TRANSIENCE AND NON-EXPLOSION OF CERTAIN STOCHASTIC NEWTONIAN SYSTEMS

: We give su–cient conditions for non-explosion and transience of the solution ( x t ; p t ) (in dimensions > 3) to a stochastic Newtonian system of the form ( where f » t g t > 0 is a d -dimensional L¶evy process, d» t is an It^o diﬁerential and c 2 C 2 ( R d ; R d ), V 2 C 2 ( R d ; R ) such that V > 0.


Introduction
This work contributes to the series of papers [13,15], [3,4], [6], [20] and [19] which are devoted to the qualitative study of the Newton equations driven by random noise. For related results see also [5], [23], [26,27], [1], [22] and the references given there. Newton equations of this type are interesting in their own right: as models for the dynamics of particles moving in random media (cf. [25]), in the theory of interacting particles (cf. [28], [29]) or in the theory of random matrices (cf. [24]), to mention but a few. On the other hand, the study of these equations fits nicely into the the larger context of (stochastic) partial differential equations, in particular Hamilton-Jacobi, heat and Schrödinger equations, driven by random noise (see [32,33] and [14,16,17,18]).
In most papers on this subject the driving stochastic process is a diffusion process with continuous sample paths, usually a standard Wiener process. Motivated by the recent growth of interest in Lévy processes, which can be observed both in mathematics literature and in applications, the present authors started in [20] and [19] the analysis of Newton systems driven by jump processes, in particular symmetric stable Lévy processes. In [20] we studied the rate of escape of a "free" particle driven by a stable Lévy process and its applications to the scattering theory of a system describing a particle driven by a stable noise and a (deterministic) external force.
In this paper we study non-explosion and transience of Newton systems of the form where ξ t = (ξ 1 t , . . . , ξ d t ) is a d-dimensional Lévy process, c ∈ C 2 (R d , R d ), V ∈ C 2 (R d ), V 0 and ∂c(xt) ∂x dξ t i := d j=1 ∂c i (xt) ∂x j dξ j t is an Itô stochastic differential. In Section 3 we give conditions under which the solutions do not explode in finite time. For symmetric α-stable driving processes ξ t = ξ (α) t we show in Section 4 that the solution process of the system (1) is always transient in dimensions d 3. We consider it as an interesting open problem to find necessary and sufficient conditions for transience and recurrence for the system (1) in dimensions d < 3. Even in the case of a driving Wiener process (white noise) only some partial results are available for d = 1, see [4,3].

Lévy Processes
The driving processes for our Newtonian system will be Lévy processes. Recall that a ddimensional Lévy process {ξ t } t 0 is a stochastic process with state space R d and independent and stationary increments; its paths t → ξ t are continuous in probability which amounts to saying that there are almost surely no fixed discontinuities. We can (and will) always choose a modification with càdlàg (i.e., right-continuous with finite left limits) paths and ξ 0 = 0. Unless otherwise stated, we will always consider the augmented natural filtration of {ξ t } t 0 which satisfies the "usual conditions". Because of the independent increment property the Fourier transform of the distribution of ξ t is of the form with the characteristic exponent ψ : R d → C which is given by the Lévy-Khinchine formula Here β ∈ R d , Q = (q ij ) ∈ R d×d is a positive semidefinite matrix and ν is a Lévy measure, i.e., a Radon measure on R d \ {0} with y =0 |y| 2 ∧ 1 ν(dy) < ∞. The Lévy-triple (β, Q, ν) can also be used to obtain the Lévy decomposition of ξ t , , is the canonical jump measure,Ñ (dy, ds) = N (dy, ds) − ν(dy) ds is the compensated jump measure, W Q t is a Brownian motion with covariance matrix Q and βt is a deterministic drift with β = E ξ 1 − s 1 ∆ξ s 1 {|∆ξs| 1} . Notice that the first two terms in the above decomposition (3) are martingales.
Lévy process whose jumps are bounded by 2R. Then where [ξ i , ξ j ] • denotes the quadratic (co)variation process.
This Lemma is a simple consequence of the well-known formula It is well-known that Lévy processes are Feller processes. The infinitesimal generator (A, D(A)) of the process (more precisely: of the associated Feller semigroup) is a pseudo-differential oper- where u(η) denotes the Fourier transform of u. The test functions C ∞ c (R d ) are an operator core. Later on, we will also use the following simple fact.
uniformly for all x ∈ R d with an absolute constant C ψ,u .
Since u ∈ C ∞ c (R d ), u is a rapidly decreasing function which means that the integral in the last line is finite.
Our standard references for the analytic theory of Lévy and Feller processes is the book [10] by Jacob, see also [11]; for stochastic calculus of semimartingales and stochastic differential equations we use Protter [30].

Non-explosion
Let (X t , P t ) = (X(t, x 0 , p 0 ), P (t, x 0 , p 0 )) be a solution of the system (1) with initial condition 0 and ∂c/∂x is uniformly bounded. Clearly, these conditions ensure local (i.e., for small times) existence and uniqueness of the solution, see e.g., [30].
The random times are stopping times w.r.t. the (augmented) natural filtration of the Lévy process {ξ t } t 0 and so is the explosion time T ∞ := sup m T m of the system (1).
Step 1. Let τ m := inf{s 0 : |P s | m} and τ ∞ := sup m τ m . It is clear that T m τ m and so T ∞ τ ∞ . Suppose that T ∞ (ω) < t < τ m (ω) τ ∞ (ω) for some t > 0 and m ∈ N. From the first equation in (1) we deduce that for every k ∈ N sup This, however, leads to a contradiction, and so τ ∞ = T ∞ .
Step 2. We will show that P The first equation in (1), dX t = P t dt, implies that X t is a continuous function; the second equation Let σ R := inf{t > 0 : |ξ t | R} be the first exit time of the process {ξ t } t 0 from the ball B R (0).
is again a stopping time and we calculate from (6) that = I + II.
Step 3. Recall that −ψ(D) is the generator of the Lévy process ξ t . We want to estimate is in the domain of the generator of ξ t , we find that is an L 2 -martingale (w.r.t. the natural filtration of {ξ t } t 0 ). The stopped process (M φ R t∧τm∧ ) t 0 is again an L 2 -martingale for fixed m, ∈ N. We can now use (8) and (9) to get we find for every t > 0 is a martingale (cf. [30], p.66 Corollary 3) and we may apply optional stopping to the bounded stopping time σ to get Therefore where we used Step 4. For the estimate of E(I ), we use Lemma 2 with u = φ to get ψ(D)φ R ∞ C ψ,φ , and also σ , so Put together, the estimates (10), (11) give Step 5. We proceed with |E(II)|. From Since we have sup s t |ξ s | R for t < σ R , the jumps |∆ξ s |, s t, cannot exceed 2R. Lemma 1 then shows Step 6. Combining (7), (12), (13) we obtain On the other hand, by Jensen's inequality, Clearly, |P τm | m and, since on {s < σ R } the driving Lévy process has jumps of size |∆ξ s | 2R, we find from (1) that Choosing m sufficiently large, say m > 2R (∂c/∂x) ∞ , we arrive at We can now combine (14) and (15) to find Letting first m → ∞ and then R → ∞ shows P(τ ∞ ) = 0 for all ∈ N, so P(τ ∞ = ∞) = 1, and the claim follows.

Transience
We will now prove that the solution {(X t , P t )} t 0 of the Newton system (1) is transient, at least if the driving noise is a symmetric stable Lévy process ξ t = ξ We restrict ourselves to presenting this particular case, but it is clear that, with minor alterations, the proof of Theorem 6 below remains valid for any driving Lévy process with rotationally symmetric Lévy measure.
Our proof is be based on the following result which extends a well-known transience criterion for diffusion processes to jump processes, see for instance [8] or [21].
Denote by {T t } t 0 the operator semigroup associated with a stochastic process and let (A, D(A)) be its generator. The full generator is the set  and full generator A. Let D ⊂ R n be a bounded Borel set and assume that there exists a sequence {u k } k∈N ⊂ C b (R n ) and some function u ∈ C(R n ), such that the following conditions are satisfied: (i) A has an extensionÃ such thatÃu k is pointwise defined, (u k ,Ãu k ) ∈ A and lim k→∞ (u k ,Ãu k ) = (u,Ãu) exists locally uniformly. (ii) u 0 and inf D u > a > 0 for some a > 0.
and from an optional stopping argument we find for any fixed T > 0 On the other hand, and because of assumption (i) we can pass to the limit k → ∞ to get where we used in the penultimate step thatÃu D c 0.
We will now turn to the task to determine the infinitesimal generator of the solution process {(X t , P t )} t 0 . The following result is, in various settings, common knowledge. We could not find a precise reference in our situation, though. Since we need some technical details of the proof, we include the standard argument.
Proof. For u = u(x, p) ∈ C 2 c (R d × R d ) we can use Itô's formula (for jump processes, now in the usual form [30, p. 70, Theorem II.32]) and get with a similar calculation to the one made in the proof of Theorem 3 Here we used the fact that the continuous martingale part of ξ t is W Q t , and so [ξ, ξ] c t = [W Q , W Q ] t = Qt. Note that we suppressed arguments in those places where no ambiguity is possible. Since P s = P s− + ∆P s = P s− − ∂c ∂x ∆ξ s we find, using the Lévy decomposition (3), The function u has compact support, and we may take expectations on both sides of the above relation and differentiate in t. Since the terms driven byÑ (dy, ds) or dW Q s are martingales, we find which is what we claimed. Notice, that the convergence is pointwise, so that it is not clear is in the domain of the generator. However, our calculation shows that which means that (u, Au) is in the full generator A.
If the driving Lévy process has no drift, no Brownian part and a rotationally symmetric Lévy measure, the form of the infinitesimal generator becomes much simpler. In this case we have for where v.p. R d f (y) ν(dy) := lim ε→0 |y|>ε f (y) ν(dy) stands for the principal value integral. It is not hard to see that v.p.
holds. The latter two representations do exist in the sense of ordinary integrals (just use a simple Taylor expansion for u up to order two) and are frequently used in the literature. For our purposes, formula (17) is better suited. Notice that all three representations extend A onto C 2 .
Proof. We want to apply Lemma 4. Take the function Moreover, we have Since {ξ t } t 0 is a symmetric α-stable process, its Lévy measure is of the form ν(dy) = c α |y| −d−α dy with c α given by (16), and (17) shows that We will see in Corollary 9 below (with B = ∂c/∂x and b = 2(V (x) − V 0 )) that we can choose γ > 0 in such a way thatÃu γ (x, p) 0. This, however, means that also condition (iv) of Lemma 4 is met.
Let χ k ∈ C ∞ c (R d ) be a cut-off function with 1 B k (0) χ k 1 B 2k (0) and set u k (x, p) := u γ (x, p)χ k (x)χ k (p). Clearly, u k ∈ C 2 c (R d × R d ) and we know from Lemma 5 that the pair (u k , Au k ) is in the full generator A. The following considerations are close to those in [31]. Write g A = g1 A ∞ . Using a Taylor expansion we find for some 0 < θ < 1 and all f ∈ and, therefore, for all compact sets K ⊂ R d and (x, Since the estimate of the local part in (17) is obvious, we find , for any f ∈ C 2 (R d × R d ) with f K×R d < ∞ and with an absolute constant C = C(K, c, V ) depending only on K, ∂c/∂x K and ∂V /∂x K . Since p → u γ (x, p) vanishes at infinity, condition (i) of Lemma 4 is satisfied for the sequence (u k , Au k ) → (u γ ,Ãu γ ).
The theorem follows now directly from Lemma 4.

Appendix
We will now give the somewhat technical proof that for some γ > 0 the function u γ (x, p) = Recall that Euler's Beta function B(x, y) is given by and satisfies the relations B(x, y) = B(y, x) and B(x, y) = x + y y B(x, y + 1), cf. Gradshteyn and Ryzhik [9, §8.38]. A change of variable in (18) according to t = s 2 yields Proof. We observe that J(v) = J(−v) and Therefore, we may assume that a = 1 and v 0. Since J(0) = ln(a) = 0, it is enough to show that J(v) is increasing. This is clear for v 1 since v → v 2 + 2vs + 1 increases for all parameter values |s| 1. For 0 < v < 1 we calculate the derivative In the case d = 3 a few lines of simple calculations give which is clearly positive. If d > 3, we use the symmetry of the measure (1 − s 2 ) d−3 2 ds and find since 2v(v 2 + 1) −1 1. The integrand can be written as Using (19) we find for all dimensions d 4 and so It is now straightforward to check that Lemma 8. Let d 3, 0 < α < 2. There exists some γ = γ(α, d) > 0 such that v.p.
holds for all p ∈ R d , λ ∈ R.
Proof. With the reasoning following Lemma 5 it is clear that the integral (21) exists. Without loss of generality we may assume that λ = 1. Denote the left-hand side of (21) by I(γ). Changing to polar coordinates we get (in the sense of an improper integral at the lower limit 0+) where Write Z(r) = |p| −2γZ (r) and observe that with v = r/|p| An application of Lemma 7 with a = 1 + |p| −2 implies and therefore Since I(0) = 0, the claim follows.
Proof. An argument similar to the one used in the proof of Lemma 8 shows that the integral (22) is well-defined for every γ > 0. Since v.p.
where we used Lemma 8 again.
Case 3: rank B = k, 1 < k < d. In this case we can find an orthogonal matrix S ∈ R d×d such that B = S B 0 0 0 S T whereB ∈ R k×k has full rank. Since the measure |y| −d−α dy is invariant under orthogonal transformations we can assume that B is already of the form B 0 0 0 ; otherwise we would make a change of variables in (22) with p = Sp in place of p. Write y = (y 1 , y 2 ) ∈ R k × R d−k , p = (p 1 , p 2 ) ∈ R k × R d−k and set b = 1 + |p 2 | 2 . Then v.p.
where we used the change of variables |y 1 |η 2 = y 2 in the last step. Since B has full rank, the claim follows from case 2.