Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes

Abstract

Suppose that S is a closed set of the unit sphere $\mathbb {S}^{d-1} = \{x\in \mathbb {R}^d: |x| =1\}$ in dimension d ≥ 2, which has positive surface measure. We construct the law of absorption of an isotropic stable Lévy process in dimension d ≥ 2 conditioned to approach S continuously, allowing for the interior and exterior of $\mathbb {S}^{d-1}$ to be visited infinitely often. Additionally, we show that this process is in duality with the unconditioned stable Lévy process. We can replicate the aforementioned results by similar ones in the setting that S is replaced by D, a closed bounded subset of the hyperplane $\{x\in \mathbb {R}^d : (x, v) = 0\}$ with positive surface measure, where v is the unit orthogonal vector and where (⋅, ⋅) is the usual Euclidean inner product. Our results complement similar results of the authors [17] in which the stable process was further constrained to attract to and repel from S from either the exterior or the interior of the unit sphere.

Appendix: Hypergeometric Identities

We work with the standard definition for the hypergeometric function,

$$\displaystyle \begin{aligned} \,{{}_2}F_1(a,b,c; z) = \sum_{n = 0}^\infty\frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!}, \qquad |z|<1. \end{aligned}$$

Of the many identities for hypergeometric functions, we need the following:

(71)

for $c-a-b \notin \mathbb {Z}.$ Hence, thanks to continuity,

(72)

We will need to apply a similar identity to (71) but for the setting that c − a − b = 0, which violates the assumption behind (71). In that case, we need to appeal to the formula

$$\displaystyle \begin{aligned} \begin{array}{rcl} {{}_2}F_1 (a,b,a+b,z) & =&\displaystyle \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \Big( \sum_{k=0}^\infty \frac{(a)_k (b)_k}{(k!)^2} (2\psi(k+1)-\psi(a+k)-\psi(b+k)) (1-z)^k \notag \\ & -&\displaystyle \log(1-z)\, {{}_2}F_1 (a,b,1,1-z)\Big), \end{array} \end{aligned} $$

(73)

for |1 − z| < 1 where the di-gamma function ψ(z) =  Γ′(z)∕ Γ(z) is defined for all $z \ne -n, n \in \mathbb {N}$.

Again, thanks to continuity, we can write

(74)

A second identity that is needed is the following combination formula, which states that for any |z| < 1, we have

(75)

Its proof can be found, for example at [1]. In the main body of the text, we use this identity for the setting that a = α∕2, b = α and c = 1 + α∕2. This gives us the identity

$$\displaystyle \begin{aligned} {{}_2}F_1\left(\frac{\alpha}{2},\alpha;1+\frac{\alpha}{2};z\right) &= \frac{\Gamma(\alpha/2)\Gamma((2+\alpha)/2)}{ \Gamma(\alpha)} (-z)^{-\alpha/2} {{}_2}F_1\left(\alpha/2,0;1-\alpha/2;\frac{1}{z}\right) \notag \\ &\qquad \qquad + \frac{\Gamma(-\alpha/2)\Gamma((2+\alpha)/2)}{\Gamma((2-\alpha)/2)\Gamma(\alpha/2)} (-z)^{-\alpha} {{}_2}F_1\left(\alpha/2,\alpha;1+\alpha/2;\frac{1}{z}\right) \notag\\ &= \frac{\Gamma(\alpha/2)\Gamma((2+\alpha)/2)}{ \Gamma(\alpha)} (-z)^{-\alpha/2} \notag \\ &\qquad \qquad - (-z)^{-\alpha} {{}_2}F_1\left(\alpha/2,\alpha;1+\alpha/2;\frac{1}{z}\right) , \notag \end{aligned} $$

where we have used the recursion formula for gamma functions twice in the final equality. This allows us to come to rest at the following useful identity

$$\displaystyle \begin{aligned} (-{z})^{-\alpha/2} {{}_2}F_1\left(\alpha/2,\alpha;1+\alpha/2;\frac{1}{z}\right) + (-z)^{\alpha/2} {{}_2}F_1\left(\frac{\alpha}{2},\alpha;1+\frac{\alpha}{2};z\right) &= \frac{\Gamma(\alpha/2)\Gamma((2+\alpha)/2)}{ \Gamma(\alpha)}. {} \end{aligned} $$

(76)

We are also interested in integral formulae, for which the hypergeometric function is used to evaluate an integral. The first is aversion of formula 3.665(2) in [15] which states that, for any 0 < |a| < r and ν > 0, as

$$\displaystyle \begin{aligned} \int_0^{\pi} \frac{\sin^{d-2} \phi }{(a^2+2a r \cos \phi + r^2)^\nu} \mathrm{d}\phi= \frac{1}{r^{2\nu}} B\Big(\frac{d-1}{2}, \frac{1}{2}\Big) \,{{}_2}F_1 \Big(\nu, \nu-\frac{d}{2}+1; \frac{d}{2}; \frac{a^2}{r^2}\Big) , {} \end{aligned} $$

(77)

where B(a, b) =  Γ(a) Γ(b)∕ Γ(a + b) is the Beta function. The second is formula 3.197.8 in [15], which states that, for Re(μ) > 0, Re(ν) > 0 and $|\arg ({u}/{\beta })|<\pi$,

$$\displaystyle \begin{aligned} \int_0^u x^{\nu-1}(u-x)^{\mu-1} (x+\beta)^\lambda \mathrm{d} x = \beta^\lambda u^{\mu+\nu-1} B(\mu,\nu) {{}_2}F_1 \left(-\lambda,\nu;\mu+\nu;-\frac{u}{\beta}\right). {} \end{aligned} $$

(78)

The third is 3.194.1 of [15] and states that, for $|\arg (1+\beta u)|>\pi$ and Re(μ) > 0, Re(ν) > 0,

$$\displaystyle \begin{aligned} \int_0^u x^{\mu -1} (1+\beta x)^{-\nu}\mathrm{d} x = \frac{u^\mu}{\mu}{{}_2}F_1(\nu, \nu-\mu; 1+\mu; -\beta u), {} \end{aligned} $$

(79)

where ₂ F ₁ in the above identity is understood as its analytic extension in the event that |βu| > 1.