key: cord-0469057-i82zpe7n
authors: D'Ovidio, Mirko
title: Non-local logistic equations from the probability viewpoint
date: 2021-05-03
journal: nan
DOI: nan
sha: 82d0b636a5e673f65aa505832515a84148fd7386
doc_id: 469057
cord_uid: i82zpe7n

We investigate the solution to the logistic equation involving non-local operators in time. In the linear case such operators lead to the well-known theory of time changes. We provide the probabilistic representation for the non-linear logistic equation with non-local operators in time. The so-called fractional logistic equation has been investigated by many researchers, the problem to find the explicit representation of the solution on the whole real line is still open. In our recent work the solution on compact sets has been written in terms of Euler's numbers.

The study of the logistic and fractional logistic growth has attracted many researchers because of the high impact in the applied sciences. Here we bring to the reader's attention some of the recently appeared papers, [1, 11, 15, 16, 17] in which the role of the logistic equation has been investigated and discussed. Concerning the the stochastic interpretation and representation, a comparison with other models of growth has been given recently in [10] . Many other works are devoted to the logistic SDEs. The literature is huge, we mention only few works here and further on in the presentation of the results.

In this short note, we discuss some aspects concerning logistic equations and random time changes. The fractional logistic equation has been investigated by many researchers in the last decade. However, the solutions has been obtained only recently in [13] . Many similar problems have been considered in order to find solutions sharing some peculiar properties with the solution to the fractional logistic equation. A discussion on this point has been given in [20] and the references therein.

We consider non-local operators more general than the Caputo-Djrbashian fractional derivative. Then, we introduce non-local logistic equations and discuss the probabilistic representation of the solutions in terms of inverses to subordinators. Thus, we use stochastic processes driven by non-local partial differential equations in order to solve non-local logistic equations.

Our presentation is based on two cases concerning respectively the logistic equation on the real line and the logistic equation on compact sets.

Let H = {H t , t ≥ 0} be a subordinator (see [5] for a detailed discussion). Then, H can be characterized by the Laplace exponent Φ, that is,

λ ≥ 0. As usual we denote by E x the expected value w.r. to P x where x is a starting point. Moreover, if Φ is the Laplace exponent of a subordinator, then there exists a unique pair (k, d) of non-negative real numbers and a unique measure Π on (0, ∞) with (1 ∧ z)Π(dz) < ∞, such that for every λ ≥ 0

The Lévy-Khintchine representation in formula (2.1) is written in terms of the killing rate k = Φ(0) and the drift coefficient

and Π is the so called tail of the Lévy measure. We recall that Φ is uniquely given by (2.1). In particular, it is a Bernstein function, then Φ is non-negative, non-decreasing and continuous. For details, we refer to the well-known book [5] . The interested reader can also consult the recent book [26] .

We define the inverse process L = {L t , t ≥ 0} to a subordinator as

We recall that H 0 = 0 and L 0 = 0. We do not consider step-processes with Π((0, ∞)) < ∞, we focus only on strictly increasing subordinators with infinite measures (then L turns out to be a continuous process). By definition of inverse process, we can write

We also denote by

the corresponding densities (see for example [21] ). Further on we use the following potentials Let M > 0 and w ≥ 0. Let M w be the set of (piecewise) continuous function on [0, ∞) of exponential order w such that | (t)| ≤ M e wt . Denote by the Laplace transform of . Then, we define the operator

where Φ is given in (2.1). Since is exponentially bounded, the integral is absolutely convergent for λ > w. The inverse Laplace transforms and D Φ t are uniquely defined. Since

the function D Φ t can be written as a convolution involving the ordinary derivative and the inverse transform of (2.2) iff ∈ M w ∩ C([0, ∞), R + ) and ∈ M w . We also observe that (Young's inequality)

where lim λ↓0 Φ(λ)/λ is finite only in some cases. For example, for d = 0 and k = 0:

ii) gamma subordinator with Φ(λ) = a ln(1 + λ/b) with ab > 0; iii) generalized stable subordinator with Φ(λ) = (λ + γ) α − γ α with γ > 0 and α ∈ (0, 1).

The operator D Φ t , in alternative and sometimes slightly different forms, it has been first considered in [25] after in [18] and recently in [9, 27] . We introduce the following notation

Remark 2.1. Let us recall a couple of special cases. i) We notice that when Φ(λ) = λ we have that H t = t and L t = t a.s. and in (2.8) the equality holds. The operator D Φ t becomes the ordinary derivative

ii) The well-known case Φ(λ) = λ α , α ∈ (0, 1) gives the Caputo-Djrbashian derivative

The corresponding processes are H t which is a stable subordinator and L t which is an inverse to a stable subordinator ( [21, 12, 22] ).

Remark 2.2. We recall the following result which will be useful below. Let us introduce the Riemann-Liouville (type) derivative

Between

where in the last step we have used the fact that ∞) ) for a constant c. The density l of the process L t solves the following problem

From the Laplace technique, by considering (2.7) and the potential (2.6) we get immediately the result. We skip the proof (the reader can consult [27] ).

The latter has the following reading

That is, the following pointwise equality holds, ∀t > 0,

where f is linear can be written in terms of the density l(t, x). This is expected in case of linear f and it is well-known in case of linear operator Av. Indeed, the latter can be included in the theory of time-changed processes first introduced in [4] for the fractional (Caputo-Djrbashian) derivative and after, in [27, 9] for a general non-local operator. For the sake of completeness we provide the following statement. 

is the unique classical solution to the non-local Cauchy problem

Proof. We have that

From the linearity of f , we can write

Let us consider Φ(λ) = λ α . The well-known case

brings to the solution

which is the Mittag-Leffler function. Thus, equation (3.1) gives v(x) = ce −ax and equation (3.2) gives (as proved in [6] )

We also notice that, from

We focus on Φ given in (2.1) with

Further on we always assume that (3.7) applies.

Let us consider

We now approach the problem to find a probabilistic representation for the fractional logistic equation. In particular, we consider the following two cases involving inverses to subordinators.

on the positive real line can be written as

Let us introduce the process v(L t ) whose realizations include plateaux according to the random time L t . We have that

where L t can be regarded as the first time the subordinator H s exits the set (0, t), that is (s < L t ) ≡ (t > H s ) under P 0 . Thus,

The λ-potential

can be associated with (4.2) only if ζ = ∞ almost surely. Formula (4.3) can be obtained from (2.5) and (2.6), see [8] for details. Let us consider ζ such that ζ = T < ∞ almost surely. From (4.3) we observe that, as λ → 0 + , we obtain for

which is finite only if the limit Φ(λ)/λ is finite. Since v is continuous and bounded, formula (4.2) is finite. In order to have a finite integral in (4.4) , the function u is obtained by extension with zero for t ≥ H T , that is for L t ≥ T .

Assume that v, u ∈ L 1 ((0, T * )) for some T * . Then, formula (4.4) can be considered in order to have a reading in terms of delayed and rushed growth (see [8] ). An helpful example of this phenomenon is presented in Figure 1 . ([0, ∞) ). Let u ∈ M 0 be the solution to

Proof. With (2.11) in mind, an integration by parts yields

From (2.10), we have that ∞) ) and therefore, by taking into account that u(0) = v 0 , we write

and this concludes the proof.

Let us consider the case Φ(λ) = λ α . With formula (4.1) at hand, from (3.4) and (3.5), we can write

This is the solution on the whole positive real line to

according to Theorem 4.1. The function (4.5) has been first considered in [29] and after in [2] in order to have a good approximation of the solution to the fractional logistic equation (with Caputo-Djrbashian derivative). In [14] the author considered shifted-Legendre polynomials in order to obtain approximate solutions of the fractional-order logistic equation. In [24] the authors considered a simple algorithm in order to obtain such solution including a numerical implementation in terms of Padè approximation. Recently, it has been also considered in [20] as the solution to a modified fractional logistic equation which is related to (4.6).

In conclusion, Theorem 4.1 extends the non-local logistic equation associated with (4.1) to a general Φ.

on the convergence set (0, r) of the real line can be written as

where E 0 = v 0 and the sequence {E k } k is given by the Euler's numbers if v 0 = 1/2. The equation

t is the Caputo-Djrbashian derivative) has been investigated in [13] . It turns out that,

, t ∈ (0, r α ) (4.8)

is written in terms of the coefficient {E α k } k which are strictly related with the numbers {E k } k . In particular, we have that E 1 k = E k , ∀ k ≥ 0 and, for the sequence {E α k } k , we have that

and

is the generating recursive formula. We termed k i α = Γ(αk + 1) Γ(αi + 1)Γ(β(k − i) + 1) (4.9) as fractional binomial coefficient because of the many similar properties shared with the binomial coefficient. For example,

Straightforward calculations show that, for any σ,

Indeed, u is defined on compacts K ⊆ (0, r). However, we are still able to establish some connection between u and the process L t . Let us consider

which is the rescaled moment of order k of L t . For a suitable sequence {E Φ k } k , we introduce the functionū

where obviously φ k (0) = 0 ∀ k > 0 and φ 0 (t) = 1 ∀ t ≥ 0. Lemma 4.2. If Φ(λ) = λ α and E Φ k = E α k for any k, then u(t) =ū(t), ∀ t ∈ (0, r α ).

Proof. The proof follows immediately by comparing (4.8) with (4.11) . Since D Φ t = D α t , the coefficients E Φ k are exactly given by E α k and, from (3.6),

This concludes the proof.

We notice that the representation (4.7) holds on compact sets. In particular, v(t), t < r implies that v(L t ) is defined as L t < r. That is, t < H r . Thus, we should consider the probabilistic representation

for the solution (4.8) where H is a stable subordinator and L is the inverse to H. Lemma 4.3. Let us consider (4.10). We have that

Proof. It follows immediately from the fact that (see formula (2.6))

from which we get the Laplace transform

This formula has been obtained in [28] . From (2.7) and the fact that φ k (0) = 0, we write

We conclude the proof.

We observe that for the sequence E Φ k = (−a) k , k ≥ 0, a > 0 we have that

where L t is an inverse to a subordinator with symbol Φ. Such a representation holds for t ≥ 0. Moreover, from Lemma 4.3 we are able to conclude that

This means thatū(t)

That is, the case in Theorem 3.2.

Concerning the extension of the result in Lemma 4.2 to a general symbol Φ, our conjecture is as follows: There exists a sequence

where H is a subordinator and L is the inverse of H.

The function (4.12) has been also studied in [18] for complete Bernstein functions and [23] for special Bernstein functions. It has been considered also in [7] in connection with the Poisson and Skellam processes. Moreover, the case E Φ k = a k has been also studied in [19] for complete Bernstein functions and in [3] in the general case. As far as we know for the moments of L t and the function (4.12) we do not have an explicit representation. Actually, this still is an open problem.

In conclusion, the above conjecture would extend the non-local logistic equation associated with (4.7) to a general Φ. t → v(t) ∈ (0, 1) given in formula (4.1) with v 0 = 0.1; Picture in the middle) a realization of L t , that is a continuous function from (0, T ) to (0, 10) where T = 8e+05. Here L t is the inverse to a stable subordinator with α = 0.5; Bottom picture) the composition v(L t ) : (0, T ) → (0, 1). The last picture shows that L t "delays" the profile of v(t). Indeed, due to the plateaux of the new time L t , from the function v(t), t ∈ (0, 10) we obtain the random function v(L t ), t ∈ (0, T ) where T >> 10. This can be regarded as a delaying effect of L t on the growth v(t).

Power series solution of the fractional logistic equation

A note on the fractional logistic equation

Abstract Cauchy problems for generalized fractional calculus

Stochastic solutions for fractional Cauchy problems

Lectures on Probability Theory and Statistics

Limit theorems for occupation times of markov processes. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete

On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators

Delayed and Rushed motions through time change

Time fractional equations and probabilistic representation

Paraggio Logistic Growth Described by Birth-Death and Diffusion Processes

The evolution of COVID-19: A discontinuous approach

On the fractional counterpart of the higher-order equations

Solutions of fractional logistic equations by Euler's numbers

A Comparative Study of Two Legendre-Collocation Schemes Applied to Fractional Logistic Equation

A Discretization Approach for the Nonlinear Fractional Logistic Equation

Solution to the fractional logistic equation by modified Eulerian numbers

A moving boundary problem with space-fractional diffusion logistic population model and densitydependent dispersal rate

General fractional calculus, evolution equations, and renewal processes. Integral Equations and Operator Theory

Growth equation of the general fractional calculus

Modified Fractional Logistic Equation

Triangular array limits for continuous time random walks. Stochastic processes and their applications

Inverse stable subordinators

Relaxation patterns and semi-markov dynamics

A new look at the fractionalization of the logistic equation

Integral equations of the first kind of Sonine type

Bernstein functions: theory and applications

Convolution-type derivatives, hitting-times of subordinators and time-changed C 0 -semigroups

Using differential equations to obtain joint moments of first-passage times of increasing Lèvy processes

Exact solution to fractional logistic equation