Time regularity of Lévy-type evolution in Hilbert spaces and of some α-stable processes

In this paper we consider the cylindrical càdlàg property of a solution to a linear equation in a Hilbert space H, driven by a Levy process taking values in a possibly larger Hilbert space U . In particular, we are interested in diagonal type processes, where processes on coordinates are functionals of independent α-stable symmetric processes. We give the equivalent characterization in this case. We apply the same techniques to obtain a sufficient condition for existence of a càdlàg version of stable processes described as integrals of deterministic functions with respect to symmetric α-stable random measures with α ∈ [1, 2).


Introduction
We first consider a linear equation in a Hilbert space H given by dX t = AX t dt, X 0 = x, where A is a generator of a C 0 semigroup (S(t)) t 0 on H. Obviously, the solution can be represented as X t = S(t)x. We may perturb the linear equation by a Lévy process Z = (Z t ) t∈T which takes values in U , where U is a Hilbert space H ⊂ U . It leads to the following equation of evolution with Lévy noise dX t = AX t dt + dZ t , t ∈ T = [0, a], X 0 = 0, a > 0.  If H ⊂ U then the solution exists in a weaker form which we discuss in a special case below. The equations of the form (1.1) were considered e.g. in [8], [10], [9], [6], [7]. One may wonder whether there exists a càdlàg version of X = (X t ) t∈T in H. This problem is described in Liu Zhai [6] and the answer is that if such a modification exists, then Z takes values in H. However, in some cases X may take values in H, even though the space, in which the noise lives, is larger. One may wonder what other kinds of regularity one may expect. In [9] several other notions of càdlàg property have been introduced, which are weaker than càdlàg in H. In the present paper we focus on the cylindrical càdlàg property. We consider the case when both Z and the equation (1.1) is of a diagonal form.
In the this paper we consider only the diagonal case with negative diagonal operator A and diagonal Lévy-type process Z, which is a much simpler question. Namely, let (e n ) ∞ n=1 be an orthonormal and complete basis in H, we assume that for any n = 1, 2, . . . vector e n belongs to the domain of A and Ae n = −γ n e n with γ n > 0. Moreover, assume that Z t = ∞ n=1 Z (n) t e n , where Z (n) are real-valued independent symmetric Lévy processes without Gaussian part and with Lévy measures µ n , respectively. Note that, in general the sum defining Z may not converge in H, but in some larger space Hilbert U . By the solution to the diagonal type evolution equation we mean the process The process X takes values in H if and only if the series ∞ n=1 (X (n) t ) 2 converges in probability (and therefore almost surely, thanks to independence). One can express this condition in terms of Lévy measures µ n (see Proposition 2.6 in [9]). An important example considered in literature is when Z (n) = σ n L (n) , where L (n) are independent standard symmetric α-stable Lévy processes and σ n 0. This will be referred to as the α-stable case. In this case, the condition for X to take values in H is ∞ n=1 σ α n 1 + γ n < ∞. (1.4) The proof of the fact can be also found in [10]. Note that in [10] it was assumed that γ n → ∞, hence condition (1.4) was written with γ n in the denominator, instead of our 1 + γ n , but the proof without the assumption that γ n tend to infinity is essentially the same (cf. Proposition 4.2 in [10]).
Moreover, by the Liu Zhai result [6] we know that in the α-stable case, there exists a càdlàg version of X = (X t ) t∈T in H if and only if Z takes values in H, which is equivalent to ∞ n=1 σ α n < ∞. (1.5) However, the intriguing situation is when (1.5) fails -which means that Z has values beyond the space H. The question is whether we can still expect some regularity of X. The regularity we analyze in this paper is the existence of a cylindrical càdlàg modification.
According to the Definition 1.1 in [9] an H-valued process X is cylindrical càdlàg if for any z ∈ H the real valued process has a càdlàg modification. Note that if X is a cylindrical càdlàg, then for any finite set of vectors z 1 , z 2 , . . . , z n ∈ H the process ( z 1 , X , z 2 , X , . . . , z n , X ) ECP 26 (2021), paper 42. has a càdlàg modification and hence the property indicates weak regularity of X as of a process in high dimensions. There are some partial results towards the question discussed in the extensive paper [9] (note that there are discussed many forms of regularity). However, the results in [9] do not completely cover even the basic question of Z (n) that are α-stable, where α ∈ (1, 2). We do propose an approach which in particular covers the question formulated as Question 4 in [9]. It should be mentioned that the case of α ∈ (0, 1] was completely solved in [7].
As it will be proved, our approach works in the much more general setting of diagonal type evolution equations implying a nice sufficient condition for the cylindrical càdlàg property for all diagonal type equations.
The process Y of (1.6) clearly depends on z ∈ H, however since z will be fixed we do not stress this dependence in the notation. Moreover, even though X does not take values in H, it is possible that the process Y is well defined and we may consider the problem of the existence of its càdlàg modification.
The key idea in the proof is to use the Poissonian representation of Lévy processes and an application of a result of [3] concerning suprema of Bernoulli processes. In this approach it is important that the Lévy processes are symmetric.
In the last part of the paper we show the usefulness of our method beyond the evolution equations. Namely, we give sufficient conditions for existence of càdlàg modifications of stable processes of the form where M is a symmetric α-stable random measure and f is a deterministic function satisfying appropriate integrability conditions. See Section 5 and Theorem 5.1 below. It is worth stressing that our condition also works in the case α ∈ (1, 2), which seems to be a difficult one. The paper is organized as follows. In Section 2 we introduce some notation and representations of the process Y given by (1.6). In Section 3 we discuss a necessary condition for existence of a càdlàg modification of the process Y . In Section 4 we provide a sufficient condition. Finally, in Section 5 we discuss the problem of càdlàg modification of stable processes of the form (1.7).

Representation of solution
For the sake of simplicity we assume that T = [0, 1]. As we have explained the solution to the evolution equation has the form (1.2). Suppose that Z (n) = σ n L (n) , where σ n 0 and L (n) , n = 1, 2, . . . are independent symmetric Lévy processes without Gaussian component and with Lévy measures ν n , respectively. That is, L (n) t has characteristic function of the form where ν n is a symmetric Borel measure on R, satisfying ν n ({0}) = 0 and R (y 2 ∧ 1)ν n (dy) < ∞.
Such processes have a càdlàg modification, and in the sequel we will always assume that L (n) , n = 1, 2, . . . are càdlàg. As described in the introduction we assume that A is a diagonal operator, and for an orthonormal basis (e n ) n of H we have Ae n = −γ n e n , with ECP 26 (2021), paper 42.
It is well known that the jump times and sizes of L (n) are points of a Poisson random measure, with intensity measure ⊗ ν n , where is the Lebesgue measure on R + . We denote this random measure by π n . Thus where the limit is a.s. Moreover, on a subsequence δ n 0 fast enough the convergence is a.s. uniform on bounded intervals (see e.g. Theorem 6.8 in [8]). Note that here we do not need to compensate, since ν n are symmetric. Also, due to symmetry π n can be represented as a sum of Dirac measures where (t n,k , y n,k ) are points of a Poisson random measure with intensity ⊗ µ n with µ n (B) = 2ν n (B ∩ R + ), which will be denoted here by π + n , andε n,k k = 1, 2, . . . are i.i.d. Rademacher random variables. In this setting the process L (n) at time t n,k has a jump of absolute value y n,k and signε n,k , i.e. ∆L (n) t n,k =ε n,k y n,k .
For n = 1, 2, . . . the corresponding Poisson random measures π + n and random signs are independent.
In this case it is well known that ν n (dy) = C α |y| α+1 dy.
We fix z ∈ H and consider existence of a càdlàg modification of where X (n) are given by (2.1), and Y t . Under a weak assumption the sum  k:y n,k δ b n ε n,k y n,k e −(t−t n,k )γn 1 t n,k t , (2.4) where b n = |σ n z, e n |, ε n,k =ε n,k sgn( z, e n ) and t n,k , y n,k ,ε n,k , n = 1, 2, . . ., i = 1, 2, . . . are as above.
and the function ψ is continuous at 0.
Proof. By (2.4), using the form of the characteristic function of integrals with respect to a Poisson random measure (see e.g. Theorem 6.6 in [8]) we have are independent, hence almost sure convergence of the series (2.3) is equivalent to its convergence in law and the result follows.
In particular, if L (n) are standard symmetric α-stable Lévy processes, then recalling x for x > 0 is bounded from above and below by 1 1+x multiplied by a constant, we see that the series (2.3) converges almost surely for any t It is clear that each of the processes Y (n) is càdlàg. Thus, using (2.4) we can use the following representation of Y The sum over k is understood as lim δ→0 k:y n,k δ .... We are ready to discuss the convergence of n Y (n) t , t ∈ T . The main idea we follow is that (Y t ) t∈T can be split into two parts according to whether b n y n,k 1 or b n y n,k < 1. The first part is a finite sum of càdlàg processes and in the second the series with respect to n, converges uniformly in L 1 , thus there is a subsequence on which the convergence is a.s. uniform on T , hence the limit is càdlàg.

Necessary condition
Recall (2.6) and (2.5). The next theorem provides a necessary condition for Y to have a càdlàg modification. This result follows from Theorem 3.4 of [7], but, as it is short, we will also present its proof, to have a full picture of our problem.   Proof of Theorem 3.1. We argue by contradiction. Suppose that (3.1) does not hold for some ε > 0 and that Y has a càdlàg modificationỸ . Fix any n and denote: Y (n,ε) t = k:y n,k ε b n ε n,k y n,k e −(t−t n,k )γn 1 t n,k t , t 0.
Then the processesỸ are càdlàg and they are independent (independence follows from the fact that π n is independently scattered). Moreover, Y (n,ε) has jumps at jump times of the Poisson process π n ([0, t] × {y : |y| ε}), t 0. Therefore, with probability one, the sample paths of the two processes defined in (3.3) must have jumps at different times. Hence, with probability one, whenever Y (n) has a jump of size ε, thenỸ has a jump of equal size and sign. Notice also, that where, for a càdlàg process Z we denote ∆Z s = Z s − Z s− .
We will show that if (3.1) does not hold then, with probability one, there are infinitely many n, such that L (n) has a jump of size ε/b n . Moreover, all L (n) are independent, hence they jump at different times. Consequently, by the argument above, this implies thatỸ must have an infinite number of jumps of size ε on [0, 1], and therefore cannot be càdlàg. This is a contradiction.
where the last equality is a consequence of the assumption of the opposite of (3.1). As ξ (n) are independent, the Borel Cantelli lemma implies that with probability 1 there are infinitely many n such that L (n) has a jump of size at least ε/b n .

Sufficient condition
We now discuss sufficient conditions for existence of càdlàg modification of Y .   Then Y has a càdlàg modification.
Before we go to the proof of the theorem we make several observations: ECP 26 (2021), paper 42. (|b n y| 2 ∧ 1)ν n (dy) < ∞ thus our result is stronger than Theorem 3.8 in [7], where |b n y| appeared with power 1 instead of the square.      Since clearly (4.4) implies that N D N = H, we derive from Baire's theorem that there must exist N and a ball B(z 0 , r) in H such that B(z, r) ⊂ D N . Consequently, sup z∈B(0,1) n | z, e n σ n | α < ∞.
On the other hand, considering equality in Hölder's inequality one can find a sequence If (4.3) fails, the latter expression tends to ∞, when N → ∞. This is contradiction.
Note that it is possible that (1.4) is satisfied and n σ α n = ∞ but (4.3) is satisfied. This means that in this case the process X is not H-càdlàg but it is cylindrically càdlàg, and for which the process Z of (1.1) does not have values in H.  Borel Cantelli lemma and the fact that each L (n) is càdlàg imply that there are only a finite number of y n,k such that b n y n,k ε.
Instead of Y it is therefore enough to consider the process where Y (n,ε) t = lim δ→0 i:δ≤y n,k <ε b n ε n,k y n,k e −γn(t−t n,k ) 1 t t n,k , by assumption (4.1). Therefore L (ε) is càdlàg.
The problem thus reduces to showing that has a càdlàg modification. We will show that with probability one the series in (4.7) converges a.s. in the supremum norm and in the topology J 1 . The property implies the existence of a càdlàg modification of the limit. Since we could not find the right reference we give a short proof below for the sake of completeness.  Then, for any t ∈ [0, 1] the process η t = ∞ n=1 η (n) t has a càdlàg modification. More precisely, ∞ n=1 η (n) converges a.s. in the Skorohod J 1 topology to someη which is the càdlàg modification of η. Moreover, the series ∞ n=1 η (n) also converges uniformly.
where Λ is the set of nondecreasing continuous functions from [0, 1] onto itself. It is known that d is a metric on D([0, 1]) inducing the Skorohod J 1 topology and such that the space D([0, 1]) with this metric is a Polish space (see [2]). Clearly, d(x, y) The space (D([0, 1]), d) is complete and that is why the series ∞ n=1 η (n) converges in probability in this space. By Theorem 1 [5] it also converges almost surely in the metric d to someη which is càdlàg. Moreover, a simple consequence of (4.8) is that η (n) ∞ converges in probability to 0 as n → ∞. Therefore, by Theorem 2 of [5], the series η = ∞ n=1 η (n) also converges a.s. in the uniform norm. Therefore, for any fixed t ∈ [0, 1] variables η t =η t a.s. It completes the proof.
We will prove the following lemma  By assumption (4.1) this implies the Cauchy condition for the series in (4.7). The proof of the theorem will be complete provided that we show Lemma 4.6, which we do presently.
Here P E is the conditional probability were we condition on all variables but (ε n,k ) n,k . Taking expectation, using the identity Eξ 2 = 2 ∞ 0 uP (|ξ| u)du and also symmetry we obtain: Letting δ → 0 we obtain (4.9).

Càdlàg modification of processes expressed as integrals with respect to symmetric stable random measures.
A large class of stable stochastic processes studied in literature are of the form where a > 0, M is an α-stable random measure defined on some measurable space (E, B) and f : [0, a] × E → R is a measurable function on the product space, satisfying appropriate integrability conditions. See e.g. [11] for a systematic treatment of stable integrals and stable processes. In this section we discuss a sufficient condition for the process of the form (5.1) to have a càdlàg modification (and hence for local boundedness of the process). Necessary and sufficient conditions for sample boundedness of processes of the form (5.1) in the case α < 1 are known. The case α > 1 seems to be more difficult (see Chapter 10 of [11]). Some more recent results on the càdlàg property of stable integrals of the form (5.1) can be found in [4] and [1]. It turns out that our methods used in the previous section can be applied also in this setting in case where M is a symmetric α-stable random measure. We assume that 0 < α < 2 and let m be a σ-finite measure on a measurable space (E, B). Let M denote a symmetric α-stable random measure on E with control measure m. That is, if we denote by E 0 := {A ∈ B : m(A) < ∞} then (M (A)) A∈E0 is a family of real valued random variables such that: (i) For any A 1 , A 2 , . . . ∈ E 0 such that A i ∩ A j = ∅ for i = j the random variables M (A 1 ), M (A 2 ), . . . are independent. Moreover, if we also have that m( a.s. Recall also, that M (A) and E f (t, x)M (dx) may be constructed using a Poisson random measure. Assume that π is a Poisson random measure on R × E with intensity measure C α |z| 1+α dzm(dx), where C α > 0 is chosen such that where the limit is in probability, and a.s.
If δ n 0 and E n ∈ B are such that m(E n ) < ∞, E n ⊂ E n+1 for all n and n E n = E, then for fixed t, the stable integral with respect to the stable random measure constructed above may be represented as  A simple, but key observation in our context is that since the Lévy measure cα |z| 1+α dz is symmetric, the Poisson random measure π may be written as π = k δ (ε k y k ,x k ) , (5.4) where π + = i δ (y k ,x k ) is a Poisson random measure with intensity measure 2cα y α+1 1 y>0 dym(dx) and ε 1 , ε 2 , . . . are i.i.d Rademacher random variables independent of π + .
We have the following theorem.  Then the process (X t ) t∈[0,a] defined by (5.1) has a càdlàg modification.
Remark 5.1. Assumptions of Theorem 5.1 essentially mean that for any x ∈ E\N the function t → f (t, x) is càdlàg and has finite variation on [0, a]. Moreover, this variation as a function of x is in L α (E, m).
Proof of Theorem 5.1. Let π be a Poisson random measure of the form (5.4) and let δ n and E n be as in (5.3). Note that π restricted to the set {|z| : z > δ n } × E n is such that the number of points (ε k y k , x k ) in this set is Poisson with parameter {y:y>δn} = An\Am y 2 f 2 (a, x) 2c α y α+1 dy m(dx) → 0.