Drifted Brownian motions governed by fractional tempered derivatives

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann--Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.


Introduction
In this paper we consider various forms of tempered fractional derivatives. For f continuous and compactly supported on the positive real line, let us consider the Marchaud type operator defined by e −ηy y α+1 dy , y > 0. (1.2) with η > 0, 0 < α < 1. Throughout the work we denote byf the Laplace transform of f . In the Fourier analysis the factor (η + iλ) α − η α is the multiplier of the Fourier transform of f ( [7]). Tempered fractional derivatives emerge in the study of the equations driving the tempered subordinators ( [1], [7]). In particular, the operator (1.1) is the generator of the subordinator H t , t > 0, with Lévy measure (1.2) and density law whose Laplace transform is given by (1.3), that is Ee −λHt = e −t((η+λ) α −η α ) = e −t ∞ 0 (1−e −λy )Π(dy) , λ > 0. The process H t is called relativistic subordinator and coincides, for η = 0, with a positively skewed Lévy process, that is a stable subordinator. Tempered stable subordinators can be viewed as the limits of Poisson random sums with tempered power law jumps ( [7]).
The fractional operator D α,η f defined in (1.1) is related to the tempered upper Weyl derivatives defined by (1.4) By combining (1.4) with the lower Weyl tempered derivatives we obtain the Riesz tempered fractional derivatives ∂ α,η f ∂|x| α of which we obtain the explicit Fourier transform in (2.5).
We consider the Dzherbashyan-Caputo derivative of order 1 2 that is The relationship between the Riemann-Liouville and the Dzherbashyan-Caputo derivative can be given as follows We remark that the problems have a unique solution given by the density law of an inverse to a stable subordinator, say L t (see for example [2, formulas 3.4 and 3.5]). It is well-known the L t (with L 0 = 0) is identical in law to a folded Brownian motion |B t | (with B 0 = 0), that is u is the unique solution to the problem Thus, by considering the theory of time changes, there exist interesting connections between fractional Cauchy problems and the domains of the generators of the base processes. In our view, concerning the drifted Brownian motion, the present paper gives new results also in this direction.
We denote by where g(t) = e ηt f (t).
Let B represents a Brownian motion starting at the origin with generator ∆. In the paper we show that the transition density u = u(x, y, t) of the 1-dimensional process A different result concerns the reflected process The fractional equation governing the iterated Brownian motion B µ2 (|B µ1 (t)|) (B µj being independent) has been studied in Iafrate and Orsingher [5] and in the special case B µ (|B(t)|) explicitly derived. For the iterated Bessel process a similar analysis is performed in [3]. A general presentation of tempered fractional calculus can be found in the paper by Sabzikar et al. [7]. Many processes like Brownian motion, iterated Brownian motion, Cauchy process have transition functions satisfying different partial differential equations and also are solutions of fractional equations of different form with various fractional derivatives. We here show that a similar situation arises when drifted reflecting Brownian motion is considered but in this case the corresponding fractional equations involve tempered Riemann-Liouville type derivatives.

A generalization of the tempered Marchaud derivative
In this section we study the tempered Weyl derivatives (upper and lower ones) and construct the Riesz tempered derivative. We are able to obtain the Fourier transform of the Riesz tempered derivatives and thus to solve some generalized fractional diffusion equation. We start by giving the explicit forms of the tempered Weyl derivative The derivativeD α,η + can be expressed in terms of D α,η as followŝ In the same way we can obtain the upper Weyl derivative in the Marchaud form as For 0 < α < 1 the Riesz fractional derivative writes In the same way we define the tempered Riesz derivative as where C α,η is a suitable constant which will be defined below. In view of the previous calculations we have that We now evaluate the Fourier transform of the tempered Riesz derivative In the last step we used the following formula (Gradshteyn and Ryzhik [4], page 490, formula 5) This means that for η → 0 we obtain from (2.5) the Fourier transform of the Riesz fractional derivative (2.3). This result shows that symmetric stable processes are governed by equations ∂u ∂t = ∂ α u ∂|x| α see for example [8], where the interplay between stable laws, including subordinators and inverse subordinators, and fractional equations is considered.

Remark 2.2. For fractional equations of the form
the Fourier transform of the solution reads +∞ −∞ e iγx u(x, t) dx = exp t C α,η 2|γ|

Fractional equations governing the drifted Brownian motion
The law of the drifted Brownian motion started at x satisfies the equations We show here that the drifted Brownian motion is related to time fractional equations with tempered derivatives. Let us consider the process The law u = u(x, y, t) of the process B µ is given by Theorem 3.1. The law of B µ solves the Cauchy problem Proof. We start by computing the Laplace-Fourier transform of the function By using the fact thatg we now compute the double transform of a(x, y) ∂g ∂x .
This implies, by inverting the Fourier transform, that We recall that This completes the proof.
The drifted Brownian motion has therefore a transition function satisfying a time fractional equation where the fractional derivative is a tempered Riemann-Liouville derivative with parameter η which is related to the drift by the relationship √ η = µ 2 .

Fractional equation governing the folded drifted Brownian motion
We here consider the process This process has distribution and therefore its transition function is for y > x and t > 0. We now prove the following theorem.