Criterion of unlimited growth of critical multidimensional stochastic models

A criterion for transience of multidimensional branching random walk in random environment

We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that \emph{Condition L} of Comets and Popov is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally.

On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes

Let $X$ be a $n$-dimensional Ornstein-Uhlenbeck process, solution of the S.D.E. $$\d X_t = AX_t \d t + \d B_t$$ where $A$ is a real $n\times n$ matrix and $B$ a Lévy process without Gaussian part. We show that when $A$ is non-singular, the law of $X_1$ is absolutely continuous in $\r^n$ if and only if the jumping measure of $B$ fulfils a certain geometric condition with respect to $A,$ which we call the exhaustion property. This optimal criterion is much weaker than for the background driving Lévy process $B$, which might be very singular and sometimes even have a one-dimensional discrete jump…

Small deviations in p-variation norm for multidimensional Levy processes

Let Z be an Rd-valued Levy process with strong finite p-variation for some p<2. We prove that the ''decompensated'' process Y obtained from Z by annihilating its generalized drift has a small deviations property in p-variation. This property means that the null function belongs to the support of the law of Y with respect to the p-variation distance. Thanks to the continuity results of T. J. Lyons/D. R. E. Williams, this allows us to prove a support theorem with respect to the p-Skorohod distance for canonical SDE driven by Z without any assumption on Z, improving the results of H. Kunita. We a…

临界情况下多维仿射递归的递归性

Recurrence of multidimensional affine recursions in the critical case

研究临界情况下多维仿射递归的递归性，通过不同自然假设，涵盖多种矩阵类型，基于“反向范数控制随机变量”的矩假设得出递归准则，证明了该递归在临界情况是递归的。

Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions

We study the Taylor expansion for the solution of a differential equation driven by a multidimensional Holder path with exponent β> 1/2. We derive a convergence criterion that enables us to write the solution as an infinite sum of iterated integrals on a nonempty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst parameter H > 1\2. We also prove that by using L_2 estimates of iterated integrals, the criterion and the speed of convergence for the stochastic Taylor expansion can be improved using Borel-Cantelli type…