Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump

We consider a one-dimensional jumping Markov process $\{X^x_t\}_{t \geq 0}$, solving a Poisson-driven stochastic differential equation. We prove that the law of $X^x_t$ admits a smooth density for $t>0$, under some regularity and non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge, our result is the first one including the important case of a non-constant rate of jump. The main difficulty is that in such a case, the map $x \mapsto X^x_t$ is not smooth. This seems to make impossible the use of Malliavin calculus techniques. To overcome this problem, we introduce a new method, in which the propagation of the smoothness of the density is obtained by analytic arguments.


Introduction
Consider a R-valued Markov process with jumps {X x t } t≥0 , starting from x ∈ R, with generator L, defined for φ : R → R sufficiently smooth and y ∈ R, by Lφ(y) = b(y)φ ′ (y) + γ(y) for some functions γ, b : R → R with γ nonnegative, for some measurable space G endowed with a nonnegative measure q, and some function h : R × G → R.
Roughly, b(y) is the drift term: between t and t + dt, X x t moves from y to y + b(y)dt. Next, γ(y)q(dz) stands for the rate at which X x t jumps from y to y + h(y, z).
We aim to investigate the smoothness of the law of X x t for t > 0. Most of the known results are based on the use of some Malliavin calculus, i.e. on a sort of differential calculus with respect to the stochastic variable ω.
The first results in this direction were obtained by Bismut [4], see also Léandre [10]. Important results are due Bichteler et al. [2]. We refer to Graham-Méléard [9], Fournier [6] and Fournier-Giet [8] for relevant applications to physic integrodifferential equations such as the Boltzmann and the coagulation-fragmentation equations. These results concern the case where q(dz) is sufficiently smooth. When q is singular, Picard [12] obtained some results using some fine arguments relying on the affluence of small (possibly irregular) jumps. Denis [5] and more recently Bally [1] also obtained some regularity results when q is singular, using the drift and the density of the jump instants, see also Nourdin-Simon [11].
All the previously cited works apply only to the case where the rate of jump γ(y) is constant. The case where γ is non constant is much more delicate. The main reason for this is that in such a case, the map x → X x t cannot be regular (and even continuous). Indeed, if γ(x) < γ(y), and if q(G) = ∞, then it is clear that for all small t > 0, X y jumps infinitely more often than X x before t. The only available results with γ not constant seem to be those of [7,8], where only the existence of a density was proved. Bally [1] considers the case where γ(y)q(dz) is replaced by something like γ(y, z)q(dz), with sup y |γ(y, z) − 1| ∈ L 1 (q): the rate of jump is not constant, but this concerns only finitely many jumps.
From a physical point of view, the situation where γ is constant is quite particular. For example in the (nonlinear) Boltzmann equation, which describes the distribution of velocities in a gas, the rate of collision between two particles heavily depends on their relative velocity (except in the so-called Maxwellian case treated in [9,6]). In a fragmentation equation, describing the distribution of masses in a system of particles subjected to breakage, the rate at which a particle splits into smaller ones will clearly almost always depend on its mass... We will show here that when q is smooth enough, it is possible to obtain some regularity results in the spirit of [2]. Compared to [2], our result is • stronger, since we allow γ to be non-constant; • weaker, since we are not able, at the moment, to study the case of processes with infinite variations, and since we treat only the one-dimensional case (our method could also apply to multidimensional processes, but our non-degeneracy conditions would be very strong).
Our method relies on the following simple ideas: (a) we consider, for n ≥ 1, the first jump instant τ n of the Poisson measure driving X x , such that the corresponding mark Z n falls in a subset G n ⊂ G with q(G n ) ≃ n; (b) using some smoothness assumptions on q and h, we deduce that X x τn has a smooth density (less and less smooth as n tends to infinity); (c) we also show that smoothness propagates with time in some sense, so that X x t has a smooth density conditionnally to {t ≥ τ n }; (d) we conclude by choosing carefully n very large in such a way that {t ≥ τ n } occurs with sufficiently great probability.
As a conclusion, we obtain the smoothness of the density using only the regularizing property of one (well-chosen) jump. On the contrary, Bichteler et al. [2] were using the regularization of infinitely many jumps, which was possible using a sort of Malliavin calculus. Surprisingly, our non-degeneracy condition does not seem to be stronger, see Subsection 2.4 for a detailed comparison in a particular (but quite typical) example.
We present our results in Section 2, and we give the proofs in Sections 3 and 4. An Appendix lies at the end of the paper.

Results
In the whole paper, N = {1, 2, ...}. Consider the one-dimensional S.D.E. where We will require some smoothness of the coefficients. For f (y) : R → R (and h(y, z) : R × G → R), we will denote by f (l) (and h (l) ) the l-th derivative of f (resp. of h with respect to y). Below, k ∈ N and p ∈ [1, ∞) are fixed.
Assumption (A k,p ): The functions b : R → R and γ : R → R + are of class C k , with all their derivatives of order 0 to k bounded. The function h : R × G → R is measurable, and for each z ∈ G, y → h(y, z) is of class C k on R. There exists η ∈ (L 1 ∩ L p )(G, q) such that for all y ∈ R, all z ∈ G, all l ∈ {0, ..., k}, |h (l) (y, z)| ≤ η(z). (1), is well-defined for all φ ∈ C 1 (R) with a bounded derivative. The following result classically holds, see e.g. [7, Section 2] for the proof of a similar statement.
,x∈R is a strong Markov process with generator L defined by (1). We will denote by p(t, x, dy) := L(X x t ) its semi-group.

Propagation of smoothness
We consider the space M(R) of finite (signed) measures on R, and we abusively write ||f || L 1 (R) := ||f || T V = R |f |(dy) for f ∈ M(R). We denote by C k b (R) (resp. C k c (R)) the set of C k -functions with all their derivatives bounded (resp. compactly supported). We introduce, for k ≥ 1, the spaceW k,1 (R) of measures f ∈ M(R) such that for all l ∈ {1, ..., k}, there exists g l ∈ M(R) such that for all φ ∈ C k c (R) (and thus for all φ ∈ C k b (R)), If so, we set f (l) = g l . Classically, for f ∈ M(R), f ∈W k,1 (R) if and only if , and in such a case, Let us finally recall that • for f ∈ C k (R), f (y)dy belongs toW k,1 (R) if and only if k 0 |f (l) | ∈ L 1 (R); • if f ∈W k,1 (R), with k ≥ 2, then f (dy) has a density of class C k−2 (R).
We now introduce a first non-degeneracy assumption (here h ′ (y, z) = ∂ y h(y, z)).
Assumption (S): There exists c 0 > 0 such that for all z ∈ G, all y ∈ R, 1 + h ′ (y, z) ≥ c 0 . Proposition 2.2 Let p ≥ k + 1 ≥ 2 be fixed, assume (I), (A k+1,p ), and (S). For t ≥ 0 and a probability measure f on R, we define p(t, f, dy) on R by p(t, f, A) = R f (dx)p(t, x, A), where p(t, x, dy) was defined in Proposition 2.1. There is C k > 0 such that for all probability measures f ∈W k,1 (R), all t ≥ 0, Assumption (S) is probably far from optimal, but something in this spirit is needed: take b ≡ 0, γ ≡ 1 and h(y, z) = −y1 A (z) + yη(z) for some A ⊂ G with q(A) < ∞ and some η ∈ L 1 (G, q). Of course, (S) is not satisfied, and one easily checks that there exists τ A exponentially distributed (with parameter q(A)) such that a.s., for all t ≥ τ A , all x ∈ R, X x t = 0. This forbids the propagation of smoothness, since then p(t, f, dy) ≥ (1 − e −q(A)t )δ 0 (dy), even if f is smooth.

Regularization
We now give the non-degeneracy condition that will provide a smooth density to our process. A generic example of application (in the spirit of [2]) will be given below. For two nonnegative measures ν,ν on G, we say that ν ≤ν if for all A ∈ G, ν(A) ≤ν(A). Here k ∈ N, p ∈ [0, ∞) and θ > 0.
Assumption (H k,p,θ ): Consider the jump kernel µ(y, du) associated to our process, defined by µ(y, A) = γ(y) G 1 A (h(y, z))q(dz) (which may be infinite) for all A ∈ B(R). There exists a (measurable) family (µ n (y, du)) n≥1,y∈R of measures on R meeting the following points: (i) for n ≥ 1, y ∈ G, 0 ≤ µ n (y, du) ≤ µ(y, du) and µ n (y, R) ≥ n; (ii) for all r > 0, n ≥ 1, sup |y|≤r µ n (y, R) < ∞; (iii) there exists C > 0 such that for all n ∈ N, y ∈ R, The principle of this assumption is quite natural: it says that at any position y, our process will have sufficiently many jumps with a sufficiently smooth density. Our main result is the following.
For any x ∈ R, p(t, x, dy) has a density y → p(t, x, y) of class C n b (R) as soon as 0 ≤ n < kt/(θ + t) − 1.

Another assumption
It might seem strange to state our regularity assumptions with the help of γ, h, q, and to our nondegeneracy conditions with the help of the jump kernel µ. However, it seems to us to be the best way to give understandable assumptions.
Let us give some conditions on γ, h, q, in the spirit of [2], which imply (H k,p,θ ).
Assumption (B k,p,θ ): G = R, and for all y ∈ R, γ(y) > 0 and there exists I(y) = (a(y), ∞) (or (−∞, a(y))) with a(y) ∈ R, with y → a(y) measurable, such that q(dz) ≥ 1 I(y) (z)dz and such that the following conditions are fulfilled: This lemma is proved in the Appendix. Let us give some examples for (4).
Observe on these examples that there is a balance between the rate of jump γ and the regularization power of jumps (given, in some sense, by lowerbounds of |h ′ z |). The more the power of regularization is small, the more the rate of jump has to be bounded from below. This is quite natural and satisfying.
As a conclusion, we have slightly less technical assumptions. About the nondegeneracy assumption, it seems that the condition in [2] and ours are very similar (when γ ≡ 1). Let us insist on the fact that this is quite surprising: one could think that since we use only the regularization of one jump, our nondegeneracy condition should be much stronger than that of [2].
We could probably state an assumption as (B k,p,θ ) for a general lowerbound of the form q(dz) ≥ 1 O (z)ϕ(z)dz, for some open subset O of R and some C ∞ function ϕ : O → R, but this would be very technical.
Finally, it seems highly probable that one may assume, instead of (S), that However, the paper is technical enough.
We prove Theorem 2.3 in Section 3 and Proposition 2.2 in Section 4.

Smoothness of the density
In this section, we assume that Proposition 2.2 holds, and we give the proof of our main result. We refer to the introduction for the main ideas of the proof.
Step 1. We first introduce some well-chosen instants of jump that will provide a density to our process. To this end, we write N = i≥1 δ (ti,ui,zi) , we consider a family of i.i.d. random variables Next, we observe, using point (ii) above and (5), that a.s., for all t ≥ 0, We thus may consider, for each n ≥ 1, the a.s. positive (H t ) t≥0 -stopping time (b) U n ≤ γ(X x τn− ) a.s. by construction; (c) conditionnally to H τn− , Z n ∼ q n (X x τn− , dz)/q n (X x τn− , G). Indeed, the triple (U n , Z n , V n ) classically follows, conditionnally to H τn− , the distribution and it then suffices to integrate over u ∈ [0, ∞) and v ∈ [0, 1] and to use that d n (y, z)q(dz) = q n (y, dz).
Step 2. By construction and due to , Hence conditionnally to H τn− , the law of X x τn is g n (ω, dy) := µ n (X x τn− , dy − X x τn− )/µ n (X x τn− , R). Indeed, for any bounded measurable function φ : R → R, using Step 1-(c) and that µ n (y, A) = γ(y) G 1 A (h(y, z))q n (y, dz), Due to assumption (H k,p,θ ), we know that for some constant C, a.s., Step 3. We now use the strong Markov property. For t ≥ 0 and n ≥ 1, for φ : R → R, with the notation of Proposition 2.2, since {t ≥ τ n } ∈ H τn− , But from Proposition 2.2 and (6), there exists a constant C t,k such that a.s.
Hence (9) becomes, using Step 1-(a) and (5), Choosing for n the integer part of k θ+t log |ξ|, we obtain, for some constant A t , Since on the other hand |p t,x (ξ)| is clearly bounded by 1, we deduce that Let finally n ≥ 0 such that n < kt θ+t − 1, which is possible if t > θ k−1 . Then (10) ensures us that |ξ| n |p t,x (ξ)| belongs to L 1 (R, dξ), which classically implies that p(t, x, dy) has a density of class C n b (R).

Propagation of smoothness
It remains to prove Proposition 2.2. It is very technical, but the principle is quite simple: we study the Fokker-Planck integro-partial-differential equation associated with our process, and show that if the initial condition is smooth, so is the solution for all times, in the sense ofW k,1 (R) spaces.
In the whole section, K is a constant whose value may change from line to line, and which depends only on k and on the bounds of the coefficients assumed in assumptions (A k+1,p ) and (S).
For functions f (y) : R → R, g(t, y) : [0, ∞) × R → R, h(y, z) : R × G → R, we will always denote by f (l) , g (l) , and h (l) the l-th derivative of f , g, h with respect to the variable y.
We consider for i ≥ 1 the approximation L i of L, recall (1), defined for all bounded and measurable φ : R → R by Here, (G i ) i≥1 is an increasing sequence of subsets of G such that ∪ i≥1 G i = G and such that for each i ≥ 1, q(G i ) < ∞.
Furthermore, f i (t) is a probability measure for all t ≥ 0.
(ii) Assume now that f i (dy) goes weakly to some probability measure f (dy) as i tends to infinity. Then for all t ≥ 0, f i (t, dy) tends weakly to p(t, f, dy) as i tends to infinity, where we use the notation of Proposition 2.2.
Proof Let us first prove the uniqueness part. We observe that for φ bounded and measurable, L i φ is also measurable and satisfies ||L i φ|| ∞ ≤ C i ||φ|| ∞ , where C i := 2i + 2||γ|| ∞ q(G i ). Hence for two solutions f i (t, dy) andf i (t, dy) to (11), an immediate computation leads us to since the total variation norm satisfies ||ν|| T V := sup ||φ||∞≤1 | R φ(y)ν(dy)|. The uniqueness of the solution to (11) follows from the Gronwall Lemma. Let us consider X 0 ∼ f independent of N , and (X x t ) t≥0,x∈R the solution to (2), associated to the Poisson measure N . Recall that p(t, f, dy) = L(X X0 t ). We introduce another Poisson measure M i (ds) on [0, ∞) with intensity measure ids, independent of N , and X i 0 ∼ f i , independent of (M i , N ). Let (X i t ) t≥0 be the (clearly unique) solution to Then one immediately checks that f i (t, dy) = L(X i t ) solves (11). This shows the existence of a solution to (11), and that this solution consists of a family of probability measures. Finally, we use the Skorokhod representation Theorem: we build X i 0 ∼ f i in such a way that X i 0 tends a.s. to X 0 . Then one easily proves that sup [0,t] |X i s − X X0 s | tends to 0 in probability, for all t ≥ 0, using repeatedly (A 1,1 ). We refer to [8, Step 1 page 653] for a similar proof. This of course implies that for all t ≥ 0, f i (t, dy) = L(X i t ) tends weakly to p(t, f, dy) = L(X X0 t ).
Point (iii). Let thus φ and g as in the statement. Then where we used the substitution y → τ i (y) (resp. y → τ (y, z)) in the first (resp. third) integral.
The following technical lemma shows that when starting with a smooth initial condition, the solution of (11) remains smooth for all times (not uniformly in i). This will enable us to handle rigorous computations. , and the associated solution f i (t, dy) to (11). Then for all t ≥ 0, f i (t, dy) has a density f i (t, y), and (t, y) → f i (t, y) belongs to Proof We will prove, using a Picard iteration, that (23) (with l = 0) admits a solution, which also solves (11), which is regular, and of which the derivatives solve (23). We omit the fixed subscript i ≥ i 0 in this part of the proof, and the initial probability measure f (dy) = f (y)dy with f ∈ C k (R) is fixed.
We now use (18) (with φ = δ n (s, .)) and (15) (with φ = γδ n (s, .)), and we easily obtain, since q(G i ) < ∞, for some constant C k,i , for all y ∈ R, the last inequality holding since l ≤ k and γ ∈ C k b (R). Taking now the supremum over y ∈ R and suming for l = 0, ..., k, we get the desired inequality.
The central part of this section consists of the following result.
We finally conclude the Proof of Proposition 2.2. We thus assume (I), (A k+1,p ) for some p ≥ k + 1 ≥ 2, and (S). Consider a probability measure f ∈W k,1 (R), and a sequence of probability measures f i ∈W k,1 (R) with densities f i ∈ C k b (R), such that f i goes weakly to f , and such that lim i ||f i ||W k,1 (R) = ||f ||W k,1 (R) . Consider the unique solution f i (t, y) to (11). Then we deduce from Proposition 4.4 that for t ≥ 0, ||f i (t, .)||W k,1 (R) ≤ ||f i ||W k,1 (R) e C k t .
The proof is finished.

Appendix
We first gather some formulae about derivatives of composed and inverse functions from R into itself. Here f (l) stands for the l-th derivative of f . Let us recall the Faa di Bruno formula. Let l ≥ 1 be fixed. The exist some coefficients a l,r i1,...,ir > 0 such that for φ : R → R and τ : R → R of class C l (R), where the sum is taken over i 1 ≥ 1, ..., i r ≥ 1 with i 1 + ... + i r = l.