Fluctuation theory for spectrally positive additive L\'evy fields

A spectrally positive additive L\'evy field is a multidimensional field obtained as the sum $\mathbf{X}_{\rm t}={\rm X}^{(1)}_{t_1}+{\rm X}^{(2)}_{t_2}+\dots+{\rm X}^{(d)}_{t_d}$, ${\rm t}=(t_1,\dots,t_d)\in\mathbb{R}_+^d$, where ${\rm X}^{(j)}={}^t (X^{1,j},\dots,X^{d,j})$, $j=1,\dots,d$, are $d$ independent $\mathbb{R}^d$-valued L\'evy processes issued from 0, such that $X^{i,j}$ is non decreasing for $i\neq j$ and $X^{j,j}$ is spectrally positive. It can also be expressed as $\mathbf{X}_{\rm t}=\mathbb{X}_{\rm t}{\bf 1}$, where ${\bf 1}={}^t(1,1,\dots,1)$ and $\mathbb{X}_{\rm t}=(X^{i,j}_{t_j})_{1\leq i,j\leq d}$. The main interest of spaLf's lies in the Lamperti representation of multitype continuous state branching processes. In this work, we study the law of the first passage times $\mathbf{T}_{\rm r}$ of such fields at levels $-{\rm r}$, where ${\rm r}\in\mathbb{R}_+^d$. We prove that the field $\{(\mathbf{T}_{\rm r},\mathbb{X}_{\mathbf{T}_{\rm r}}),{\rm r}\in\mathbb{R}_+^d\}$ has stationary and independent increments and we describe its law in terms of this of the spaLf $\mathbf{X}$. In particular, the Laplace exponent of $(\mathbf{T}_{\rm r},\mathbb{X}_{\mathbf{T}_{\rm r}})$ solves a functional equation leaded by the Laplace exponent of $\mathbf{X}$. This equation extends in higher dimension a classical fluctuation identity satisfied by the Laplace exponents of the ladder processes. Then we give an expression of the distribution of $\{(\mathbf{T}_{\rm r},\mathbb{X}_{\mathbf{T}_{\rm r}}),{\rm r}\in\mathbb{R}_+^d\}$ in terms of the distribution of $\{\mathbb{X}_{\rm t},{\rm t}\in\mathbb{R}_+^d\}$ by the means of a Kemperman-type formula, well-known for spectrally positive L\'evy processes.