Analysis of an Inhomogeneous Random Walk with Spatial Decay of Transition Probabilities and Parameter Renewal per Excursion

Abstract: In this paper, we propose and analyze a novel one-dimensional inhomogeneous random walk model that combines spatial decay of transition probabilities with a temporal renewal structure for each excursion. In this model, the probability of moving to the right from each state creats a spatial inhomogeneity that causes a stronger pull-back toward the origin as the process moves farther away. Furthermore, it features a random environment aspect where the parameter of each transition probability is independently resampled from a uniform distribution at the beginning of each excursion. We rigorously derive the hitting probability to an upper boundary using a scale function. Furthermore, by solving linear difference equations, we provide the probability generating function of the first hitting time, the expected occupation time for each state during an excursion (discrete Green's function), and the distribution and expectation of the maximum penetration depth.

• The paper derives closed-form expressions for hitting probabilities using a novel scale function method.
• It employs a renewal framework that resets transition probabilities at each excursion from the origin.
• The study rigorously characterizes occupation times and excursion maxima, offering insights for adaptive stochastic processes.

Analysis of an Inhomogeneous Random Walk with Spatial Decay and Parameter Renewal

Introduction and Model Architecture

The paper introduces an analytically tractable one-dimensional inhomogeneous random walk defined on the non-negative integers, featuring two essential mechanisms: a spatial decay of transition probabilities and random renewal of the transition parameter at the onset of each excursion from the origin. Specifically, at the start of every excursion, a parameter $p$ is sampled i.i.d. from a uniform distribution on $(0,1)$ and governs the transition probabilities during the excursion. The rightward transition probability from site $k$ is given by $r^k p$ for a spatial decay parameter $r\in (0,1]$, leading to an exponentially decreasing drift away from the origin as $k$ increases. Upon return to zero or absorption at an upper barrier $N$, a new excursion is initialized with a freshly sampled $p$; this engenders a random environment renewal at each excursion, ensuring inter-excursion independence.

This construction generalizes earlier random walk models that modulate transition laws at each return to the origin [pilipenko2017limit], and subsumes dynamics akin to the impatient random walk [englander2019impatient] by embedding inhomogeneity in both space (by the $r^k$ factor) and time (by environmental renewal).

Main Theoretical Results

The study rigorously analyzes several key functionals associated with excursions and hitting behaviors of this process, utilizing scale function methods, difference equations, and renewal-geometry arguments.

Hitting Probability and Scale Function Analysis

The central tool is a scale function $S(k;p)$ defined as:

$(0,1)$0

By solving associated difference equations, the probability that, starting from zero, an excursion is absorbed at the upper barrier $(0,1)$1 before returning to the origin is obtained explicitly as:

$(0,1)$2

Conversely, the probability of returning to zero without reaching $(0,1)$3 is $(0,1)$4.

Renewal Structure and First Hitting Time

Given the renewal at each excursion, the total number of excursions until reaching $(0,1)$5 is geometrically distributed. For $(0,1)$6, the probability that exactly $(0,1)$7 failed excursions (i.e., excursions returning to $(0,1)$8 before reaching $(0,1)$9) occur before absorption is:

$k$0

This explicit geometric law is a direct consequence of the environment resampling at each excursion.

The probability generating function (PGF) for the first hitting time of $k$1 is constructed through a renewal sum, leveraging solutions to tridiagonal boundary-value problems corresponding to successful and unsuccessful excursions. Specifically, the PGF is given by:

$k$2

Here, $k$3, $k$4 solve recurrences expressing the PGFs of the return time to $k$5 and the hitting time to $k$6 before $k$7, respectively, given $k$8.

By differentiating the PGF at $k$9, an explicit formula for the unconditional expected time to reach $r^k p$0 is derived:

$r^k p$1

where $r^k p$2 is computable via the associated occupation time analysis.

Occupation Time and Green's Function

An explicit Green's function expression for the expected number of visits to site $r^k p$3 before absorption, conditional on the initial $r^k p$4, is derived as:

$r^k p$5

The total expected occupation time of a state across excursions is then integrated over the environmental randomness.

Excursion Maximums

The cumulative distribution and expectation of the maximal site reached during an excursion (penetration depth) are exactly characterized. The tail law for the maximum $r^k p$6 in an excursion is:

$r^k p$7

and the expectation is the infinite sum over these tail probabilities.

Discussion: Implications and Significance

The explicit analytic description of absorption and occupation statistics for this inhomogeneous random walk is notable for several reasons:

• Novelty of dynamic random environment: The model generalizes static inhomogeneous or random-environment walks by introducing renewal of randomness at each excursion, opening avenues for more realistic modeling of environment fluctuations and adaptive systems.
• Enhanced drift control: The spatial decay factor $r^k p$8 generates a non-linear restoring force towards the origin, yielding richer behavior than homogeneous or even simple reflecting walks.
• Geometric decomposability: The renewal of $r^k p$9 at excursion boundaries renders the analysis particularly tractable via geometric and generating function techniques, facilitating closed-form results for first hitting times and occupation statistics.
• Interplay with classical excursion theory: The framework extends the paradigm of local time decompositions and excursion dynamics [CsakiHu2003], and is positioned to contribute to applications in quantitative finance (excursion risk) [FujitaYoshida2023_JSIAML], non-parametric testing [BaronRukhin1998], and controlled stochastic processes.

Future Directions

Several directions naturally arise from these results:

• Limit laws and scaling: Examination of scaling limits as $r\in (0,1]$0 (critical regime) or $r\in (0,1]$1 may yield new universal behaviors or connections to diffusions in random environments.
• Extensions to higher dimension or general graphs: Adapting the parameter renewal approach to more general state spaces could advance our understanding of random walks with environment switching.
• Applications to adaptive systems: Since the environment updates at return times, analogous mechanisms may be relevant for learning algorithms or adaptive control, where parameter updates are triggered by returns to a baseline or critical state.
• Large deviations and rare events: Given the explicit tail laws for maxima, further studies into large deviation principles and excursion extremes may be feasible.

Conclusion

This paper presents a comprehensive analytical treatment of a random walk in a dynamically renewed random environment with spatially decaying transition probabilities. By leveraging scale functions and explicit recurrence solutions, the study provides closed-form expressions for hitting probabilities, occupation measures, and excursion maxima distributions. The model’s tractability, coupled with its novel environmental renewal mechanism, makes it a significant addition to the theory of inhomogeneous random walks and excursion processes, with broad potential for application in stochastic modeling and statistical physics.

---

References:

• "On a limit behavior of a random walk with modifications upon each visit to zero" [pilipenko2017limit]
• "Impatient random walk" [englander2019impatient]
• "An introduction to excursion risk through discrete-time excursions" [FujitaYoshida2023_JSIAML]
• Further references as cited in the original paper (2604.22644).