In this paper we consider various forms of tempered fractional derivatives. For a function f continuous and compactly supported on the positive real line, let us consider the Marchaud type operator defined by (1.1) (Dα,ηf)(x)=∫0∞(f(x)−f(x−y))Π(dy) where (1.2) Π(dy)=αΓ(1−α)e−ηyyα+1dy,y>0, 0,\]]]> with η>0,0<α<1 0,0<\alpha <1$]]>. The operator (1.1) coincides with the classical Marchaud derivative for η=0 .

The Laplace transform of the fractional operator (1.1) reads (1.3) ∫0∞e−λx(Dα,ηf)(x)dx=(∫0∞(1−e−λy)Π(dy))f˜(λ)=((η+λ)α−ηα)f˜(λ). Throughout the work we denote by f˜ the Laplace transform of f. In the Fourier analysis the factor (η+iλ)α−ηα is the multiplier of the Fourier transform of f [7]. Tempered fractional derivatives emerge in the study of equations driving the tempered subordinators [1, 7]. In particular, the operator (1.1) is the generator of the subordinator Ht,t>0 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−ηα)=e−t∫0∞(1−e−λy)Π(dy),λ>0. 0.\end{aligned}\]]]> The process Ht 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 Dα,ηf defined in (1.1) is related to the tempered upper Weyl derivatives defined by (1.4) (Dˆ+α,ηf)(x)=1Γ(1−α)ddx∫−∞xf(t)(x−t)αe−η(x−t)dt. By combining (1.4) with the lower Weyl tempered derivatives we obtain the Riesz tempered fractional derivatives ∂α,ηf∂|x|α from which we obtain the explicit Fourier transform in (2.5).

We consider the Dzherbashyan–Caputo derivative of order 12 , that is, (1.5) (D12f)(t)=1π∫0tf′(s)(t−s)−12ds with the Laplace transform ∫0∞e−λt(D12f)(t)dt=λ12f˜(λ)−λ12−1f(0),λ>0. 0.\]]]> The relationship between the Riemann–Liouville and the Dzherbashyan–Caputo derivative can be given as follows, (1.6) (D12f)(t)=(D12f)(t)+t12−1Γ(12)f(0), from which we observe that (1.7) ∫0∞e−λt(D12f)(t)dt=λ12f˜(λ). We remark that the problems (D12u)(t)=−∂u∂y,t>0,y>0u(0,y)=δ(y)and(D12u)(t)=−∂u∂y,t>0,y>0u(0,y)=δ(y)u(t,0)=1πt,t>0 0,y>0\\ {} u(0,y)=\delta (y)\end{array}\right.\hspace{1em}\text{and}\hspace{1em}\left\{\begin{array}{l@{\hskip10.0pt}l}\big({\mathcal{D}^{\frac{1}{2}}}u\big)(t)=-\displaystyle \frac{\partial u}{\partial y},& t>0,y>0\\ {} u(0,y)=\delta (y)\\ {} u(t,0)=\displaystyle \frac{1}{\sqrt{\pi t}},& t>0\end{array}\right.\]]]> have a unique solution given by the density law of an inverse to a stable subordinator, say Lt (see for example [2, formulas 3.4 and 3.5]). It is well known that Lt (with L0=0 ) is identical in law to a folded Brownian motion |Bt| (with B0=0 ), that is, u is the unique solution to the problem ∂u∂t=∂2u∂y2,t>0,y>0,u(0,y)=δ(y),∂u∂y(t,0)=0. 0,\hspace{0.1667em}y>0,\\ {} u(0,y)=\delta (y),\\ {} \displaystyle \frac{\partial u}{\partial y}(t,0)=0.\end{array}\right.\]]]> 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 (1.8) Dt12,ηf:=e−ηtDt12(eηtf)−ηf the tempered Riemann–Liouville type derivative. The equality between definitions (1.8) and (1.1) can be verified by comparing the corresponding Laplace transforms. Indeed, from (1.7), ∫0∞e−λtDt12,ηfdt=∫0∞e−(λ+η)tDt12(g)dt−ηf˜(λ)=λ+η∫0∞e−(λ+η)tg(t)dt−ηf˜(λ) where g(t)=eηtf(t) .

Let B represent a Brownian motion starting at the origin with generator Δ. In the paper we show that the transition density u=u(x,y,t) of the 1-dimensional process Bμ(t)=B(t)+μt+x,μ>0,x∈R, 0,\hspace{0.1667em}x\in \mathbb{R},\]]]> satisfies the fractional equation on (0,∞)×R2 (1.9) Dt12,ηu+ηu=a(x,y)(∂u∂x+ηu)=−a(x,y)(∂u∂y−ηu),u(x,y,0)=δ(x−y) where a(x,y)=1(−∞,y](x)−1(y,∞)(x) and η=μ24.

A different result concerns the reflected process |Bμ(t)+μt|+x=|Bˆ|μ(t) whose transition density v=v(x,y,t) satisfies the equation (1.10) Dt12,ηv+ηv=∂v∂x+ηtanh(η(y−x))v,t>0,y>x>0, 0,\hspace{0.2778em}y>x>0,\]]]> with initial and boundary conditions v(x,y,0)=δ(y−x),v(x,x,t)=e−ηtπt,t>0, 0,\end{aligned}\]]]> and η=μ24. The fractional equation governing the iterated Brownian motion Bμ2(|Bμ1(t)|) ( Bμj being independent) has been studied in [6] and in the special case Bμ(|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 [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 forms 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.

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 (2.1) (Dˆ+α,ηf)(x)=1Γ(1−α)ddx∫−∞xf(t)(x−t)αe−η(x−t)dt=1Γ(1−α)ddx∫0∞f(x−t)tαe−ηtdt=1Γ(1−α)ddx∫0∞f(x−t)e−ηt∫t∞αw−α−1dwdt=1Γ(1−α)∫0∞αw−α−1dw∫0wf′(x−t)e−ηtdt=1Γ(1−α)∫0∞αw−α−1dw∫x−wxf′(t)e−η(x−t)dt=1Γ(1−α)∫0∞αw−α−1e−ηx{f(t)eηt|x−wx−η∫x−wxf(t)eηtdt}dw=1Γ(1−α)∫0∞αw−α−1e−ηx[f(x)eηx−f(x−w)eη(x−w)]dw−ηΓ(1−α)∫0∞αw−α−1∫x−wxf(t)e−η(x−t)dwdt=1Γ(1−α)∫0∞αw−α−1[f(x)−f(x−w)e−ηw]dw−ηΓ(1−α)∫0∞αw−α−1∫0wf(x−t)e−ηtdwdt=1Γ(1−α)∫0∞αw−α−1[f(x)+f(x)e−ηw−f(x)e−ηw−f(x−w)e−ηw]dw−ηΓ(1−α)∫0∞f(x−t)e−ηt∫t∞αw−α−1dwdt=1Γ(1−α)∫0∞(f(x)−f(x−w))αe−ηwwα+1dwdt+f(x)Γ(1−α)∫0∞αw−α−1(1−e−ηw)dw−ηΓ(1−α)∫0∞f(x−t)e−ηttαdt=∫0∞(f(x)−f(x−w))Π(dw)+η∫0∞(f(x)−f(x−w))e−ηwwαΓ(1−α)dw The derivative Dˆ+α,η can be expressed in terms of Dα,η as follows: Dˆ+α,ηf=Dα,ηf−ηDα−1,ηf. In the same way we can obtain the upper Weyl derivative in the Marchaud form as (2.2) (Dˆ−α,ηf)(x)=1Γ(1−α)ddx∫x∞f(t)(x−t)αe−η(x−t)dt=∫0∞αw−α−1Γ(1−α){e−ηwf(x+w)−f(x)}dw+η∫0∞eηwf(x+w)Γ(1−α)wαdw=1Γ(1−α)∫0∞[f(x+w)−f(x)]αe−ηwwα+1+f(x)Γ(1−α)∫0∞αw−α−1(e−ηw−1)dw+η∫0∞f(x+t)e−ηwΓ(1−α)tαdt=∫0∞[f(x+w)−f(x)]Π(dw)+η∫0∞[f(x+w)−f(x)]e−ηwΓ(1−α)wαdw. For 0<α<1 the Riesz fractional derivative writes (2.3) ∂αf∂|x|α=−12cosαπ2Γ(1−α)∫−∞+∞f(t)|x−t|αdt=−12cosαπ2Γ(1−α)[ddx∫−∞xf(t)(x−t)αdt−ddx∫x∞f(t)(t−x)αdt]. In the same way we define the tempered Riesz derivative as ∂α,ηf∂|x|α=Cα,η[ddx∫−∞xf(t)(x−t)αe−η(x−t)Γ(1−α)dt−ddx∫x∞f(t)(t−x)αe−η(t−x)Γ(1−α)dt] where Cα,η is a suitable constant which will be defined below. In view of the previous calculations we have that (2.4) ∂α,ηf∂|x|α=Cα,η[∫0∞(f(x)−f(x−w))αe−ηwdwΓ(1−α)wα+1+η∫0∞(f(x)−f(x−w))e−ηwdwΓ(1−α)wα−∫0∞(f(x+w)−f(x))αe−ηwdwΓ(1−α)wα+1−η∫0∞(f(x+w)−f(x))e−ηwdwΓ(1−α)wα]=Cα,η[∫0∞(2f(x)−f(x−w)−f(x+w))αe−ηwdwΓ(1−α)wα+1+η∫0∞(2f(x)−f(x−w)−f(x+w))e−ηwdwΓ(1−α)wα]. We now evaluate the Fourier transform of the tempered Riesz derivative (2.5) ∫−∞+∞eiγx∂α,ηf∂|x|αdx=Cα,η{Fˆ(γ)∫0∞(1−eiγw)αe−ηwdwΓ(1−α)wα+1+ηFˆ(γ)∫0∞(1−eiγw)e−ηwdwΓ(1−α)wα−Fˆ(γ)∫0∞(eiγw−1)αe−ηwdwΓ(1−α)wα+1−ηFˆ(γ)∫0∞(eiγw−1)e−ηwdwΓ(1−α)wα}=Cα,ηFˆ(γ){2∫0∞(1−cosγw)αe−ηwdwΓ(1−α)wα+1+2η∫0∞(1−cosγw)e−ηwdwΓ(1−α)wα}=Cα,ηFˆ(γ){−2w−α(1−cosγw)e−ηwΓ(1−α)|0∞−2η∫0∞(1−cosγw)e−ηwdwwαΓ(1−α)+2γ∫0∞e−ηwsinγwwαΓ(1−α)dw+2η∫0∞(1−cosγw)e−ηwwαΓ(1−α)dw}=Cα,ηFˆ(γ)2|γ|∫0∞e−ηwsin|γ|wwαΓ(1−α)dw=Cα,ηFˆ(γ)2|γ|(η2+γ2)1−α2sin((1−α)arctan|γ|η). In the last step we used the following formula ([5], p. 490, formula 5) ∫0∞xμ−1e−βxsinδxdx=Γ(μ)(β2+δ2)μ2sin(μarctanδμ) with Reμ>−1 -1$]]>, Reβ≥Imδ . Remark 1.

For η→0 we have that limη→0sin((1−α)arctan|γ|η)=cos(πα2). Therefore limη→0∫−∞+∞eiγx∂α,ηf∂|x|αdx=2Cα,0|γ|αcos(πα2)Fˆ(γ) and thus the normalizing constant must be Cα,0=−(2cosπα2)−1 .

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=∂αu∂|x|α see, for example, [4], where the interplay between stable laws, including subordinators and inverse subordinators, and fractional equations is considered.

The law of the drifted Brownian motion started at x satisfies the equations ∂u∂t=∂2u∂y2−μ∂u∂y,t>0,y∈R, 0,y\in \mathbb{R},\]]]> and ∂u∂t=∂2u∂x2+μ∂u∂x,t>0,x∈R. 0,\hspace{0.2778em}x\in \mathbb{R}.\]]]> We show here that the drifted Brownian motion is related to time fractional equations with tempered derivatives. Let us consider the process (3.1) Bμ(t)=B(t)+μt+x,μ∈R,x∈R. The law u=u(x,y,t) of the process Bμ is given by (3.2) u(x,y,t)=e−(y−x−μt)24t4πt=e−(y−x)24t4πte−μ2t4+μ2(y−x),t>0,x,y∈R. 0,\hspace{0.2778em}x,y\in \mathbb{R}.\]]]>

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=μ2 .

We here consider the process (4.1) |B(t)+μt|+x=|Bμ(t)|+x,x>0. 0.\]]]> This process has distribution P(|B(t)+μt|+x<y)=P(x−y−μt<B(t)<y−x−μt)=∫x−y−μty−x−μte−w24t4πtdw and therefore its transition function is (4.2) P(|Bμ(t)|+x∈dy)/dy=e−(y−x−μt)24t4πt+e−(y−x+μt)24t4πt=e−(y−x)24t4πte−μ2t4[e−μ2(y−x)+eμ2(y−x)]=v(x,y,t) for y>x x$]]> and t>0 0$]]>. We now prove the following theorem.