Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II

Let us denote by P-0 the Weiner measure defined on the canonical space (Omega=C(R+,R), (X-t)(t >= 0), (F-t)(t >= 0)), and let (S-t) (resp. (I-t)), be the one-sided-maximum (resp. minimum), (L-t(0)) the local time at 0, and (D-t) the number of down-crossings from b to a (with b > a). Let f : RxRd -> ]0, +infinity[ be a Dorel function, and (A(t)) be a process chosen within the set {(St); (St, t); (Lt0); (Dt)}, which consists of 5 elements. We prove a penalization result: under suitable assumptions on f, there exists a positive ((F-t), P-0) - martingale (M-t(f)), starting at 1, such that [GRAPHICS] We determine the law of (Xt) under the p.m. Q(0)(f) defined on (Omega, F-infinity) by (0.1). For the 1(st), 3(rd), and 5(th) elements of the set, we prove first that Q(f)(0) (A(infinity) < infinity) = 1, and more generally Q(f)(0) (0 < g < infinity) = 1 where g = sup {(8) > 0, A(8) = A(infinity)} (with the convention sup 0 = 0). Secondly, we split the trajectory of (X-t) into two parts: (X-t)(0 <= t < g) and (Xt+g)(t >= 0), and we describe their laws under Q(0)(f), conditionally on A(infinity). For the 2nd and 4th elements, a similar result holds replacing A(infinity) by resp. S-infinity and S-infinity V I-infinity.