Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach

Abstract

In this paper, we continue our understanding of the stable process from the perspective of the theory of self-similar Markov processes in the spirit of [10,13]. In particular, we turn our attention to the case of d-dimensional isotropic stable process, for d ≥ 2. Using a completely new approach we consider the distribution of the point of closest reach. This leads us to a number of other substantial new results for this class of stable processes. We engage with a new radial excursion theory, never before used, from which we develop the classical Blumenthal-Getoor-Ray identities for first entry/exit into a ball, cf. [3], to the setting of n-tuple laws. We identify explicitly the stationary distribution of the stable process when reflected in its running radial supremum. Moreover, we provide a representation of the Wiener-Hopf factorisation of the MAP that underlies the stable process through the Lamperti-Kiu transform.where the two terms either side of the multiplication sign constitute the two Wiener-Hopf factors. See e.g. Chapter VI in [2] for background. Recall that if Ψ is the characteristic exponent of any Lévy process, then there exist two Bernstein functions κ andκ (see [18] for a definition) such that, up to a multiplicative constant,Identity (1.7) is what we refer to as the Wiener-Hopf factorisation. The left-hand factor codes the range of the running maximum and the right-hand factor codes the range of the running 2 infimum of ξ. It can be checked that both belong to the class of so-called beta subordinators (see [8], as well as some of the discussion later in this paper) and, in particular, have infinite activity. This implies that ξ is regular for both the upper and lower half-lines, which in turn, means that any sphere of radius r > 0 is regular for both its interior and exterior for X. This and the fact that X has càdlàg paths ensures that, denotingthe quantity X G(t) is well defined as the point of closest reach to the origin up to time t in the sense that X G(t)− = X G(t) andThe process (G(t), t ≥ 0) is monotone increasing and hence there is no problem defining G(∞) = lim t→∞ G(t) almost surely. Moreover, as X is transient in the sense of (1.3), it is also clear that, almost surely, G(∞) = G(t) for all t sufficiently large and thatOur first main result provides explicitly the law of X G(∞) .Theorem 1.1 (Point of Closest Reach to the origin). The law of the point of closest reach to the origin is given by P x (X G(∞) ∈ dy) = π −d/2 Γ (d/2) 2 Γ ((d − α)/2) Γ (α/2) (|x| 2 − |y| 2 ) α/2 |x − y| d |y| α dy, 0 < |y| < |x|.

Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls

“…as the distribution of the furthest reach from the origin immediately before exit time admits a density by Theorem 1.3-(ii) of Kyprianou et al (2020). Similarly, letting B c (0, R) := R d \ B(0, R), we have…”

“…As we are dealing with continuous viscosity solutions, we need to ensure that the random variable H(T x ) is sufficiently integrable, so that the expected value E[H(T x )] is a continuous function of x ∈ R d , see Lemma 3.3. For this, in Lemma 3.2 we show, using results of Kyprianou et al (2020), that the exit time of a stable process from the ball B(0, R) is almost surely continuous with respect to its initial condition x ∈ B(0, R). This allows us to show the uniform integrability of (H(T x )) x∈B(0,R) using the fractional laplacian ∆ s = −(−∆) s and its associated stable process.…”

Fluctuation theory for Lévy processes with completely monotone jumps

“…Stable Lévy processes, which are prime examples of Lévy processes with completely monotone jumps, have been studied in this context since the pioneering work of Darling [15]; other classical references are [6,16,30]. For a sample of recent works on that subject, see [4,13,19,22,23,31,44,45,49,50,51,62,67]. Related fluctuation identities for more general classes of Lévy processes are provided, for example, in [18,20,29,43,46,52,61]; see also the references therein.…”

Probabilistic representations of solutions of nonlinear PDEs

“…as the distribution of the furthest reach from the origin immediately before exit time admits a density by Theorem 1.3-piiq of [52]. Similarly, letting s B c p0, Rq :" R d z s Bp0, Rq, we have…”

“…As we are dealing with continuous viscosity solutions, we need to ensure that the random variable HpT x q is sufficiently integrable, so that the expected value ErHpT x qs is a continuous function of x P R d , see Lemma 3.3.3. For this, in Lemma 3.3.2 we show, using results of [52], that the exit time of a stable process from the ball Bp0, Rq is almost surely continuous with respect to its initial condition x P Bp0, Rq. This allows us to show the uniform integrability of pHpT x qq xPBp0,Rq using the fractional laplacian Δ s " ´p´Δq s and its associated stable process.…”