论文标题
将两个本地时间的联合分布用于扩散过程，并应用于构建各种条件过程
Joint distribution of two Local Times for diffusion processes with the application to the construction of various conditioned processes
论文作者
论文摘要
对于扩散过程$ x（t）$ drift $μ（x）$和扩散系数$ d = 1/2 $，我们研究了两个局部时间的联合分布$ a（t）= \ int_ {0}^{t}dτδ（x（τ）） $ x = 0 $和$ x = l $，以及其总和$σ（t）= a（t）+b（t）$的简单统计信息。他们在很大时间$ t \至 + \ infty $的渐近统计数据涉及两个非常不同的情况：（i）当扩散过程$ x（t）$是短暂的，两个局部时间$ [a（t）; b（t）; b（t）$仍然是有限的随机变量$ [a^*（\ infty），b^*（\ infty），b^*（\ infty）$，我们分配了他们的限制性分配; （ii）当扩散过程$ x（t）$经常性时，我们描述了两个密集的局部时间的较大偏差属性$ a = \ frac {a（t）} {t} $和$ b = \ frac {b（t）} {t）{t）{t} $ untgendion sum $ $ $ $ $ $ $ $σ= = \ frac b $ b $ b $ b $然后，这些属性用于构建各种条件过程$ [x^*（t），a^*（t），b^*（t）] $满足涉及两个当地时间的某些约束，从而概括了我们先前的工作[arxiv：2205.15818]，这些工作涉及与单个局部时间$ a（t）$ a（t）$有关的条件。特别是对于无限的时间范围$ t \ to +\ infty $，我们考虑对有限的渐近值$ [a^*（\ infty），b^*（\ infty）] $或$σ^*（\ infty）$的条件，以及针对密集的条件，以及plucigation $ [a^a^*a^a^*$''''关于当地时代的生成功能，在大偏差的制度中。然后将这种通用结构应用于最简单的情况下，无条件扩散是均匀漂移$μ$的布朗运动。
For a diffusion process $X(t)$ of drift $μ(x)$ and of diffusion coefficient $D=1/2$, we study the joint distribution of the two local times $A(t)= \int_{0}^{t} dτδ(X(τ)) $ and $B(t)= \int_{0}^{t} dτδ(X(τ)-L) $ at positions $x=0$ and $x=L$, as well as the simpler statistics of their sum $ Σ(t)=A(t)+B(t)$. Their asymptotic statistics for large time $t \to + \infty$ involves two very different cases : (i) when the diffusion process $X(t)$ is transient, the two local times $[A(t);B(t)]$ remain finite random variables $[A^*(\infty),B^*(\infty)]$ and we analyze their limiting joint distribution ; (ii) when the diffusion process $X(t)$ is recurrent, we describe the large deviations properties of the two intensive local times $a = \frac{A(t)}{t}$ and $b = \frac{B(t)}{t}$ and of their intensive sum $σ= \frac{Σ(t)}{t}=a+b$. These properties are then used to construct various conditioned processes $[X^*(t),A^*(t),B^*(t)]$ satisfying certain constraints involving the two local times, thereby generalizing our previous work [arXiv:2205.15818] concerning the conditioning with respect to a single local time $A(t)$. In particular for the infinite time horizon $T \to +\infty$, we consider the conditioning towards the finite asymptotic values $[A^*(\infty),B^*(\infty)]$ or $Σ^*(\infty) $, as well as the conditioning towards the intensive values $[a^*,b^*] $ or $σ^*$, that can be compared with the appropriate 'canonical conditioning' based on the generating function of the local times in the regime of large deviations. This general construction is then applied to the simplest case where the unconditioned diffusion is the Brownian motion of uniform drift $μ$.
