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0;T Il 2 / C g B C u 2 0 (11) 36 V. Ortiz-L´opez and M. ˝ Œ0; T /I H/. For any N > 0, we define HTN D fh 2 HT W khkHT Ä N g; N PH D fv 2 PH W v 2 HTN ; a:s:g; and we consider HTN endowed with the weak topology of HT . s; //; ek . 3 that this equation has a unique solution and that u";v 2 E˛ with ˛ 2 I. Consider the following conditions: (a) The set fV h ; h 2 HTN g is a compact subset of E˛ , where V h is the solution of (9). N PH that converges in distribution as " ! 0 in distribution, as E˛ -valued random variables.

The next section is devoted to the proof of Proposition 3. 3 Approximation of Xt and Integration by Parts Formula In order to bound the Fourier transform of the process Xt solution of (28), we will apply the differential calculus developed in Sect. 2. The first step consists in an approximation of Xt by a random variable XtN which can be viewed as an element of our basic space S 0 . 0; t/ is finite and consequently we may consider the random variable XtN as a function of these jump times and apply the methodology proposed in Sect.

We will follow the programme of Sect. s; /iH ds of (12) (with " D 1) that did not appear in [7]. This consists of the following steps. 3 in [7]. This refers to an approximation of the localized version of (12) on a light cone. In the approximating sequence, the fundamental solution G of the wave equation is replaced by a smoothed version Gn defined in (21). Going through the proof of that Proposition, we see that for the required extension, the term 44 V. Ortiz-L´opez and M. t // Á ; a where we have used the notation introduced in (17), (19).

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