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Densité en temps petit d’un processus de sauts. (Density in small times of a jump process). (French) Zbl 0616.60078
Sémin. probabilités XXI, Lect. Notes Math. 1247, 81-99 (1987).
[For the entire collection see Zbl 0606.00022.]
Under certain conditions, it can be proved using Malliavin calculus that the law of a jump process \(x_ t(x)\) starting from x has got a smooth density \(p_ t(x,y)\). Denoting by g(x,y-x) the density of the Lévy- measure of \(x_ t(x)\), we show that \(p_ t(x,y)\sim tg(x,y-x)\) when \(t\to 0\). That means that the best way in small time to go from x to y is to jump from x to y in only one jump (it is rather natural). As a main tool, we use Malliavin calculus for jump processes.

MSC:
60J75 Jump processes (MSC2010)
60F10 Large deviations
60J35 Transition functions, generators and resolvents
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