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subordinator, we find the transition density by solving
an evolution equation with the (weak) Riemann  Liouville fractional derivative and show that the
Weyl fractional derivative is the negative of the adjoint to the Riemann  Liuoville (weak) fractional
derivative.
Keywords:evolution semigroups, fundamental solution, fractional derivative, subordinator
Introduction
In the series of papers
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[2,3,4,5]
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a technique to construct evolution semigroups(Tt)t≥0generated
by some operatorsLwas developed. In the frame of the suggested technique the following fact
was used: the identity
∫∞
s
Tt−s[ξ′(t) +Lξ(t)]dt=−ξ(s)(1)
is true for each \testfunction"ξ:R→Dom(L)and eachs∈R.
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1957
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The object(Tt)t≥0, satisfying the identity (1) with a given operatorLwas called
fundamental solutionof∂t+L. It was shown in the paper [1, Th. 4.1] that this object(Tt)t≥0is
indeed the semigroup generated byLand there are no other candidates except∂t+Lto fulfill
(1) with the given(Tt)t≥0. The technique of
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[2,3,4,5]
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was used in [1] in particular to discuss
evolution semigroups generated by additive perturbations of the(1/2)stable subordinator, i.e. of
the operatorLequal to the Weyl fractional derivative of order1/2.
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1979
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It was shown in the paper [1, Th. 4.1] that this object(Tt)t≥0is
indeed the semigroup generated byLand there are no other candidates except∂t+Lto fulfill
(1) with the given(Tt)t≥0. The technique of [2,3,4,5] was used in
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[1]
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in particular to discuss
evolution semigroups generated by additive perturbations of the(1/2)stable subordinator, i.e. of
the operatorLequal to the Weyl fractional derivative of order1/2.
This note is supposed to be an addition to the discussion of [1].
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2230
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The technique of [2,3,4,5] was used in [1] in particular to discuss
evolution semigroups generated by additive perturbations of the(1/2)stable subordinator, i.e. of
the operatorLequal to the Weyl fractional derivative of order1/2.
This note is supposed to be an addition to the discussion of
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[1]
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. The aim of this note is to clarify
the connection between the notion of fundamental solution presented above and the traditional
notion used in the Theory of Partial Differential Equations and in Functional Analysis (cf. [7])
Science and Education of the Bauman MSTU42
and to outline the interplay between transitional probabilities of stochastic processes, evolution
semigroups, evolution equation
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2454
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The aim of this note is to clarify
the connection between the notion of fundamental solution presented above and the traditional
notion used in the Theory of Partial Differential Equations and in Functional Analysis (cf.
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[7]
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)
Science and Education of the Bauman MSTU42
and to outline the interplay between transitional probabilities of stochastic processes, evolution
semigroups, evolution equations and their fundamental solutions.
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To this end
we show that the Weyl fractional derivative is the negative of the adjoint to the Riemann{Liuoville
(weak) fractional derivative and again arrive at the identity (1).
1. Notations and definitions
Below we will use standard techniques of Fourier analysis and the Schwarz theory of distributions, for which we refer the reader to
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[7]
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.
LetD(Rd) :=C∞c(Rd)be the space of test functions, i.e. infinitely smooth functions with
compact supports. LetD′(Rd)be the space of all generalized functions (distributions) onRd, i.e.
the space dual toD(Rd)taken with the standard topology.
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Science and Education of the Bauman MSTU46
Heref0∈ S(R)and the problem is wellposed inL(R). The theory of evolution semigroups
provides the solution of (3) in the form
f(t,x) =T∗tf0(x).
This classical Cauchy problem (3) can be also transformed into the generalized one in a standard
way (see
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[7]
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): letf(t,x)be a solution of (3). For eacht <0and allx∈Rddefinef(t,x) := 0
and consider the functionfas an element ofS′(R×Rd). Then the weak derivative∂tfoffwith
respect to the variabletis calculated as follows:
∂tf(t,x) =
∂f
∂t
(t,x) +f0(x)δ(t),
here
∂f
∂t
is the classical derivative andf0is the initial data of the Cauchy problem (3).
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And hence
the solution of the classical Cauchy problem (3) solves the equation
∂tf(t,x)−L∗f(t,x) =f0(x)δ(t)(4)
inS′(R×Rd).
Assume that a fundamental solutionEof the operator∂t−L∗inS′(R×Rd)exists. IfL∗is
a local operator than our assumption is true (see
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[7]
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) and the solution of (4) is given then by the
Duhamel formula
f(t,x) = [E(t,x)]∗[f0(x)δ(t)] = [E(t,·)∗f0](x).(5)
Proposition 2.The Duhamel formula (5) is also true in the case whenL∗is the generator of
the Levy processXt, whose symbolψis of classC∞(Rd).
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