Entrance and exit at infinity for stable jump diffusions
Entrance and exit at infinity for stable jump diffusions
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DOI：
10.1214/19-aop1389
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发表时间：
2018-02
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通讯作者：
L. Doering;A. Kyprianou
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文献类型：
--
作者：
L. Doering;A. Kyprianou
在 20 世纪 50 年代的开创性工作中，William Feller 根据其进入边界（费勒爆炸测试）和从边界进入内部的能力，对 $-\infty\leq a<b\leq \infty$ 上的所有一维扩散进行了分类。 Feller 的技术仅限于扩散过程，因为相应的微分生成器允许显式计算和 Hille-Yosida 理论的使用。在本文中，我们研究从无穷大的退出和进入，以获得最自然的概括，即由 $\alpha\in (0,2)$ 的稳定 L\'evy 过程驱动的形式为 \[ dZ_t=\sigma(Z_{t-})\,dX_t, \] 的跳跃扩散。跳跃扩散的许多结果已经得到证明，采用了费勒工作后开发的各种技术，但无限边界的出口和入口长期以来一直保持开放。我们表明，跳跃的存在意味着在没有漂移的扩散环境中看不到的特征。对于 $\alpha\in (0,1)$ 来说，有限时间爆炸是可能的，而对于 $\alpha\in [1,2)$ 来说，从不同类型的无穷大进入是可能的。我们推导出 $\sigma$ 上的充分必要条件，以便 (i) 存在非爆炸解，并且 (ii) 相应的过渡半群延伸到“无穷大”处的入口点。我们的证明基于稳定过程路径变换的最新发展，通过 Lamperti-Kiu 表示和新的 Wiener-Hopf 分解用于其中的 L'evy 过程。这些论点汇集了使用 Riesz-Bogdan--\.Zak 变换、自相似马尔可夫过程的入口定律、L\'evy 过程的永积分和涨落理论的结果的原始和复杂的应用，这些在 SDE 环境中以前没有使用过，从而允许我们采用经典理论，例如 Hunt-Nagasawa 对偶性和 Getoor 的瞬态和递归表征。
In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on $-\infty\leq a<b\leq \infty$ in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form \[ dZ_t=\sigma(Z_{t-})\,dX_t, \] driven by stable L\'evy processes for $\alpha\in (0,2)$. Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller's work but exit and entrance from infinite boundaries has long remained open. We show that the presence of jumps implies features not seen in the diffusive setting without drift. Finite time explosion is possible for $\alpha\in (0,1)$, whereas entrance from different kinds of infinity is possible for $\alpha\in [1,2)$. We derive necessary and sufficient conditions on $\sigma$ so that (i) non-exploding solutions exist and (ii) the corresponding transition semigroup extends to an entrance point at `infinity'. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti-Kiu representation and new Wiener-Hopf factorisations for L\'evy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz-Bogdan--\.Zak transformation, entrance laws for self-similar Markov processes, perpetual integrals of L\'evy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt-Nagasawa duality and Getoor's characterisation of transience and recurrence.