From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Abstract

Let be a positive integer, a positive constant and ( ) ≥1 be a sequence of independent identically distributed pseudorandom variables. We assume that the 's take their values in the discrete set {− , − + 1, . . . , − 1, } and that their common pseudodistribution is characterized by the (positive or negative) real numbers P{ = } = 0 + (−1) −1 ( 2 + ) for any ∈ {− , − + 1, . . . , − 1, }. Let us finally introduce ( ) ≥0 the associated pseudorandom walk defined on Z by 0 = 0 and = ∑ =1 for ≥ 1. In this paper, we exhibit some properties of ( ) ≥0 . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for ( ) ≥0 as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation / = (−1) −1 2 / 2 . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.

First exit time from a bounded interval for pseudo-processes driven by the equation∂/∂t=(−1)N−1Lachal1 2014Stochastic Processes and their ApplicationsSelf Cite2020View full textAdd to dashboardCite

“…So, we state the heuristic that an analogous boundary value problem should hold for the Feynman-Kac functional related to τ ab . The results obtained here through this approach coincide with limiting results deduced from a suitable pseudo-random walk studied in [17]. Moreover, when taking the limit as a goes to −∞ or b goes to +∞ in the present results, we retrieve the pseudo-distribution of (τ a , X τ a ) obtained in [14].…”

“…An explanation of this curious fact should be found in considering a linear pseudorandom walk with 2N consecutive neighbours around each sites. Indeed, after suitably normalizing such a walk, the neighbours cluster into a single site and form a multipole; see the draft [17].…”

Estimates for Functionals of Solutions to Higher-Order Heat-Type Equations with Random Initial Conditions

“…Many of the papers mentioned above study the distributions of functionals of the pseudoprocess X t , such as the sojourn time Γ t = t 0 I [0,∞) (X s )ds, the local times and others. In some cases functionals like max 0≤s≤t X s , X(T a ), X(T a,b ), where T a = inf{s : X s ≥ a}, T a,b = inf{s : (X s ≥ b) ∪ (X s ≤ a)}, permit us to go deeper into the sample behavior of the pseudoprocesses (see, for example, some papers by Lachal [30]- [32], Nishioka [39]- [41]).…”

“…More recently pseudoprocesses related to higher-order equations have been analysed and functionals connected with them have been evaluated by Orsingher [42], Hochberg and Orsingher [15], [16], see also [5], and, in more systematic way, by Lachal [30]- [32] among others. Fractional versions of the equation (1.6) with space derivatives of the Riemann-Liouville type have been studied by Orsingher and Toaldo [44], Smorodina, Faddeev, Platonova [17], [46], [45].…”

“…Pseudoprocesses constructed as limit of pseudo random walks have been proposed recently by Nakajima and Sato [38] and before by Lachal [32].…”