On a Fokker-Planck equation for wealth distribution

We study here a Fokker-Planck equation with variable coefficient of diffusion and boundary conditions which appears in the study of the wealth distribution in a multi-agent society [ 2 , 10 , 22 ]. In particular, we analyze the large-time behavior of the solution, by showing that convergence to the steady state can be obtained in various norms at different rates.


1.
Introduction. Among the mathematical models introduced in recent years to study the evolution of wealth distribution in a multi-agent society [22], Fokker-Planck type equations play an important role. Let f (v, t) denote the density of agents with personal wealth v ≥ 0 at time t ≥ 0. The prototype of these Fokker-Planck equations reads where λ and σ denote two positive constants related to essential properties of the trade rules of the agents. Equation (1) has been first derived by Bouchaud and Mezard [2] through a mean field limit procedure applied to a stochastic differential equation for the wealth density. The same equation was subsequently obtained by one of the present authors with Cordier and Pareschi [10] via an asymptotic procedure from a Boltzmann-type kinetic model for trading agents. This procedure also furnished the existence (without uniqueness) of a weak solution to equation (1). One of the main features of equation (1) is that it possesses a unique stationary solution of unit mass, given by the (inverse) Γ-like distribution [2,10] This stationary distribution, as predicted by the observations of the Italian economist Vilfredo Pareto [23], exhibits a power-law tail for large values of the wealth variable. Equation (1) differs from the classical Fokker-Planck equation in two important points. First, the domain of the wealth variable v takes only values in R + . Second, the coefficient of diffusion depends on the wealth variable. This makes the analysis of the large-time behavior of the solution to equation (1) very different from the analogous one studied in [24] for the classical Fokker-Planck equation.
Indeed, Fokker-Planck equations with variable coefficients and in presence of boundary conditions have been rarely studied. Maybe the first result in this direction can be found in a paper by Feller [13], who treated the case v ∈ R + and coefficient of diffusion v, with a general drift term (cf. also the book [14] for a general view about boundary conditions for diffusion equations). In particular, the importance of the boundary conditions has been shown in [13] to be related to the action of the drift term.
More recently, the pure initial value problem for Fokker-Planck type equations with almost general coefficient of diffusions has been studied by Le Bris and Lions in [19]. However, even if the coefficients of equation (1) are included in their analysis, the results of [19] are not directly applicable to the initial-boundary value problem.
As far as the large-time behavior is concerned, the main argument in the standard Fokker-Planck equation is to resort to entropy decay, and to logarithmic Sobolev inequalities [1]. Unlikely, as discussed in [20], this type of inequalities do not seem available in presence of variable diffusion coefficients.
A semi-formal approach to the large-time behavior of Fokker-Planck type equations appearing when studying problems related to socio-economic sciences has been recently dealt with in [15]. There, a possible argument to prove convergence to equilibrium for the solution to (1) has been identified in Chernoff type inequalities [9,18].
In the following, by resorting in part to the strategy outlined in [15], we will give a rigorous answer to some of the questions which have been left open in [2,10], which include a precise relationship between the solution of the kinetic model considered in [10] and the solution to the Fokker-Planck equation (1), and an exhaustive study of its large-time behavior. As we shall see, various properties of the solution to equation (1) can in fact be extracted from the limiting relationship between the Fokker-Planck description and its kinetic level, given by the bilinear Boltzmanntype equation introduced in [10]. We will discuss this aspect in Section 2, by means of a detailed Fourier analysis. In particular, we will show that, at least for some range of the parameters λ and σ, the Fokker-Planck equation (1) can be rigorously obtained from the bilinear kinetic model in the asymptotic limit procedure known as quasi-invariant trade limit. Then, convergence to equilibrium will be discussed in Section 3. The essential argument here will be to resort to an inequality of Chernoff type [9,18], recently revisited in [15], that allows to prove different types of convergence. In particular, we will prove that the solution to equation (1) converges to equilibrium with exponential rate only in the case of initial data sufficiently close to equilibrium (in the sense of a weighted L 2 -norm), and to a polynomial rate in L 1 -norm (at least 1/t) for a large class of initial data.

Kinetic model and Fokker-Planck equation.
2.1. Main properties of the Fokker-Planck equation. In this introductory section, we will briefly recall some existing results about the Fokker-Planck equation (1) and its kinetic counterpart [15]. To start with the analysis of the initial value problem for the Fokker-Planck equation (1), it is essential to consider, together with a suitable decay of the solution at infinity, physical boundary conditions at the point v = 0. A clear way to understand the role of these conditions is to evaluate the evolution of the observable quantities φ = φ(v). Let φ(v) be a smooth function, bounded with its first derivative at v = 0. A simple computation shows that Then, a universal law for the evolution of the observables requires the vanishing of the boundary term. While the vanishing at infinity follows by choosing initial data with a smooth and rapid decay, at the boundary v = 0 it is enough to impose that and Condition (3) is automatically satisfied for a sufficiently regular density f . On the contrary condition (4) requires an exact balance between the so-called advective and diffusive fluxes on the boundary v = 0. This condition is usually referred to as the no-flux boundary condition. If both conditions (3) and (4) hold, the physical observables evolve in time according to Equation (5) is usually known as weak form of the Fokker-Planck equation (1). By choosing φ(v) = 1, v shows that the solution to (1) satisfies Therefore, if the (nonnegative) initial value ϕ(v) of equation (1) is a density function satisfying the normalization conditions the solution to (1) still satisfies conditions (6). In other words, if the initial datum is a probability density with unit mean, then the solution at any subsequent time remains a probability density with unit mean. For n ∈ N + let us define An elementary computation shows that, if ϕ satisfies conditions (6) and its second moment is bounded, the second moment of the solution follows the law Hence, the value of the second moment stays bounded when σ < 2λ, while it diverges in the opposite case. In the former case, solving equation (7) we obtain which implies It is clear that the principal moments of the solution to the Fokker-Planck equation can be obtained recursively, and explicitly evaluated at the price of increasing length of computations. Since it will be useful in the following, we evaluate here the third moment M 3 (t). We obtain Then, if the initial density ϕ(v) has the third moment bounded, the evolution law for M 3 (t) is given by Using (8), the third moment is evaluated as Therefore the third moment is uniformly bounded in time if σ < λ and it grows to +∞ in the opposite case. Last, choosing φ(v) = e −iξv we obtain the Fourier transformed version of the Fokker-Planck equation (1) where, as usual g(ξ) denotes the Fourier transform of g(v), v ∈ R + . In this case

2.2.
The kinetic model. The basic model discussed in this section has been introduced in [10] within the framework of classical models of wealth distribution in economy. As shown in [10], the Fokker-Planck equation (1) is strongly related to a bilinear kinetic model of Boltzmann type, modelling the evolution of wealth in a multi-agent society in which agents interact through binary trades [22]. This model belongs to a class of models in which the interacting agents are indistinguishable. The agents' state at any instant of time t ≥ 0 is completely characterized by their current wealth v ≥ 0 [11,12]. When two agents encounter in a trade, their pre-trade wealths v, w change into the post-trade wealths v * , w * according to the rule [6,7,8] v The interaction coefficients p i and q i are non-negative random variables. While q 1 denotes the fraction of wealth of the second agent transferred to the first agent, the difference p 1 − q 2 is the relative gain (or loss) of wealth of the first agent due to market risks. It is usually assumed that p i and q i have fixed laws, which are independent of v and w, and of time. This means that the amount of wealth an agent contributes to a trade is (on the average) proportional to the respective agent's wealth.
In [10] the trade has been modelled to include the idea that wealth changes hands for a specific reason: one agent intends to invest his wealth in some asset, property etc. in possession of his trade partner. Typically, such investments bear some risk, and either provide the buyer with some additional wealth, or lead to the loss of wealth in a non-deterministic way. An easy realization of this idea consists in coupling the saving propensity parameter [6,7] with some risky investment that yields an immediate gain or loss proportional to the current wealth of the investing agent. The interactions rules for this model are obtained by fixing where 0 ≤ λ ≤ 1 is the parameter which identifies the saving propensity, namely the intuitive behavior which prevents the agent to put in a single trade the whole amount of his money, while ε is a small positive parameter, which measures the quantity of money exchanged in a single trade. The coefficients η ε ,η ε are independent and identically distributed random parameters, such that always η ε ,η ε ≥ ελ − 1. This clearly implies v * , w * ≥ 0. Therefore, the collision rule in [10] reads Remark 1. In the rest of the paper, the mean value of a random quantity θ will be denoted by θ . A simple way to characterize the ε-dependence of the random parameters is to define η ε andη ε as independent copies of a random variable η with finite variance σ, with η ε =η ε = √ εη. If the random parameters are even, so that implying conservation of the average wealth. In the remaining cases, it is immediately seen that the mean wealth is not preserved, but it increases or decreases exponentially (see the computations in [10]). Various specific choices for the random parameters have been discussed in [21]. Note that, when η = 0, one has η 2 ε = η 2 ε = εσ. Owing to classical arguments of kinetic theory [22], it has been shown in [10] that the evolution of the wealth density consequent to the binary interactions (13) obeys a Boltzmann-type equation. To outline the dependence on ε, let us denote with h ε (v, τ ) the distribution of the agents wealth v ≥ 0 at time τ > 0. Then, the equation for the evolution of h ε can be fruitfully written in weak form. It corresponds to say that, for any smooth function φ, h ε satisfies the equation Existence and uniqueness of the solution to equation (15) has been proven in [21]. We will detail later on some of these results for their connection with the Fokker-Planck equation (1). The weak form (15) allows to evaluate moments of the solution in a closed form. The choice φ(v) = 1 immediately gives mass conservation. In ad- (14) one obtains that the mean value of the solution is preserved in time. Therefore, if the initial value satisfies the normalization conditions (6) it follows that the solution h ε (v, τ ) still satisfies the same conditions at any subsequent time τ > 0.
The evolution law of the second moment of h ε (v, τ ) depends explicitly on ε, and clearly changes with ε. This is due to the fact that changing the value of ε in the binary collision (13) we change the quantity of wealth which is involved into the trade. In the limit case ε → 0, we have a trade in which the post interaction wealths are left unchanged. This suggests to scale time in such a way to maintain an effective law of evolution of the second moment even in the limit ε → 0. This can be easily done by setting t = ετ , while h ε (v, τ ) = f ε (v, t) [10]. One then obtains that f ε (v, t) satisfies where we defined (φ, Qε(fε, fε)) = 1 2ε In this case, (16) changes into Remark 2. Note that the conservation of the mean value is not modified by this scaling. However, in (19) the dependence on ε remains in the factor λ−ελ 2 . One can easily eliminate this dependence by choosing in (13) a different value (ε-dependent) of the saving propensity λ. This can be obtained by the choicẽ which is such thatλ − ελ 2 = λ. Moreover Clearly, definition (20) requires to choose ε small enough.
2.3. The invariant trade limit of the Boltzmann equation. The close relation between the kinetic equation (17) and the Fokker-Planck equation (5) has been outlined in [10], where it was shown that in the limit ε → 0 a subsequence of solutions f ε (v, t) to (17) converges to f (v, t), solution of (5). In this section we aim in improving these results.
In the rest, we will fix a time T > 0, and we will consider both equations (17) and (1) in the time interval 0 ≤ t ≤ T . In addition, let the even random variable η which defines the random part of the trade possess the third moment bounded, and let us set |η| 3 = σ 3 . Analogously, let |η| = σ 1 . Using a Taylor's formula at the second order, one can write φ(v * ) as where, for some α such that 0 ≤ α ≤ 1,ṽ = αv + (1 − α)v * . Then, by (13) and the properties of the random variable η it holds In particular, (17) can be rewritten as where the last integral (the remainder) accounts for all the higher order ε-dependent terms in the expansion Therefore, for any given (smooth function) φ(v) and density f (v), v ∈ R + , we have the identity Remark 3. In reason of the fact that the solution to the kinetic model (17) satisfies conditions (6) for all times t ≥ 0, it follows from (22) that In the sequel, we will work with various distances, which have been found very useful when dealing with the spatially homogeneous Boltzmann equation for Maxwell molecules [16,25]. The first one is a family of metrics that has been introduced in the paper [16] to study the trend to equilibrium of solutions to the space homogeneous Boltzmann equation for Maxwell molecules, and subsequently applied to a variety of problems related to kinetic models of Maxwell type. For a more detailed description, we address the interested reader to the lecture notes [4] and to [5].
Definition 2.1. Given s > 0 and two probability densities f 1 and f 2 , their Fourier based distance d s (f 1 , f 2 ) is given by the quantity The distance is finite, provided that f 1 and f 2 have the same moments up to order [s], where, if s / ∈ N + , [s] denotes the entire part of s, or up to order s − 1 if s ∈ N + . Moreover d s is an ideal metric. Its main properties are the following 1. Let X 1 , X 2 , X 3 , with X 3 independent of the pair X 1 , X 2 be random variables with probability distributions f 1 , f 2 , f 3 . Then where the symbol * denotes convolution; 2. Define for a given nonnegative constant a the dilation Then, given two probability densities f 1 and f 2 , for any positive constant a 3. Let d s (f 1 , f 2 ) be finite for some s > 0. Then the following interpolation property holds [4] d for any 0 < p < s.

Definition 2.2.
For m ∈ N + , let C m (R + ) be the set of m-times continuously differentiable functions, endowed with its natural norm · m . Then for f = f (v), v ∈ R + , let us define Remark 4. As proven in [25], the metric d 2 is equivalent to · * m , m ∈ N + , that is Let m ≥ 3. Thanks to (21) , whenever f (v) is a probability density with the third moment bounded  (5), for where the constant C ε (T ) depends on moments of f (v, t) up to the order three. Note that the equation for the Fourier transform of the density f ε (v, t), solution of the kinetic equation (17) in reason of (24) we can write where Since the initial datum of the Fokker-Planck equation has moments bounded up to the order three, (29) coupled with the equivalence of · m and d 2 implies that, if f (v, t) is the corresponding solution to the Fokker-Planck equation, in the time interval 0 ≤ t ≤ T , for a suitable constant D ε where lim ε→0 r ε (T ) = 0.
Using the expression (31) for the Fokker-Planck operator in (5), we obtain that the difference between the solution f ε of (30) and the solution f to the Fokker-Planck equation (11) satisfies Let f ε (v, t) and f (v, t) solutions departing from initial valuef 0 and f 0 satisfying conditions (6) and such that their distance d 2 (f 0 , f 0 ) is finite. Let us set Consider that, if σ < 2λ, for ε sufficiently small If this is the case, h ε (t) satisfies Proceeding as in [25], Theorem 5, we conclude by Gronwall inequality that (36) implies Letting ε going to 0 we obtain Hence, if we start with the same initial valuef Analogous reasoning can be used to prove uniqueness of the solution to the Fokker-Planck equation (5). By resorting to the approximation (31) of the Fokker-Planck operator in (5), we obtain that the difference between two solutions f (v, t) and g(v, t) to the Fokker-Planck equation (11), for any given small value of ε satisfies ∂ ∂t Let f (v, t) and g(v, t) solutions departing from initial value f 0 and g 0 satisfying conditions (6) and such that their distance d 2 (f 0 , g 0 ) is finite. If the third moments of f 0 and g 0 are finite, proceeding as before we conclude with the bound that clearly implies uniqueness of the solution. We can resume the previous results in the following. Theorem 2.3. Let f 0 (v) be a probability density in R + satisfying conditions (6), and such that its third moment is finite. Assume moreover that the random part η in the binary collision (13) is even, and such that |η| 3 is finite. Then, for any finite time T , as ε → 0, the unique solution f ε (v, t) to the kinetic model (17) with initial datum f 0 converges to the solution f (v, t) of the Fokker-Planck equation (5) with the same initial datum f 0 , and Moreover, the solution to the Fokker-Planck equation is unique.

2.4.
A regularity result. Theorem 2.3 shows that, for any given datum with a suitable decay at infinity, the Fokker-Planck equation (5) possesses a unique solution. However, if the initial datum belongs toḢ p (R + ), we can conclude that the solution maintains the same regularity for any subsequent positive time. The following result holds Theorem 2.4. Let f 0 (v) be a probability density in R + that belongs toḢ p (R + ). Then, theḢ p −norm of the solution f (v, t) to the Fokker-Planck equation (1), for t ≤ T , still belongs toḢ p (R + ), and Proof. Let f (ξ, t) = a(ξ, t) + ib(ξ, t). Starting from the Fourier transform version of the Fokker-Planck equation (11), let us split it into the real and imaginary part.
Multiplying equations (40) respectively by 2a and 2b and summing up, we get Hence, multiplying by |ξ| p and integrating over R with respect to ξ, we obtain the evolution equation of theḢ p/2 −norm of f (v, t), where, as usual, the homogeneous Sobolev spaceḢ s , for s > 0 is defined by the norm We have Integrating by parts the two integrals, it results The last integral in (43) can be estimated from below as follows Indeed, for all µ > 0 it holds Optimizing over µ we get An analogous computation works for b.
The inequality (44) implies that if the initial data has finiteḢ p −norm, then for all finite t > 0, theḢ p −norm of the solution remains finite and inequality (39) holds.
3.1. An equivalent formulation. In this section, we will be concerned with the study of the large-time behaviour of the Fokker-Planck equation (1). The main argument here will be the study of the time evolution of various Lyapunov functionals, starting from Shannon entropy of the solution f (v, t) relative to the steady state f ∞ (v). We recall that the relative Shannon entropy of the two probability density functions f and g is defined by the formula As a first step in this analysis, we will introduce in the following equivalent formulations of the Fokker-Planck equation, that result to be very useful to justify rigorously the behaviour of these Lyapunov functionals. Indeed, equation (1) admits many equivalent formulations, each of them well adapted to different purposes [15]. To this extent, recall that the equilibrium distribution f ∞ defined in (2) or, equivalently Then, for v > 0 Hence, we can write the Fokker-Planck equation (1) in the equivalent form which enlightens the role of the logarithm of the quotient f /f ∞ , and In particular, owing to (46), the form (49) allows us to obtain the evolution equation which shows that F satisfies the equation If mass conservation is imposed on equation (1), we obtain at v = 0 the boundary conditions (3) and (4). In analogous way, the boundary conditions of the equivalent form (50) are now written in the form and In view of the decay property at v = 0 of the steady state f ∞ , the boundary conditions (51) and (52) are satisfied any time the solution to equation (50) is bounded together with its derivative at zero.

3.2.
A regularized problem. In order to apply in a rigorous way the strategy outlined in [15], for any given initial density f 0 (v) and positive constant δ 1, let us consider a regular approximation f δ 0 (v) satisfying the conditions Then, in a time interval (0, T ) the (unique) solution F δ (v, t) of the initial-boundary value problem for equation (50) corresponding to the initial value F δ where we denoted Indeed, a solution constant in the interval v ≤ δ satisfies both the boundary conditions (51) and (52), and equation (50), and converges to the right initial value as t → 0. Analogous conclusion can be drawn in the interval v ≥ 1/δ. Consider now the solution to equation (50) in a bounded interval (v − , v + ), where v − ≤ δ and v + ≥ 1/δ, with boundary conditions F (v − , t) = 1 and F (v + , t) = 1 for t ≤ T , and initial value F δ 0 = f δ 0 /f ∞ . Since in this interval the coefficient of the second-order term is strictly positive, the second condition in (55) follows from the maximum principle for the solution to a uniformly parabolic equation.
The previous discussion shows that, since the initial datum F δ 0 satisfies for some positive constants m δ < M δ , the same condition holds at any subsequent time t ≤ T , so that m δ ≤ F δ (v, t) ≤ M δ .
(57) As remarked in [15], condition (57) allows to recover rigorously the time decay of various Lyapunov functionals. To this extent, let us recall, for the sake of clarity, some results from [15]. The first of these results reads Theorem 3.1. Let the smooth function Φ(x), x ∈ R + be convex. Then, if F (v, t) is the solution to equation (50) in R + , and c ≤ F (v, t) ≤ C for some positive constants c < C, the functional Θ(F (t)) = I f ∞ (v)Φ(F (v, t)) dv is monotonically decreasing in time, and the following equality holds d dϑ Θ(F (t)) = −I Θ (F (t)), where I Θ denotes the nonnegative quantity Theorem 3.1 of [15] can be coupled with the so-called Chernoff inequality with weight [9,18], in the version recently proven in [15] (cf. Theorem 3.3). In our case, this result reads Theorem 3.2. Let X be a random variable distributed with density f ∞ (v), v ∈ R + , where the probability density function f ∞ satisfies the differential equality If the function φ is absolutely continuous on R + and φ(X) has finite variance, then with equality if and only if φ(X) is linear in X.
3.3. L 1 -convergence. Choose now Φ(x) = x log x, x ≥ 0 in Theorem 3.1. Then, Θ(F δ (t)) coincides with the entropy of f δ relative to f ∞ . If the relative entropy is finite at time t = 0, by Theorem 3.1 it decays, and its rate of decay is given by the expression If we apply inequality (61) with φ(v) = F δ (v, t) with fixed t > 0 we get On the other hand, as remarked in [17], whenever f and g are probability density functions, the square of their Hellinger distance The last inequality in (65) follows by Cauchy-Schwartz inequality. Finally, for t > 0