On Tanaka formulae for \((\alpha, d,\beta)\)-superprocesses
This work treats \((\alpha, d, \beta)\)-superprocesses, and the aim of this paper is to establish Tanaka formulae for \(( \alpha, d, \beta)\)-superprocesses in dimensions where the local time exists, under the optimal initial condition. More precisely, let \({\mathcal M}(\mathbb R^d)\) be the set of Radon measures on \(\mathbb R^d\), and \({\mathcal M}_r(\mathbb R^d)\) the whole set of all elements \(\mu \in {\mathcal M}(\mathbb R^d)\) such that \(\langle \mu, \phi_r \rangle < \infty\) with \(\varphi_r(x)=( 1 + | x |^2)^{- r}\), \(x\in \mathbb R^d\) for \(r > d/2\). \({\mathcal B}_b\) denotes the family of all bounded Borel measurable functions on \(\mathbb R^d\). A càdlàg \({\mathcal M}_r(\mathbb R^d)\)-valued process \(X =( X_t)_{t \geq 0}\) is called an \(( \alpha, d, \beta)\)-superprocess if it is a time homogeneous and strongly Markov process such that for all \(\varphi, \psi\in {\mathcal B}_b^+\), \[ {\mathbb E} \left\{ \exp \left. \left( - \langle X_t, \varphi \rangle - \int_0^t \langle X_s, \psi \rangle \,ds \right) \,\right|\, X_0 = \mu \right\} = \exp ( - \langle \mu, V_t ( \varphi, \psi ) \rangle ), \tag{1} \] where \(\mu \in {\mathcal M}_r(\mathbb R^d)\), \(\langle \mu, f \rangle\) denotes integral of \(f\) over \(\mathbb R^d\) with respect to measure \(\mu\), and \(V_t(\varphi, \psi)\) is the unique non-negative solution of \[ v_t = S_t^{\alpha} \varphi + \int_0^t S_s^{\alpha} \psi\, ds - \int_0^t S_{t-s}^{\alpha}( (v_s)^{1 + \beta})\, ds, \quad t \geq 0; \tag{2} \] see, e.g., [\textit{I. Iscoe}, Probab. Theory Relat. Fields 71, 85--116 (1986; Zbl 0555.60034)]. If the weighted occupation time \(Y_t(dx):=\int_0^t X_s\, ds\), \(\forall t \geq 0\), is absolutely continuous with respect to Lebesgue measure on \(\mathbb R^d\), then the local time of \(X\) at \(x \in \mathbb R^d\) is defined as \[ L_t^x := \begin{cases}\lim\limits_{n \to \infty} \langle Y_t, \varphi_{n,x} \rangle, &\text{if it exists}, \\ 0, & \text{otherwise}, \end{cases}\tag{3} \] where \(\{ \varphi_{n,x} \}_{n \geq 1}\) is a sequence of non-negative functions on \(\mathbb R^d\) such that \(\varphi_{n,x}(y) dy\) converges weakly to the Dirac measure \(\delta_x\) at \(x\) as \(n \to \infty\). For the \(( \alpha, d, \beta)\)-superprocess \(X\) with \(X_0 \not\equiv 0\), only when \(d < \alpha + \alpha / \beta\), the local time \(( L_t^x)\) of \(X\) at \(x \in \mathbb R^d\) exists, and then its distribution does not depend on the choice of \(\{ \varphi_{n, x} \}_{n \geq 1}\), and for any \(t > 0\), \(\langle Y_t, \varphi_{n, x} \rangle\to L_t^x\) in probability as \(n \to \infty\). Moreover, when \(X_0\) is the Lebesgue measure or \(X_0(dx)\) \(=\) \(h(x) dx\) \(\not= 0\) with \(h \in {\mathcal B}_b^+\), then the \(( \alpha, d, \beta)\)-superprocess \(X\) with \(d < \alpha + \alpha / \beta\) has local time \(( L_t^x)_{t \geq 0}\) for any \(x \in \mathbb R^d\) [cf. \textit{K. Fleischmann}, Math.\ Nachr.\ 135, 131--147 (1988; Zbl 0655.60071) and \textit{E. B. Dynkin}, [Astérisque, 157-158, 147--171 (1988; Zbl 0659.60105); see also \textit{R. J. Adler}, and \textit{M. Lewin}, Stochastic Processes\ Appl.\ 41, No.~1, 45--67 (1992; Zbl 0754.60086) and \textit{S. M. Krone}, Ann.\ Probab.\ 21, No.~3, 1599--1623 (1993; Zbl 0778.60056)]. Let \(p_t^{\alpha}( \cdot)\) be the transition density function of an \(\alpha\)-stable process with generator \(\Delta_{\alpha}\) \(=\) \(- ( - \Delta)^{\alpha / 2}\). Then the author assumes: (A.1): For fixed \(x \in \mathbb R^d\), \(X_0 \in {\mathcal M}_r(\mathbb R^d) \setminus \{ 0 \}\) satisfies \[ \int_0^1 \int_{\mathbb R^d} p_s^{\alpha}(x-y)\, X_0 ( dy) \,ds < \infty.\tag{4} \] Then the local time \(L_t^x\) satisfies \[ {\mathbb E}_{\mu}^{\alpha, \beta} [ L_t^x] = \int_0^t \int_{\mathbb R^d} p_s^{\alpha} (x -y)\, \mu(dy) \,ds, \] and \(L_t^x\) is continuous in \(t\), \({\mathbb P}_{\mu}^{\alpha, \beta}\)-a.s. [cf. \textit{E. B. Dynkin}, and \textit{S. E. Kuznetsov}, Ann.\ Probab.\ 25, No.~2, 640--661 (1997; Zbl 0880.60079)], where \(\Omega = D_{ {\mathcal M}_r(\mathbb R^d) }\) is the space of càdlàg \({\mathcal M}_r(\mathbb R^d)\)-valued paths with Skorokhod topology, and \({\mathbb P}_{\mu}^{\alpha, \beta}\) denotes the law of \(( \alpha, d, \beta)\)-superprocess \(X\) on \(\Omega\) with \(X_0 = \mu\in {\mathcal M}_r(\mathbb R^d)\). Define \[ G_{\alpha}^{\lambda}(y) := \int_0^{\infty} \exp \{ - \lambda \tau \} p_t^{\alpha} (y)\, dt, \quad \forall \lambda > 0, \quad y \in \mathbb R^d. \] Then \((-\Delta_{\alpha} + \lambda) G_{\alpha}^{\lambda}( \cdot - y) = \delta_y\) holds in the distribution sense for all \(y \in \mathbb R^d\). The following is the main result of this paper, which asserts that the Tanaka formulae of \((\alpha, d, \beta)\)-superprocesses \(X\) hold for all \(t \in [0, \infty)\). Theorem. For \(0 < \beta \leqslant 1\), \(0 < \alpha \leqslant 2\) and \(d < \alpha + \alpha / \beta\), let \(X =( X_t)_{t \geq 0}\) be an \(( \alpha, d, \beta)\)-superprocess with \(X_0 = \mu\) satisfying (A.1). Then for any \(\lambda \in (0, \infty)\), \[ \begin{aligned} \langle X_t, G_{\alpha}^{\lambda} ( \cdot - x) \rangle = \langle X_0, G_{\alpha}^{\lambda} ( \cdot - x) \rangle & + \lambda \int_0^t \langle X_s, G_{\alpha}^{\lambda} ( \cdot - x) \rangle \,ds\tag{5}\\ + M_t( G_{\alpha}^{\lambda}( \cdot - x)) & - L_t^x, \quad \forall t \geq 0, \quad {\mathbb P}_{\mu}^{\alpha, \beta}-a.s.\end{aligned} \] holds, where \(M_t( G_{\alpha}^{\lambda}( \cdot - x))\) is a càdlàg \({\mathcal F}_t\)-measurable, and for any \(T \in [0, \infty)\) and any \(\theta \in [0, \beta)\), \[ \begin{aligned} {\mathbb E}_{\mu}^{\alpha, \beta} \left\{ \sup_{t \leqslant T} | \langle X_t, G_{\alpha}^{\lambda}( \cdot - x) \rangle |^{1 + \theta} \right\} < \infty, \qquad & {\mathbb E}_{\mu}^{\alpha, \beta} \left\{ \sup_{t \leqslant T} | M_t( G_{\alpha}^{\lambda}( \cdot - x)) |^{1 + \theta} \right\} < \infty, \\ \text{and} \qquad {\mathbb E}_{\mu}^{\alpha, \beta} \left\{ \sup_{ t \leqslant T} ( L_t^x)^{ 1 + \theta} \right\} & = {\mathbb E}_{\mu}^{\alpha, \beta} \left\{ (L_T^x)^{1 + \theta} \right\} < \infty \end{aligned} \] hold, where \(( {\mathcal F}_t )\) is a filtration given by \({\mathcal F}_t = \bigcap_{s > t} \sigma( X_u, u \leqslant s)\).
scientific article; zbMATH DE number 5157273