The fractional Poisson measure in infinite dimensions Habib - PDF document

The fractional Poisson measure in infinite dimensions Habib Ouerdiane Department of Mathematics Faculty of Sciences of Tunis University of Tunis El-Manar Tunis, Tunisia Symposium on Probability and Analysis . Institute of Mathematics, Academia Sinica, August 10-12, 2010, Taipei,Taiwan. 1

1 Introduction The Poisson measure π in R is given by π ( A ) = e − σ � σ n n ! n ∈ A the parameter σ being called the intensity . The Laplace transform of π is ∞ � � e λ · � σ n n ! e λn = e σ ( e λ − 1 ) = e − σ l π ( λ ) = E n =0 For n -tuples of independent Poisson variables one would have � σ k ( e λk − 1 ) l π ( λ ) = e Continuing λ k to imaginary arguments λ k = if k , the characteristic function is � σ k ( e ifk − 1 ) C π ( λ ) = e (1) Looked at as a renewal process, P ( X = n ) = e − σ σ n n ! would be the probability of n events occurring in the time interval σ . The survival probability, that is, the probability of no event is Ψ ( σ ) = e − σ which satisfies the first order differential equation d dσ Ψ ( σ ) = − Ψ ( σ ) (2) d Replacing in (2) the derivative dσ by the (Caputo) fractional derivative we obtain: � σ Ψ ′ ( τ ) 1 D α Ψ ( σ ) = ( σ − τ ) α dτ = − Ψ ( σ ) (0 < α < 1) Γ (1 − α ) 0 2

one has the solution Ψ ( σ ) = E α ( − σ α ) with E α being the Mittag-Leffler function of parameter α ∞ � z n E α ( z ) = Γ ( αn + 1) , z ∈ C (3) n =0 ( α > 0). One then obtains a fractional Poisson process [3], [13] with the probability of n events P ( X = n ) = σ αn n ! E ( n ) α ( − σ α ) E ( n ) denoting the n -th derivative of the Mittag-Leffler α function. In contrast with the Poisson case ( α = 1), this pro- cess has power law asymptotics rather than exponential, which implies that it is not anymore Markovian. The characteristic function of this process is given by � σ α � �� e iλ − 1 C α ( λ ) = E α In the next we develop an infinite-dimensional general- ization of the fractional Poisson measure and its analysis. 3

2 Complete monotonicity of the Mittag-Leffler function for complex arguments A positive C ∞ -function f is said to be completely mono- tone if for each k ∈ N 0 ( − 1) k f ( k ) ( t ) ≥ 0 , ∀ t > 0 According to Bernstein’s theorem (see e.g. [4, Chapter XIII.4 Theorem 1]), for functions f such that f (0 + ) = 1 the complete monotonicity property is equivalent to the existence of a probability measure ν on R + 0 such that � ∞ e − tτ dν ( τ ) < ∞ , f ( t ) = ∀ t > 0 0 H. Pollard in [16] proved the complete monotonicity of E α , 0 < α < 1, for non-positive real arguments showing that � ∞ e − tτ dν α ( τ ) , E α ( − t ) = ∀ t ≥ 0 (4) 0 for ν α being the probability measure on R + 0 dν α ( τ ) := α − 1 τ − 1 − 1 /α f α ( τ − 1 /α ) dτ (5) where f α is the α -stable probability density given by � ∞ e − tτ f α ( τ ) dτ = e − t α , 0 < α < 1 0 The complete monotonicity property and the integral representation (4) of E α may be extended to complex arguments. Lemma 1 For any z ∈ C such that Re( z ) ≥ 0 , the fol- lowing representation holds � ∞ e − zτ dν α ( τ ) , E α ( − z ) = 0 < α ≤ 1 0 4

Proof. According to [16], for each 0 < α < 1 fixed, for all t ≥ 0 one has � ∞ e − tτ dν α ( τ ) , E α ( − t ) = 0 � ∞ ∞ � ( − t ) n τ n dν α ( τ ) = (6) n ! 0 n =0 Comparing (6) with the Taylor expansion (3) of E α , one concludes that the moments of the measure ν α are given by � ∞ n ! τ n dν α ( τ ) = m n ( ν α ) := n ∈ N 0 Γ( αn + 1) , 0 For complex values z let � ∞ e − zτ dν α ( τ ) I ( − z ) := 0 which is finite provided Re( z ) ≥ 0. For each z ∈ C such that Re( z ) ≥ 0 one then obtains �� ∞ � ∞ ∞ ∞ � ( − z ) n � ( − z ) n � ( − z ) n τ n dν α ( τ ) I ( − z ) = = m n ( ν α ) = n ! n ! Γ( αn + 1) 0 n =0 n =0 n =0 =E α ( − z ) , leading to the integral representation � ∞ e − zτ dν α ( τ ) E α ( − z ) = 0 for all z ∈ C such that Re( z ) ≥ 0. � 5

3 Infinite-dimensional fractional Poisson mea- sures For the Poisson measure ( α = 1) an infinite-dimensional generalization is obtained by generalizing (1) to � ( e iϕ ( x ) − 1 ) dµ ( x ) C ( ϕ ) = e (7) for test functions ϕ ∈ D ( M ), D ( M ) being the space of C ∞ -functions of compact support in a manifold M (fixed from the very beginning), and then using the Bochner- Minlos theorem to show that C is the Fourier transform of a measure on the distribution space D ′ ( M ). Because the Mittag-Leffler function is a “natural” gener- alization of the exponential function one conjectures that an infinite-dimensional version of the fractional Poisson measure would have a characteristic functional �� � ( e iϕ ( x ) − 1) dµ ( x ) C α ( ϕ ) := E α , ϕ ∈ D ( M ) (8) with µ a positive intensity measure fixed on the underly- ing manifold M . However, a priori it is not obvious that this is the Fourier transform of a measure on D ′ ( M ) nor that it corresponds to independent processes because the Mittag-Leffler function does not satisfy the factorization properties of the exponential. Similarly to the Poisson case, to carry out our con- struction and analysis in detail we always assume that M is a geodesically complete connected oriented (non- compact) Riemannian C ∞ -manifold, where we fix the corresponding Borel σ -algebra B ( M ), and µ is a non- atomic Radon measure, which we assume to be non- 6

degenerate (i.e., µ ( O ) > 0 for all non-empty open sets O ⊂ M ). Having in mind the most interesting applica- tions, we also assume that µ ( M ) = ∞ . Theorem 2 For each 0 < α ≤ 1 fixed, the functional C α in Eq. (8) is the characteristic functional of a probability µ on the distribution space D ′ ( M ) . measure π α Proof. That C α is continuous and C α (0) = 1 follows easily from the properties of the Mittag-Leffler function. To check the positivity one uses the complete monotonic- ity of E α , 0 < α < 1, which by Appendix A (Lemma 1) implies the integral representation � ∞ e − τz dν α ( τ ) E α ( − z ) = (9) 0 for any z ∈ C such that Re ( z ) ≥ 0, ν α being the proba- bility measure (5). Hence by (9) � ∞ � � � M dµ ( x ) ( 1 − e i ( ϕa − ϕb ) ) z ∗ C α ( ϕ a − ϕ b ) z ∗ e − τ a z b = dν α ( τ ) a z b 0 a,b a,b (10) Each one of the terms in the integrand corresponds to the characteristic function of a Poisson measure. Thus, for each τ the integrand is positive and therefore the spectral integral (10) is also positive. From the Bochner- Minlos theorem it then follows that C α is the character- istic functional of a probability measure π α µ on the mea- ′ ( M ) , C σ ( D ′ ( M ))), C σ ( D ′ ( M )) being the surable space ( D σ -algebra generated by the cylinder sets. For the α = 1 case see e.g. [6]. � 7

Introducing the fractional Poisson measure by the above ′ ( M ) , C σ ( D ′ ( M ))). approach yields a probability measure on ( D The next step is to find an appropriate support for the fractional Poisson measure. Using the analyticity of the Mittag-Leffler function one may informally rewrite (8) as � � � �� � n ∞ E ( n ) � − dµ ( x ) α e iϕ ( x ) dµ ( x ) C α ( ϕ ) = n ! n =0 � � � � ∞ E ( n ) � − dµ ( x ) α e i ( ϕ ( x 1 )+ ϕ ( x 2 )+ ··· + ϕ ( x n )) dµ ⊗ n = n ! n =0 � � � For the Poisson case ( α = 1) instead of E ( n ) − dµ ( x ) α � � � one would have exp − dµ ( x ) for all n , the rest being the same. Therefore one concludes that the main dif- ference in the fractional case ( α � = 1) is that a different weight is given to each n -particle space, but that a con- figuration space [1], [2] is also the natural support of the fractional Poisson measure. The explicit construction is made in next Section . Notice however that the different weights, multiplying the n -particle space measures, are physically quite signif- icant in that they have decays, for large volumes, much smaller than the corresponding exponential factor in the Poisson measure. 8

3.1 Fractional Poisson Measure as a mixture of a classical Poisson measures Using now the spectral representation (9) of the Mittag- Leffler function one may rewrite (8) as � ∞ � � � ( e iϕ ( x ) − 1) dµ ( x ) C α ( ϕ ) = exp τ dν α ( τ ) 0 with the integrand being the characteristic function of the Poisson measure π τµ , τ > 0. In other words, the char- acteristic functional (8) coincides with the characteristic � ∞ functional of the measure 0 π τµ dν α ( τ ). By uniqueness, this implies the following result: Theorem 3 The fractional measure π α µ admits the inte- gral decomposition � ∞ π α µ = π τµ dν α ( τ ) 0 i.e, the measure π α µ is an mixture of classical Poisson measures π τµ , τ > 0 . 9