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On the First-Passage Area of a One-Dimensional Diffusion Process with Stochastic Resetting

For a one-dimensional diffusion process with stochastic resetting , obtained from an underlying diffusion X(t), we study the statistical properties of its first-passage time through zero, when starting from x > 0, and its first-passage area, i.e. the random area swept out by until its first-passage time through zero. By making use of solutions of certain associated ODEs, we find explicit expressions for the Laplace transforms of the first-passage time and the first-passage area, and their single and joint moments.

1.1. Formulation of the problem and general results

This note deals with the first-passage area (FPA) of a diffusion process with stochastic resetting. It is a continuation of Abundo (2013, 2023b), Abundo and Del Vescovo (2017) and Abundo and Furia (2019), regarding the FPA of jump-diffusions, drifted Brownian motion, Le?vy process and Ornstein-Uhlenbeck process. In fact, here we considered a one-dimensional diffusion process in the presence of stochastic resetting , obtained from an underlying diffusion X(t); this kind of process is treated, for example, in den Hollander et al. (2019) and Evans et al. (2020) (see also the references in Abundo 2023a). We studied the statistical properties of the first-passage time (FPT) through zero of , starting from x > 0, and its FPA, namely the random area swept out by until its FPT through zero. In some cases, we explicitly obtained the Laplace transform of the FPT and FPA, and their single and joint moments. Moreover, we provided the distribution of the maximum displacement of .

In the case that the underlying diffusion X(t) is Brownian motion without drift, the FPA was already studied in Singh and Pal (2022), although the results found therein were obtained using special functions. In contrast, here we used nothing but elementary functions: our arguments were based on classical results for one-dimensional diffusions. In fact, the study of the distributions of the FPT and FPA was carried out via solutions of certain associated ODEs. We focused on the case when the underlying diffusion X(t) is a Wiener process, i.e. a Brownian motion with or without drift, but the results can be extended to other processes.

The FPT and FPA of a diffusion process with stochastic resetting have important applications in many areas, for example, in biology, in the ambit of stochastic models for the activity of a neuron subject to resetting (see, for example, Nobile et al. 1985 and the references contained therein). Other important applications are found in queuing theory, where the first hitting time to zero can be identified with the busy period, i.e. the first instant at which the queue is empty, and the FPA is the total waiting time of the "customers" during a busy period (see, for example, Kearney 2004).

Now, we precisely describe the process with stochastic resetting.

We consider a one-dimensional temporally homogeneous diffusion process X(t) driven by the SDE:

and starting from the position X(0) = x > 0, where the drift µ(·) and diffusion coefficient s(·) are regular enough functions, such that there exists a unique strong solution of the SDE [1.1] for a given starting point, and Bt is the standard Brownian motion. We also assume that the FPT of the diffusion X(t) below zero is finite with probability one.

We assume that resetting events can occur according to a homogeneous Poisson process with rate r > 0. Until the first resetting event , the process coincides with X(t), and it evolves according to [1.1] with . When the reset occurs, is set instantly to a position xR > 0. After that, evolves again according to [1.1], starting afresh (independently of the past history) from xR, until the next resetting event occurs, etc. The inter-resetting times turn out to be independent and exponentially distributed random variables with parameter r. In other words, in any time interval (t, t + ?t), with ?t 0+, the process can pass from to the position xR with probability r?t + o(?t), or it can continue its evolution according to [1.1] with probability 1-r?t+o(?t). The process so obtained is called diffusion with stochastic resetting. For any C2 function f(x), its infinitesimal generator is given by (see, for example, Abundo 2013):

where is the "diffusion part" of the generator, i.e. regarding the diffusion process X(t).

For an initial position x > 0, we are concerned with the FPT of through zero, namely:

and the corresponding FPA

which is the area enclosed between the time axis and the path of the process up to the FPT through zero. We assume that both t(x) and A(x) are finite with probability one, for any x > 0. Note that

where tX(x) is the first-hitting time to zero of X(t) starting from x > 0 and s is an exponentially distributed random variable with parameter r > 0.

In fact, we limit ourselves to study the case when the underlying process X(t) is a Wiener process, i.e. Brownian motion with or without drift.

The main qualitative difference between the FPT of the process and the FPT of the underlying diffusion X(t) is that, for the process , the moments of the FPT are finite, while for the second one, they may be infinite. This is, for example, the case of Brownian motion starting from x > 0, where, as is well known, the first-hitting time to zero is finite with probability 1, but it has infinite expectation.

For ? > 0, let us consider the Laplace transform (LT) of , U(x) = ax + b (a, b = 0) being a polynomial of degree one, i.e.

Taking U(x) = 1, we obtain the LT of the FPT t(x), while for U(x) = x, we get the LT of the FPA A(x). The following holds (see Abundo 2023a):

PROPOSITION 1.1.-

The LT M?(x) of satisfies the differential problem:

where L denotes the infinitesimal generator of the underlying diffusion X(t), which is given, for any C2 function f, by

and f' and f? denote the first and second derivatives of f.

REMARK 1.1.-

Proposition 1.1 was already proved in Singh and Pal (2022) in the case when X(t) is Brownian motion. Note that, for r = 0 (i.e. when no resetting is allowed), we obtain equation (2.12) of Abundo (2013), provided that the jump part in the infinitesimal generator is set to zero, while the second boundary condition is M?(+8) = 0.

If the n-th order moment of exists finite, it is provided by:

By calculating the n-th derivative with respect to ?, at ? = 0, of both members of [1.7], we obtain that, setting T0(x) = 1, the n-th order moments Tn(x) satisfy the ODEs:

with the constraint Tn(0) = 0 and the addition of an appropriate further condition (indeed, we need two conditions to obtain the unique solution of [1.10]). Note that for r = 0, [1.10] becomes equation (2.19) of Abundo (2013). In particular, for U(x) = 1, [1.10] is nothing but the celebrated Darling and Siegert's equation (1953) for the moments of the FPT of a diffusion without resetting.

As regards the joint moments of t(x) and A(x), we consider the joint LT of t(x) and A(x), i.e. :

As easily seen, we get:

and

Applying the same reasoning as before and taking U(x) = ?1 + ?2x, we obtain that solves the problem

Then, by applying and calculating it for ?1 = ?2 = 0, we obtain that V(x) ? E[t(x)A(x)] is the solution of the problem:

with a suitable additional condition.

Note that for r = 0, [1.15] becomes the analogous equation, respectively, obtained in Abundo and Del...