log |V φ(i) -t| = lim inf i→+∞ - 1 α(V φ(i) ) log |Γ φ(i) | log |V φ(i) -t| .
Since α is continuous, φ(i) tends to infinity, the sequences (V φ(i) ) i converges to t, and almost surely ( Γ i i ) i tends to 1 when i tends to infinity, one obtains

h Y (t) ≤ 1 α(t) lim inf i→+∞ - log |φ(i)| log |V φ(i) -t| .
This inequality holds for all sequences V φ(i) ∈ R t , and thus, ∀t

/ ∈ S, h Y (t) ≤ δ(t) α(t)
Proof of Lemma 5: Since g is càdlàg, h g is non negative for all t. Integration increases pointwise regularity by at least one, and thus h f (t) ≥ 1 for all t. An alternative direct proof goes as follows: let t ∈ (0, 1), and h > 0. One computes

f (t + h) -f (t) h -g(t + ) = 1 h t+h t (g(x) -g(t + ))dx = 1 0 (g(t + sh) -g(t + ))ds.
∀s ∈ (0, 1), lim h→0 + (g(t + sh) -g(t + )) = 0. Since g is càdlàg, it is bounded and thus:

lim h→0 + f (t + h) -f (t) h = g(t + ).
Likewise, for h < 0,

f (t + h) -f (t) h -g(t -) =
f (t + h) -f (t) h = g(t -).
This entails h f (t) ≥ 1

Proof of Lemma 6: Theorem 7 of [START_REF] Guével | On two multistable extensions of stable Lévy motion and their semimartingale representation[END_REF] states that:

D(t) = A(t) + B(t),
where

A(t) = t 0 +∞ i=1 γ i d C 1/α(.) α(.) Γ -1/α(.) i dt (s)1 [0,s) (V i ) ds.
In addition, Lemma 8 of [START_REF] Guével | On two multistable extensions of stable Lévy motion and their semimartingale representation[END_REF] entails that

N i=1 γ i d C 1/α(.) α(.) Γ -1/α(.) i dt (s)1 [0,s) (V i ) N con- verges to +∞ i=1 γ i d C 1/α(.) α(.) Γ -1/α(.) i dt (s)1 [0,s) (V i ) uniformly on [0,
W (t) = +∞ i=1 γ i C 1/α(V i ) α(V i ) Γ -1/α(V i ) i -i -1/α(V i ) 1 (V i ≤t) (6)
and

Z(t) = +∞ i=1 γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 (V i ≤t) . (7)
Set

I k,m = [ k 2 m , k+1 2
m ), N m,j,k = Card {V i , i = 2 j , ..., 2 j+1 -1, V i ∈ I k,m }, d k,m = max u∈I k,m α(u)
and

C 0 = max t∈[0,1] C 1/α(t) α(t) . Define M m,j,k t = 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [0,t]∩I k,m (V i ).
It is easily seen that

sup (s,t)∈I 2 k,m 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) ≤ 2 sup t∈[0,1] M m,j,k t . ( 8
) For t ∈ (0, 1), let α n (t) = √ n 1 n n i=1
1 V i ≤t -t denote the empirical process, and

w n (a) = sup |t-s|≤a
|α n (t) -α n (s)| denote the oscillation modulus of α n . We apply Lemma 2.4

of [START_REF] Stute | The oscillation behavior of empirical processes[END_REF] with

m ≥ 3, a = 2 -m , s = m 1/4 j, n = 2 j , δ = 1 2 .
This yields that there exists

M 0 ∈ N such that ∀ m ≥ M 0 , ∃ j(m) with m ≤ j(m) ≤ 2m such that ∀ j ≥ j(m) : m 1/4 j ≤ √ 2 j-m and P sup 0≤|t-s|≤ 1 2 m |α 2 j (t) -α 2 j (s)| > m 1/4 √ j √ 2 m ≤ 256 × 2 m e -j √ m 64 . (9)
We need to estimate N m,j,k and sup

(s,t)∈I 2 k,m 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) for k = 0, ..., 2 m - 1 and j ≥ m. • Study of N m,j,k for m ≤ j ≤ j(m) -1: Set X i = 1 k 2 m ≤V i ≤ k+1 2 m and n = 2 j . (X i ) i is an i.i.d. sequence of Bernoulli random variables with parameter p = 1 2 m . For m ≤ j ≤ j(m) -1, one has √ mj ≥ 2 j-m = np, and
thus, for a > 0, using a classical bound on the sum of i.i.d. Bernoulli random variables,

P N m,j,k > a √ mj ≤ P   2 j+1 -1 i=2 j X i ≥ a2 j-m   = P   2 j+1 -1 i=2 j X i ≥ anp   ≤ 1 a n ≤ 1 a j .
As a consequence,

P   j(m)-1 j=m 2 m -1 k=0 N m,j,k > a √ mj   ≤ 2m j=m 2 m -1 k=0 1 a j ≤ 2 m +∞ j=m a -j ≤ a a -1 2 a m . Choosing a = 3, Borel Cantelli lemma entails that P   lim sup m→+∞ j(m)-1 j=m 2 m -1 k=0 N m,j,k > 3 √ mj   = 0.
• Study of N m,j,k for j ≥ j(m):

P N m,j,k > 2.2 j-m = P √ 2 j α 2 j k + 1 2 m -α 2 j k 2 m + 2 j-m ≥ 2.2 j-m ≤ P √ 2 j w 2 j ( 1 2 m ) ≥ 2 j-m = P w 2 j ( 1 2 m ) ≥ √ 2 j-m √ 2 m ≤ P w 2 j ( 1 2 m ) ≥ m 1/4 √ j √ 2 m
≤ 256 × 2 m e -j √ m 64 using ( 9). As a consequence,

P   +∞ j=j(m) 2 m -1 k=0 N m,j,k > 22 j-m   ≤ +∞ j=m 2 m (256 × 2 m e -j √ m 64 ) ≤ 256.4 m . e -m √ m 64 1 -e - √ m

64

.

Borel Cantelli lemma yields

P   lim sup m→+∞ +∞ j=j(m) 2 m -1 k=0 N m,j,k ≥ 2.2 j-m   = 0. • Study of sup (s,t)∈I 2 k,m 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) for m ≤ j ≤ j(m) -1:
Almost surely, there exists m 0 such that, for m ≥ m 0 and for m

≤ j ≤ j(m) -1, ∀k = 0, ..., 2 m -1, N m,j,k ≤ 3 √ mj, and ∀(s, t) ∈ I 2 k,m , ∀i = 2 j , ..., 2 j+1 -1, γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) ≤ C 0 2 j d k,m thus sup (s,t)∈I 2 k,m 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) ≤ 3 √ mjC 0 2 j d k,m . • Study of sup (s,t)∈I 2 k,m 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) for j ≥ j(m):
We consider the events

E m,j,k = N m,j,k ≤ 2.2 j-m , F m,j,k =    sup (s,t)∈I 2 k,m 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) ≥ jC 0 √ 2.2 j-m 2 j d k,m    and G m,j,k = sup t∈[0,1] M m,j,k t ≥ jC 0 √ 2.2 j-m 2.2 j d k,m . Relation (8) entails that P (F m,j,k ∩ E m,j,k ) ≤ P (G m,j,k ∩ E m,j,k ) .
In the following computation, l corresponds to the number of terms V i belonging to I k,m , and n corresponds to the number of those V i among them that contribute to the supremum of G m,j,k .

G m,j,k ∩E m,j,k ⊂ 2.2 j-m l=1 l 1 ,...,l l ∈[2 j ,2 j+1 -1]   G m,j,k ∩   i∈{l 1 ,...l l } V i ∈ I k,m   ∩   i / ∈{l 1 ,...l l } V i / ∈ I k,m     .
Using independence of the V i ,

P (G m,j,k ∩ E m,j,k ) ≤ 2.2 j-m l=1 l 1 ,...,l l ∈[2 j ,2 j+1 -1] P   G m,j,k ∩ i∈{l 1 ,...l l } V i ∈ I k,m   P   i / ∈{l 1 ,...l l } V i / ∈ I k,m   . Now, P i / ∈{l 1 ,...l l } V i / ∈ I k,m = 1 -1 2 m
2 j -l . Let us fix an order on the V i belonging to I k,m : there are l! possibilities, all equiprobable, and thus

P   G m,j,k ∩ i∈{l 1 ,...l l } V i ∈ I k,m   = (l!)P G m,j,k ∩ ( k 2 m < V l 1 < ... < V l l < k + 1 2 m ) . Let A l k,m denote the event { k 2 m < V l 1 < ... < V l l < k+1 2 m }. Then P G m,j,k ∩ A l k,m ≤ l n=1 P A l k,m ∩ n i=1 γ l i C 1/α(V l i ) α(V l i ) l -1/α(V l i ) i ≥ jC 0 √ 2.2 j-m 2.2 j d k,m ≤ l n=1 k 2 m <x 1 <...<x l < k+1 2 m P n i=1 γ l i C 1/α(x i ) α(x i ) l -1/α(x i ) i ≥ jC 0 √ n 2.2 j d k,m dx 1 ...dx l .
The probability inside the integral above may be estimated with the help of Lemma 1.5 in [START_REF] Ledoux | Probability in Banach spaces[END_REF]:

P   n i=1 γ l i C 1/α(x i ) α(x i ) C 0 2 j d k,m l 1/α(x i ) i ≥ j √ n 2   ≤ 2e -j 2 8 .
One then computes

P (G m,j,k ∩ E m,j,k ) ≤ 2.2 j-m l=1 l 1 ,...,l l ∈[2 j ,2 j+1 -1] (1 - 1 2 m ) 2 j -l (l!) l n=1 2e -j 2 8 k 2 m <x 1 <...<x l < k+1 2 m dx 1 ...dx l ≤ 2.2 j-m l=1 (2 j )! (l!)(2 j -l)! (1 - 1 2 m ) 2 j -l (l!)2le -j 2 8 1 2 ml 1 l! ≤ 2e -j 2 8 2.2 j-m l=1 l (2 j )! (l!)(2 j -l)! (1 - 1 2 m ) 2 j -l 1 2 ml ≤ 4.2 j-m e -j 2 8 2.2 j-m l=1 (2 j )! (l!)(2 j -l)! (1 - 1 2 m ) 2 j -l 1 2 ml ≤ 4.2 j-m e -j 2 8 .
As a consequence,

P   +∞ j=j(m) 2 m -1 k=0 (F m,j,k ∩ E m,j,k )   ≤ 4 +∞ j=m 2 j e -j 2 8 ≤ +∞ j=m e -j
for m ≥ 100. Borel Cantelli lemma entails that

P   lim sup m→+∞ +∞ j=j(m) 2 m -1 k=0 F m,j,k   = 0.
• Computation of the Hölder exponent: Let t ∈ (0, 1), t / ∈ S and let U be an open interval of (0, 1) containing t. Denote d U = max t∈U α(t). If δ(t) = 0, then h Y (t) = 0 and the formula holds. Suppose now δ(t) > 0.

Let ε > 0 be such that δ(t) > ε and

1 d U -1 2 > ε. There exists i 0 ∈ N such that ∀i ≥ i 0 , |t -V i | ≥ 1 i 1 δ(t)-ε . Choose m large enough so that 1 2 m < min{|t -V i |, i = 1, ..., i 0 }. Let j 0 = [m(δ(t) -ε)].
Increasing m if necessary, we may and will assume that i 0 ≤ 2 j 0 , and ∀j ≥ j 0 , j ≤ 2 jε . Let s ∈ (0, 1) be such that 1 2 m+2 ≤ |t -s| < 1 2 m+1 . There exists k ∈ {0, ..., 2 m -1} such that (t, s) ∈ I 2 k,m . Increasing again m if necessary, we may assume that I k,m ⊂ U. Then d k,m ≤ d U , and, for i ≤ 2 j 0 , 1 [s,t) (V i ) = 0. One computes:

|Z(t) -Z(s)| = +∞ i=1 γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) = +∞ j=j 0 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) ≤ j(m) j=j 0 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) + +∞ j=j(m) 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) i -1/α(V i ) 1 [s,t) (V i ) ≤ j(m) j=j 0 3 √ mj 2 j d k,m + +∞ j=j(m) jC 0 √ 2.2 j-m 2 j d k,m ≤ 2(3 + √ 2C 0 ) √ m +∞ j=j 0 j √ 2 j-m 2 j d k,m ≤ 6(1 + C 0 ) √ m √ 2 m +∞ j=j 0 2 j(ε+ 1 2 -1 d U ) ≤ 6(1 + C 0 ) √ m √ 2 m 2 j 0 (ε+ 1 2 -1 d U ) 1 -2 ε+ 1 2 -1 d U ≤ K U,ε,C 0 √ m2 m -1 2 +(δ(t)-ε)(ε+ 1 2 -1 d U ) ≤ K U,ε,C 0 | log |t -s|||t -s| (δ(t)-ε)( 1 d U -ε)+ 1 2 -1 2 (δ(t)-ε) . Since 1 2 -1 2 (δ(t) -ε) > 0, |Z(t) -Z(s)| ≤ K U,ε,C 0 | log |t -s|||t -s| (δ(t)-ε)( 1 d U -ε) .
Let us now study W (recall (6)): there exists i 0 such that, for all i ≥ i 0 ,

Γ -1/α(V i ) i -i -1/α(V i ) ≤ K c,d i 1 d k,m + 1 2 ≤ K c,d i 1 d U + 1 2 .
One then computes:

|W (t) -W (s)| = +∞ j=j 0 2 j+1 -1 i=2 j γ i C 1/α(V i ) α(V i ) (Γ -1/α(V i ) i -i -1/α(V i ) )1 [s,t) (V i ) ≤ j(m) j=j 0 3K c,d C 0 √ mj 2 j( 1 2 + 1 d U ) + +∞ j=j(m) K c,d C 0 2.2 j-m 2 j( 1 2 + 1 d U ) ≤ K √ m +∞ j=j 0 2 j(ε-1 2 -1 d U ) + K 2 m +∞ j=j 0 2 j( 1 2 -1 d U ) ≤ K | log |t -s|| 2 m(δ(t)-ε)(ε-1 2 -1 d U ) 1 -2 ε-1 2 -1 d U + K|t -s| 2 m(δ(t)-ε)( 1 2 -1 d U ) 1 -2 1 2 -1 d U ≤ K | log |t -s|||t -s| (δ(t)-ε)( 1 2 + 1 d U -ε) + K|t -s| 1+(δ(t)-ε)( 1 d U - 1
2 ) .

Gathering our results, we have shown that:

|B(t) -B(s)| ≤ K | log |t -s|||t -s| (δ(t)-ε)( 1 d U -ε) .
In other words, h B (t) ≥ δ(t) d U for any open interval U containing t. Letting the diameter of U go to 0, one gets h B (t) ≥ δ(t) α(t) .

Proof of Lemma 7:

Lemma 2 entails that Y is almost surely a càdlàg process. Thus, for all t ∈ (0, 1), h Y (t) ≥ 0.

Proof of Lemma 8:

We seek to compute the Hausdorff dimension of F 0 = {t ∈ [0, 1]\S : δ(t) = 0} ∪ S.

Let E γ = lim sup j→+∞ 2 j+1 -1 i=2 j V i -i -1 γα(V i ) , V i + i -1 γα(V i ) . Since E γ ⊂ {t : h Y (t) ≤ γ}, {t ∈ [0, 1]\S : δ(t) = 0} ⊂ γ>0 E γ . Now, i≥1 V i -i -1 γd , V i + i -1 γd i is a covering of E γ , and thus dim H (E γ ) ≤ γd. As a consequence, dim H ({t ∈ [0, 1]\S : δ(t) = 0}) = 0. Since dim H (S) = 0, we find that f H (0) = 0.
Proof of Lemma 9: Following [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF], set λ i = 1 1+Γ i . For the system of points P = {(V i , λ i )} i≥1 and t ∈ [0, 1], define the approximation rate of t by P as δ t (P) = sup{δ ≥ 1 : t belongs to an infinite number of balls B(V i , λ δ i )}. Let us show that δ t (P) = 1 δ(t) . In that view, note first that, since lim i→+∞

Γ i i = 1 almost surely, δ(t) = inf (V φ(n) )n∈Rt lim inf i∞ - log |1 + Γ φ(i) | log |V φ(i) -t| .
Let δ ≥ 1 be such that t belongs to an infinite number of balls B(V i , λ δ i ). There exists

φ : N → N such that (V φ(n) ) n ∈ R t and |t -V φ(i) | ≤ λ δ φ(i) , i.e. - log |1+Γ φ(i) | log |V φ(i) -t| ≤ 1 δ . As a consequence δ(t) ≤ 1 δ .
Taking the supremum over all admissible δ, one gets δ t (P) ≤ 1 δ(t) . For the reverse inequality, consider two cases:

• δ(t) = 1: since δ t (P) ≥ 1, one gets δ t (P) ≥ 1 δ(t) ;

• δ(t) < 1: choose ε > 0 such that δ(t) + ε ≤ 1. There exists φ : N → N such that for all large enough i ∈ N, -

log |1+Γ φ(i) | log |V φ(i) -t| ≤ δ(t) + ε, i.e. |t -V φ(i) | ≤ 1 (1+Γ φ(i) ) 1 δ(t)+ε
. By definition, this entails δ t (P) ≥ 1 δ(t)+ε , and finally δ t (P) ≥ 1 δ(t) by letting ε go to 0.

We now apply Theorem 21 of [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF]: since hd ≤ 1, one has hα(t) ≤ 1 for all t ∈ (0, 1). The function t → 1 hα(t) is continuous and thus dim H ({t ∈ (0, 1) :

δ t (P) = 1 hα(t) }) = sup{hα(t), t ∈ (0, 1)} = hd i.e. f H (h) = hd
Proof of Lemma 10: Since hc < 1 < hd and α is C 1 , there exist (t 0 , t 1 ) ∈ (0, 1) 2 such that hα(t 0 ) = 1 and hα(t) ≤ 1 for all t ∈ I := (t 1 , t 0 ) or (t 0 , t 1 ).

Define g : t → min(1, 1 hα(t) ). Theorem 21 in [START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF] yields:

dim H ({t ∈ I : δ t (P) = g(t)}) = sup{ 1 g(t) , t ∈ I} = sup{hα(t), t ∈ I} = hα(t 0 ) = 1.

One gets that f

H (h) ≥ dim H ({t ∈ I : δ(t) = hα(t)}) = 1, i.e. f H (h) = 1.

Proof of Lemma 11:

By definition and Lemma 3,

F 1/c = {t ∈ [0, 1] : δ(t) = α(t) c } = {t ∈ [0, 1] : α(t) = c} ∩ {t ∈ [0, 1] : δ(t) = 1}. Set E = {t ∈ [0, 1] : α(t) = c}, E 0 = {t ∈ [0, 1] : δ(t) < 1} and E 1 = {t ∈ [0, 1] : δ(t) = 1}. Then [0, 1] = E 0 ∪ E 1 , F 1/c = E ∩ E 1 and thus f H ( 1 c ) ≤ dim H (E). Now, if dim H (E) = 0,
the lemma holds true since F 1/c is not empty and thus f H (1/c) ≥ 0. Suppose then that dim H (E) > 0. Choose s < dim H (E). This implies that H s (E) = +∞, and Theorem 4.10 in [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF] entails that there exist a compact set E c ⊂ E such that 0 < H s (E c ) < +∞.

Set µ s (.) = H s (E c ∩ .). This is a finite and positive Borel measure on [0, 1]. Theorem 3.7 in [START_REF] Guével | Incremental moments and Hölder exponents of multifractional multistable processes[END_REF], along with Lemmas 4 and 6, entail that for all t ∈ (0, 1), P(t ∈ E 1 ) = 1 and thus P(t ∈ E c ∩ E 1 ) = 1 t∈Ec .

One computes:

E [µ s (E 0 )] = E 1 0 1 t∈E 0 µ s (dt) = 1 0 E [1 t∈E 0 ] µ s (dt) = Ec P(t ∈ E 0 )µ s (dt) = 0.
Thus, µ s (E 0 ) is a positive random variable with vanishing expectation: almost surely,

µ s (E 0 ) = 0. Since µ s (E c ) = µ s (E c ∩ E 0 ) + µ s (E c ∩ E 1 ), one obtains that, almost surely, µ s (E c ) = µ s (E c ∩ E 1 ). Now, H s (E ∩ E 1 ) ≥ H s (E c ∩ E 1 ) = H s (E c ) > 0. Thus dim H (E ∩ E 1 ) ≥ s, ∀s ≤ dim H (E) and dim H (E ∩ E 1 ) ≥ dim H (E).

Proof of Lemma 12:

For all t ∈ [0, 1], α(t) ≥ c and almost surely, for all t ∈ [0, 1],

δ(t) ≤ 1, thus, almost surely, for all t ∈ [0, 1], h Y (t) ≤ 1 c . As a consequence, for h > 1 c , F h = ∅ and f H (h) = -∞.

Large deviation and Legendre multifractal spectra

We compute in this section the large deviation and Legendre multifractal spectra of the process B on an interval. Recall that we consider the process on [0, 1], and that the large deviation multifractal spectrum of a process X on [0, 1] is the (random) function f g defined on R by

f g (β) = lim ε→0 + lim inf n→+∞ log N ε n (β) log n ,
where, for a positive integer n and ε > 0,

N ε n (β) = #{j ∈ {0, . . . , n -1} : β -ε ≤ log |X( j+1 n ) -X( j n )| -log n ≤ β + ε}.
Other large deviation multifractal spectra can be defined by replacing the increments X( j+1 n ) -X( j n ) by other measures of the variation of X, such as its oscillations, but we will not consider these in this work. We do not recall the definition of the Legendre multifractal spectrum, and refer the reader to [START_REF] Falconer | Fractal Geometry: Mathematical Foundations and Applications[END_REF][START_REF] Véhel | Multifractal Analysis of Choquet Capacities: Preliminary Results[END_REF] instead.

We shall denote

P j n = P β -ε ≤ log |Y ( j+1 n ) -Y ( j n )| -log n ≤ β + ε = P 1 n β+ε ≤ |Y ( j + 1 n ) -Y ( j n )| ≤ 1 n β-ε .
Set also

X j = 1 { 1 n β+ε ≤|B( j+1 n )-B( j n )|≤ 1 n β-ε }
which follows a Bernoulli law with parameter P j n . Clearly,

N ε n (β) = n j=1 X j .
For U an open interval of (0, 1), we write

J n (U) = {j : j n ∈ U}.
There exists a constant K U > 0 such that, for n large enough,

#J n (U) ≥ K U n.
Finally, we will make use of the characteristic function of B, which reads [START_REF] Guével | On two multistable extensions of stable Lévy motion and their semimartingale representation[END_REF]:

E exp i m j=1 θ j B(t j ) = exp   - m j=1 θ j 1 [0,t j ] (s) α(s) ds   . ( 10
)
where

m ∈ N, (θ 1 , . . . , θ m ) ∈ R m , (t 1 , . . . , t m ) ∈ R m .

Main result

The large deviation and Legendre multifractal spectra of B are described by the following theorem:

Theorem 13. With probability one, the large deviation and Legendre multifractal spectra of B satisfy:

f g (β) = f l (β) =                -∞ for β < 0; βd for β ∈ [0, 1 d ]; 1 for β ∈ ( 1 d , 1 c ]; 1 + 1 c -β for β ∈ ( 1 c , 1 + 1 c ]; -∞ for β > 1 c . (11)
The fact that f l = f g stems from the general result that f l is always the concave hull of f g when the set {β : f g (β) ≥ 0} is bounded. The part concerning f g in Theorem 13 follows from a series of lemmas that are proven in the next sections. We note in passing that, comparing with Theorem 1, we see that the weak multifractal formalism holds for B, but the strong one does not, that is, f H ≤ f g and f H = f g . The decreasing part with slope -1 for "large" exponents present in f g but not in f H is a common phenomenon when variations are measured with increments.

In order to prove Theorem 13, we will first show in each case of (11) that the equality holds true for any given β with probability one. Permuting "for all β" and "almost surely" will then often be achieved thanks to the two following general simple but useful lemmas on the large deviation spectrum, which are of independent interest. Lemma 14. The large deviation spectrum of any real function is an upper semicontinuous function.

Proof. Let f g be the large deviation spectrum of a real function. Consider β ∈ R, (x j ) j≥1 a sequence such that lim j→+∞ x j = β, and set

ε j = sup k≥j |β -x k |. Note first that lim ε→0 lim inf n→+∞ log N ε n (β) log n = lim j→+∞ lim inf n→+∞ log N ε j n (β) log n .
For all j ≥ 1 and all l ≥ j, N

2ε j n (β) ≥ N ε j n (x j ) ≥ N ε l n (x j ). As a consequence, lim inf n→+∞ log N 2ε j n (β) log n ≥ lim inf n→+∞ log N ε l n (x j ) log n .
Letting l tend to infinity, one gets lim inf

n→+∞ log N 2ε j n (β) log n ≥ f g (x j
) and letting j tend to infinity one finally obtains

f g (β) ≥ lim sup j→+∞ f g (x j ).
Lemma 15. Assume that there exist four functions h, h, g, g with lim u→0 g(u) = lim u→0 g(u) = 0 such that, for all β in some interval I and all sufficiently small ε > 0, almost surely

h(β) + g(ε) ≤ lim inf n→+∞ log N ε n (β) log n ≤ h(β) + g(ε).
Then, almost surely, for all β in I, h(β) ≤ f g (β) ≤ h(β).

Proof. Define

M n (β 1 , β 2 ) = #{j ∈ {0, . . . , n -1} : β 1 ≤ log |X( j+1 n ) -X( j n )| -log n ≤ β 2 }.
Then, for

β 1 < β < β 2 , N (β 2 -β)∨(β-β 1 ) n (β) ≤ M n (β 1 , β 2 ) ≤ N (β 2 -β)∧(β-β 1 ) n (β)
and

f g (β) = inf (β 1 ,β 2 )∈Q 2 ,β 1 <β<β 2 lim inf n→+∞ log M n (β 1 , β 2 ) log n . The set A = (β 1 ,β 2 )∈Q 2 M(β 1 , β 2 )
is countable, and, thus f g is obtained by a countable infimum for all β ∈ I. This yields the result.

Preliminary lemmas

Statements

Define

H(t) = H λ,p (t) = λt -log(1 -p + pe t ). Lemma 16. If 0 < p < λ < 1, then sup t>0 H(t) = λ log( λ p ) + (1 -λ) log( 1 -λ 1 -p ). Lemma 17. If 0 < λ < p < 1, then sup t<0 H(t) = λ log( λ p ) + (1 -λ) log( 1 -λ 1 -p ). Lemma 18. If p = p(n) = Kn b and λ = λ(n) = n a
, where K > 0 and 0 > a > b, then there exists n 0 ∈ N such that, for all n ≥ n 0 ,

sup t>0 H(t) ≥ (a -b) 2 n a log n. Lemma 19. If p = p(n) = K 1 n b and λ = λ(n) = K 2 n a , where K 1 > 0, K 2 > 0 and 0 > b > a, then there exists n 0 ∈ N such that, for all n ≥ n 0 , sup t<0 H(t) ≥ K 1 4 n b . Lemma 20. If p = p(n) = Kn b and λ = λ(n) = n a
, where K > 0 and 0 > a > b, then there exists n 0 ∈ N such that, for all n ≥ n 0 ,

inf t>0 e -λtn (1 -p + pe t ) n ≤ e -(a-b) 2 n 1+a log n . Lemma 21. If p = p(n) = K 1 n b and λ = λ(n) = K 2 n a , where K 1 > 0, K 2 > 0 and 0 > b > a, then there exists n 0 ∈ N such that, for all n ≥ n 0 , inf t<0 e -λtn (1 -p + pe t ) K U n ≤ e -K 1 K U 4 n 1+b .
Lemma 22. Assume there exist b ∈ (-1, 0) and K > 0 such that, for all n ≥ n 0 and for all j ∈ 1, n , P j n ≤ Kn b . Then, almost surely,

lim inf n→+∞ log N ε n (β) log n ≤ 1 + b.
Lemma 23. Assume there exist an open interval U, a real b ∈ (-1, 0) and K > 0 such that, for all n ≥ n 0 and for all j ∈ J n (U), P j n ≥ Kn b . Then, almost surely,

lim inf n→+∞ log N ε n (β) log n ≥ 1 + b.

Proofs

Proof of Lemma 16. sup t>0

H(t) = H(t 0 ) where t 0 = log λ(1-p) p(1-λ) .
Proof of Lemma 17. sup t<0 H(t) = H(t 0 ) where t 0 = log λ(1-p) p(1-λ) .

Proof of Lemma 18. With t 0 = log λ(1-p) p(1-λ) , one has, for n large enough,

H(t 0 ) = n a log n a-b K + (1 -n a ) log 1 -n a 1 -Kn b = n a log n a-b K + (1 -n a ) log 1 + Kn b -n a 1 -Kn b ≥ n a log n a-b K + 2(1 -n a ) Kn b -n a 1 -Kn b = (a -b)n a log n + n a 2(1 -n a ) Kn b-a -1 1 -Kn b -log K . Since lim n→+∞ 2(1 -n a ) Kn b-a -1 1-Kn b -log K = -2 -log K, there exists n 0 ∈ N such that, for all n ≥ n 0 , n -a H(t 0 ) log n ≥ (a -b) + 1 log n 2(1 -n a ) Kn b-a -1 1 -Kn b -log K ≥ (a -b) 2 .
Proof of Lemma 19. With t 0 = log λ(1-p) p(1-λ) , one has, for n large enough,

H(t 0 ) = K 2 n a log K 2 n a-b K 1 + (1 -K 2 n a ) log 1 + K 1 n b -K 2 n a 1 -K 1 n b ≥ K 2 n a log K 2 n a-b K 1 + 1 2 (1 -K 2 n a ) K 1 n b -K 2 n a 1 -K 1 n b = n b K 2 n a-b log K 2 n a-b K 1 + 1 2 (1 -K 2 n a ) K 1 -K 2 n a-b 1 -K 1 n b which yields the result since lim n→+∞ K 2 n a-b log( K 2 n a-b K 1 ) + 1 2 (1 -K 2 n a ) K 1 -K 2 n a-b 1-K 1 n b = K 1 2 .
Proof of Lemma 20. One has

inf t>0 e -λtn (1 -p + pe t ) n = inf t>0 e -nH(t) = e -n sup t>0 H(t)
.

Lemma 18 then implies that, for n ≥ n 0 ,

inf t>0 e -λtn (1 -p + pe t ) n ≤ e -(a-b) 2 n 1+a log n . Proof of Lemma 21. Write inf t<0 e -λtn (1 -p + pe t ) K U n = e -K U n sup t<0 H(t)
where

H(t) = λ K U t -log(1 -p + pe t ). Lemma 19 ensures that, for n ≥ n 0 , inf t<0 e -λtn (1 -p + pe t ) n ≤ e -K 1 K U 4 n 1+b .
Proof of Lemma 22. Fix a ∈ (b, 0). Then, for all t > 0,

P N ε n (β) ≥ n 1+a = P e t n j=1 X j ≥ e tn 1+a ≤ e -ntλ E n j=1 e tX j
where λ = n a . The X j are independent and E e tX j = 1 -P j n + P j n e t , thus

P N ε n (β) ≥ n 1+a ≤ e -ntλ n j=1
(1 -P j n + P j n e t ), ∀t > 0.

For t > 0, the function p → 1 -p + pe t is increasing and so, by assumption on P j n ,

P N ε n (β) ≥ n 1+a ≤ e -ntλ (1 -p + pe t ) n ,
where p = Kn b . Minimizing over t > 0 and using Lemma 20, one gets

∀n ≥ n 0 , P N ε n (β) ≥ n 1+a ≤ e -(a-b)
2 n 1+a log n and thus

n∈N P N ε n (β) ≥ n 1+a < +∞.
The Borel-Cantelli lemma then ensures that, almost surely,

∃n 0 ∈ N, ∀n ≥ n 0 , N ε n (β) ≤ n 1+a or lim inf n→+∞ log N ε n (β) log n ≤ 1 + a.
Since this inequality holds true for any a ∈ (b, 0) one has indeed that, almost surely,

lim inf n→+∞ log N ε n (β) log n ≤ 1 + b.

Proofs

Proof of Lemma 24. Set µ j = α(t j )β + α(t j )ε -1. Using the truncation inequality [21, Section 13, p. 209], one computes

P j n = P 1 n β+ε ≤ |Y (t j+1 ) -Y (t j )| ≤ 1 n β-ε ≤ P |Y (t j+1 ) -Y (t j )| ≥ 1 n β+ε ≤ 7 n β+ε n β+ε 0 1 -e - t j+1 t j |ξ| α(x) dx dξ = 7 1 0 1 -e - t j+1 t j n (β+ε)α(x) |v| α(x) dx dv ≤ 7 1 0 t j+1 t j n (β+ε)α(x) |v| α(x) dxdv ≤ 7 t j+1 t j n (β+ε)α(x) dx = 7n µ j t j+1 t j n 1+(β+ε)(α(x)-α(t j )) dx.
Since α is C 1 , there exists a constant K such that, for all x ∈ (t j , t j+1 ),

|β + ε||α(x) -α(t j )| ≤ K|x -t j | ≤ K n .
As a consequence,

P j n ≤ 7n µ j t j+1 t j n 1+ K n dx = 7n µ j n K n ≤ K 1 n µ j for a constant K 1 . Proof of Lemma 25. Set µ = 1 2 1 n β+ε + 1 n β-ε and σ = 1 2 1 n β-ε -1 n β+ε .
Choose a function ϕ that satisfies the following properties:

1. supp(ϕ) ⊂ [-1, 1]. 2. ϕ is even. 3. ϕ is C 4 . 4. ∀x ∈ R, 0 ≤ ϕ(x) ≤ 1.
5. ϕ is not identically 0.

These properties imply in particular that φ is real and even. In addition, for all x ∈ R:

ϕ - x + µ σ + ϕ x -µ σ ≤ 1 [-n -β+ε ,-n -β-ε ] (x) + 1 [n -β-ε ,n -β+ε ] (x).
Since the Fourier transform of ϕ x-µ σ is σ exp(-iµξ) φ(σξ), Parseval formula yields

P j n ≥ R ϕ(- x + µ σ ) + ϕ( x -µ σ ) P(Y (t j+1 ) -Y (t j ) ∈ dx) ≥ 1 π σ R cos(µξ) φ(σξ)e - t j+1 t j |ξ| α(x) dx dξ = 2 π +∞ 0 cos( µ σ η) φ(η)e - t j+1 t j | η σ | α(x) dx dη. Now, R cos( µ σ η) φ(η)dη = ϕ( µ σ ) = 0
since µ > σ. One may thus write:

P j n ≥ 2 π +∞ 0 cos( µ σ η) φ(η)(e - t j+1 t j | η σ | α(x) dx -1)dη =: 2 π n α(t j )β-α(t j )ε-1 I j n ,
where I j n = I j n,1 + I j n,2 + I j n,3 with

I j n,1 = +∞ 0 φ(η) cos( µ σ η) -cos(η) n 1-α(t j )β+εα(t j ) e - t j+1 t j | η σ | α(x) dx -1 dη, I j n,2 = +∞ 0 φ(η) cos(η)n 1-α(t j )β+εα(t j ) e - t j+1 t j | η σ | α(x) dx -1 + t j+1 t j | η σ | α(x) dx dη, I j n,3 = - +∞ 0 φ(η) cos(η)n 1-α(t j )β+εα(t j ) t j+1 t j | η σ | α(x) dxdη.
Let us show that there exists K U > 0 such that, for all j ∈ J n (U), n 1-α(t j )β+εα(t j )

t j+1 t j | η σ | α(x) dx ≤ K U (η c + η d ).
Since 1 σ = 2n β+ε n 2ε -1 and, there exits K such that, for all x ∈ (t j , t j+1 ), |α(x) -α(t j )| ≤ K n , one has |σ α(t j )-α(x) | ≤ K and n 1-α(t j )β+εα(t j )

t j+1 t j | η σ | α(x) dx = n -α(t j )β+εα(t j ) σ α(t j ) n t j+1 t j |η| α(x) σ α(t j )-α(x) dx ≤ 2 α(t j ) n 2εα(t j ) (n 2ε -1) α(t j ) (η c + η d )n t j+1 t j σ α(t j )-α(x) dx ≤ K(η c + η d ).
As a consequence,

|I j n,1 | ≤ +∞ 0 | φ(η)|| µ σ -1|n 1-α(t j )β+εα(t j ) t j+1 t j | η σ | α(x) dxdη ≤ K| µ σ -1| +∞ 0 | φ(η)|(η c + η d )dη.
One finally obtains that sup j∈Jn(U )

|I j n,1 | ≤ K| 2 n 2ε -1 | and lim n→+∞ ( sup j∈Jn(U ) |I j n,1 |) = 0.
Let us now deal with I j n,2 .

|I j n,2 | ≤ +∞ 0 | φ(η)|n 1-α(t j )β+εα(t j ) 1 2 t j+1 t j | η σ | α(x) dx 2 dη ≤ K 2 U 2 +∞ 0 | φ(η)|n 1-α(t j )β+εα(t j ) (η c + η d ) 2 n 2α(t j )β-2-2εα(t j ) dη ≤ K U n α(t j )(β-ε-1 α(t j ) ) ≤ K U n d U (β-ε-1 d U )
and thus lim n→+∞ ( sup

j∈Jn(U ) |I j n,2 |) = 0. Fubini's theorem implies that I j n,3 = t j+1 t j n 1-α(t j )β+εα(t j ) σ α(x)   - +∞ 0 φ(η) cos(η)η α(x) dη   dx = t j+1 t j n 1-α(t j )β+εα(t j ) σ α(x) F (α(x))dx
where

F (δ) = - +∞ 0 φ(η) cos(η)η δ dη.
The appendix contains a proof that min δ∈[c,d] F (δ) > 0. As a consequence,

I j n,3 ≥ ( min δ∈[c,d] F (δ)) n -α(t j )β+εα(t j ) σ α(t j ) n t j+1 t j σ α(t j )-α(x) dx = ( min δ∈[c,d] F (δ)) n 2ε n 2ε -1 α(t j ) n t j+1 t j e (α(t j )-α(x)) log σ dx ≥ K U n 2ε n 2ε -1 c n t j+1 t j (e (α(t j )-α(x)) log σ -1)dx + 1 . Now lim n→+∞ n 2ε n 2ε -1 = 1 and lim n→+∞ ( sup j∈Jn(U ) |n t j+1 t j (e (α(t j )-α(x)) log σ -1)dx|) = 0. As a consequence, ∃K U > 0, ∃n 0 ∈ N : ∀n ≥ n 0 , inf j∈Jn(U ) I j n,3 ≥ K U > 0 and thus inf j∈Jn(U ) (n 1-α(t j )β+εα(t j ) P j n ) > 0.
Proofs of Lemma 26 and Lemma 27. Parseval formula yields

P j n = 1 π R sin( ξ n β-ε ) -sin( ξ n β+ε ) ξ e - t j+1 t j |ξ| α(x) dx dξ. Set µ j = 1 α(t j ) + ε -β with ε ∈ (0, β -1 c U ).
Using the change of variable ξ = n 1/α(t j ) v, one gets

P j n = 1 π R sin(n µ j v) -sin(n µ j -2ε v) v e - t j+1 t j |v| α(x) n α(x) α(t j ) dx dv. Define I j n = R e - t j+1 t j |v| α(x) n α(x) α(t j ) dx dv.
Since α is Lipschitz and c ≤ α(x) ≤ d, one deduces that

∃K 1 U > 0, ∃K 2 U > 0, ∃n 0 ∈ N, K 1 U ≤ inf j∈Jn(U ) I j n ≤ sup j∈Jn(U ) I j n ≤ K 2 U .
Now,

P j n = n µ j I j n    1 + 1 I j n R ( sin(n µ j v) -sin(n µ j -2ε v) n µ j v -1)e - t j+1 t j |v| α(x) n α(x) α(t j ) dx dv    = n µ j I j n (1 + L j n ).
Lemma 24 implies that, for n large enough,

P(N ε n (β) ≥ 1) ≤ 1 -(1 -Kn dβ+dε-1 ) n ≤ 1 -e -3 2 Kn dβ+dε ≤ 3 2 Kn dβ+dε ,
and thus lim n→+∞ P(N ε n (β) ≥ 1) = 0. Since N ε n (β) tends to 0 in probability when n tends to infinity, there exists a subsequence σ(n) such that N ε σ(n) (β) tends to 0 almost surely. This implies that, almost surely, lim inf n→+∞ log N ε n (β) log n = -∞. We have proved that:

∀β < 0, ∀ε ∈ E β , almost surely, lim inf n→+∞ log N ε n (β) log n = -∞. (12)
Let Ω β,ε = {ω : ( 12

There exists an open interval U such that, for all t ∈ U, α(t) ≥ d -ε. Using Lemma 25, there exist K > 0 and n 0 ∈ N such that, for all n ≥ n 0 and all j ∈ J n (U),

P j n ≥ Kn (d-ε)(β-ε)-1 ,
and Lemma 23 then implies that, almost surely,

lim inf n→+∞ log N ε n (β) log n ≥ (d -ε)(β -ε).
We thus have proved that, for all β ∈ (0, 1 d ) and all ε ∈ E β , almost surely,

(d -ε)(β -ε) ≤ lim inf n→+∞ log N ε n (β) log n ≤ dβ + dε. (14)
Then Lemma 15 ensures that almost surely, for all β ∈ (0, 1 d ), f g (β) = dβ.

Proof of Lemma 30. We obtain Inequality [START_REF] Falconer | Multifractional, multistable, and other processes with prescribed local form[END_REF] for β = 0 by applying Lemma 24 and Lemma 22. This yields that, almost surely,

lim inf n→+∞ log N ε n (0) log n ≤ dε.
As a consequence, f g (0) ≤ 0.

Then apply Lemma 14 and Lemma 29 to obtain Proof of Lemma 32. Let β ∈ ( 1 c , 1 + 1 c ). Denote E β = (0, min(β -1 c , 1 + 1 c -β)). Fix ε ∈ E β . By Lemma 26, there exist K > 0 and n 0 ∈ N such that, for all n ≥ n 0 and all j ∈ 1, n , P j n ≤ Kn 1 c +ε-β .

Lemma 22 then implies that, almost surely,

lim inf n→+∞ log N ε n (β) log n ≤ 1 + 1 c + ε -β. ( 15
)
Choose an open interval U such that 1 c -2ε ≤ 1 d U ≤ β -ε. By Lemma 27, there exist K > 0 and n 0 ∈ N such that, for alln ≥ n 0 and all j ∈ J n (U),

P j n ≥ Kn 1 d U +ε-β .
Lemma 23 then implies that, almost surely,

lim inf n→+∞ log N ε n (β) log n ≥ 1 + 1 d U + ε -β ≥ 1 + 1 c -ε -β.
We have proved that, for all β ∈ ( 1 c , 1 + 1 c ) and all ε ∈ E β , almost surely,

1 + 1 c -ε -β ≤ lim inf n→+∞ log N ε n (β) log n ≤ 1 + 1 c + ε -β. ( 16
)
The result then follows from Lemma 15.

Proof of Lemma 33. We obtain Inequality (15) for β = 1 + 1 c by applying Lemma 26 and Lemma 22. This yields that, almost surely, (1 -P j n ).

1 0(

 1 g(t + sh) -g(t -))ds, and lim h→0 -

  ) holds}, Ω β = ε∈E β ∩Q Ω β,ε and Ω = β∈(-∞,0)∩Q Ω β . Note that Ω has probability 1, and consider ω ∈ Ω. Choose β < 0 and j ∈ N. Set β j =[βj] j . For j large enough, N1/j n (β) ≤ N 2/j n (β j ). In addition lim inf n→∞ log N 2/j n (β j ) log n = -∞. As a consequence, lim inf n→∞ log N 1/j n (β) log n = -∞ and, almost surely, for all β < 0, f g (β) = -∞. Proof of Lemma 29. Fix β ∈ (0, 1 d ). Denote E β = {ε ∈ (0, min(β, 1 d -β)) such that (dε)(β -ε) ∈ (0, 1)}. Choose ε ∈ E β . ByLemma 24, there exists K > 0 and n 0 ∈ N such that, for all n ≥ n 0 and all j ∈ 1, n , P j n ≤ Kn dβ+dε-1 . Lemma 22 then implies that, almost surely, lim inf n→+∞ log N ε n (β) log n ≤ dβ + dε.

≥

  Proof of Lemma 31.Let β ∈ ( 1 d , 1 c ) and ε ∈ (0, 1). Choose an open interval U such that β > 1 c U and β < 1 d U + 2ε.Lemma 27 then ensures that there exist K > 0 and n 0 ∈ N such that, for all n ≥ n 0 and all j ∈ J n (U), Kn -ε . By Lemma 23, for all β ∈ ( 1 d , 1 c ), almost surely,

  As a consequence, f g (1 + 1 c ) ≤ 0. Then, Lemma 14 and Lemma 32 lead tof g Proof of Lemma 34. Let β > 1 + 1 c and denote E β = (0, β -1 -1 c ). For ε ∈ E β ,

Proof of Lemma 23. Fix a < b. For all t < 0,

≤ e -tn 1+a j∈Jn(U )

(1 -P j n + P j n e t ).

When t < 0, the function p → 1 -p + pe t is decreasing. As a consequence, by assumption on P j n and with p = Kn b , λ = n a , one has, for n large enough,

Minimizing over t < 0 and using Lemma 21, one gets

and thus

As in the proof of Lemma 22, this leads to ∀a < b, almost surely, lim inf

and finally, almost surely,

Estimates of P j n

For U an open interval, denote c U = inf t∈U α(t) and d U = sup t∈U α(t). Set also t j = j n .

Lemmas

Lemma 24. Assume β < 1 d . Then, ∃K > 0 , ∃n 0 ∈ N such that ∀n ≥ n 0 , ∀j ∈ 1, n ,

One computes:

There exist K 3 U ∈ R and n 0 ∈ N such that, for all n ≥ n 0 ,

As a consequence, lim n→+∞ ( sup

3.4 Estimates for the number of increments and determination of the spectrum

Lemmas

Lemma 28. Almost surely, ∀β < 0, f g (β) = -∞.

Lemma 29. Almost surely, ∀β ∈ (0, 1 d ), f g (β) = βd. Lemma 30. Almost surely, f g (0) = 0.

Proofs

Proof of Lemma 28. Fix β < 0.

Lemma 26 ensures that, for n large enough,

Thus, lim n→+∞ P(N ε n (β) ≥ 1) = 0, which implies that N ε n (β) tends to 0 in probability when n tends to infinity. There exists a subsequence σ(n) such that N ε σ(n) (β) tends to 0 almost surely. As a consequence, for all β > 1 + 1 c and all ε ∈ E β , almost surely,

We conclude as in the last part of the proof of Lemma 28.

Appendix

The following result is due to R. Schelling [START_REF] Schelling | [END_REF]:

Lemma 35. For all β > 0,

Proof. The Lévy-Khintchine formula yields

where c is a positive constant. By definition, ϕ is smooth and supported on[-1, 1], thus ϕ(1) = 0. As a consequence, we find that This is inequality [START_REF] Guével | Incremental moments and Hölder exponents of multifractional multistable processes[END_REF].

It is easy to see that the function β → F (β) is continuous. As a consequence, min δ∈[c,d] F (δ) > 0.