arXiv subjects

Wei Sun

Publications and source records attributed to Wei Sun.

472 records · Page 27Linked to original sources

arXiv2010

On the generalized Feynman-Kac transformation for nearly symmetric Markov processes

Suppose $X$ is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on $L^{2}(E;m)$. For $u\in D(\mathcal{E})$, we have Fukushima's decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$. In this paper, we investigate the strong continuity of the generalized Feynman-Kac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$. Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$. Denote by $J_1$ the dissymmetric part of the jumping measure $J$ of $(\mathcal{E},D(\mathcal{E}))$. Under the assumption that $J_1$ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant $α_0\ge 0$ such that $\|P^{u}_{t}\|_2\leq e^{α_0 t}$ for every $t>0$. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on $L^{2}(E;m)$. If $X$ is equipped with a differential structure, then this result also holds without assuming that $J_1$ is finite.

math.PR↗

arXiv2009

Some Diffusion Processes Associated With Two Parameter Poisson-Dirichlet Distribution and Dirichlet Process

The two parameter Poisson-Dirichlet distribution $PD(α,θ)$ is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman's Poisson-Dirichlet distribution. The two parameter Dirichlet process $Π_{α,θ,ν_0}$ is the law of a pure atomic random measure with masses following the two parameter Poisson-Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures $PD(α,θ)$ and $Π_{α,θ,ν_0}$. The methods used come from the theory of Dirichlet forms.

math.PR↗

arXiv2006

Generalized Mehler Semigroups and Catalytic Branching Processes with Immigration

Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein-Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is established through fluctuation limits, providing a rich class of non-differentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein-Uhlenbeck processes in different topologies is also discussed.

math.PR↗

arXiv2006

Rates for branching particle approximations of continuous-discrete filters

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that $t\to X_t$ is a Markov process and we wish to calculate the measure-valued process $t\toμ_t(\cdot)\doteq P\{X_t\in \cdot|σ\{Y_{t_k}, t_k\leq t\}\}$, where $t_k=kε$ and $Y_{t_k}$ is a distorted, corrupted, partial observation of $X_{t_k}$. Then, one constructs a particle system with observation-dependent branching and $n$ initial particles whose empirical measure at time $t$, $μ_t^n$, closely approximates $μ_t$. Each particle evolves independently of the other particles according to the law of the signal between observation times $t_k$, and branches with small probability at an observation time. For filtering problems where $ε$ is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of $ε$. We analyze the algorithm on Lévy-stable signals and give rates of convergence for $E^{1/2}\{\|μ^n_t-μ_t\|_γ^2\}$, where $\Vert\cdot\Vert_γ$ is a Sobolev norm, as well as related convergence results.

math.PR↗