alpha-Wiener bridges: singularity of induced measures and sample path properties

General alpha-Wiener bridges

An alpha-Wiener bridge is a one-parameter generalization of the usual Wiener bridge, where the parameter alpha>0 represents a mean reversion force to zero. We generalize the notion of alpha-Wiener bridges to continuous functions $α:[0,T)\to R$. We show that if the limit $\lim_{t\uparrow T}α(t)$ exists and is positive, then a general alpha-Wiener bridge is in fact a bridge in the sense that it converges to 0 at time T with probability one. Further, under the condition $\lim_{t\uparrow T}α(t)\ne 1$ we show that the law of the general alpha-Wiener bridge can not coincide with the law of any non t…

Karhunen-Loeve expansions of alpha-Wiener bridges

We study Karhunen-Loeve expansions of the process $(X_t^{(α)})_{t\in[0,T)}$ given by the stochastic differential equation $dX_t^{(α)} = -\fracα{T-t} X_t^{(α)} dt+ dB_t,$ $t\in[0,T),$ with an initial condition $X_0^{(α)}=0,$ where $α>0,$ $T\in(0,\infty)$ and $(B_t)_{t\geq 0}$ is a standard Wiener process. This process is called an $α$-Wiener bridge or a scaled Brownian bridge, and in the special case of $α=1$ the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of $X^{(α)}$. As applications, we calculate the Laplace transform and the distribution function of the…

Weiner型准则用于热方程基本奇点的可移除性及其后果

Wiener-type Criterion for the Removability of the Fundamental Singularity for the Heat Equation and its Consequences

本文提出新的Weiner型准则，用于任意开集在奇点处的细拓扑薄度条件下对热方程基本奇点可移除性的几何刻画。

Large deviations for Hilbert space valued Wiener processes: a sequence space approach

Ciesielski's isomorphism between the space of alpha-Hölder continuous functions and the space of bounded sequences is used to give an alternative proof of the large deviation principle for Wiener processes with values in Hilbert space.

Hitting distributions of alpha-stable processes via path censoring and self-similarity

In this paper we return to the problem of Blumenthal-Getoor-Ray, published in 1961, which gave the law of the position of first entry of a symmetric alpha-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional alpha-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between alpha-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of…

Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation

We consider a class of Gibbs measures defined with respect to increments $\{ω(t)-ω(s)\}_{s<t}$ of $d$-dimensional Wiener measure, with the underlying Hamiltonian carrying interactions of the form $H(t-s,ω(t)-ω(s))$ that are invariant under uniform translations of paths. In such interactions we allow {\it{long-range}} dependence in the time variable (including power law decay up to $t\mapsto (1+t)^{-(2+\varepsilon)}$ for $\varepsilon>0$) and unbounded (singular) interactions (including singularities of the form $x\mapsto 1/|x|^p$ in $d\geq 3$ or $x\mapsto δ_0(x)$ in $d=1$) attached to the space…