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

Let P0 denote the Wiener measure defined on the canonical space ( Ω = C(R+,R), (Xt)t≥0, (Ft)t≥0 ) , and (St) (resp. (It)), be the one sided-maximum (resp. minimum), (L 0 t ) the local time at 0, and (Dt) the number of down-crossings from b to a (with b > a). Let f : R×Rd −→]0,+∞[ be a Borel function, and (At) be a process chosen within the set : { (St); (St, t); (L 0 t ); (St, It, L 0 t ); (Dt) } , which consists of 5 elements. We prove a penalization result : under some suitable assumptions on f , there exists a positive ( (Ft), P0 ) -martingale (M t ), starting at 1, such that : lim t→∞ E0 [ 1Γsf(Xt, At) ] E0[f(Xt, At)] = Qf0 (Γs) := E0 [ 1ΓsM f s ] , ∀Γs ∈ Fs and s ≥ 0. (0.1) We determine the law of (Xt) under the p.m. Q f 0 defined on ( Ω,F∞ ) by (0.1). For the 1, 3 and 5 elements of the set, we prove first that Qf0(A∞ <∞) = 1, and more generally Qf0 (0 < g <∞) = 1 where g = sup{s > 0, As = A∞} (with the convention sup ∅ = 0). Secondly, we split the trajectory of (Xt) in two parts : (Xt)0≤t≤g and (Xt+g)t≥0, and we describe their laws under Q f 0 , conditionally on A∞. For the 2 and 4 elements, a similar result holds replacing A∞ by resp. S∞, S∞ ∨ I∞.