Sistema de Clima

The literature concerning the time inversion property on Rn for n ≥ 2 is sparse. Unlike the

time inversion property in one dimension, it was not considered by Watanabe [1975] and was only explored eventually as a generalisation in Gallardo and Yor [2005]. This is probably the reason that few examples of the time inversion property on Rn have been studied. However,

despite the literature being meagre, Gallardo and Yor [2005] and Lawi [2008] were able to show that, under some minor conditions on the Markov process, the semigroup density derived onR

also serves as a necessary and sufficient condition for the time inversion property to hold onRn.

This is covered in much greater detail in Section 1.2. The semigroup density derived by Lawi, Gallardo et al. also permitted them to list several examples of processes with the time inversion property on Rn, but up to this point the list was restricted to Brownian motion, generalised

Dunkl processes and the Wishart processes.

In a parallel direction, Vuolle-Apiala [2012] also studied processes with the time inversion property onRn. However, Vuolle-Apiala only considered processes with almost sure continuous

paths inRn(a subset of the processes considered by Gallardo and Yor [2005]) that were polar at

the origin. In this restricted class, he was able to show that the restriction of rotation invariance, given by

(RI) Rt under Px has the same finite dimensional distributions as T−1(Rt) underPT(x) for all

rotationsT ∈O(n),

combined with 2-self-similarity were sufficient restrictions on a diffusion inRn to mean that it

was guaranteed to enjoy the time inversion property. To clarify, we note that in (RI) we take all rotations that are also a member of the set O(n) - the set of orthogonal matrices onRn. This

led to the possibility of several more classes of processes with the time inversion property onC

by taking the skew product representation, see Section 1.6, with a Bessel process as the radial part guaranteeing self-similarity.

In this chapter, we extend the work of Gallardo and Yor [2005] and Lawi [2008] to go some way to completely determining the class of all processes satisfying Lawi’s restrictions that enjoy the time inversion property onRn.

This chapter is laid out as follows. In Section 3.2, we review the time inversion property on Rn and recall the bijective change of coordinates to n-spherical notation which lends itself

more readily to the self-similar property; a key property with strong links to time inversion. As an extension of the work in the area of continuous paths by Watanabe [1975] and Vuolle-Apiala [2012], we consider the jumps associated with processes enjoying the time inversion property. Consequently, Section 3.3 explicitly determines the jumps that are permitted by a process with the time inversion property, given in Definition 4, under the assumptions (H1-3). We then look at a relationship between processes with the time inversion property and the Bessel process in Section 3.4, proving that 2ρ(R·) is a squared Bessel process. The results of these sections culminates in a characterisation of processes with the time inversion property in terms of the infinitesimal generator in Section 3.5. Finally, Section 3.6 considers a subset of processes with the time inversion property and investigates a link between these processes and the skew product representation that will be the subject of Chapter 4.

3.2

Preliminaries and Notation

In this section, we first state some assumptions that we use throughout the chapter alongside some preliminary material. This includes a review of the Lawi semigroup density for a process enjoying the time inversion property together with the assumptions we use to make this a necessary and sufficient condition up to anh-transform. We also review the spherical coordinates notation inn-dimensions, which is more compatible with the self-similar property than Cartesian coordinates.

For the remainder of this chapter, we use the definition of the time inversion property given in Definition 4 and follow the restrictions laid down by Lawi to define the semigroup density, but extended to n dimensions. Let R := {(Rt : t > 0),Px} be a Feller process on a

state space S = {_Rn _{for some n≥ 2} that satisfies the assumptions (H1), (H2’) and (H3) of}

the previous chapter extended to the state space Rn. Cones which are strict subsets of Rn or Rn\ {0} are excluded from this study.

In addition to these analogous assumptions, we also make the additional assumption on the functionρ(x):

(H4) The function ρ in (3.1) is continuous and positive for all x ∈_Rn_{\ {0} vanishing only at}

the origin.

On account of this, we now use the notation (H1-4’) to refer to assumptions (H1), (H2’), (H3) and (H4).

We also assume that this process is on the probability space (Ω,F,P) and generates

the right continuous filtration FR t

to invoke the work of Gallardo and Yor [2005] and Lawi [2008]. Expressly, this means that the process R given above satisfies the time inversion property of degree one if and only if its semigroup density pt(x, y) is given by

pt(x, y) = 1 tn2 Φ x t12 , y t12 θ y t12 exp −ρ x t12 −ρ y t12 (3.1) for the same restrictions on the functions given by (2.3), (2.4) and (2.5) in the previous chapter, or it is inh-transform with a such a process. However, we also assume that β >1−n in (2.4)

θ(λy) =λβθ(y) (3.2) and so we omit any processes with β <1−n orh-transforms of these. As these processes did not satisfy (H3) on Rin Chapter 2, we do not feel that this is too strong an assumption.

Initially, at least, we only burden ourselves with processes that have semigroup densities of this form since the h-transforms of Feller processes are well understood and for more detail we refer the reader to Doob [1957] and [Revuz and Yor, 2005, Chapter VIII.3].

The Spherical Coordinates Notation

The self-similar property and many of the restrictions on the semigroup density in Lawi [2008], given by (2.3), (2.4) and (2.5) are with respect to a scalar variable (λ > 0) and are therefore challenging to apply to Cartesian coordinates, which do not generally satisfy the scalar prop- erties in higher dimensions. For this reason, in the sequel, we would like to move to spherical coordinates so we recall this change of variables here.

We state our notation here, but for more detail we refer the reader to Appendix B. For any point y∈_Rn_{, in spherical notation we refer to this asy=r}

yg(φ_y) whereg:Rn−1 →Sn−1

and Sn−1 _{is the (n−1)-dimensional sphere on}

Rn. The function g gives the angular part of

the process. Moreover, the bijective nature of the spherical coordinates construction allows us to make an integral substitution, which we refer to as a function h satisfying dy1. . .dyn =

rn−1h(φ

y)drdφy. The construction of g and h are also described more fully in Appendix B.