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Green Measures for a Class of Non-markov Processes

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01 April 2024

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Abstract

In this paper, we investigate the Green measure for a class of non-Gaussian processes in $\mathbb{R}^{d}$. These measures are associated with the family of generalized grey Brownian motions $B_{\beta,\alpha}$, $0

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

The goal of this paper (see Theorem 1 and Corollary 1 below) is to prove the existence of the Green measure for a class of non-Gaussian processes in

R^{d}

, called generalized grey Brownian motion (ggBm for short). We denote this family of processes by

B_{β, α}

with parameters

0 < β \leq 1

and

0 < α \leq 2

. More precisely, for a Borel function

f : R^{d} ⟶ R

, the potential of f (see [1,2] for details) is defined as

V_{β, α} (f, x) = \int_{0}^{\infty} E [f (x + B_{β, α} (t))] d t, x \in R^{d} .

(1)

We would like to investigate the class of functions f for which the potential of f has the representation

V_{β, α} (f, x) = \int_{R^{d}} f (y) G_{β, α} (x, d y),

(2)

where

G (x, \cdot) : = G_{β, α} (x, \cdot)

is a Radon measure on

R^{d}

called Green measure corresponding to the ggBm

B_{β, α}

, see Definition 2 below. First, we establish the existence of the perpetual integral (cf. Theorem 1)

\int_{0}^{\infty} f (x + B_{β, α} (t)) d t

with probability 1. This leads to an explicit representation of the Green measure for ggBm, namely (cf. Corollary 1)

G_{β, α} (x, d y) = \frac{D}{{| x - y |}^{d - 2 / α}} d y, d α > 2, 1 < α \leq 2,

where D is a constant that depends on

β, α

, and the dimension d; see Theorem 1 for the explicit expression. Note that as

d α > 2

and

1 < α \leq 2

, the Green measure

G_{β, α} (x, \cdot)

exists for

d \geq 2

, since

d > 2 / α \in [1, 2)

. The Brownian case (

α = 1)

is covered only for

d \geq 3

We emphasize that the existence of the Green measure for a given process X is not always guaranteed. As an example, the d-dimensional Brownian motion (Bm) starting at

x \in R^{d}

has a density given by

p_{t} (x, y) = {(2 π t)}^{- d / 2} exp ({- | x - y |}^{2} / (2 t))

y \in R^{d}

. It is not difficult to see that

\int_{0}^{\infty} p_{t} (x, y) d t

does not exist for

d = 1, 2

. This implies the non-existence of the Green measure of Bm for

d = 1, 2

. On the other hand, for

d \geq 3

, the Green measure of Bm on

R^{d}

exists and is given by

G (x, d y) = {C (d) | x - y |}^{2 - d} d y

, where

C (d)

is a constant depending on the dimension d; see [3] and the references therein for more details. In a two-dimensional space, the Green measure of ggBm is determined by the parameter

α

. The Green measure of ggBm for

d = 1

requires further analysis (for Bm see [4], Ch. 4) which we will postpone for a future paper.

The paper is organized as follows. In Section 2 we recall the definition and main properties of ggBm that will be needed later. In Section 3 we show the existence of the perpetual integral with probability 1, which leads to the explicit formula for the Green measure for ggBm. In Section 4, we discuss the results obtained, connect them with other topics, and draw conclusions.

2. Generalized Grey Brownian Motion

We recall the class of non-Gaussian processes, called the generalized grey Brownian motion, which we study below. This class of processes was first introduced by Schneider [5,6], and was generalized by Mura et al. (see [7,8]) as a stochastic model for slow/fast anomalous diffusion described by the time fractional diffusion equation.

2.1. Definition and Properties

For

0 < β \leq 1

the Mittag-Leffler (entire) function

E_{β}

is defined by the Taylor series

E_{β} (z) : = \sum_{n = 0}^{\infty} \frac{z^{n}}{Γ (β n + 1)}, z \in C,

(3)

where

Γ (z) = \int_{0}^{\infty} t^{z - 1} e^{- t} d t, z \in C, Re (z) \geq 0

is the Euler gamma function.

The M-Wright function is a special case of the class of Wright functions

W_{λ, μ}

λ > - 1

μ \in C

via

M_{β} (z) : = W_{- β, 1 - β} (- z) = \sum_{n = 0}^{\infty} \frac{{(- z)}^{n}}{n! Γ (- β n + 1 - β)} .

The special choice

β = 1 / 2

yields the Gaussian density on

[0, \infty)

M_{1 / 2} (z) = \frac{1}{\sqrt{π}} \exp (- \frac{z^{2}}{4}) .

(4)

The Mittag-Leffler function

E_{β}

is the Laplace transform of the M-Wright function, that is,

E_{β} (- s) = \int_{0}^{\infty} e^{- s τ} M_{β} (τ) d τ .

(5)

The generalized moments of the density

M_{β}

of order

δ > - 1

are finite and are given (see [7]) by

\int_{0}^{\infty} τ^{δ} M_{β} (τ) d τ = \frac{Γ (δ + 1)}{Γ (β δ + 1)} .

(6)

Definition 1.

Let

0 < β \leq 1

and

0 < α \leq 2

be given. A d-dimensional continuous stochastic process

B_{β, α} = \{B_{β, α} (t), t \geq 0\}

starting at

0 \in R^{d}

and defined on a complete probability space

(Ω, F, P)

, is a ggBm in

R^{d}

(see [8] for

d = 1

) if:

$P (B_{β, α} (0) = 0) = 1$ , that is, $B_{β, α}$ starts at zero $P$ -almost surely ( $P$ -a.s.).
Any collection $\{B_{β, α} (t_{1}), \dots, B_{β, α} (t_{n})\}$ with $0 \leq t_{1} < t_{2} < \dots < t_{n} < \infty$ has a characteristic function given, for any $θ = (θ_{1}, \dots, θ_{n}) \in {(R^{d})}^{n}$ with $θ_{k} = (θ_{k, 1}, \dots θ_{k, d})$ , $k = 1, \dots, n$ , by

$E [\exp (i \sum_{k = 1}^{n} {(θ_{k}, B_{β, α} (t_{k}))}_{R^{d}})] = E_{β} [- \frac{1}{2} \sum_{j = 1}^{d} {(θ_{., j}, γ_{α} θ_{., j})}_{R^{n}}],$

(7)

where $E$ denotes the expectation w.r.t. $P$ and

$γ_{α} : = γ_{α, n} : = {(t_{k}^{α} + t_{j}^{α} - {| t_{k} - t_{j} |}^{α})}_{k, j = 1}^{n} .$
The joint probability density function of $(B_{β, α} (t_{1}), \dots, B_{β, α} (t_{n}))$ is equal to

$ρ_{β} (θ, γ_{α}) = \frac{{(2 π)}^{- \frac{n d}{2}}}{{(\det γ_{α})}^{d / 2}} \int_{0}^{\infty} τ^{- \frac{n d}{2}} e^{- \frac{1}{2 τ} \sum_{j = 1}^{d} {(θ_{\cdot, j}, γ_{α}^{- 1} θ_{\cdot, j})}_{R^{n}}} M_{β} (τ) d τ .$

(8)

The following are the most important key properties of ggBm:

(P1).: For each $t \geq 0$ , the moments of any order of $B_{β, α} (t)$ are given by

$\{\begin{matrix} E [| B_{β, α} {(t) |}^{2 n + 1}] & = 0, \\ E [| B_{β, α} {(t) |}^{2 n}] & = \frac{(2 n)!}{2^{n} Γ (β n + 1)} t^{α n} . \end{matrix}$
(P2).: The covariance function has the form

$E [(B_{β, α} (t), B_{β, α} (s))] = \frac{d}{2 Γ (β + 1)} (t^{α} + s^{α} - {| t - s |}^{α}), t, s \geq 0 .$

(9)
(P3).: For each $t, s \geq 0$ , the characteristic function of the increments is

$E [e^{i (k, B_{β, α} (t) - B_{β, α} (s))}] = E_{β} (- \frac{{| k |}^{2}}{2} {| t - s |}^{α}), k \in R^{d} .$

(10)
(P4).: The process $B_{β, α}$ is non-Gaussian, $α / 2$ -self-similar with stationary increments.
(P5).: The ggBm is not a semimartingale. Furthermore, $B_{α, β}$ cannot be of finite variation in $[0, 1]$ and, by scaling and stationarity of the increment, on any interval in $R^{+}$ .
(P5).: For $n = 1$ , the density $ρ_{β} (x, t)$ , $x \in R^{d}$ , $t > 0$ , is the fundamental solution of the following fractional differential equation (see [9])

$D_{t}^{2 β} ρ_{β} (x, t) = Δ_{x} ρ_{β} (x, t),$

where $Δ_{x}$ is the d-dimensional Laplacian in x and $D_{t}^{2 β}$ is the Caputo-Dzherbashian fractional derivative; see [10] for the definition and properties.

2.2. Representations of Generalized Grey Brownian Motion

The ggBm admits different representations in terms of well-known processes. It follows from (7) that ggBm has an elliptical distribution, see Section 2 in [11] or Section 3 in [12]. On the other hand, ggBm is also given as a product (see [7] for

d = 1

) of two processes as follows

\{B_{β, α} (t), t \geq 0\} \overset{L}{=} \{\sqrt{Y_{β}} B^{α / 2} (t), t \geq 0\} .

(11)

Here,

\overset{L}{=}

means equality in law, the nonnegative random variable

Y_{β}

has density

M_{β}

and

B^{α / 2}

is a d-dimensional fBm with Hurst parameter

α / 2

and independent of

Y_{β}

We give another representation of ggBm

B_{β, α}

as a subordination of fBm (see Prop. 2.14 in [13] for

d = 1

) which is used below. For completeness, we give the short proof.

Proposition 1.

The ggBm has the following representation

\{B_{β, α} (t), t \geq 0\} \overset{L}{=} \{B^{α / 2} (t Y_{β}^{1 / α}), t \geq 0\} .

(12)

Proof.

We must show that both representations (11) and (12) have the same finite-dimensional distribution. For every

θ = (θ_{1}, \dots, θ_{n}) \in {(R^{d})}^{n}

, we have

\begin{matrix} E [exp (i \sum_{k = 1}^{n} (θ_{k}, B^{α / 2} (t_{k} Y_{β}^{1 / α})))] & = & \int_{0}^{\infty} E [exp (i \sum_{k = 1}^{n} (θ_{k}, B^{α / 2} (t_{k} y^{1 / α})))] M_{β} (y) d y \\ = & \int_{0}^{\infty} E [exp (i \sum_{k = 1}^{n} (θ_{k}, y^{1 / 2} B^{α / 2} (t_{k})))] M_{β} (y) d y \\ = & E [exp (i \sum_{k = 1}^{n} (θ_{k}, Y_{β}^{1 / 2} B^{α / 2} (t_{k})))] . \end{matrix}

In the second equality, we used the

α / 2

-self-similarity of fBm. This completes the proof. □

3. The Green Measure for Generalized Grey Brownian Motion

In this section we show the existence of the Green measure for the ggBm, see (1) and (2). Let us begin by discussing the existence of the Green measure for a general stochastic process X.

Let

X = {X (t), t \geq 0}

be a stochastic process in

R^{d}

starting from

x \in R^{d}

. If

X (t)

t \geq 0

, has a probability distribution

ρ_{X (t)} (x, \cdot)

, then Eq. (1) becomes

V_{X} (x, f) = \int_{0}^{\infty} \int_{R^{d}} f (y) ρ_{X (t)} (x, d y) d t .

(13)

Then, applying the Fubini theorem, the Green measure

G_{X} (x, \cdot)

of X is given by

G_{X} (x, d y) = \int_{0}^{\infty} ρ_{X (t)} (x, d y) d t,

assuming the existence of

G_{X} (x, \cdot)

as a Radon measure on

R^{d}

. That is, for every bounded Borel set

B \in B_{b} (R^{d})

we have

G_{X} (x, B) = \int_{0}^{\infty} ρ_{X (t)} (x, B) d t < \infty .

If the probability distribution

ρ_{X (t)} (x, \cdot)

is also absolutely continuous with respect to the Lebesgue measure, say

ρ_{X (t)} (x, d y) = ρ_{t} (x, y) d y

, then the function

g_{X} (x, y) : = \int_{0}^{\infty} ρ_{t} (x, y) d t, \forall y \in R^{d},

(14)

is called the Green function of the stochastic process X. Moreover, the Green measure in this case is given by

G_{X} (x, d y) = g_{X} (x, y) d y

This leads us to the following definition of the Green measure of a stochastic process

X .

Definition 2.

Let

X = {X (t), t \geq 0}

be a stochastic process on

R^{d}

starting from

x \in R^{d}

and

ρ_{X (t)} (x, \cdot)

be the probability distribution of

X (t)

t \geq 0

. The Green measure of X is defined as a Radon measure on

R^{d}

G_{X} (x, B) : = \int_{0}^{\infty} ρ_{X (t)} (x, B) d t, B \in B_{b} (R^{d}),

\int_{R^{d}} f (y) G_{X} (x, d y) = \int_{R^{d}} f (y) \int_{0}^{\infty} ρ_{X (t)} (x, d y) d t, f \in C_{0} (R^{d})

whenever these integrals exist.

In other words,

G_{X} (x, B)

is the expected length of time the process remains in B.

In order to state the main theorem which establishes the existence of the Green measure for ggBm, first, we introduce a proper Banach space of functions

f : R^{d} ⟶ R

such that

\int_{0}^{\infty} f (x + B_{β, α} (t)) d t

is finite

P

-a.s. Without loss of generality, we may assume that

f \geq 0

above. We define the space

C L (R^{d})

of continuous real-valued on

R^{d}

C L (R^{d}) : = \{f : R^{d} \to R ∣ f is continuous, bounded and f \in L^{1} (R^{d})\} .

The space

C L (R^{d})

becomes a Banach space with the norm

{∥ f ∥}_{C L} : = {∥ f ∥}_{\infty} + {∥ f ∥}_{1}, \forall f \in C L (R^{d}),

where

{∥ \cdot ∥}_{\infty}

denotes the sup-norm and

{∥ \cdot ∥}_{1}

is the norm in

L^{1} (R^{d}

). The choice of

C L (R^{d})

allows us to show that the family of random variables (also known as perpetual integral functionals)

\int_{0}^{\infty} f (x + B_{β, α} (t)) d t, f \in C L (R^{d})

have finite expectations

P

-a.s.

Theorem 1.

Let

f \in C L (R^{d})

and

x \in R^{d}

be given and consider ggBm

B_{β, α}

with

d α > 2

and

1 < α \leq 2

. Then, the perpetual integral functional

\int_{0}^{\infty} f (x + B (t)) d t

is finite

P

-a.s. and its expectation equals

\begin{matrix} E [\int_{0}^{\infty} f (x + B_{β, α} (t)) d t] = D \int_{R^{d}} \frac{f (x + y)}{{| y |}^{d - 2 / α}} d y, \end{matrix}

(15)

where

D = D (β, α, d) = \frac{1}{α} 2^{- 1 / α} π^{- \frac{d}{2}} Γ (\frac{d}{2} - \frac{1}{α}) \frac{Γ (1 - \frac{1}{α})}{Γ (1 - \frac{β}{α})}

Proof.

Given

x \in R^{d}

and

f \in C L (R^{d})

non-negative, let

ρ_{β} (\cdot, t^{α})

denote the density of

B_{β, α} (t)

t \geq 0

, given by (see (8) with

n = 1

)

ρ_{β} (y, t^{α}) = \frac{1}{{(2 π t^{α})}^{d / 2}} \int_{0}^{\infty} τ^{- d / 2} e^{- \frac{{| y |}^{2}}{2 t^{α} τ}} M_{β} (τ) d τ, y \in R^{d} .

First, we show the equality (15). It follows from the above considerations that

\begin{matrix} E [\int_{0}^{\infty} f (x + B_{β, α} (t)) d t] = \int_{0}^{\infty} \int_{R^{d}} f (x + y) ρ_{t}^{β, α} (y) d y d t . \\ = \int_{0}^{\infty} \int_{R^{d}} f (x + y) \frac{1}{{(2 π t^{α})}^{d / 2}} \int_{0}^{\infty} τ^{- d / 2} M_{β} (τ) e^{- \frac{{| y |}^{2}}{2 t^{α} τ}} d τ d y d t . \end{matrix}

Using Fubini’s Theorem, we first compute the t-integral and use the assumption

d α > 2

. We obtain

\int_{0}^{\infty} \frac{1}{{(2 π t^{α} τ)}^{d / 2}} e^{- \frac{{| y |}^{2}}{2 t^{α} τ}} d t = C (α, d) \frac{τ^{- \frac{1}{α}}}{{| y |}^{d - 2 / α}},

where

C (α, d) : = \frac{1}{α} 2^{- 1 / α} π^{- \frac{d}{2}} Γ (\frac{d}{2} - \frac{1}{α}) .

Next we compute the

τ

-integral using (6) so that

\int_{0}^{\infty} τ^{- 1 / α} M_{β} (τ) d τ = \frac{Γ (1 - \frac{1}{α})}{Γ (1 - \frac{β}{α})}, α > 1 .

Combining gives

E [\int_{0}^{\infty} f (x + B_{β, α} (t)) d t] = D \int_{R^{d}} \frac{f (x + y)}{{| y |}^{d - 2 / α}} d y,

where

D = D (β, α, d) = C (α, d) \frac{Γ (1 - \frac{1}{α})}{Γ (1 - \frac{β}{α})}

. Therefore, the equality (15) is shown.

Now we show that the right-hand side of (15) is finite for every non-negative

f \in C L (R^{d})

. To see this, we may use the local integrability of

{| y |}^{d - 2 / α}

in y and obtain

\begin{matrix} \int_{R^{d}} \frac{f (x + y)}{{| y |}^{d - 2 / α}} d y & = & \int_{{| y | \leq 1}} \frac{f (x + y)}{{| y |}^{d - 2 / α}} d y + \int_{{| y | > 1}} \frac{f (x + y)}{{| y |}^{d - 2 / α}} d y \\ \leq & C_{1} {∥ f ∥}_{\infty} + C_{2} {∥ f ∥}_{1} \leq C {∥ f ∥}_{C L} . \end{matrix}

Therefore, the integral in (15) is, in fact, well defined. In other words, the integral

\int_{0}^{\infty} f (x + B_{β, α} (t)) d t

exists with probability 1. This completes the proof. □

As a consequence of the above theorem, we immediately obtain the Green measure of ggBm

B_{β, α}

, that is, comparing (2) and (15).

Corollary 1.

The Green measure of ggBm

B_{β, α}

for

d α > 2

is given by

G_{β, α} (x, d y) = \frac{D}{{| x - y |}^{d - 2 / α}} d y .

Remark 1.

It is possible to show that given $f \neq 0$ , the perpetual integral $\int_{0}^{\infty} f (x + B_{β, α} (t)) d t$ is a non-constant random variable. As a consequence, for $f \geq 0$ the variance of $\int_{0}^{\infty} f (x + B_{β, α} (t)) d t$ is strictly positive. The proof uses the notion of conditional full support of ggBm. We will not provide a detailed explanation of this result that closely follows the ideas of Theorem 2.2 in [3] to which we address the interested readers.
Note also that the functional in (1)

$V_{β, α} (\cdot, x) : C L (R^{d}) ⟶ R$

is continuous. In fact, from the proof of Theorem 1 for any $f \in C L (R^{d})$ yields

$| V_{β, α} {(f, x) | \leq K ∥ f ∥}_{C L},$

where K is a constant depending on the parameters $β, α$ , and d.

4. Discussion and Conclusions

We have derived the Green measure for the class of stochastic processes called the generalized grey Brownian motion in Euclidean space

R^{d}

for

d \geq 2

. This class includes, in particular, fractional Brownian motion and other non-Gaussian processes. To address the case where

d = 1

, a renormalization process is needed. However, this will be postponed to future work. For

β = α = 1

ggBm

B_{1, 1}

is nothing but a Brownian motion. In this case, the Green measure exists for

d \geq 3

The relationship between the Green measure and the local time of the ggBm can be described as follows. For any

T > 0

and a continuous function

f : R^{d} ⟶ R

, the integral functional

\int_{0}^{T} f (B_{β, α} (t)) d t

(16)

is well-defined. For

d = 1

the integral (16) with

f \in L^{1} (R)

is represented as

\int_{0}^{T} f (B_{β, α} (t)) d t = \int_{R} f (x) L_{β, α} (T, x) d x,

where

L_{β, α} (T, x)

is the local time of ggBm up to time T at the point x, see [11,12]. The Green measure corresponds to the asymptotic behaviour in T of the expectation of local time

L_{β, α} (T, x)

. The existence of this asymptotic depends on the dimension d and the transient or recurrent properties of the process.

Author Contributions

Methodology, H.P.S. and J.L.S.; Investigation, H.P.S. and J.L.S. Both authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by FCT-Fundação para a Ciência e a Tecnologia, Portugal grant number UIDB/MAT/04674/2020, https://doi.org/10.54499/UIDB/04674/2020 through the Center for Research in Mathematics and Applications (CIMA) related to the Statistics, Stochastic Processes and Applications (SSPA) group.

Conflicts of Interest

The authors declare no conflicts of interest.

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