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3 below, we will use a smoothed version of the fundamental solution G defined as follows. x/dx D 1. t; x/ D . 0/. 2. t; x/. t; x/ 2 Œ0; T  R3 coincide. 1 in [7] for the path-wise t integral G;Z . 40 V. Ortiz-L´opez and M. 1. Fix q 2 3; 1Œ and a bounded domain O R3 . ˇ; ı/ ^ Œ. Ft /-adapted, mean-square continuous stochastic process. t s/ / ds: (25) Proof. 4. We recall that this proposition is devoted t to prove an analogue property for the stochastic integral vG;Z . 5 in [7]. Then, as in [7], (24) is obtained from (23) by applying Fatou’s lemma.

It was observed by Itˆo that the solution is a Brownian motion, that is the infinitesimal generator of the solutions is 12 . X i / vanishes. In [16], this observation is used to prove a Bismut type formula for differential forms related to gradient Brownian flow, in [18] to obtain an effective criterion for strong 1-completeness, and in [17] to obtain moment estimates for the derivative flow T t of the gradient SDEs. vt /dt: i D1 where ==t . / W T 0 M ! T t M denotes the stochastic parallel translation corresponding to the Levi-Civita connection along a path which is defined almost surely for almost all continuous paths.

Bally and E. Cl´ement where the last inequality follows from Gronwall lemma. s/ analogous to (39). XtN / solution of (30). We recall that F N D XtN and we will check that F N satisfies the hypotheses of Theorem 1. Lemma 4. (i) We assume that H1 holds. s; a/. 1En 1=˛/ 1 WD n and (16) holds. tl 1 ; tl /, that is ml 1, 8l 2f1; : : : ; Lg. Moreover, we have Ln f n > 0g. Proof. The proof of (i) is a straightforward consequence of Lemma 3, replacing PJtN c. c/ by pD1 p / and observing that Z tZ N Jt X pD1 c.

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