论文标题
根据原点处的本地时间进行调节扩散过程
Conditioning diffusion processes with respect to the local time at the origin
论文作者
论文摘要
当无条件的过程是一个扩散过程时，漂移$μ（x）$和扩散系数$ d = 1/2 $，当地时间$ a（t）= \ int_ {0}^{t}^{t}dτδ（x（τ））$ in ENIRAM $ $ x = 0 $是最重要的是，最重要的是一个最重要的time-time-time-time-time-time-dime-dime-time-dime-dime-dime-dime-dime-dime-dime-dime-dime-dime-dime-dime-distdddive。我们构建各种条件流程$ [x^*（t），a^*（t）] $涉及当地时间$ a^*（t）$在Time Horizon $ t $中。当Horizon $ t $有限时，我们将考虑对最终位置的条件$ x^*（t）$以及最终的当地时间$ a^*（t）$，以及针对最终当地时间$ a^*（t）$单独使用的最终位置$ x^*（t）$的条件。在无限的时间范围$ t \至 +\ infty $的限制下，我们考虑对有限渐近的当地时间$ a _ _ {\ infty}^*< +< +\ iffty $，以及与广泛的当地时间$ a^*$相关的$ a_t \ simeq t a^*$的条件，以相比，该$ a^*$'当地时间在大偏差方面的功能。然后，将这种一般结构应用于生成具有不同复发/瞬态特性的三个无条件扩散的各种约束随机轨迹：（i）瞬态扩散的最简单例子对应于均匀的严格正面漂移$μ（x）=μ> 0 $; （ii）扩散的最简单示例是通过漂移$μ（x）= - μ\，{\ rm sgn}（x）$ of参数$μ> 0 $给出了平衡的最简单示例。 （iii）不趋向于平衡的反复扩散的最简单示例是布朗运动，而没有漂移$μ= 0 $。
When the unconditioned process is a diffusion process $X(t)$ of drift $μ(x)$ and of diffusion coefficient $D=1/2$, the local time $A(t)= \int_{0}^{t} dτδ(X(τ)) $ at the origin $x=0$ is one of the most important time-additive observable. We construct various conditioned processes $[X^*(t),A^*(t)]$ involving the local time $A^*(T)$ at the time horizon $T$. When the horizon $T$ is finite, we consider the conditioning towards the final position $X^*(T)$ and towards the final local time $A^*(T)$, as well as the conditioning towards the final local time $A^*(T)$ alone without any condition on the final position $X^*(T)$. In the limit of the infinite time horizon $T \to +\infty$, we consider the conditioning towards the finite asymptotic local time $A_{\infty}^*<+\infty$, as well as the conditioning towards the intensive local time $a^* $ corresponding to the extensive behavior $A_T \simeq T a^*$, that can be compared with the appropriate 'canonical conditioning' based on the generating function of the local time in the regime of large deviations. This general construction is then applied to generate various constrained stochastic trajectories for three unconditioned diffusions with different recurrence/transience properties : (i) the simplest example of transient diffusion corresponds to the uniform strictly positive drift $μ(x)=μ>0$; (ii) the simplest example of diffusion converging towards an equilibrium is given by the drift $μ(x)=- μ\, {\rm sgn}( x)$ of parameter $μ>0$; (iii) the simplest example of recurrent diffusion that does not converge towards an equilibrium is the Brownian motion without drift $μ=0$.
