Local moduli of continuity for permanental processes that are zero at zero
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913231052734464 |
|---|---|
| author | Marcus, Michael B. Rosen, Jay |
| author_facet | Marcus, Michael B. Rosen, Jay |
| contents | Let $u(s,t)$ be a continuous potential density of a symmetric Lévy process or diffusion with state space $T$ killed at $T_{0}$, the first hitting time of $0$, or at $λ\wedge T_{0}$, where $λ$ is an independent exponential time. Let
\[
f(t)=\int_{T} u(t,v)\,dμ(v), \]
where $μ$ is a finite positive measure on $T$. Let $X_α=\{X_α(t),t\in T \}$ be an $α-$permanental process with kernel
\[
v(s,t)=u(s,t)+f(t). \]
Then when $\lim_{t\to 0}u(t,t)=0$, \[
\limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{a.s.} \] and
\[
\limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\le 1+C_{u,h} ,\qquad \text{a.s.} \] where $C_{u,μ}\le |μ|$ is a constant that depends on both $u$ and $μ$, which is given explicitly, and is different in the different examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local moduli of continuity for permanental processes that are zero at zero Marcus, Michael B. Rosen, Jay Probability 60E07, 60G15, 60G17, 60G99 Let $u(s,t)$ be a continuous potential density of a symmetric Lévy process or diffusion with state space $T$ killed at $T_{0}$, the first hitting time of $0$, or at $λ\wedge T_{0}$, where $λ$ is an independent exponential time. Let \[ f(t)=\int_{T} u(t,v)\,dμ(v), \] where $μ$ is a finite positive measure on $T$. Let $X_α=\{X_α(t),t\in T \}$ be an $α-$permanental process with kernel \[ v(s,t)=u(s,t)+f(t). \] Then when $\lim_{t\to 0}u(t,t)=0$, \[ \limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{a.s.} \] and \[ \limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\le 1+C_{u,h} ,\qquad \text{a.s.} \] where $C_{u,μ}\le |μ|$ is a constant that depends on both $u$ and $μ$, which is given explicitly, and is different in the different examples. |
| title | Local moduli of continuity for permanental processes that are zero at zero |
| topic | Probability 60E07, 60G15, 60G17, 60G99 |
| url | https://arxiv.org/abs/2402.07074 |