Local time and Tanaka formulae for super Brownian and super stable processes
Consider \(X=(X_ t, t\geq 0)\), a branching measure-valued process with initial measure \(X_ 0=\) Lebesgue measure on \(\mathbb{R}^ d\), such that the underlying spatial motion is that of a symmetric stable process of index \(\alpha\in(0,2]\) and the branching mechanism is in the domain of attraction of a stable law of index \(1+\beta\), \(\beta\in (0,1]\). The first part of the paper is concerned with integrability properties of \(X\). In particular, the authors prove that every function \(\Phi\in L^ p(\mathbb{R}^ d,X_ 0)\) for \(1\leq p<\infty\) is a.s. in \(L^ p(\mathbb{R}^ d,X_ t)\) for almost every \(t>0\). In the case \(\beta=1\), if \(\Phi\) is in the intersection of Sobolev spaces \(W^{2,2}\cap W^{2,1}\), then the real-valued process \(\langle\Phi,X_ t\rangle\) is a.s. continuous, and an Itô's formula is derived. In dimension \(d<2\alpha\) and for \(\beta=1\), the process \(X\) has local times \(L^ x_ t\). That is \[ L^ x_ t=\lim_{\varepsilon\to 0}\text{vol}(B_{\varepsilon,x})^{-1}\int^ t_ 0\langle\Phi_{\varepsilon,x},X_ s\rangle ds, \] where \(B_{\varepsilon,x}\) denotes the ball centered at \(x\) with radius \(\varepsilon\), \(\text{vol}(B_{\varepsilon,x})\) its volume and \(\Phi_{\varepsilon,x}\) its characteristic function. The preceding Itô's formula yields the following Tanaka's formula for the local time \(L^ 0_ t\): \[ L^ 0_ t=\langle G^ \lambda_ \alpha, X_ 0\rangle-\langle G^ \lambda_ \alpha, X_ t\rangle+\int^ t_ 0\langle G^ \lambda_ \alpha, X_ s\rangle ds+\int^ t_ 0\langle G^ \lambda_ \alpha,X^ \alpha(ds)\rangle. \] Here, \(G^ \lambda_ \alpha\) is the Green's function associated with the underlying stable motion, and \(X^ \alpha(\cdot,ds)\) is the martingale measure that appears in the semimartingale decomposition of \(X_ t\). This Tanaka's formula is then applied to derive the existence of a jointly continuous version of the local times \(L^ x_ t\).
scientific article; zbMATH DE number 90021
scientific article