Asymptotic behavior of the transition density for jump type processes in small time (Q1345464)

Let \(x_ t\) be the Markov process of pure jump type which satisfies the following stochastic differential equation \[ x_ t(x) = x + \sum_{s \leq t} \gamma \bigl( x_{s-} (x),\;\Delta z(s) \bigr), \] where \(z(t)\) is an \(R^ d\)-valued Lévy process of pure jump type with the Lévy measure \(h(d \zeta)\), \(\Delta z(t) = Z(t) - z(t-)\) and \(\gamma(x,\zeta)\) is a nondegenerate bounded function from \(R^ d \times R^ d\) to \(R^ d\). Under some assumptions on the Lévy measure \(h(d \zeta)\), this paper provides an estimate on the density \(p_ t (x,y)\) of the Markov process \(x_ t\) when the time parameter is small. More precisely, it is proven that as \(t \to 0\), \[ p_ t (x,y) \sim C \bigl( x,y, \alpha (x,y) \bigr) t^{\alpha (x,y)}, \] where \(\alpha (x,y)\) can be interpreted roughly as the minimum number of jumps by which the trajectory can reach \(y\) from \(x\) \((x \neq y)\). This result is a refinement of an earlier results by \textit{R. Léandre} [Sémin. probabilités XXI, Lect. Notes Math. 1247, 81-99 (1987; Zbl 0616.60078)]. This paper makes use of the theory of Malliavin calculus of jump type and the technique of Léandre's paper.