# The 10 reference contexts in paper Ya. Butko A., Я. Бутко А. (2016) “О фундаментальных решениях, переходных вероятностях и дробных производных // On fundamental solutions, transition probabilities and fractional derivatives” / spz:neicon:technomag:y:2015:i:1:p:42-52

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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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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 \test-function"ξ:R→Dom(L)and eachs∈R.
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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 \test-function"ξ:R→Dom(L)and eachs∈R. Here Dom(L)is the domain of the generatorL. The object(Tt)t≥0, satisfying the identity (1) with a given operatorLwas called fundamental solutionof∂t+L. It was shown in the paper
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[1, Th. 4.1]
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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 [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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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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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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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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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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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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. 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]) and to outline the interplay between transitional probabilities of stochastic processes, evolution semigroups, evolution equations and their fundamental solutions.
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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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) and to outline the interplay between transitional probabilities of stochastic processes, evolution semigroups, evolution equations and their fundamental solutions. To make the picture as clear as possible we restrict the discussion to the case of L-evy processes with infinitely smooth symbol, e.g. with compactly supported L-evy measure, and to a class of subordinators.
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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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. 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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negative definite function, i.e.ψ is given by the Levy | Khinchine formula ψ(p) =a(q) +i`(q)·p+p·Q(q)p+ ∫ y6=0 ( 1−eip·y+ ip·y 1 +|y|2 ) N(dy), p∈Rd, where, for each fixedq∈Rd,`(q)∈Rd,Q(q)is a positive semidefinite symmetric matrix and ν(dy)is a measure kernel onRd\{0}such that ∫ y6=0 |y|2 1 +|y|2 N(dy)<∞. Note, thatψgrows at infinity with all its derivatives not faster than a polynomial
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[6, Th. 3.7.13]
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. A pseudo-differential operator with the symbolψis defined onS(Rd)as a compositionF−1ψF. Note also thatF−1ψ(ξ)F=Fψ(−ξ)F−1. The extension of this operator to the spaceS′(Rd)is defined by 〈F−1ψFf, φ〉:=〈f,F[ψ(F−1φ)]〉, φ∈S(Rd), f∈S′(Rd).
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Consider now the Cauchy problem inRd ∂f ∂t (t,x) =L∗f(t,x), f(0,x) =f0(x). (3) Heref0∈ S(Rd)and the problem is well-posed inL2(Rd). 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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): 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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) 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 L-evy processXt, whose symbolψis of classC∞(Rd).
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