Approximation of integrals over asymptotic sets with by Barbe P. PDF

By Barbe P.

This booklet is the 1st of a bigger undertaking that i could attempt to entire. A moment quantity could be dedicated to the asymptotic research of multivariate integrals over small wedges and their functions. a 3rd one may still expand the various result of the 1st volumes to the limitless dimensional environment, the place there are a few in all likelihood remarkable functions within the research of stochastic techniques.

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Use Fubini’s theorem and the change of variable u = v + c to obtain I[c,u] I(x) dx e−u du = e−c L(c) = e−c v 0 v 0 e−v |Γc+v \ Γc | dv e−v |Γc+v | dv . Since the function I is convex, nonnegative and tends to infinity with its argument, its graph x, I(x) : x ∈ Rd ⊂ Rd+1 is included in a cone with vertex the point (0, . . , 0, −1) and passing trough a ball centered at 0 in Rd × { 0 }. , |Γt | C1 (1 + t)d for some positive constant C1 . Thus, L(c) C1 e−c e−v (c + v + 1)d dv v 0 3d C1 e−c e−v (cd + v d + 1)dv v 0 2.

T∗ dt We now have all the informations we need on the normal flow. Its integral curves define a foliation of Rd \ { 0 }. We will obtain an expression for A e−I(x) dx by integrating first on ΛI(A) , and then along the leaves. The virtue of the normal flow is that computation of the Jacobian in the change of variables is easy, because the normal vector field and the level sets of the function I are orthogonal at every point of Rd . Before rewriting the integral, we need to parameterize the boundary ∂A in the flow coordinate system.

2. The normal flow and the normal foliation 39 uniformly in v in the unit sphere STp π−1 (p) (0, 1) of Np DA . 3) expp (tv) −1 (p), where µp is the Riemannian measure over the unit sphere of Tp πA centered at the origin. This last expression looks quite complicated. However, we are almost done, and the intuition goes as follows. Roughly, we want to choose DA as ΛI(A) ∩ ∂A, so that I is minimal in A over DA — but we will actually need to have a little bit of freedom for some applications and make a slightly more subtle choice.

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