y − x + α z + D Moreover, the second derivative D z −

Proof of corollary 3.6 Thanks to proposition 3.4, D satisfies the assumptions of theorem 3.3. Thus equation 2 has a unique strong solution X with local time L and reflection direction n. Using proposition 3.4 again, there are non-unique coefficients c i ω, s ∈ [0, 1 β ] for each normal vector in the reflection term to be written as a convex combination : n ω, s = X ∂ D i ∋Xω,s c i ω, sn i Xω, s 3 Let us prove that there exists a measurable choice of the c i ’s : The map n resp. n i X is only defined for ω, s such that Xω, s ∈ ∂ D resp. Xω, s ∈ ∂ D i . We extend these maps by zero to obtain measurable maps on Ω × [0, T ] for an arbitary positive T . Note that equality 3 holds for the extended maps too. For each ω, s, we define the map f ω,s on R p by f ω,s c = | P p i=1 c i n i Xω, s − nω, s|. For a positive integer k, let R k = {0, 1 k , 2 k , . . . , 1 k ⌊ k β ⌋} p denote the 1 k -lattice on [0, 1 β ] p endowed with lexicographic order. The smaller point in R k for which f ω,s reaches its minimum value is c k ω, s = X c ∈R k c Y c ′ ∈R k ,c ′ 6=c 1I f ω,s c ′ f ω,s c + 1I f ω,s c ′ = f ω,s c1I c ′ c c k is a measurable map on Ω × [0, T ] and 3 implies that | f ω,s c k ω, s| ≤ p k . Taking coordi- nate after coordinate the limsup of the sequence of c k k , we obtain a measurable process c ∞ satisfying 3. Finally, let L i = Z · 1I ∂ D i Xsc i s dLs. L i has bounded variations on each [0, T ] since the c i ’s are bounded, and it is a local time in the sense of remark 2.1. Here, strong uniqueness holds for process X and for the reflection term p X i=1 Z t n i Xsd L i s. The uniqueness of this term does not imply uniqueness of the L i ’s, because the c i ’s are not unique. „ Proof of proposition 3.4 Let x ∈ ∂ D. Since D = T p i=1 D i , the set {i s.t. x ∈ ∂ D i } is not empty. Since D i satisfies U ES α i restricted to D, for each y ∈ D i y − x.n i x + 1 2 α i |y − x| 2 ≥ 0, and consequently : ∀i s.t. ∂ D i ∋ x ∀y ∈ D y − x.n i x + 1 2 min ∂ D j ∋x α j |y − x| 2 ≥ 0 Summing over i for non-negative c i ’s such that P ∂ D i ∋x c i ≤ 1 β we obtain : ∀y ∈ D X ∂ D j ∋x c i n i x.y − x + 1 2 β min ∂ D i ∋x α j |y − x| 2 ≥ 0 151 which implies that P ∂ D j ∋x c i n i x belongs to the set N

x,

α of normal vectors on the boundary of D, for α = β min ∂ D j ∋x α j . Thanks to remark 3.5, this proves the inclusion N ′ x :=    n ∈ S m , n = X ∂ D i ∋x c i n i x with c i ≥ 0    ⊂ N

x, α

Let us prove the converse inclusion. For i such that ∂ D i ∋ x, the boundary ∂ D i is at least C 2 in

D, hence there exist a closed ball Bx, η

x i with η x i 0, and a C 2 function f x i : B x, η x i −→ R with non-vanishing derivative, such that B x, η x i ∩ D i = Bx, η x i ∩ {y ∈ R m , f x i y ≥ 0} there even exists an orthonormal coordinate system in which f x i y is the difference between the last coordinate of y and a C 2 function of the other coordinates, but we do not need this precise form here. The Taylor-Lagrange formula provides : ∀z s.t. |z| ≤ η x i f x i x + z = f x i x + ∇ f x i x.z + 1 2

z.D

2 f x i z ∗ z for some z ∗ ∈ Bx, η x i depending on z. By definition of f x i , x ∈ ∂ D i implies f x i x = 0 and ∇ f x i x = |∇ f x i

x|n

i

x. Moreover, the second derivative D

2 f x i is continuous hence bounded on the closed ball by some constant ||D 2 f x i || ∞ : ∀z s.t. |z| ≤ η x i f x i x + z ≥ |∇ f x i

x|n

i

x.z −

||D 2 f x i || ∞ 2 |z| 2 For any ǫ 0 and for δ i ǫ x = min ‚ η x i , 2 |∇ f x i x| ||D 2 f x i || ∞ ǫ Œ we obtain : |z| ≤ δ i ǫ x and z.n i x ≥ ǫ|z| =⇒ f x i x + z ≥ 0 =⇒ x + z ∈ D i Consequently, for N ǫ = \ ∂ D i ∋x {z, n i x.z ≥ ǫ|z|} we have : {x + z, z ∈ N