Stochastic equations of super-Lévy processes with general branching mechanism

Abstract

In this work, the process of distribution functions of a one-dimensional super-Lévy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time-space Gaussian white noises and Poisson random measures. This generalizes the recent work of Xiong (2013), where the result for a super-Brownian motion with binary branching mechanism was obtained.

Backward doubly stochastic equations with jumps and comparison theorems

“…In the sequel of this section, we will show that comparison theorem still hold for the 1/2-Hölder continuous case which is studied in He et al (2014).…”

“…Recently, some results about stochastic control problems of BDSDEs have been obtained by Han et al (2010) and Bahlali and Gherbal (2010). This work is motivated by Xiong (2013) and He et al (2014) which mainly studied the distribution function valued process of super-Brownian motions and super-Lévy processes characterized as the pathwise unique solution to a SPDE. For any super-Lévy process with transition semigroup (Q t ) t≥0 defined by (1.4) in He et al (2014), they proved that its distribution function valued process solves the following stochastic integral equation: for any x ∈ R,…”

“…As inHe et al (2014), we make the convention that the stochastic integral of a progressive process always refers to that of a predictable version of the integrand. Here we emphasis that the integrals in (2.2) denote by W T ( ← − ds, du),Ñ T 0 ( ← − ds, du) and N T 1 ( ← − ds, du) are backward ones, which can be defined as the time-reversal of the corresponding forward stochastic integrals; seeHe et al (2014) for more precise explanations. Of course, the integrals with respect to dB(s) and M (ds, du) in (2.2) are forward ones.…”

A class of super-Lévy processes in random environment

“…连续状态分枝过程的无穷维形式称为超过程, 是描述微观粒子在某个空间中繁衍和迁徙的随机过 程模型. He 等 [12] 证明了具有一般分枝机制的一维超 Lévy 过程的分布函数过程是如下随机偏微分方 程 (stochastic partial differential equation, SPDE) 的轨道唯一非负解:…”

Pathwise uniqueness for an SPDE with Hölder continuous coefficient driven by $\alpha $-stable noise

“…A more precise analysis on the regularity of the solution was given in Mytnik and Wachtel (2015) [20]. He et al (2014) [11] showed that another jump-type and (1.2) related SPDE on the distribution-function-valued process is pathwise unique. For p = 1, the uniqueness of solution (including the weak uniqueness) to SPDE (1.2) and the regularities of the solution X t (·) at a fixed time t are also left as open problems; see [19,Remark 5.9].…”