Abstract
These are the handouts of an undergraduate minicourse at the Università di Bari (see Fig. 1), in the context of the 2017 INdAM Intensive Period “Contemporary Research in elliptic PDEs and related topics”. Without any intention to serve as a throughout epitome to the subject, we hope that these notes can be of some help for a very initial introduction to a fascinating field of classical and modern research.
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1 The Laplace Operator
The operator mostly studied in partial differential equations is likely the so-called Laplacian, given by
Of course, one may wonder why mathematicians have a strong preference for such kind of operators—say, why not studying
Since historical traditions, scientific legacies or impositions from above by education systems would not be enough to justify such a strong interest in only one operator (plus all its modifications), it may be worth to point out a simple geometric property enjoyed by the Laplacian (and not by many other operators). Namely, Eq. (1.1) somehow reveals that the fact that a function is harmonic (i.e., that its Laplace operator vanishes in some region) is deeply related to the action of “comparing with the surrounding values and reverting to the averaged values in the neighborhood”.
Working hard (and profitably) in Bari
To wit, the idea behind the integral representation of the Laplacian in formula (1.1) is that the Laplacian tries to model an “elastic” reaction: the vanishing of such operator should try to “revert the value of a function at some point to the values nearby”, or, in other words, from a “political” perspective, the Laplacian is a very “democratic” operator, which aims at levelling out differences in order to make things as uniform as possible. In mathematical terms, one looks at the difference between the values of a given function u and its average in a small ball of radius r, namely
In the smooth setting, a second order Taylor expansion of u and a cancellation in the integral due to odd symmetry show that 
is quadratic in r, hence, in order to detect the “elastic”, or “democratic”, effect of the model at small scale, one has to divide by r
2 and take the limit as 
. This is exactly the procedure that we followed in formula (1.1).
Other classical approaches to integral representations of elliptic operators come in view of potential theory and inversion operators, see e.g. [96].
This tendency to revert to the surrounding mean suggests that harmonic equations, or in general equations driven by operators “similar to the Laplacian”, possess some kind of rigidity or regularity properties that prevents the solutions to oscillate too much (of course, detecting and establishing these properties is a marvelous, and technically extremely demanding, success of modern mathematics, and we do not indulge in this set of notes on this topic of great beauty and outmost importance, and we refer, e.g. to the classical books [62, 71,72,73]).
Interestingly, the Laplacian operator, in the perspective of (1.1), is the infinitesimal limit of integral operators. In the forthcoming sections, we will discuss some other integral operators, which recover the Laplacian in an appropriate limit, and which share the same property of averaging the values of the function. Differently from what happens in (1.1), such averaging procedure will not be necessarily confined to a small neighborhood of a given point, but will rather tend to comprise all the possible values of a certain function, by possibly “weighting more” the close-by points and “less” the contributions coming from far.
2 Some Fractional Operators
We describe here the basics of some different fractionalFootnote 1 operators. The fractional exponent will be denoted by s ∈ (0, 1). For more exhaustive discussions and comparisons see e.g. [24, 49, 81, 82, 84, 91, 104, 107, 108]. For simplicity, we do not treat here the case of fractional operators of order higher than 1 (see e.g. [3,4,5, 50]).
2.1 The Fractional Laplacian
A very popular nonlocal operator is given by the fractional Laplacian
Here above, the notation “ P.V. ” stands for “in the Principal Value sense”, that is
The definition in (2.1) differs from others available in the literature since a normalizing factor has been omitted for the sake of simplicity: this multiplicative constant is only important in the limits as 
and 
, but plays no essential role for a fixed fractional parameter s ∈ (0, 1).
The operator in (2.1) can be also conveniently written in the form
The expression in (2.2) reveals that the fractional Laplacian is a sort of second order difference operator, weighted by a measure supported in the whole of \(\mathbb {R}^n\) and with a polynomial decay, namely
Of course, one can give a pointwise meaning of (2.1) and (2.2) if u is sufficiently smooth and with a controlled growth at infinity (and, in fact, it is possible to set up a suitable notion of fractional Laplacian also for functions that grow polynomially at infinity, see [59]). Besides, it is possible to provide a functional framework to define such operator in the weak sense (see e.g. [106]) and a viscosity solution approach is often extremely appropriate to construct general regularity theories (see e.g. [31]).
We refer to [49] for a gentle introduction to the fractional Laplacian.
From the point of view of the Fourier Transform, denoted, as usual, by \(\widehat {\cdot }\) or by \({\mathcal {F}}\) (depending on the typographical convenience), an instructive computation (see e.g. Proposition 3.3 in [49]) shows that
for some c > 0. An appropriate choice of the normalization constant in (2.1) (also in dependence of n and s) allows us to take c = 1, and we will take this normalization for the sake of simplicity (and with the slight abuse of notation of dropping constants here and there). With this choice, the fractional Laplacian in Fourier space is simply the multiplication by the symbol |ξ|2s, consistently with the fact that the classical Laplacian corresponds to the multiplication by |ξ|2. In particular, the fractional Laplacian recoversFootnote 2 the classical Laplacian as 
. In addition, it satisfies the semigroup property, for any s, s′∈ (0, 1) with s + s′⩽1,
that is
As a special case of (2.4), when s = s′ = 1∕2, we have that the square root of the Laplacian applied twice produces the classical Laplacian, namely
This observation gives that if \(U:\mathbb {R}^n\times [0,+\infty )\to \mathbb {R}\) is the harmonic extensionFootnote 3 of \(u:\mathbb {R}^n\to \mathbb {R}\), i.e. if
then
See Appendix A for a confirmation of this. In a sense, formula (2.7) is a particular case of a general approach which reduces the fractional Laplacian to a local operator which is set in a halfspace with an additional dimension and may be of singular or degenerate type, see [30].
As a rather approximative “general nonsense”, we may say that the fractional Laplacian shares some common feature with the classical Laplacian. In particular, both the classical and the fractional Laplacian are invariant under translations and rotations. Moreover, a control on the size of the fractional Laplacian of a function translates, in view of (2.3), into a control of the oscillation of the function (though in a rather “global” fashion): this “democratic” tendency of the operator of “averaging out” any unevenness in the values of a function is indeed typical of “elliptic” operators—and the classical Laplacian is the prototype example in this class of operators, while the fractional Laplacian is perhaps the most natural fractional counterpart.
To make this counterpart more clear, we will say that a function u is s-harmonic in a set Ω if (− Δ)s u = 0 at any point of Ω (for simplicity, we take this notion in the “strong” sense, but equivalently one could look at distributional definitions, see e.g. Theorem 3.12 in [18]).
For example, constant functions in \(\mathbb {R}^n\) are s-harmonic in the whole space for any s ∈ (0, 1), as both (2.1) and (2.2) imply.
Another similarity between classical and fractional Laplace equations is given by the fact that notions like those of fundamental solutions, Green functions and Poisson kernels are also well-posed in the fractional case and somehow similar formulas hold true, see e.g. Definitions 1.7 and 1.8, and Theorems 2.3, 2.10, 3.1 and 3.2 in [22] (and related formulas hold true also for higher-order fractional operators, see [3,4,5, 50]).
In addition, space inversions such as the Kelvin Transform also possess invariant properties in the fractional framework, see e.g. [19] (see also Lemma 2.2 and Corollary 2.3 in [63], and in addition Proposition A.1 on page 300 in [97] for a short proof). Moreover, fractional Liouville-type results hold under various assumptions, see e.g. [64] and [59].
Another interesting link between classical and fractional operators is given by subordination formulas which permit to reconstruct fractional operators from the heat flow of classical operators, such as
see [11].
In spite of all these similarities, many important structural differences between the classical and the fractional Laplacian arise. Let us list some of them.
Difference 2.1 (Locality Versus Nonlocality)
The classical Laplacian of u at a point x only depends on the values of u in B r(x), for any r > 0.
This is not true for the fractional Laplacian. For instance, if \(u\in C^\infty _0(B_2, \,[0,1])\) with u = 1 in B 1, we have that, for any \(x\in \mathbb {R}^n\setminus B_4\),
while of course Δu(x) = 0 in this setting.
It is worth remarking that the estimate in (2.8) is somewhat optimal. Indeed, if u belongs to the Schwartz space (or space of rapidly decreasing functions)
we have that, for large |x|,
See Appendix B for the proof of this fact.
Difference 2.2 (Summability Assumptions)
The pointwise computation of the classical Laplacian on a function u does not require integrability properties on u. Conversely, formula (2.1) for u can make sense only when
which can be read as a local integrability complemented by a growth condition at infinity. This feature, which could look harmless at a first glance, can result problematic when looking for singular solutions to nonlinear problems (as, for example, in [1, 66] where there is an unavoidable integrability obstruction on a bounded domain) or in “blow-up” type arguments (as mentioned in [59], where the authors propose a way to outflank this restriction).
Difference 2.3 (Computation Along Coordinate Directions)
The classical Laplacian of u at the origin only depends on the values that u attains along the coordinate directions (or, up to a rotation, along a set of n orthogonal directions).
This is not true for the fractional Laplacian. As an example, let \(u\in C^\infty _0( B_2 (4e_1+4e_2), \,[0,1])\), with u = 1 in B 1(4e 1 + 4e 2). Let also R j be the straight line in the jth coordinate direction, that is
see Fig. 2. Then
for each j ∈{1, …, n}, and so u(te j) = 0 for all \(t\in \mathbb {R}\) and j ∈{1, …, n}. This gives that Δu(0) = 0.
Coordinate directions not meeting a bump function
On the other hand,
which says that (− Δ)s u(0)≠0.
Difference 2.4 (Harmonic Versus s-Harmonic Functions)
If Δu(0) = 1, \(\|u-v\|{ }_{C^2(B_1)}\leqslant \varepsilon \) and ε > 0 is sufficiently small (see Fig. 3) then Δv(0)⩾1 − const ε > 0, and in particular Δv(0)≠0.
A function v which is “close to u”
Quite surprisingly, this is not true for the fractional Laplacian. More generally, in this case, as proved in [55], for any ε > 0 and any (bounded, smooth) function \(\bar u\), we can find v ε such that
A proof of this fact in dimension 1 for the sake of simplicity is given in [112] (the original paper [55] presents a complete proof in any dimension). See also [70, 99, 100] for different approaches to approximation methods in fractional settings which lead to new proofs, and very refined and quantitative statements.
We also mention that the phenomenon described in (2.11) (which can be summarized in the evocative statement that all functions are locally s-harmonic (up to a small error)) is very general, and it applies to other nonlocal operators, also independently from their possibly “elliptic” structure (for instance all functions are locally s-caloric, or s-hyperbolic, etc.). In this spirit, for completeness, in Sect. 5 we will establish the density of fractional caloric functions in one space variable, namely of the fact that for any ε > 0 and any (bounded, smooth) function \(\bar u=\bar u(x,t)\), we can find v ε = v ε(x, t) such that
We also refer to [58] for a general approach and a series of general results on this type of approximation problems with solutions of operators which are the superposition of classical differential operators with fractional Laplacians. Furthermore, similar results hold true for other nonlocal operators with memory, see [23]. See in addition [36, 37, 79] for related results on higher order fractional operators.
Difference 2.5 (Harnack Inequality)
The classical Harnack Inequality says that if u is harmonic in B 1 and u⩾0 in B 1 then
for a suitable universal constant, only depending on the dimension.
The same result is not true for s-harmonic functions. To construct an easy counterexample, let \(\bar u(x)=|x|{ }^2\) and, for a small ε > 0, let v ε be as in (2.11). Notice that, if x ∈ B 1 ∖ B 1∕4
if ε is small enough, while
These observations imply that v ε(0) < v ε(x) for all x ∈ B 1 ∖ B 1∕4 and therefore the infimum of v ε in B 1 is taken at some point \(\bar x\) in the closure of B 1∕4. Then, we define
Notice that u ε is s-harmonic in B 1, since so is v ε, and u ε⩾0 in B 1. Also, u ε is strictly positive in B 1 ∖ B 1∕4. On the other hand, since \(\bar x\in B_{1/2}\)
which implies that u ε cannot satisfy a Harnack Inequality as the one in (2.13).
In any case, it must be said that suitable Harnack Inequalities are valid also in the fractional case, under suitable “global” assumptions on the solution: for instance, the Harnack Inequality holds true for solutions that are positive in the whole of \(\mathbb {R}^n\) rather than in a given ball. We refer to [75, 76] for a comprehensive discussion on this topic and for recent developments.
Difference 2.6 (Growth from the Boundary)
Roughly speaking, solutions of Laplace equations have “linear (i.e. Lipschitz) growth from the boundary”, while solutions of fractional Laplace equations have only Hölder growth from the boundary. To understand this phenomenon, we point out that if u is continuous in the closure of B 1, with Δu = f in B 1 and u = 0 on ∂B 1, then
Notice that the term (1 −|x|) represents the distance of the point x ∈ B 1 from ∂B 1. See e.g. Appendix C for a proof of (2.14).
The case of fractional equations is very different. A first example which may be useful to keep in mind is that the function
For an elementary proof of this fact, see e.g. Section 2.4 in [24]. Remarkably, the function in (2.15) is only Hölder continuous with Hölder exponent s near the origin.
Another interesting example is given by the function
which satisfies
A proof of (2.17) based on extension methods and complex analysis is given in Appendix D.
The identity in (2.17) is in fact a special case of a more general formula, according to which the function
satisfies
For this formula, and in fact even more general ones, see [61]. See also [69] for a probabilistic approach.
Interestingly, (2.15) can be obtained from (2.19) by a blow-up at a point on the zero level set.
Notice also that
therefore, differently from the classical case, u s does not satisfy an estimate like that in (2.14).
It is also interesting to observe that the function u s is related to the function \(x_+^{s}\) via space inversion (namely, a Kelvin transform) and integration, and indeed one can also deduce (2.19) from (2.15): this fact was nicely remarked to us by Xavier Ros-Oton and Joaquim Serra, and the simple but instructive proof is sketched in Appendix E.
Difference 2.7 (Global (Up to the Boundary) Regularity)
Roughly speaking, solutions of Laplace equations are “smooth up to the boundary”, while solutions of fractional Laplace equations are not better than Hölder continuous at the boundary. To understand this phenomenon, we point out that if u is continuous in the closure of B 1,
then
See e.g. Appendix F for a proof of this fact.
The case of fractional equations is very different since the function u s in (2.18) is only Hölder continuous (with Hölder exponent s) in B 1, hence the global Lipschitz estimate in (2.21) does not hold in this case. This phenomenon can be seen as a counterpart of the one discussed in Difference 2.6. The boundary regularity for fractional Laplace problems is discussed in details in [97].
Difference 2.8 (Explosive Solutions)
Solutions of classical Laplace equations cannot attain infinite values in the whole of the boundary. For instance, if u is harmonic in B 1, then
Indeed, by the Mean Value Property for harmonic functions, for any ρ ∈ (0, 1),
from which (2.22) plainly follows (another proof follows by using the Maximum Principle instead of the Mean Value Property). On the contrary, and quite remarkably, solutions of fractional Laplace equations may “explode” at the boundary and (2.22) can be violated by s-harmonic functions in B 1 which vanish outside B 1.
For example, for
one has
and, of course, (2.22) is violated by u −1∕2. The claim in (2.24) can be proven starting from (2.17) and by suitably differentiating both sides of the equation: the details of this computation can be found in Appendix G. For completeness, we also give in Appendix H another proof of (2.24) based on complex variable and extension methods.
A geometric interpretation of (2.24) is depicted in Fig. 4 where a point x ∈ (−1, 1) is selected and the graph of u −1∕2 above the value u −1∕2(x) is drawn with a “dashed curve” (while a “solid curve” represents the graph of u −1∕2 below the value u −1∕2(x)): then, when computing the fractional Laplacian at x, the values coming from the dashed curve, compared with u −1∕2(x), provide an opposite sign with respect to the values coming from the solid curve. The “miracle” occurring in (2.24) is that these two contributions with opposite sign perfectly compensate and cancel each other, for any x ∈ (−1, 1).
The function u −1∕2 and the cancellation occurring in (2.24)
More generally, in every smooth bounded domain \(\Omega \subset \mathbb {R}^n\) it is possible to build s-harmonic functions exploding at ∂ Ω at the same rate as dist(⋅, ∂ Ω)s−1. A phenomenon of this sort was spotted in [66], and see [1] for the explicit explosion rate. See [1] also for a justification of the boundary behavior, as well as the study of Dirichlet problems prescribing a singular boundary trace.
Concerning this feature of explosive solutions at the boundary, it is interesting to point out a simple analogy with the classical Laplacian. Indeed, in view of (2.15), if s ∈ (0, 1) and we take the function \(\mathbb {R}\ni x\mapsto x_+^s\), we know that it is s-harmonic in (0, +∞) and it vanishes on the boundary (namely, the origin), and these features have a clear classical analogue for s = 1. Then, since for all s ∈ (0, 1] the derivative of \(x_+^s\) is \(x_+^{s-1}\), up to multiplicative constants, we have that the latter is s-harmonic in (0, +∞) and it blows-up at the origin when s ∈ (0, 1) (conversely, when s = 1 one can do the same computations but the resulting function is simply the characteristic function of (0, +∞) so no explosive effect arises).
Similar computations can be done in the unit ball instead of (0, +∞), and one simply gets functions that are bounded up to the boundary when s = 1, or explosive when s ∈ (0, 1) (further details in Appendices G and H).
Difference 2.9 (Decay at Infinity)
The Gaussian \(e^{-|x|{ }^2}\) reproduces the classical heat kernel. That is, the solution of the heat equation with initial datum concentrated at the origin, when considered at time t = 1∕4, produces the Gaussian (of course, the choice t = 1∕4 is only for convenience, any time t can be reduced to unit time by scaling the equation).
The fast decay prescribed by the Gaussian is special for the classical case and the fractional case exhibits power law decays at infinity. More precisely, let us consider the heat equation with initial datum concentrated at the origin, that is
and set
By taking the Fourier Transform of (2.25) in the x variable (and possibly neglecting normalization constants) one finds that
hence
and consequently
being \({\mathcal {F}}^{-1}\) the anti-Fourier Transform of the Fourier Transform \({\mathcal {F}}\). When s = 1, and neglecting the normalizing constants, the expression in (2.28) reduces to the Gaussian (since the Gaussian is the Fourier Transform of itself). On the other hand, as far as we know, there is no simple explicit representation of the fractional heat kernel in (2.28), except in the “miraculous” case s = 1∕2, in which (2.28) provides the explicit representation
See Appendix I for a proof of (2.29) using Fourier methods and Appendix J for a proof based on extension methods.
We stress that, differently from the classical case, the heat kernel \({\mathcal {G}}_{1/2}\) decays only with a power law. This is in fact a general feature of the fractional case, since, for any s ∈ (0, 1), it holds that
and, for |x|⩾1 and s ∈ (0, 1), the heat kernel \({\mathcal {G}}_s (x)\) is bounded from below and from above by \(\frac {\,{\mathrm {const}}\,}{|x|{ }^{n+2s}}\).
We refer to [78] for a detailed discussion on the fractional heat kernel. See also [13] for more information on the fractional heat equation. For precise asymptotics on fractional heat kernels, see [15, 17, 47, 95].
The decay of the heat kernel is also related to the associated distribution in probability theory: as we will see in Sect. 4.2, the heat kernel represents the probability density of finding a particle at a given point after a unit of time; the motion of such particle is driven by a random walk in the classical case and by a random process with long jumps in the fractional case and, as a counterpart, the fractional probability distribution exhibits a “long tail”, in contrast with the rapidly decreasing classical one.
Another situation in which the classical case provides exponentially fast decaying solutions while the fractional case exhibits polynomial tails is given by the Allen-Cahn equation (see e.g. Section 1.1 in [65] for a simple description of this equation also in view of phase coexistence models). For concreteness, one can consider the one-dimensional equation
For s = 1, the system in (2.31) reduces to the pendulum-like system
The solution of (2.32) is explicit and it has the form
as one can easily check. Also, by inspection, we see that such solution satisfies
Conversely, to the best of our knowledge, the solution of (2.31) has no simple explicit expression. Also, remarkably, the solution of (2.31) decays to the equilibria ± 1 only polynomially fast. Namely, as proved in Theorem 2 of [92], we have that the solution of (2.31) satisfies
and the estimates in (2.35) are optimal, namely it also holds that
See Appendix K for a proof of (2.36). In particular, (2.36) says that solutions of fractional Allen-Cahn equations such as the one in (2.31) do not satisfy the exponential decay in (2.34) which is fulfilled in the classical case.
The estimate in (2.36) can be confirmed by looking at the solution of the very similar equation
Though a simple expression of the solution of (2.37) is not available in general, the “miraculous” case s = 1∕2 possesses an explicit solution, given by
That (2.38) is a solution of (2.37) when s = 1∕2 is proved in Appendix L. Another proof of this fact using (2.29) is given in Appendix M.
The reader should not be misled by the similar typographic forms of (2.33) and (2.38), which represent two very different behaviors at infinity: indeed
and the function in (2.38) satisfies the slow decay in (2.36) (with s = 1∕2) and not the exponentially fast one in (2.34).
Equations like the one in (2.31) naturally arise, for instance, in long-range phase coexistence models and in models arising in atom dislocation in crystals, see e.g. [52, 110].
A similar slow decay also occurs in the study of fractional Schrödinger operators, see e.g. [38] and Lemma C.1 in [68]. For instance, the solution of
satisfies, for any |x|⩾1,
A heuristic motivation for a bound of this type can be “guessed” from (2.39) by thinking that, for large |x|, the function Γ should decay more or less like (− Δ)s Γ, which has “typically” the power law decay described in (2.10).
If one wishes to keep arguing in this heuristic way, also the decays in (2.30) and (2.36) may be seen as coming from an interplay between the right and the left side of the equation, in the light of the decay of the fractional Laplace operator discussed in (2.10). For instance, to heuristically justify (2.30), one may think that the solution of the fractional heat equation which starts from a Dirac’s Delta, after a unit of time (or an “infinitesimal unit” of time, if one prefers) has produced some bump, whose fractional Laplacian, in view of (2.10), may decay at infinity like \(\frac 1{|x|{ }^{n+2s}}\). Since the time derivative of the solution has to be equal to that, the solution itself, in this unit of time, gets “pushed up” by an amount like \(\frac 1{|x|{ }^{n+2s}}\) with respect to the initial datum, thus justifying (2.30).
A similar justification for (2.36) may seem more tricky, since the decay in (2.36) is only of the type \(\frac 1{|t|{ }^{2s}}\) instead of \(\frac 1{|t|{ }^{1+2s}}\), as the analysis in (2.10) would suggest. But to understand the problem, it is useful to consider the derivative of the solution \(v:=\dot u\) and deduce from (2.31) that
That is, for large |t|, the term 1 − 3u 2 gets close to 1 − 3 = −2 and so the profile at infinity may locally resemble the one driven by the equation (− Δ)s v = −2v. In this range, v has to balance its fractional Laplacian, which is expected to decay like \(\frac 1{|t|{ }^{1+2s}}\), in view of (2.10). Then, since u is the primitive of v, one may expect that its behavior at infinity is related to the primitive of \(\frac 1{|t|{ }^{1+2s}}\), and so to \(\frac 1{|t|{ }^{2s}}\), which is indeed the correct answer given by (2.36).
We are not attempting here to make these heuristic considerations rigorous, but perhaps these kinds of comments may be useful in understanding why the behavior of nonlocal equations is different from that of classical equations and to give at least a partial justification of the delicate quantitative aspects involved in a rigorous quantitative analysis (in any case, ideas like these are rigorously exploited for instance in Appendix K).
See also [21] for decay estimates of ground states of a nonlinear nonlocal problem.
We also mention that other very interesting differences in the decay of solutions arise in the study of different models for fractional porous medium equations, see e.g. [33, 34, 48].
Difference 2.10 (Finiteness Versus Infiniteness of the Mean Squared Displacement)
The mean squared displacement is a useful notion to measure the “speed of a diffusion process”, or more precisely the portion of the space that gets “invaded” at a given time by the spreading of the diffusive quantity which is concentrated at a point source at the initial time. In a formula, if u(x, t) is the fundamental solution of the diffusion equation related to the diffusion operator 
, namely
being δ
0 the Dirac’s Delta, one can define the mean squared displacement relative to the diffusion process 
as the “second moment” of u in the space variables, that is
For the classical heat equation, by Fourier Transform one sees that, when 
, the fundamental solution of (2.41) is given by the classical heat kernel
and thereforeFootnote 4 in such case, the substitution \(y:=\frac {x}{2\sqrt {t}}\) gives that
for some C > 0. This says that the mean squared displacement of the classical heat equation is finite, and linear in the time variable.
On the other hand, in the fractional case in which 
, by (2.27) the fractional heat kernel is endowed with the scaling property
with \({\mathcal {G}}_s\) being as in (2.25) and (2.26). Consequently, in this case, the substitution \(y:= \frac {x}{t^{\frac 1{2s}} }\) gives that
Now, from (2.30), we know that
and therefore we infer from (2.44) that
This computation shows that, when s ∈ (0, 1), the diffusion process induced by − (− Δ)s does not possess a finite mean squared displacement, in contrast with the classical case in (2.43).
Other important differences between the classical and fractional cases arise in the study of nonlocal minimal surfaces and in related fields: just to list a few features, differently than in the classical case, nonlocal minimal surfaces typically “stick” at the boundary, see [25, 53, 56], the gradient bounds of nonlocal minimal graphs are different than in the classical case, see [26], nonlocal catenoids grow linearly and nonlocal stable cones arise in lower dimension, see [45, 46], stable surfaces of vanishing nonlocal mean curvature possess uniform perimeter bounds, see Corollary 1.8 in [42], the nonlocal mean curvature flow develops singularity also in the plane, see [41], its fattening phenomena are different, see [40], and the self-shrinking solutions are also different, see [39], and genuinely nonlocal phase transitions present stronger rigidity properties than in the classical case, see e.g. Theorem 1.2 in [60] and [67]. Furthermore, from the probabilistic viewpoint, recurrence and transiency in long-jump stochastic processes are different from the case of classical random walks, see e.g. [6] and the references therein.
We would like to conclude this list of differences with one similarity, which seems to be not very well-known. There is indeed a “nonlocal representation” for the classical Laplacian in terms of a singular kernel. It reads as
This one is somehow very close to (2.2) with one important modification: the difference operator in the numerator of the integrand has been increased in order, in such a way that it is able to compensate the singularity of the kernel in 0. We include in Appendix N a computation proving (2.46) when u is C 2, α around x. For a complete proof, involving Fourier transform techniques and providing the explicit value of the constant, we refer to [3].
2.2 The Regional (or Censored) Fractional Laplacian
A variant of the fractional Laplacian in (2.1) consists in restricting the domain of integration to a subset of \(\mathbb {R}^n\). In this direction, an interesting operator is defined by the following singular integral:
We remark that when \(\Omega :=\mathbb {R}^n\) the regional fractional Laplacian in (2.47) boils down to the standard fractional Laplacian in (2.1).
In spite of the apparent similarity, the regional fractional Laplacian and the fractional Laplacian are structurally two different operators. For instance, concerning Difference 2.4, we mention that solutions of regional fractional Laplace equations do not possess the same rich structure of those of fractional Laplace equations, and indeed
A proof of this observation will be given in Appendix O.
Interestingly, the regional fractional Laplacian turns out to be useful also in a possible setting of Neumann-type conditions in the nonlocal case, as presentedFootnote 5 in [54]. Related to this, we mention that it is possible to obtain a regional-type operator starting from the classical Laplacian coupled with Neumann boundary conditions (details about it will be given in formula (2.52) below).
2.3 The Spectral Fractional Laplacian
Another natural fractional operator arises in taking fractional powers of the eigenvalues. For this, we write
where ϕ k is the eigenfunction corresponding to the kth eigenvalue of the Dirichlet Laplacian, namely
with \(0<\lambda _0<\lambda _1\leqslant \lambda _2\leqslant \dots \). We normalize the sequence ϕ k to make it an orthonormal basis of L 2( Ω) (see e.g. page 335 in [62]). In this setting, we define
We refer to [109] for extension methods for this type of operator. Furthermore, other types of fractional operators can be defined in terms of different boundary conditions: for instance, a spectral decomposition with respect to the eigenfunctions of the Laplacians with Neumann boundary data naturally leads to an operator \((-\Delta )^s_{N,\Omega }\) (and such operator also have applications in biology, see e.g. [90] and [57]).
It is also interesting to observe that the spectral fractional Laplacian with Neumann boundary conditions can also be written in terms of a regional operator with a singular kernel. Namely, given an open and bounded set \(\Omega \subset \mathbb {R}^n\), denoting by ΔN,Ω the Laplacian operator coupled with Neumann boundary conditions on ∂ Ω, we let \({\{(\mu _j,\psi _j)\}}_{j\in \mathbb {N}}\) the pairs made up of eigenvalues and eigenfunctions of − ΔN,Ω, that is
with \(0=\mu _0<\mu _1\leqslant \mu _2\leqslant \mu _3\leqslant \dots \).
We define the following operator by making use of a spectral decomposition
Comparing with (2.50), we can consider \((-\Delta )^s_{N,\Omega }\) a spectral fractional Laplacian with respect to classical Neumann data. In this setting, the operator \((-\Delta )^s_{N,\Omega }\) is also an integrodifferential operator of regional type, in the sense that one can write
for a kernel K(x, y) which is comparable to \(\frac 1{|x-y|{ }^{n+2s}}\). We refer to Appendix P for a proof of this.
Interestingly, the fractional Laplacian and the spectral fractional Laplacian coincide, up to a constant, for periodic functions, or functions defined on the flat torus, namely
See e.g. Appendix Q for a proof of this fact.
On the other hand, striking differences between the fractional Laplacian and the spectral fractional Laplacian hold true, see e.g. [91, 107].
Interestingly, it is not true that all functions are s-harmonic with respect to the spectral fractional Laplacian, up to a small error, that is
A proof of this will be given in Appendix R. The reader can easily compare (2.54) with the setting for the fractional Laplacian discussed in Difference 2.4.
Remarkably, in spite of these differences, the spectral fractional Laplacian can also be written as an integrodifferential operator of the form
for a suitable kernel K and potential β, see Lemma 38 in [2] or Lemma 10.1 in [20]. This can be proved with analogous computations to those performed in the case of the regional fractional Laplacian in the previous paragraph.
2.4 Fractional Time Derivatives
The operators described in Sections in 2.1, 2.2, and 2.3 are often used in the mathematical description of anomalous types of diffusion (i.e. diffusive processes which produce important differences with respect to the classical heat equation, as we will discuss in Sect. 4): the main role of such nonlocal operators is usually to produce a different behavior of the diffusion process with respect to the space variables.
Other types of anomalous diffusions arise from non-standard behaviors with respect to the time variable. These aspects are often the mathematical counterpart of memory effects. As a prototype example, we recall the notion of Caputo fractional derivative, which, for any t > 0 (and up to normalizing factors that we omit for simplicity) is given by
We point out that, for regular enough functions u,
Though in principle this expression takes into account only the values of u(t) for t⩾0, hence u does not need to be defined for negative times, as pointed out e.g. in Section 2 of [7], it may be also convenient to constantly extend u in (−∞, 0). Hence, we take the convention for which u(t) = u(0) for any t⩽0. With this extension, one has that, for any t > 0,
Hence, one can write (2.57) as
This type of formulas also relates the Caputo derivative to the so-called Marchaud derivative, see e.g. [104].
In the literature, one can also consider higher order Caputo derivatives, see e.g. [85, 89] and the references therein.
Also, it is useful to consider the Caputo derivative in light of the (unilateral) Laplace Transform (see e.g. Chapter 2.8 in [94], and [86])
With this notation, up to dimensional constants, one can write (for a smooth function with exponential control at infinity) that
see Appendix S for a proof.
In this way, one can also link equations driven by the Caputo derivative to the so-called Volterra integral equations: namely one can invert the expression \( \partial ^s_{C,t} u=f\) by
for some normalization constant C > 0, see Appendix S for a proof.
It is also worth mentioning that the Caputo derivative of order s of a power gives, up to normalizing constants, the “power minus s”: more precisely, by (2.56) and using the substitution 𝜗 := τ∕t, we see that, for any r > 0,
for some C > 0.
Moreover, in relation to the comments on page 17, we have that
See Appendix U for a proof of this.
The Caputo derivatives describes a process “with memory”, in the sense that it “remembers the past”, though “old events count less than recent ones”. We sketch a memory effect of Caputo type in Appendix V.
Due to its memory effect, operators related to Caputo derivatives have found several applications in which the basic parameters of a physical system change in time, in view of the evolution of the system itself: for instance, in studying flows in porous media, when time goes, the fluid may either “obstruct” the holes of the medium, thus slowing down the diffusion, or “clean” the holes, thus making the diffusion faster, and the Caputo derivative may be a convenient approach to describe such modification in time of the diffusion coefficient, see [35].
Other applications of Caputo derivatives occur in biology and neurosciences, since the network of neurons exhibit time-fractional diffusion, also in view of their highly ramified structure, see e.g. [51] and the references therein.
We also refer to [24, 113, 115] and to the references therein for further discussions on different types of anomalous diffusions.
3 A More General Point of View: The “Master Equation”
The operators discussed in Sects. 2.1, 2.2, 2.3, and 2.4 can be framed into a more general setting, that is that of the “master equation”, see e.g. [32].
Master equations describe the evolution of a quantity in terms of averages in space and time of the quantity itself. For concreteness one can consider a quantity u = u(x, t) and describe its evolution by an equation of the kind
for some \(c\in \mathbb {R}\) and a forcing term f, and the operator L has the integral form
for a suitable measure μ (with the integral possibly taken in the principal value sense, which is omitted here for simplicity; also one can consider even more general operators by taking actions different than translations and more general ambient spaces).
Though the form of such operator is very general, one can also consider simplifying structural assumptions. For instance, one can take μ to be the space-time Lebesgue measure over \(\mathbb {R}^n\times (0,+\infty )\), namely
Another common simplifying assumption is to assume that the kernel is induced by an uncorrelated effect of the space and time variables, with the product structure
The fractional Laplacian of Sect. 2.1 is a particular case of this setting (for functions depending on the space variable), with the choice, up to normalizing constants,
More generally, for \(\Omega \subseteq \mathbb {R}^n\), the regional fractional Laplacian in Sect. 2.2 comes from the choice
Finally, in view of (2.58), for time-dependent functions, the choice
produces the Caputo derivative discussed in Sect. 2.4.
We recall that one of the fundamental structural differences in partial differential equations consists in the distinction between operators “in divergence form”, such as
and those “in non-divergence form”, such as
This structural difference can also be recovered from the master equation. Indeed, if we consider a (say, for the sake of concreteness, strictly positive, bounded and smooth) matrix function \(M:\mathbb {R}^n\to {\text{Mat }}(n\times n)\), we can take into account the master spatial operator induced by the kernel
that is, in the notation of (3.1),
Then, up to a normalizing constant, if
then
A proof of this will be given in Appendix W.
It is interesting to observe that condition (3.6) says that, if we set z := x − y, then
and so the kernel in (3.4) is invariant by exchanging x and z. This invariance naturally leads to a (possibly formal) energy functional of the form
We point out that condition (3.8) translates, roughly speaking, into the fact that the energy density in (3.9) “charges the variable x as much as the variable z”.
The study of the energy functional in (3.9) also drives to a natural quasilinear generalization, in which the fractional energy takes the form
for a suitable Φ, see e.g. [80, 114] and the references therein for further details on quasilinear nonlocal operators. See also [113] and the references therein for other type of nonlinear fractional equations.
Another case of interest (see e.g. [14]) is the one in which one considers the master equation driven by the spatial kernel
that is, in the notation of (3.1),
Then, up to a normalizing constant, if
then
A proof of this will be given in Appendix X.
We recall that nonlocal linear operators in non-divergence form can also be useful in the definition of fully nonlinear nonlocal operators, by taking appropriate infima and suprema of combinations of linear operators, see e.g. [83] and the references therein for further discussions about this topic (which is also related to stochastic games).
We also remark that understanding the role of the affine transformations of the spaces on suitable nonlocal operators (as done for instance in (3.10) and (3.10)) often permits a deeper analysis of the problem in nonlinear settings too, see e.g. the very elegant way in which a fractional Monge-Ampère equation is introduced in [29] by considering the infimum of fractional linear operators corresponding to all affine transformations of determinant one of a given multiple of the fractional Laplacian.
As a general comment, we also think that an interesting consequence of the considerations given in this section is that classical, local equations can also be seen as a limit approximation of more general master equations.
We mention that there are also many other interesting kernels, both in space and time, which can be taken into account in integral equations. Though we focused here mostly on the case of singular kernels, there are several important problems that focus on “nice” (e.g. integrable) kernels, see e.g. [8, 43, 88] and the references therein.
As a technical comment let us point out that, in a sense, the nice kernels may have computational advantages, but may provide loss of compactness and loss of regularity issues: roughly speaking, convolutions with smooth kernel are always smooth, thus any smoothness information on a convolved function gives little information on the smoothness of the original function—viceversa, if the convolution of an “object” with a singular kernel is smooth, then it means that the original object has a “good order of vanishing at the origin”. When the original object is built by the difference of a function and its translation, such vanishing implies some control of the oscillation of the function, hence opening a door towards a regularity result.
4 Probabilistic Motivations
We provide here some elementary, and somewhat heuristic, motivations for the operators described in Sect. 2 in view of probability and statistics applications. The treatment of this section is mostly colloquial and not to be taken at a strictly rigorous level (in particular, all functions are taken to be smooth, some uniformity problems are neglected, convergence is taken for granted, etc.). See e.g. [74] for rigorous explanations linking pseudo-differential operators and Markov/Lévy processes. See also [9, 12, 16, 101, 111] for other perspectives and links between probability and fractional calculus and [77] for a complete survey on jump processes and their connection to nonlocal operators.
The probabilistic approach to study nonlocal effects and the analysis of distributions with polynomial tails are also some of the cornerstones of the application of mathematical theories to finance, see e.g. [87, 93], and models with jump process for prices have been proposed in [44].
4.1 The Heat Equation and the Classical Laplacian
The prototype of parabolic equations is the heat equation
for some c > 0. The solution u may represent, for instance, a temperature, and the foundation of (4.1) lies on two basic assumptions:
-
the variation of u in a given region \(U\subset \mathbb {R}^n\) is due to the flow of some quantity \(v:\mathbb {R}^n\to \mathbb {R}^n\) through U,
-
v is produced by the local variation of u.
The first ansatz can be written as
where ν denotes the exterior normal vector of U and \({\mathcal {H}}^{n-1}\) is the standard (n − 1)-dimensional surface Hausdorff measure.
The second ansatz can be written as v = c∇u, which combined with (4.2) and the Divergence Theorem gives that
Since U is arbitrary, this gives (4.1).
Let us recall a probabilistic interpretation of (4.1). The idea is that (4.1) follows by taking suitable limits of a discrete “random walk”. For this, we take a small space scale h > 0 and a time step
We consider the random motion of a particle in the lattice \(h\mathbb {Z}^n\), as follows. At each time step, the particle can move in any coordinate direction with equal probability. That is, a particle located at \(h\bar k\in h\mathbb {Z}^n\) at time t is moved to one of the 2n points \(h\bar k\pm he_1\), …, \(h\bar k\pm h e_n\) with equal probability (here, as usual, e j denotes the jth element of the standard Euclidean basis of \(\mathbb {R}^n\)).
We now look at the expectation to find the particle at a point \(x\in h\mathbb {Z}^n\) at time \(t\in \tau \mathbb {N}\). For this, we denote by u(x, t) the probability density of such expectation. That is, the probability for the particle of lying in the spatial region B r(x) at time t is, for small r, comparable with
Then, the probability of finding a particle at the point \(x\in h\mathbb {Z}^n\) at time t + τ is the sum of the probabilities of finding the particle at a closest neighborhood of x at time t, times the probability of jumping from this site to x. That is,
Also,
Thus, subtracting u(x, t) to both sides in (4.4), dividing by τ, recalling (4.3), and taking the limit (and neglecting any possible regularity issue), we formally find that
which is (4.1).
4.2 The Fractional Laplacian and the Regional Fractional Laplacian
Now we consider an open set \(\Omega \subseteq \mathbb {R}^n\) and a discrete random process in \(h\mathbb {Z}^n\) which can be roughly speaking described in this way. The space parameter h > 0 is linked to the time step
A particle starts its journey from a given point \(h\bar k\in \Omega \) of the lattice \(h\mathbb {Z}^n\) and, at each time step τ, it can reach any other point of the lattice hk, with \(k\ne \bar k\), with probability
then the process continues following the same law. Notice that the above probability density does not allow the process to leave the domain Ω, since P h vanishes in the complement of Ω (in jargon, this process is called “censored”).
In (4.6), the constant C > 0 is needed to normalize to total probability and is defined by
We let
and
Notice that, for any \(\bar k\in \mathbb {Z}^n\), it holds that
hence, for a fixed h > 0 and \(\bar k\in \mathbb {Z}^n\), this aggregate probability does not equal to 1: this means that there is a remaining probability \(p_h(\bar k)\geqslant 0\) for which the particle does not move (in principle, such probability is small when so is h, but, for a bounded domain Ω, it is not negligible with respect to the time step, hence it must be taken into account in the analysis of the process in the general setting that we present here).
We define u(x, t) to be the probability density for the particle to lie at the point \(x\in \Omega \cap (h\mathbb {Z}^n)\) at time \(t\in \tau \mathbb {N}\). We show that, for small space and time scale, the function u is well described by the evolution of the nonlocal heat equation
for some normalization constant c > 0. To check this, up to a translation, we suppose that x = 0 ∈ Ω and we set c h := c h(0) and p h := p h(0). We observe that the probability of being at 0 at time t + τ is the sum of the probabilities of being somewhere else, say at \(hk\in h\mathbb {Z}^n\), at time t, times the probability of jumping from hk to the origin, plus the probability of staying put: that is
Thus, recalling (4.7),
So, we divide by τ and, in view of (4.5), we find that
We write this identity changing k to − k and we sum up: in this way, we obtain that
Now, for small y, if u is smooth enough,
and therefore, if we write
we (formally) have that
for small |y|.
Now, we fix δ > 0 and use the Riemann sum representation of an integral to write (for a bounded Riemann integrable function \(\varphi :\mathbb {R}^n\setminus B_\delta \to \mathbb {R}\)),
If, in addition, (4.11) is satisfied, one has that, for small δ,
From this and (4.12) we have that
Also, in view of (4.11),
Hence, (4.13) boils down to
and so, taking δ arbitrarily small,
Therefore, recalling (4.10),
This confirms (4.9).
As a final comment, in view of these calculations and those of Sect. 4.1, we may compare the classical random walk, which leads to the classical heat equation, and the long-jump random walk which leads to the nonlocal heat equation and relate such jumps to an “infinitely fast” diffusion, in the light of the computations of the associated mean squared displacements (recall (2.43) and (2.45)).
4.3 The Spectral Fractional Laplacian
Now, we briefly discuss a heuristic motivation for the fractional heat equation run by the spectral fractional Laplacian, that is
for some normalization constant c > 0. To this end, we consider a bounded and smooth set \(\Omega \subset \mathbb {R}^n\) and we define a random motion of a “distribution of particles” in Ω. For any x ∈ Ω and t⩾0, the function u(x, t) denotes the “number of particles” present at the point x at the time t. No particles lie outside Ω and we write u as a suitable superposition of eigenfunctions {ϕ k}k⩾1 of the Laplacian with Dirichlet boundary data (this is a reasonable assumption, given that such eigenfunctions provide a basis of L 2( Ω), see e.g. page 335 in [62]). In this way, we write
Namely, in the notation in (2.49), the evolution of the particle distribution u is defined on each spectral component u k and it is taken to follow a “classical” random walk, but the space/time scale is supposed to depend on k as well: namely, spectral components relative to high frequencies will move slower than the ones relative to low frequencies (namely, the time step is taken to be longer if the frequency is higher).
More precisely, for any \(k\in \mathbb {N}\), we suppose that each of the u k particles of the kth spectral component undergo a classical random walk in a lattice \(h_k\mathbb {Z}^d\), as described in Sect. 4.1, but with time step
We suppose that h k and τ k are “small space and time increments”. Namely, after a time step τ k, each of these u k(t) ϕ k(x) particles will move, with equal probability \(\frac 1{2n}\), to one of the points x ± h k e 1, …, x ± h k e n (for simplicity, we are imaging here u k to be positive; the case of negative u k represents a “lack of particles”, which is supposed to diffuse with the same law). Hence, the number of particles at time t + τ k which correspond to the kth frequency of the spectrum and which lie at the point x ∈ Ω is equal to the sum of the number of the particles at time t which lie somewhere else times the probability of jumping to x in this time step, that is, in formula,
Moreover,
Consequently, from this and (4.16),
Hence, with a formal computation, dividing by τ
k, using (4.15) and sending h
k, 
(for a fixed k), we obtain
Hence, from (2.49) (and neglecting converge issues in k), we have
that is (4.14).
4.4 Fractional Time Derivatives
We consider a model in which a bunch of people is supposed to move along the real line (say, starting at the origin) with some given velocity f, which depends on time. We consider the case in which the environment surrounding the moving people is “tricky”, and some of them risk to get stuck for some time, and they are able to “exit the trap” only by overcoming their past velocity. Concretely, we fix a function φ : [0, +∞) → [0, +∞) with
Then we define
and we notice that
Then, we denote by u(t) the position of the “generic person” at time t, with u(0) = 0. We suppose that some people, say a proportion p 1 of the total population, move with the prescribed velocity for a unit of time, after which their velocity is the difference between the prescribed velocity at that time and the one at the preceding time with respect to the time unit. In formulas, this says that there is a proportion p 1 of the total people who travels with velocity
After integrating, we thus obtain that there is a proportion p 1 of the total people whose position is described by the function
For instance, if f is constant, then the position u 1 grows linearly for a unit of time and then remains put (this would correspond to consider “stopping times” in the motion, see Fig. 5).
The motions u k described in Sect. 4.4 when the velocity field f is constant
Similarly, a proportion p 2 of the total population evolves with prescribed velocity f for two units of time, after which its velocity becomes the difference between the prescribed velocity at that time and the one at the preceding time with respect to two time units, namely
In this case, an integration gives that there is a proportion p 2 of the total people whose position is described by the function
Repeating this argument, we suppose that for each \(k\in \mathbb {N}\) we have a proportion p k of the people that move initially with the prescribed velocity f, but, after k units of time, get their velocity changed into the difference of the actual velocity field and that of k units of time before (which is indeed a “memory effect”). In this way, we have that a proportion p k of the total population moves with law of motion given by
The average position of the moving population is then given by
We now specialize the computation above for the case
with s ∈ (0, 1). Notice that the quantity in (4.17) is finite in this case, and we can denote it simply by C s. In addition, we will consider long time asymptotics in t and introduce a small time increment h which is inversely proportional to t, namely
In this way, recalling that the motion was supposed to start at the origin (i.e., u(0) = 0) and using the substitution η := 𝜗∕t, we can write (4.18) as
where we have recognized a Riemann sum in the last line.
We also point out that the conditions
are equivalent to
and, furthermore,
Therefore we use the substitution ξ := 1 − η and we exchange the order of integrations, to deduce from (4.19) that
The substitution τ := tξ then gives
which, comparing with (2.61) and possibly redefining constants, gives that \( \partial ^s_{C,t} u=f\).
Of course, one can also take into account the case in which the velocity field f is induced by a classical diffusion in space, i.e. f = Δu, and in this case one obtains the time fractional diffusive equation \(\partial ^s_{C,t} u=\Delta u\).
4.5 Fractional Time Diffusion Arising from Heterogeneous Media
A very interesting phenomenon to observe is that the geometry of the diffusion medium can naturally transform classical diffusion into an anomalous one. This feature can be very well understood by an elegant model, introduced in [10] (see also [105] and the references therein for an exhaustive account of the research in this direction) consisting in random walks on a “comb”, that we briefly reproduce here for the facility of the reader. Given ε > 0, the comb may be considered as a transmission medium that is the union of a “backbone” \({\mathcal {B}}:=\mathbb {R}\times \{0\}\) with the “fingers” \({\mathcal {P}}_k:=\{\varepsilon k\}\times \mathbb {R}\), namely
see Fig. 6.
The comb \({\mathcal {C}}_\varepsilon \)
Harmonic extension in the halfplane of the function \( \mathbb {R}\ni x\mapsto (1-x^2)_+^{1/2}\)
We suppose that a particle experiences a random walk on the comb, starting at the origin, with some given horizontal and vertical speeds. In the limit, this random walk can be modeled by the diffusive equation along the comb \({\mathcal {C}}_\varepsilon \)
with d
1, d
2 > 0. The case d
1 = d
2 corresponds to equal horizontal and vertical speeds of the random walk (and this case is already quite interesting). Also, in the limit as 
, we can consider the Riemann sum approximation
and \({\mathcal {C}}_\varepsilon \) tends to cover the whole of \(\mathbb {R}^2\) when ε gets small. Accordingly, at least at a formal level, as the fingers of the comb become thicker and thicker, we can think that
and reduce (4.20) to the diffusive equation in \(\mathbb {R}^2\) given by
The very interesting feature of (4.21) is that it naturally induces a fractional time diffusion along the backbone. The quantity that experiences this fractional diffusion is the total diffusive mass at a point of the backbone. Namely, one sets
and we claim that
Equation (4.23) reveals the very relevant phenomenon that a diffusion governed by the Caputo derivative may naturally arise from classical diffusion, only in view of the particular geometry of the domain.
To check (4.23), we first point out that
Then, we observe that, if a, \(b\in \mathbb {C}\), and
To check this let \(\varphi \in C^\infty _0(\mathbb {R})\). Then, integrating twice by parts,
thus proving (4.25).
We also remark that, in the notation of (4.25), we have that δ 0(y)g(y) = δ 0(y)g(0) = bδ 0(y), and so, for every \(c\in \mathbb {R}\),
Now, taking the Fourier Transform of (4.21) in the variable x, using the notation \(\hat u(\xi ,y,t)\) for the Fourier Transform of u(x, y, t), and possibly neglecting normalization constants, we get
Now, we take the Laplace Transform of (4.27) in the variable t, using the notation w(ξ, y, ω) for the Laplace Transform of \(\hat u(\xi ,y,t)\), namely \( w(\xi ,y,\omega ):={\mathcal {L}} \hat u(\xi ,y,\omega )\). In this way, recalling that
and therefore
we deduce from (4.27) that
That is, setting
we see that
and hence we can write (4.28) as
In light of (4.26), we know that this equation is solved by taking w = g, that is
As a consequence, by (4.22),
This and (4.24) give that
that is
Transforming back and recalling (2.60), we obtain (4.23), as desired.
5 All Functions Are Locally s-Caloric (Up to a Small Error): Proof of (2.12)
We let \((x,t)\in \mathbb {R}\times \mathbb {R}\) and consider the operator 
. One defines
and for any \(J\in \mathbb {N}\), we define
Notice that \({\mathcal {V}}_J\) is a linear subspace of \(\mathbb {R}^{N+1}\), for some \(N\in \mathbb {N}\). The core of the proof is to establish the maximal span condition
To this end, we argue for a contradiction and we suppose that \({\mathcal {V}}_J\) is a linear subspace strictly smaller than \(\mathbb {R}^{N+1}\): hence, there exists
such that
One considers ϕ to be the first eigenfunctions of (− Δ)s in (−1, 1) with Dirichlet data, normalized to have unit norm in \(L^2(\mathbb {R})\). Accordingly,
for some λ > 0.
In view of the boundary properties discussed in Difference 2.6, one can prove that
with o(1) infinitesimal as 
. So, fixed ε, τ > 0, we define
This function is smooth for any x in a small neighborhood of the origin and any \(t\in \mathbb {R}\), and, in this domain,
This says that \(h_{\varepsilon ,{\tau }}\in {\mathcal {V}}\) and therefore
This, together with (5.3), implies that, for any fixed and positive τ and y,
Hence, fixed τ > 0, this identity and (5.4) yield that
with o(1) infinitesimal as 
.
We now take \(\bar \alpha _x\) be the largest integer α x for which there exists an integer α t such that \(\bar \alpha _x+\alpha _t\in [0,J]\) and \(\nu _{(\bar \alpha _x,\alpha _t)}\ne 0\). Notice that this definition is well-posed, since not all the \(\nu _{(\alpha _x,\alpha _t)}\) can vanish, due to (5.2). Then, (5.5) becomes
since the other coefficients vanish by definition of \(\bar \alpha _x\).
Thus, we multiply (5.6) by \(\varepsilon ^{\bar \alpha _x-s} {\tau }^{-\frac {\bar \alpha _x}{2s}}\) and we take the limit as 
: in this way, we obtain that
Since this is valid for any τ > 0, by the Identity Principle for Polynomials we obtain that
and thus \(\nu _{(\bar \alpha _x,\alpha _t)}=0\), for any integer α t for which \(\bar \alpha _x+\alpha _t\in [0,J]\). But this is in contradiction with the definition of \(\bar \alpha _x\) and so we have completed the proof of (5.1).
From this maximal span property, the proof of (2.12) follows by scaling (arguing as done, for instance, in [112]).
〈〈The longest appendix measured 26cm (10.24in) when it was removed from 72-year-old Safranco August (Croatia) during an autopsy at the Ljudevit Jurak University Department of Pathology, Zagreb, Croatia, on 26 August 2006.〉〉
(Source: http://www.guinnessworldrecords.com/world-records/largest-appendix-removed)
Notes
- 1.
The notion (or, better to say, several possible notions) of fractional derivatives attracted the attention of many distinguished mathematicians, such as Leibniz, Bernoulli, Euler, Fourier, Abel, Liouville, Riemann, Hadamard and Riesz, among the others. A very interesting historical outline is given in pages xxvii–xxxvi of [104].
- 2.
We think that it is quite remarkable that the operator obtained by the inverse Fourier Transform of \( \,|\xi |{ }^{2} \,\widehat u\), the classical Laplacian, reduces to a local operator. This is not true for the inverse Fourier Transform of \( \,|\xi |{ }^{2s} \,\widehat u\). In this spirit, it is interesting to remark that the fact that the classical Laplacian is a local operator is not immediate from its definition in Fourier space, since computing Fourier Transforms is always a nonlocal operation.
- 3.
Some care has to be used with extension methods, since the solution of (2.6) is not unique (if U solves (2.6), then so does U(x, y) + cy for any \(c\in \mathbb {R}\)). The “right” solution of (2.6) that one has to take into account is the one with “decay at infinity”, or belonging to an “energy space”, or obtained by convolution with a Poisson-type kernel. See e.g. [24] for details.
Also, the extension method in (2.6) and (2.7) can be related to an engineering application of the fractional Laplacian motivated by the displacement of elastic membranes on thin (i.e. codimension one) obstacles, see [28]. The intuition for such application can be grasped from Figs. 7, 10, and 12. These pictures can be also useful to develop some intuition about extension methods for fractional operators and boundary reaction-diffusion equations.
- 4.
See Appendix A in [103] for a very nice explanation of the dimensional analysis and for a throughout discussion of its role in detecting fundamental solutions.
- 5.
Some colleagues pointed out to us that the use of R and r in some steps of formula (5.5) of [54] are inadequate. We take this opportunity to amend such a flaw, presenting a short proof of (5.5) of [54]. Given ε > 0, we notice that
where the constants are also allowed to depend on Ω and u. Furthermore, if we define Ωε to be the set of all the points in Ω with distance less than ε from ∂ Ω, the regularity of ∂ Ω implies that the measure of Ωε is bounded by const ε, and therefore
These observations imply that
Taking ε as small as we wish, we obtain formula (5.5) in [54].
- 6.
From the geometric point of view, one can also take radial coordinates, compute the derivatives of K along the unit sphere and use scaling.
- 7.
The difficulty in proving (G.1) is that the function u 1∕2 is not differentiable at ± 1 and the derivative taken inside the integral might produce a singularity (in fact, formula (G.1) exactly says that such derivative can be performed with no harm inside the integral). The reader who is already familiar with the basics of functional analysis can prove (G.1) by using the theory of absolutely continuous functions, see e.g. Theorem 8.21 in [98]. We provide here a direct proof, available to everybody.
- 8.
As a historical remark, we recall that e −|ξ| is sometimes called the “Abel Kernel” and its Fourier Transform the “Poisson Kernel”, which in dimension 1 reduces to the “Cauchy-Lorentz, or Breit-Wigner, Distribution” (that has also classical geometric interpretations as the “Witch of Agnesi”, and so many names attached to a single function clearly demonstrate its importance in numerous applications).
- 9.
Let us mention another conceptual simplification of nonlocal problems: in this setting, the integral representation often allows the formulation of problems with minimal requirements on the functions involved (such as measurability and possibly minor pointwise or integral bounds). Conversely, in the classical setting, even to just formulate a problem, one often needs assumptions and tools from functional analysis, comprising e.g. Sobolev differentiability, distributions or functions of bounded variations.
- 10.
In complex variables, one can also interpret the function U in terms of the principal argument function
$$\displaystyle \begin{aligned}{\mathrm{Arg}}(r e^{i\varphi})=\varphi\in(-\pi,\pi],\end{aligned}$$with branch cut along the nonpositive real axis. Notice indeed that, if z = x + iy and y > 0,
$$\displaystyle \begin{aligned}{\mathrm{Arg}}(z+i)=\frac\pi2-\arctan\frac{x}{y+1}=\frac\pi2\left(1-U(x,y) \right).\end{aligned}$$This observation would also lead to (L.1).
- 11.
- 12.
A slightly different approach as that in (3.7) is to consider the energy functional in (3.9) and prove, e.g. by Taylor expansion, that it converges to the energy functional
$$\displaystyle \begin{aligned}\,{\mathrm{const}}\, \int_{\mathbb{R}^n}a_{ij}(x) \,\partial_i u(x)\,\partial_j u(x)\,dx.\end{aligned}$$On the other hand, a different proof of (3.7), that was nicely pointed out to us by Jonas Hirsch (who has also acted as a skilled cartoonist for Fig. 13) after a lecture, can be performed by taking into account the weak form of the operator in (3.5), i.e. integrating such expression against a test function \(\varphi \in C^\infty _0(\mathbb {R}^n)\), thus finding
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle (1-s)\,\iint_{\mathbb{R}^n\times\mathbb{R}^n} \frac{ \big(u(x)-u(x-y)\big)\,\varphi(x)}{ |M(x-y,y)\, y|{}^{n+2s}}\,dx\,dy \\ &\displaystyle =&\displaystyle (1-s)\,\iint_{\mathbb{R}^n\times\mathbb{R}^n} \frac{ \big(u(x)-u(z)\big)\,\varphi(x)}{ |M(z,x-z)\,(x-z)|{}^{n+2s}}\,dx\,dz \\ \noalign{} &\displaystyle =&\displaystyle (1-s)\,\iint_{\mathbb{R}^n\times\mathbb{R}^n} \frac{ \big(u(z)-u(x)\big)\,\varphi(z)}{ |M(x,z-x)\,(x-z)|{}^{n+2s}}\,dx\,dz \\ &\displaystyle =&\displaystyle -(1-s)\,\iint_{\mathbb{R}^n\times\mathbb{R}^n} \frac{ \big(u(x)-u(z)\big)\,\varphi(z)}{ |M(z,x-z)\,(x-z)|{}^{n+2s}}\,dx\,dz ,\end{array} \end{aligned} $$where the structural condition (3.6) has been used in the last line. This means that the weak formulation of the operator in (3.5) can be written as
$$\displaystyle \begin{aligned}\frac{1-s}{2}\,\iint_{\mathbb{R}^n\times\mathbb{R}^n} \frac{ \big(u(x)-u(z)\big)\,\big(\varphi(x)-\varphi(z)\big)}{ |M(z,x-z)\,(x-z)|{}^{n+2s}}\,dx\,dz.\end{aligned}$$So one can expand this expression and take the limit as
, to obtain $$\displaystyle \begin{aligned}\,{\mathrm{const}}\, \int_{\mathbb{R}^n}a_{ij}(x) \,\partial_i u(x)\,\partial_j\varphi(x)\,dx,\end{aligned}$$which is indeed the weak formulation of the classical divergence form operator.
, to obtain References
N. Abatangelo, Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete Contin. Dyn. Syst. 35(12), 5555–5607 (2015). https://doi.org/10.3934/dcds.2015.35.5555. MR3393247
N. Abatangelo, L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian. Ann. Inst. H. Poincaré Anal. Non Linéaire 34(2), 439–467 (2017). https://doi.org/10.1016/j.anihpc.2016.02.001. MR3610940
N. Abatangelo, S. Jarohs, A. Saldaña, Positive powers of the Laplacian: from hypersingular integrals to boundary value problems. Commun. Pure Appl. Anal. 17(3), 899–922 (2018). https://doi.org/10.3934/cpaa.2018045. MR3809107
N. Abatangelo, S. Jarohs, A. Saldaña, Green function and Martin kernel for higher-order fractional Laplacians in balls. Nonlinear Anal. 175, 173–190 (2018). https://doi.org/10.1016/j.na.2018.05.019. MR3830727
N. Abatangelo, S. Jarohs, A. Saldaña, On the loss of maximum principles for higher-order fractional Laplacians. Proc. Am. Math. Soc. 146(11), 4823–4835 (2018). https://doi.org/10.1090/proc/14165. MR3856149
E. Affili, S. Dipierro, E. Valdinoci, Decay estimates in time for classical and anomalous diffusion. arXiv e-prints (2018), available at 1812.09451
M. Allen, L. Caffarelli, A. Vasseur, A parabolic problem with a fractional time derivative. Arch. Ration. Mech. Anal. 221(2), 603–630 (2016). https://doi.org/10.1007/s00205-016-0969-z. MR3488533
F. Andreu-Vaillo, J.M. Mazón, J.D. Rossi, J.J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, vol. 165 (American Mathematical Society, Providence, 2010); Real Sociedad Matemática Española, Madrid, 2010. MR2722295
D. Applebaum, Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, 2nd edn., vol. 116 (Cambridge University Press, Cambridge, 2009). MR2512800
V.E. Arkhincheev, É.M. Baskin, Anomalous diffusion and drift in a comb model of percolation clusters. J. Exp. Theor. Phys. 73, 161–165 (1991)
A.V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them. Pac. J. Math. 10, 419–437 (1960). MR0115096
R. Bañuelos, K. Bogdan, Lévy processes and Fourier multipliers. J. Funct. Anal. 250(1), 197–213 (2007). https://doi.org/10.1016/j.jfa.2007.05.013. MR2345912
B. Barrios, I. Peral, F. Soria, E. Valdinoci, A Widder’s type theorem for the heat equation with nonlocal diffusion. Arch. Ration. Mech. Anal. 213(2), 629–650 (2014). https://doi.org/10.1007/s00205-014-0733-1. MR3211862
R.F. Bass, D.A. Levin, Harnack inequalities for jump processes. Potential Anal. 17(4), 375–388 (2002). https://doi.org/10.1023/A:1016378210944. MR1918242
A. Bendikov, Asymptotic formulas for symmetric stable semigroups. Expo. Math. 12(4), 381–384 (1994). MR1297844
J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121 (Cambridge University Press, Cambridge, 1996). MR1406564
R.M. Blumenthal, R.K. Getoor, Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960). https://doi.org/10.2307/1993291. MR0119247
K. Bogdan, T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Stud. Math. 133(1), 53–92 (1999). MR1671973
K. Bogdan, T. Żak, On Kelvin transformation. J. Theor. Probab. 19(1), 89–120 (2006). MR2256481
M. Bonforte, A. Figalli, J.L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Anal. PDE 11(4), 945–982 (2018). https://doi.org/10.2140/apde.2018.11.945. MR3749373
L. Brasco, S. Mosconi, M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality. Calc. Var. Partial Differ. Equ. 55(2), 23, 32 (2016). https://doi.org/10.1007/s00526-016-0958-y. MR3461371
C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework. Commun. Pure Appl. Anal. 15(2), 657–699 (2016). https://doi.org/10.3934/cpaa.2016.15.657. MR3461641
C. Bucur, Local density of Caputo-stationary functions in the space of smooth functions. ESAIM Control Optim. Calc. Var. 23(4), 1361–1380 (2017). https://doi.org/10.1051/cocv/2016056. MR3716924
C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20 (Springer, Cham, 2016); Unione Matematica Italiana, Bologna. MR3469920
C. Bucur, L. Lombardini, E. Valdinoci, Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter. Ann. Inst. H. Poincaré Anal. Non Linéaire 36(3), 655–703 (2019)
X. Cabré, M. Cozzi, A gradient estimate for nonlocal minimal graphs. Duke Math. J. 168(5), 775–848 (2019)
X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. 367(2), 911–941 (2015). https://doi.org/10.1090/S0002-9947-2014-05906-0. MR3280032
L.A. Caffarelli, Further regularity for the Signorini problem. Commun. Partial Differ. Equ. 4(9), 1067–1075 (1979). https://doi.org/10.1080/03605307908820119. MR542512
L. Caffarelli, F. Charro, On a fractional Monge-Ampère operator. Ann. PDE 1(1), 4, 47 (2015). MR3479063
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007). https://doi.org/10.1080/03605300600987306. MR2354493
L. Caffarelli, L. Silvestre, Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009). MR2494809
L. Caffarelli, L. Silvestre, Hölder regularity for generalized master equations with rough kernels, in Advances in Analysis: The Legacy of Elias M. Stein. Princeton Mathematical Series, vol. 50 (Princeton University Press, Princeton, 2014), pp. 63–83. MR3329847
L.A. Caffarelli, J.L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete Contin. Dyn. Syst. 29(4), 1393–1404 (2011). MR2773189
L. Caffarelli, F. Soria, J.L. Vázquez, Regularity of solutions of the fractional porous medium flow. J. Eur. Math. Soc. (JEMS) 15(5), 1701–1746 (2013). https://doi.org/10.4171/JEMS/401. MR3082241
M. Caputo, Linear models of dissipation whose Q is almost frequency independent. II. Fract. Calc. Appl. Anal. 11(1), 4–14 (2008). Reprinted from Geophys. J. R. Astr. Soc. 13(1967), no. 5, 529–539. MR2379269
A. Carbotti, S. Dipierro, E. Valdinoci, Local Density of Solutions to Fractional Equations. Graduate Studies in Mathematics (De Gruyter, Berlin, 2019)
A. Carbotti, S. Dipierro, E. Valdinoci, Local density of Caputo-stationary functions of any order. Complex Var. Elliptic Equ. (to appear). https://doi.org/10.1080/17476933.2018.1544631
R. Carmona, W.C. Masters, B. Simon, Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91(1), 117–142 (1990). https://doi.org/10.1016/0022-1236(90)90049-Q. MR1054115
A. Cesaroni, M. Novaga, Symmetric self-shrinkers for the fractional mean curvature flow. ArXiv e-prints (2018), available at 1812.01847
A. Cesaroni, S. Dipierro, M. Novaga, E. Valdinoci, Fattening and nonfattening phenomena for planar nonlocal curvature flows. Math. Ann. (to appear). https://doi.org/10.1007/s00208-018-1793-6
E. Cinti, C. Sinestrari, E. Valdinoci, Neckpinch singularities in fractional mean curvature flows. Proc. Am. Math. Soc. 146(6), 2637–2646 (2018). https://doi.org/10.1090/proc/14002. MR3778164
E. Cinti, J. Serra, E. Valdinoci, Quantitative flatness results and BV-estimates for stable nonlocal minimal surfaces. J. Differ. Geom. (to appear)
J. Coville, Harnack type inequality for positive solution of some integral equation. Ann. Mat. Pura Appl. 191(3), 503–528 (2012). https://doi.org/10.1007/s10231-011-0193-2. MR2958346
J.C. Cox, The valuation of options for alternative stochastic processes. J. Finan. Econ. 3(1–2), 145–166 (1976). https://doi.org/10.1016/0304-405X(76)90023-4
M. Cozzi, E. Valdinoci, On the growth of nonlocal catenoids. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. (to appear)
J. Dávila, M. del Pino, J. Wei, Nonlocal s-minimal surfaces and Lawson cones. J. Differ. Geom. 109(1), 111–175 (2018). https://doi.org/10.4310/jdg/1525399218. MR3798717
C.-S. Deng, R.L. Schilling, Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations, ArXiv e-prints (2018), available at 1803.11435
A. de Pablo, F. Quirós, A. Rodríguez, J.L. Vázquez, A fractional porous medium equation. Adv. Math. 226(2), 1378–1409 (2011). https://doi.org/10.1016/j.aim.2010.07.017. MR2737788
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012). https://doi.org/10.1016/j.bulsci.2011.12.004. MR2944369
S. Dipierro, H.-C. Grunau, Boggio’s formula for fractional polyharmonic Dirichlet problems. Ann. Mat. Pura Appl. 196(4), 1327–1344 (2017). https://doi.org/10.1007/s10231-016-0618-z. MR3673669
S. Dipierro, E. Valdinoci, A simple mathematical model inspired by the Purkinje cells: from delayed travelling waves to fractional diffusion. Bull. Math. Biol. 80(7), 1849–1870 (2018). https://doi.org/10.1007/s11538-018-0437-z. MR3814763
S. Dipierro, G. Palatucci, E. Valdinoci, Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting. Commun. Math. Phys. 333(2), 1061–1105 (2015). https://doi.org/10.1007/s00220-014-2118-6. MR3296170
S. Dipierro, O. Savin, E. Valdinoci, Graph properties for nonlocal minimal surfaces. Calc. Var. Partial Differ. Equ. 55(4), 86, 25 (2016). https://doi.org/10.1007/s00526-016-1020-9. MR3516886
S. Dipierro, X. Ros-Oton, E. Valdinoci, Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33(2), 377–416 (2017). https://doi.org/10.4171/RMI/942. MR3651008
S. Dipierro, O. Savin, E. Valdinoci, All functions are locally s-harmonic up to a small error. J. Eur. Math. Soc. (JEMS) 19(4), 957–966 (2017). https://doi.org/10.4171/JEMS/684. MR3626547
S. Dipierro, O. Savin, E. Valdinoci, Boundary behavior of nonlocal minimal surfaces. J. Funct. Anal. 272(5), 1791–1851 (2017). https://doi.org/10.1016/j.jfa.2016.11.016. MR3596708
S. Dipierro, N. Soave, E. Valdinoci, On stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data. Indiana Univ. Math. J. 67(1), 429–469 (2018). https://doi.org/10.1512/iumj.2018.67.6282. MR3776028
S. Dipierro, O. Savin, E. Valdinoci, Local approximation of arbitrary functions by solutions of nonlocal equations. J. Geom. Anal. 29(2), 1428–1455 (2019)
S. Dipierro, O. Savin, E. Valdinoci, Definition of fractional Laplacian for functions with polynomial growth. Rev. Mat. Iberoam (to appear)
S. Dipierro, J. Serra, E. Valdinoci, Improvement of flatness for nonlocal phase transitions. Amer. J. Math. (to appear)
B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian. Fract. Calc. Appl. Anal. 15(4), 536–555 (2012). https://doi.org/10.2478/s13540-012-0038-8. MR2974318
L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998). MR1625845
M.M. Fall, T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems. J. Funct. Anal. 263(8), 2205–2227 (2012). https://doi.org/10.1016/j.jfa.2012.06.018. MR2964681
M.M. Fall, T. Weth, Liouville theorems for a general class of nonlocal operators. Potential Anal. 45(1), 187–200 (2016). https://doi.org/10.1007/s11118-016-9546-1. MR3511811
A. Farina, E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions (World Scientific Publishing, Hackensack, 2009), pp. 74–96. https://doi.org/10.1142/9789812834744_0004. MR2528756
P. Felmer, A. Quaas, Boundary blow up solutions for fractional elliptic equations. Asymptot. Anal. 78(3), 123–144 (2012). MR2985500
A. Figalli, J. Serra, On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4 + 1, preprint at arXiv:1705.02781 (2017, submitted)
R.L. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 69(9), 1671–1726 (2016). https://doi.org/10.1002/cpa.21591. MR3530361
R.K. Getoor, First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90 (1961). https://doi.org/10.2307/1993412. MR0137148
T. Ghosh, M. Salo, G. Uhlmann, The Calderón problem for the fractional Schrödinger equation. ArXiv e-prints (2016), available at 1609.09248
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics (Springer, Berlin, 2001). Reprint of the 1998 edition. MR1814364
E. Giusti, Direct Methods in the Calculus of Variations (World Scientific Publishing, River Edge, 2003). MR1962933
Q. Han, F. Lin, Elliptic Partial Differential Equations. Courant Lecture Notes in Mathematics, 2nd edn., vol. 1 (Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, 2011). MR2777537
N. Jacob, Pseudo-Differential Operators and Markov Processes. Mathematical Research, vol. 94 (Akademie Verlag, Berlin, 1996). MR1409607
M. Kaßmann, Harnack-Ungleichungen Für nichtlokale Differentialoperatoren und Dirichlet-Formen (in German). Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 336 (Universität Bonn, Mathematisches Institut, Bonn, 2001). Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2000. MR1941020
M. Kaßmann, A new formulation of Harnack’s inequality for nonlocal operators. C. R. Math. Acad. Sci. Paris 349(11–12), 637–640 (2011). https://doi.org/10.1016/j.crma.2011.04.014 (English, with English and French summaries). MR2817382
M. Kaßmann, Jump processes and nonlocal operators, in Recent Developments in Nonlocal Theory (De Gruyter, Berlin, 2018), pp. 274–302. MR3824215
V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. 80(3), 725–768 (2000). https://doi.org/10.1112/S0024611500012314. MR1744782
N.V. Krylov, On the paper “All functions are locally s-harmonic up to a small error” by Dipierro, Savin, and Valdinoci. ArXiv e-prints (2018), available at 1810.07648
T. Kuusi, G. Mingione, Y. Sire, Nonlocal equations with measure data. Commun. Math. Phys. 337(3), 1317–1368 (2015). https://doi.org/10.1007/s00220-015-2356-2. MR3339179
M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7–51 (2017). https://doi.org/10.1515/fca-2017-0002. MR3613319
N.S. Landkof, Foundations of Modern Potential Theory (Springer, New York, 1972). Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR0350027
H.C. Lara, G. Dávila, Regularity for solutions of non local parabolic equations. Calc. Var. Partial Differ. Equ. 49(1–2), 139–172 (2014). https://doi.org/10.1007/s00526-012-0576-2. MR3148110
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity (Imperial College Press, London, 2010). An introduction to mathematical models. MR2676137
F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153–192 (2001). MR1829592
F. Mainardi, P. Paradisi, R. Gorenflo, Probability distributions generated by fractional diffusion equations. preprint at arXiv:0704.0320v1 (2007, submitted)
B. Mandelbrot, The variation of certain speculative prices. J. Bus. 36, 394 (1963)
J.M. Mazón, J.D. Rossi, J. Toledo, The heat content for nonlocal diffusion with non-singular kernels. Adv. Nonlinear Stud. 17(2), 255–268 (2017). https://doi.org/10.1515/ans-2017-0005. MR3641640
R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37(31), R161–R208 (2004). https://doi.org/10.1088/0305-4470/37/31/R01. MR2090004
E. Montefusco, B. Pellacci, G. Verzini, Fractional diffusion with Neumann boundary conditions: the logistic equation. Discrete Contin. Dyn. Syst. Ser. B 18(8), 2175–2202 (2013). https://doi.org/10.3934/dcdsb.2013.18.2175. MR3082317
R. Musina, A.I. Nazarov, On fractional Laplacians. Commun. Partial Differ. Equ. 39(9), 1780–1790 (2014). https://doi.org/10.1080/03605302.2013.864304. MR3246044
G. Palatucci, O. Savin, E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm. Ann. Mat. Pura Appl. 192(4), 673–718 (2013). https://doi.org/10.1007/s10231-011-0243-9. MR3081641
V. Pareto, Cours D’économie Politique, vol. I/II (F. Rouge, Lausanne, 1896)
I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198 (Academic Press, San Diego, CA, 1999). An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR1658022
G. Pólya, On the zeros of an integral function represented by Fourier’s integral. Messenger 52, 185–188 (1923)
M. Riesz, L’intégrale de Riemann-Liouville et le problème de Cauchy (French). Acta Math. 81, 1–223 (1949). MR0030102
X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101(3), 275–302 (2014). https://doi.org/10.1016/j.matpur.2013.06.003 (English, with English and French summaries). MR3168912
W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966). MR0210528
A. Rüland, M. Salo, The fractional Calderón problem: low regularity and stability. ArXiv e-prints (2017), available at 1708.06294
A. Rüland, M. Salo, Quantitative approximation properties for the fractional heat equation. ArXiv e-prints (2017), available at 1708.06300
L.A. Sakhnovich, Levy Processes, Integral Equations, Statistical Physics: Connections and Interactions. Operator Theory: Advances and Applications, vol. 225 (Birkhäuser/Springer, Basel, 2012). MR2963050
L. Saloff-Coste, The heat kernel and its estimates, in Probabilistic Approach to Geometry. Advanced Studies in Pure Mathematics, vol. 57 (Mathematical Society of Japan, Tokyo, 2010), pp. 405–436. MR2648271
S. Salsa, Equazioni a Derivate Parziali. Metodi, Modelli e Applicazioni (Italian), 2nd edn. (Springer, Milano, 2010)
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives (Gordon and Breach Science Publishers, Yverdon, 1993). Theory and applications; Edited and with a foreword by S. M. Nikol′skiı̆; Translated from the 1987 Russian original; Revised by the authors. MR1347689
T. Sandev, A. Schulz, H. Kantz, A. Iomin, Heterogeneous diffusion in comb and fractal grid structures. Chaos Solitons Fractals 114, 551–555 (2018). https://doi.org/10.1016/j.chaos.2017.04.041. MR3856678
R. Servadei, E. Valdinoci, Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389(2), 887–898 (2012). https://doi.org/10.1016/j.jmaa.2011.12.032. MR2879266
R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators. Proc. R. Soc. Edinburgh Sect. A 144(4), 831–855 (2014). https://doi.org/10.1017/S0308210512001783. MR3233760
E.M. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30 (Princeton University Press, Princeton, 1970). MR0290095
P.R. Stinga, J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35(11), 2092–2122 (2010). https://doi.org/10.1080/03605301003735680. MR2754080
J.F. Toland, The Peierls-Nabarro and Benjamin-Ono equations. J. Funct. Anal. 145(1), 136–150 (1997). https://doi.org/10.1006/jfan.1996.3016. MR1442163
E. Valdinoci, From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SeMA 49, 33–44 (2009). MR2584076
E. Valdinoci, All functions are (locally) s-harmonic (up to a small error)—and applications, in Partial Differential Equations and Geometric Measure Theory. Lecture Notes in Mathematics, vol. 2211 (Springer, Cham, 2018), pp. 197–214. MR3790948
J.L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete Contin. Dyn. Syst. Ser. S 7(4), 857–885 (2014). https://doi.org/10.3934/dcdss.2014.7.857. MR3177769
J.L. Vázquez, The Dirichlet problem for the fractional p-Laplacian evolution equation. J. Differ. Equ. 260(7), 6038–6056 (2016). https://doi.org/10.1016/j.jde.2015.12.033. MR3456825
J.L. Vázquez, The mathematical theories of diffusion: nonlinear and fractional diffusion, in Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Lecture Notes in Mathematics, vol. 2186 (Springer, Cham, 2017), pp. 205–278. MR3588125
Acknowledgements
It is a great pleasure to thank the Università degli Studi di Bari for its very warm hospitality and the Istituto Nazionale di Alta Matematica for the strong financial and administrative support which made this minicourse possible. And of course special thanks go to all the participants, for their patience in attending the course, their competence, empathy and contagious enthusiasm. This work was supported by INdAM and ARC Discovery Project N.E.W. Nonlocal Equations at Work.
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Appendices
Appendix A: Confirmation of (2.7)
We write Δx to denote the Laplacian in the coordinates \(x\in \mathbb {R}^n\). In this way, the total Laplacian in the variables \((x,y)\in \mathbb {R}^n\times (0,+\infty )\) can be written as
Given a (smooth and bounded, in the light of footnote 3 on page 5) \(u:\mathbb {R}^n\to \mathbb {R}\), we take U := E u be (smooth and bounded) as in (2.6).
We also consider the operator
and we take V (x, y) := −∂ y U(x, y). Notice that ΔV = −∂ y ΔU = 0 in \(\mathbb {R}^n\times (0,+\infty )\) and V (x, 0) = Lu(x) for any \(x\in \mathbb {R}^n\). In this sense, V is the harmonic extension of Lu and so we can write V = E Lu and so, in the notation of (A.2), and recalling (2.6) and (A.1), we have
This gives that L 2 = − Δ, which is consistent with L = (− Δ)1∕2, thanks to (2.5).
Appendix B: Proof of (2.10)
Let \(u\in {\mathcal {S}}\). By (2.9), we can write
Fixed \(x\in \mathbb {R}^n\) (with |x| to be taken large), recalling the notation in (2.3), we consider the map y↦δ u(x, y) and we observe that
Hence, if \(|Y|\leqslant |x|/2\) we have that \(|x\pm Y|\geqslant |x|-|Y|\geqslant |x|/2\), and thus
thanks to (B.1).
Therefore, a second order Taylor expansion of δ u in the variable y gives that, if \(|y|\leqslant |x|/2\),
Consequently,
Moreover, by (B.1),
This and (B.2), recalling (2.3), establish (2.10).
Appendix C: Proof of (2.14)
Let \(M:=\frac 1{2n}\,\left (1+\sup _{B_1}|f|\right )\) and v(x) := M(1 −|x|2) − u(x). Notice that v = 0 along ∂B 1 and
in B 1. Consequently, v⩾0 in B 1, which gives that \(u(x)\leqslant M(1-|x|{ }^2)\).
Arguing similarly, by looking at \(\tilde v(x):= M(1-|x|{ }^2)+u(x)\), one sees that \(-u(x) \leqslant M(1-|x|{ }^2)\). Accordingly, we have that
This proves (2.14).
Appendix D: Proof of (2.17)
The idea of the proof is described in Fig. 7. The trace of the function in Fig. 7 is exactly the function u 1∕2 in (2.16). The function plotted in Fig. 7 is the harmonic extension of u 1∕2 in the halfplane (like an elastic membrane pinned at the halfcircumference along the trace). Our objective is to show that the normal derivative of such extended function along the trace is constant, and so we can make use of the extension method in (2.6) and (2.7) to obtain (2.17).
In further detail, we use complex coordinates, identifying \((x,y)\in \mathbb {R}\times (0,+\infty )\) with \(z:=x+iy\in \mathbb {C}\) with ℑ(z) > 0. Also, as customary, we define the principal square root in the cut complex plane
by defining, for any \(z=re^{{i\varphi }}\in \mathbb {C}_\star \),
see Fig. 8 (for typographical convenience, we distinguish between the complex and the real square root, by using the symbols √(⋅) and \(\sqrt {\cdot }\) respectively).
Real and imaginary part of the complex principal square root
The principal square root function is defined using the nonpositive real axis as a “branch cut” and
Moreover,
To check these facts, we take \(z\in \mathbb {C}_\star \): since \(\mathbb {C}_\star \) is open, we have that \(z+w\in \mathbb {C}_\star \) for any \(w\in \mathbb {C}\setminus \{0\}\) with small module. Consequently, by (D.2), we obtain that
Dividing by w and taking the limit, we thus find that
Since \(\mathbb {C}_\star \subseteq \mathbb {C}\setminus \{0\}\), we have that z≠0, and thus √(z)≠0. As a result, we can divide (D.5) by 2√(z) and conclude that
which establishes, at the same time, both (D.3) and (D.4), as desired.
We also remark that
To check this, if z = x + iy with y > 0, we observe that
Hence, if 1 − z 2 lies on the real axis, we have that xy = 0, and so x = 0. Then, the real part of 1 − z 2 in this case is equal to 1 + y 2 which is strictly positive. This proves (D.6).
Thanks to (D.6), for any \(z\in \mathbb {C}\) with ℑ(z) > 0 we can define the function √(1 − z 2). From (D.7), we can write
Notice that
As a consequence,
This says that, if x 2 > 1 then
while if x 2 < 1 then
On this account, we deduce that
and therefore, recalling (D.1),
This implies that
Now we define
The function U 1∕2 is the harmonic extension of u 1∕2 in the halfplane, as plotted in Fig. 7. Indeed, from (D.10),
Furthermore, from (D.3), we have that U 1∕2 is the real part of a holomorphic function in the halfplane and so it is harmonic.
These considerations give that U 1∕2 solves the harmonic extension problem in (2.6), hence, in the light of (2.7),
Now, recalling (D.4), we see that, for any x ∈ (−1, 1) and small y > 0,
We stress that the latter denominator does not vanish when x ∈ (−1, 1) and y > 0 is small. So, using that \(\Re (ZW)=\Re Z\Re W-\Im Z\Im W\) for any Z, \(W\in \mathbb {C}\), we obtain that
From (D.9), for any x ∈ (−1, 1) we have that
This and the fact that ∂ y √(1 − z 2) is bounded (in view of (D.12)) give that, for any x ∈ (−1, 1),
This, (D.9) and (D.13) imply that, for any x ∈ (−1, 1),
and therefore
Plugging this information into (D.11), we conclude the proof of (2.17), as desired.
Appendix E: Deducing (2.19) from (2.15) Using a Space Inversion
From (2.15), up to a translation, we know that
We let w s be the space inversion of v s induced by the Kelvin transform in the fractional setting, namely
By (E.1), see Corollary 2.3 in [63], it follows that w s(x) is s-harmonic in (0, 1). Consequently, the function
is also s-harmonic in (0, 1). We thereby conclude that the function
is also s-harmonic in (0, 1). See Fig. 9 for a picture of w s and \(W^\star _s\) when s = 1∕2. Let now
and notice that U s is the primitive of \(s W^\star _s\). Since the latter function is s-harmonic in (0, 1), after an integration we thereby deduce that (− Δ)s U s = const in (0, 1). This and the fact that
imply (2.19).
The functions w 1∕2 and \(W^\star _{1/2}\)
Harmonic extension in the halfplane of the function \( \mathbb {R}\ni x\mapsto (1-x^2)^{-1/2}_+\)
Appendix F: Proof of (2.21)
Fixed \(y\in \mathbb {R}^n\setminus \{0\}\) we let \({\mathcal {R}}^y\) be a rotation which sends \(\frac {y}{|y|}\) into the vector e 1 = (1, 0, …, 0), that is
for any i ∈{1, …, n}. We also denote by
the so-called Kelvin Transform. We recall that for any i, j ∈{1, …, n},
and so, by (F.1),
This says that \( {\mathcal {R}}^y\;(DK(y))\;({\mathcal {R}}^y)^{-1}\) is a diagonalFootnote 6 matrix, with first entry equal to \(-\frac {1}{|y|{ }^2}\) and the others equal to \(\frac {1}{|y|{ }^2}\).
As a result,
The Kelvin Transform is also useful to write the Green function of the ball B 1, see e.g. formula (41) on p. 40 and Theorem 13 on p. 35 of [62]. Namely, we take n⩾3 for simplicity, and we write
and, for a suitable choice of the constant, for any x ∈ B 1 we can write the solution of (2.20) in the form
see e.g. page 35 in [62].
On this account, we have that, for any x ∈ B 1,
Notice that here we have used the transformations ζ := x − y and η := K(y), exploiting also (F.2). The claim in (2.21) is thus established.
Appendix G: Proof of (2.24) and Probabilistic Insights
We give a proof of (2.24) by taking a derivative of (2.17). To this aim, we claimFootnote 7 that
To this end, we fix x ∈ (−1, 1) and \(h\in \mathbb {R}\). We define
In the sequel, we will take |h| as small as we wish in order to compute incremental quotients, hence we can assume that
We also define
Since I x(h) ⊆ (x−1−2|h|, x−1+2|h|)∪(x+1−2|h|, x+1+2|h|)∪(1−x−2|h|, 1−x+2|h|)∪(−1−x−2|h|, −1−x+2|h|), we have that
Furthermore,
To check this, let y ∈ I x(h). Then, by (G.3), there exist σ 1,x,y, σ 2,x,y ∈{−1, 1} such that
and therefore
where the last inequality is a consequence of (G.2), and this establishes (G.5).
Now, we introduce the following notation for the incremental quotient
and we observe that, since u 1∕2 is globally Hölder continuous with exponent 1∕2, it holds that
for any x, \(y\in \mathbb {R}\). Consequently, recalling (G.4) and (G.5), we conclude that
Now we take derivatives of u 1∕2. For this, we observe that, for any ξ ∈ (−1, 1),
Since the values outside (−1, 1) are trivial, this implies that
Now, by (G.3), we know that if \(y\in \mathbb {R}\setminus I_x(h)\) we have that \(x+y+t\in \mathbb {R}\setminus \{-1,1\}\) for all \(t\in \mathbb {R}\) with |t| < |h| and therefore we can exploit (G.7) and find that
Similar arguments show that, for any \(y\in \mathbb {R}\setminus I_x(h)\),
Consequently, for any \(y\in \mathbb {R}\setminus I_x(h)\),
Now we set
and we claim that
for a suitable C x > 0, possibly depending on x. For this, we first observe that if |y|⩾3 then |x ± y|⩾1 and also |x ± y + h|⩾1. This implies that if |y|⩾3, then u 1∕2(x ± y) = u 1∕2(x ± y + h) = 0 and therefore
This and the fact that u 1∕2 is smooth in the vicinity of the fixed x ∈ (−1, 1) imply that (G.9) holds true when |y|⩾3. Therefore, from now on, to prove (G.9) we can suppose that
We will also distinguish two regimes, the one in which \(|y|\leqslant \frac {\ell _x}4\) and the one in which \(|y|>\frac {\ell _x}4\).
If \(|y|\leqslant \frac {\ell _x}4\) and \(|t|\leqslant h\), we have that
due to (G.2), and similarly \(|(x-y+t)-1|\geqslant \frac {\ell _x}2\). This implies that
for some C x > 0 that depends on ℓ x. Consequently, we find that if \(|y|\leqslant \frac {\ell _x}4\) then
Conversely, if \(y\in \mathbb {R}\setminus I_x(h)\), with \(|y|>\frac {\ell _x}4\), then we make use of (G.7) and (G.10) to see that
Also, if \(y\in \mathbb {R}\setminus I_x(h)\) we deduce from (G.3) that \(|1\pm (x+y)|\geqslant 2 |h|\) and therefore, if \(|\tau |\leqslant |h|\), then
Therefore
Hence, we insert this information into (G.12) and we conclude that
Similarly, one sees that
In view of (G.13) and (G.14), we get that, for any \(y\in \mathbb {R}\setminus I_x(h)\) with \(|y|>\frac {\ell _x}4\),
Combining this with (G.11), we obtain (G.9), up to renaming constants.
Now, we point out that the right hand side of (G.9) belongs to \(L^1(\mathbb {R})\). Accordingly, using (G.9) and the Dominated Convergence Theorem, and recalling also (G.7), it follows that
where the last identity is a consequence of (G.8).
From this and (G.6), the claim in (G.1) follows, as desired.
Now, we rewrite (G.1) as
We claim that
This follows plainly for x = 0, since u −1∕2 is even. Hence, from here on, to prove (G.16) we assume without loss of generality that x ∈ (0, 1). Moreover, by changing variable y↦ − y, we see that
and therefore
Now, we apply the change of variable
We observe that when z ranges in (−1, 1), then ξ ranges therein as well. Moreover,
thus, by (G.17),
We now apply another change of variable
which gives
where
Now we notice that
Inserting this identity into (G.18), we obtain (G.16), as desired.
Then, from (G.15) and (G.16) we get that
that is
From this and (2.17) we infer that x (− Δ)1∕2 u −1∕2 = 0 and so (− Δ)1∕2 u −1∕2 = 0 in (−1, 1).
These consideration establish (2.24), as desired. Now, we give a brief probabilistic insight on it. In probability—and in stochastic calculus—a measurable function \(f:\mathbb {R}^n\to \mathbb {R}\) is said to be harmonic in an open set \(D\subset \mathbb {R}^n\) if, for any D 1 ⊂ D and any x ∈ D 1,
Notice that, since W t has (a.s.) continuous trajectories, then (a.s.) \(W_{\tau _{D_1}}\in \partial D_1\). This notion of harmonicity coincides with the analytic one.
If one considers a Lévy-type process X t in place of the Brownian motion, the definition of harmonicity (with respect to this other process) can be given in the very same way. When X t is an isotropic (2s)-stable process, the definition amounts to having zero fractional Laplacian (− Δ)s at every point of D and replace (G.19) by
In this identity, we can consider a sequence \(\{D_j:D_j\subset D,j\in \mathbb {N}\}\), with 
, and equality
When f = 0 in \(\mathbb {R}^n\setminus D\), the right-hand side of (G.20) can be not 0 (since \(X_{\tau _{D_j}}\) may also end up in D ∖ D j), and this leaves the possibility of finding f which satisfies (G.20) without vanish identically (an example of this phenomenon is exactly given by the function u −1∕2 in (2.24)).
It is interesting to observe that if f vanishes outside D and does not vanish identically, then, the only possibility to satisfy (G.20) is that f diverges along ∂D. Indeed, if \(|f|\leqslant \kappa \), since \(f(X_{\tau _{D_j}})\neq 0\) only when x ∈ D ∖ D
j and 
as j →∞, we would have that
and (G.20) would imply that f must vanish identically.
Of course, the function u −1∕2 in (2.23) embodies exactly this singular boundary behavior.
Appendix H: Another Proof of (2.24)
Here we give a different proof of (2.24) by using complex analysis and extension methods. We use the principal complex square root introduced in (D.2) and, for any \(x\in \mathbb {R}\) and y > 0 we define
where z := x + iy.
The function U −1∕2 is plotted in Fig. 10. We recall that the function U −1∕2 is well-defined, thanks to (D.6). Also, the denominator never vanishes when y > 0 and so U −1∕2 is harmonic in the halfplane, being the real part of a holomorphic function in such domain.
Furthermore, in light of (D.9), we have that
and therefore
This gives that U −1∕2 is the harmonic extension of u −1∕2 to the halfplane. Therefore, by (2.6), (2.7), and (D.14), for any x ∈ (−1, 1) we have that
that is (2.24).
Appendix I: Proof of (2.29) (Based on Fourier Methods)
When n = 1, we use (2.28) to find thatFootnote 8
This proves (2.28) when n = 1.
Let us now deal with the case n⩾2. By changing variable Y := 1∕y, we see that
Therefore, summing up the left hand side to both sides of this identity and using the transformation \(\eta :=y-\frac 1y\),
As a result,
where the substitution t := |ξ| y 2 has been used.
Accordingly, by (2.28), the Gaussian Fourier transform and the change of variable τ := t(1 + |x|2),
This establishes (2.29).
Appendix J: Another Proof of (2.29) (Based on Extension Methods)
The idea is to consider the fundamental solution in the extended space and take a derivative (the time variable acting as a translation and, to favor the intuition, one may keep in mind that the Poisson kernel is the normal derivative of the Green function). Interestingly, this proof is, in a sense, “conceptually simpler”, and “less technical” than that in Appendix I, thus demonstrating that, at least in some cases, when appropriately used, fractional methods may lead to cultural advantagesFootnote 9 with respect to more classical approaches.
For this proof, we consider variables \(X:=(x,y)\in \mathbb {R}^n\times (0,+\infty )\subset \mathbb {R}^{n+1}\) and fix t > 0. We let Γ be the fundamental solution in \(\mathbb {R}^{n+1}\), namely
By construction Δ Γ is the Delta Function at the origin and so, for any t > 0, we have that \(\tilde \Gamma (X;t)=\tilde \Gamma (x,y;t):=\Gamma (x,y+t)\) is harmonic for \((x,y)\in \mathbb {R}^n\times (0,+\infty )\). Accordingly, the function \(U(x,y;t):=\partial _y\tilde \Gamma (x,y;t)\) is also harmonic for \((x,y)\in \mathbb {R}^n\times (0,+\infty )\). We remark that
This function is plotted in Fig. 11 (for the model case in the plane). We observe that
As a consequence, by (2.6) and (2.7) (and noticing that the role played by the variables y and t in the function U is the same),
This shows that u solves the fractional heat equation, with u approaching a Delta function when 
. Hence
that is (2.29).
Harmonic extension in the halfplane of the function \( \mathbb {R}\ni x\mapsto\frac {1}{1+|x|{ }^2}\)
Harmonic extension in the halfplane of the function \( \mathbb {R}\ni x\mapsto\frac 2\pi \,\arctan x\)
Appendix K: Proof of (2.36)
First, we construct a useful barrier. Given A > 1, we define
We claim that if A is sufficiently large, then
To prove this, fix t⩾3 (the case \(t\leqslant -3\) being similar). Then, if |ξ − t| < 1, we have that
As a consequence, if |τ − t| < 1,
This implies that
On the other hand,
In addition, if |τ|⩽1 then \(|\tau -t|\geqslant t-\tau \geqslant 3-1>1\), hence
Accordingly,
We also observe that if |τ|⩽1 then \(|t-\tau |\leqslant t+1\leqslant 2t\) and therefore
So, we plug this information into (K.4), assuming A > 1 and we obtain that
Thus, gathering together the estimates in (K.2), (K.3) and (K.5), we conclude that
as long as A is sufficiently large. This proves (K.1).
Now, to prove (2.36), we define \(v:=\dot u>0\). From (2.40), we know that
Given ε > 0, we define
We claim that
Indeed, for large ε, it holds that w ε < 0 < v and so (K.7) is satisfied. In addition, for any ε > 0,
Suppose now that ε ⋆ > 0 produces a touching point between \(w_{\varepsilon _\star }\) and v, namely \(w_{\varepsilon _\star }\leqslant v\) and \(w_{\varepsilon _\star }(t_\star )=v(t_\star )\) for some \(t_\star \in \mathbb {R}\). Notice that, if |τ|⩽3,
and therefore |t ⋆| > 3. Accordingly, if we set \(v_\star :=v-w_{\varepsilon _\star }\), using (K.1) and (K.6), we see that
Since the latter integrand is nonnegative, we conclude that v ⋆ must vanish identically, and thus \(w_{\varepsilon _\star }\) must coincide with v. But this is in contradiction with (K.8) and so the proof of (K.7) is complete.
Then, by sending 
in (K.7) we find that \(v\geqslant \frac \iota {A}\,w\), and therefore, for t⩾1 we have that \(\dot u(t)= v(t)\geqslant \kappa t^{-1-2s}\), for all t > 1, for some κ > 0.
Consequently, for any t > 1,
and a similar estimates holds for 1 + u(t) when t < −1.
These considerations establish (2.36), as desired.
Appendix L: Proof of (2.38)
Here we prove that (2.38) is a solution of (2.37). The idea of the proof, as showed in Fig. 12, is to consider the harmonic extension of the function \(\mathbb {R}\ni x\mapsto \frac 2\pi \,\arctan x\) in the halfplane \(\mathbb {R}\times (0,+\infty )\) and use the method described in (2.6) and (2.7).
We let
The function U is depictedFootnote 10 in Fig. 12. Of course, it coincides with u when y = 0 and, for any \(x\in \mathbb {R}\) and y > 0,
Hence, the setting in (2.6) is satisfied and so, in light of (2.7). we have
Also, by the trigonometric Double-angle Formula, for any \(\theta \in \left (-\frac \pi 2,\frac \pi 2\right )\),
Hence, taking \(\theta :=\arctan x\),
This and (L.2) show that (2.38) is a solution of (2.37).
Appendix M: Another Proof of (2.38) (Based on (2.29))
This proof of (2.38) is based on the fractional heat kernel in (2.29). This approach has the advantage of being quite general (see e.g. Theorem 3.1 in [27]) and also to relate the two “miraculous” explicit formulas (2.29) and (2.38), which are available only in the special case s = 1∕2.
For this, we let P = P(x, t) the fundamental solution of the heat flow in (2.25) with n = 1 and s = 1∕2. Notice that, by (2.29), we know that
with
Also, by scaling,
For any \(x\in \mathbb {R}\) and any t > 0, we define
In light of (M.2), we see that
which is bounded in \(\mathbb {R}\times [0,+\infty )\), and infinitesimal as t → +∞ for any fixed \(x\in \mathbb {R}\).
Notice also that
by (2.5). As a consequence,
where the last identity follows from (M.2).
Besides, from (M.2) we have that
and so
In view of this, we have that
Accordingly, from (M.4) and (M.5), using the extension method in (2.6) and (2.7) (with the variable y called t here), we conclude that, if
then
We remark that, by (M.1) and (M.3),
This, (M.1) and (M.6) give that
that is (2.38), as desired.
Appendix N: Proof of (2.46)
Due to translation invariance, we can reduce ourselves to proving (2.46) at the origin. We consider a measurable \(u:\mathbb {R}^n\to \mathbb {R}\) such that
Assume first that
for some r > 0. As a matter of fact, under these assumptions on u, the right-hand side of (2.46) vanishes at 0 regardless the size of r. Indeed,
This proves (2.46) under the additional assumption in (N.1), that we are now going to remove. To this end, for r ∈ (0, 1), denote by χ r the characteristic function of B r, i.e. χ r(x) = 1 if x ∈ B r and χ r(x) = 0 otherwise. Consider now u ∈ C 2, α(B r), for some α ∈ (0, 1), with
for simplicity (note that one can always modify u by considering \(\tilde u(x)=u(x)-u(0)-\nabla u(0)\cdot x\) and without affecting the operators in (2.46)). Then, the right hand side of (2.46) becomes in this case
The second addend is trivial for any r ∈ (0, 1), in view of the above remark, since u(1 − χ r) is constantly equal to 0 in B r. For the first one, we have
Now, we recall (N.2) and we notice that
which in turn implies that
On the other hand, a Taylor expansion and (N.2) yield
in view of (1.1), for some \(\eta :B_r\to \mathbb {R}\) such that \(|\eta (x)|\leqslant c|x|{ }^\alpha \). From this, (N.3) and (N.4) we deduce that
which finally justifies (2.46).
It is interesting to remark that the main contribution to prove (2.46) comes in this case from the “intermediate ring” in (N.5).
Appendix O: Proof of (2.48)
Take for instance Ω to be the unit ball and \(\bar u=1-|x|{ }^2\). Suppose that \(\|\bar u-v_\varepsilon \|{ }_{C^2(\Omega )}\leqslant \varepsilon \). Then, for small ε, if \(x\in \mathbb {R}^n\setminus B_{1/2}\) it holds that
while
This implies that there exists \(x_\varepsilon \in \overline {B_{1/2}}\) such that
As a result,
This says that (− Δ)s v ε cannot vanish at x ε and so (2.48) is proved.
Appendix P: Proof of (2.52)
Let us first notice that the identity
holds for any λ > 0 and s ∈ (0, 1), because
We also observe that when \(u\in C^\infty _0(\Omega )\), the coefficients \(\hat u_j\) decay fast as j →∞: indeed
Therefore, applying equality (P.1) to the μ j’s in (2.51) we obtainFootnote 11
where \(\{ e^{t \Delta _{N,\Omega } }\}_{t>0}\) stands for the heat semigroup associated to ΔN,Ω. i.e. \(e^{t{\Delta _{N,\Omega }}}u\) solves
To check (P.2), it is sufficient to notice that
and that
Under suitable regularity assumptions on Ω, write now the heat semigroup in terms of the heat kernel \(p_{N}^\Omega \) as
where the following two-sided estimate on \(p_{{N}}^\Omega \) holds (see, for example, [102, Theorem 3.1])
Recall also that \(p_{N}^\Omega (t,x,y)=p_{N}^\Omega (t,y,x)\) for any t > 0 and x, y ∈ Ω, and that
which follows from noticing that, for any \(u\in C^\infty _0(\Omega )\),
and therefore
By (P.5), for any x ∈ Ω and t > 0,
and, exchanging the order of integration in (P.2) (see below for a justification of this passage), one gets
where, in view of (P.4), we have
These considerations establish (2.52). Note however that in the above computations there is a limit exiting the integral in the t variable, namely:
To properly justify this we are going to build an integrable majorant in t and independent of ε of the integrand
To this end, first of all we observe that, by the boundedness of u and (P.5),
which is integrable at infinity. So, to obtain an integrable bound for (P.7), we can now focus on small values of t, say t ∈ (0, 1). For this, we denote by p the heat kernel in \(\mathbb {R}^N\) and we write
We first manipulate A. We reformulate u as
so that
In the last expression, the first integral on the right-hand side is 0 by odd symmetry, while for the second one
As for the last integral in (P.8), we have that
Equations (P.9) and (P.10) imply that
which is integrable for t ∈ (0, 1).
We turn now to the estimation of B which we rewrite as
where χ U stands for the characteristic function of a set \(U\subset \mathbb {R}^n\). By definition, B solves the heat equation in Ω with zero initial condition. Moreover, since u is supported in a compact subset K of Ω, B is satisfying the (lateral) boundary condition
for some c 1, c 2 > 0, in view of (P.4) and that, for x ∈ ∂ Ω and y ∈ K, \(|x-y|\geqslant \text{dist}(K,\partial \Omega )>0\). Then, by the parabolic maximum principle (see, for example, Section 7.1.4 in [62]),
which again is integrable for t ∈ (0, 1). These observations provide an integrable bound for the integrand in (P.8), thus completing the justification of the claim in (P.6), as desired.
Appendix Q: Proof of (2.53)
If u is periodic, we can write it in Fourier series as
and the Fourier basis is also a basis of eigenfunctions. We have that
and this shows (2.53).
Appendix R: Proof of (2.54)
We fix \(\bar k\in \mathbb {N}\). We consider the \(\bar k\)th eigenvalue \(\lambda _{\bar k}>0\) and the corresponding normalized eigenfunction \(\phi _{\bar k}=:\bar u\). We argue by contradiction and suppose that for any ε > 0 we can find v ε such that \( \|\bar u-v_\varepsilon \|{ }_{C^2(B_1)}\leqslant \varepsilon \), with \((-\Delta )^s_{D,\Omega } v_\varepsilon =0\) in B 1.
Using the notation in (2.49), we have that \(\bar u_k=\delta _{k\bar k}\) and therefore
Furthermore
Comparing this with (R.1), we obtain that \(\lambda _{\bar k}^{2s}\leqslant \,{\mathrm {const}}\,\varepsilon \), which is a contradiction for small ε. Hence, the proof of (2.54) is complete.
Appendix S: Proof of (2.60)
Let
Then, by (2.59) and writing 𝜗 := ω (t − τ), we see that
where Γ denotes here the Euler’s Gamma Function. This and (2.56) give (2.60), up to neglecting normalizing constants, as desired.
It is also worth pointing out that, as 
, formula (2.60) recovers the classical derivative, since, by (2.59),
which is (2.60) when s = 1.
Appendix T: Proof of (2.61)
First, we compute the Laplace Transform of the constant function. Namely, by (2.59), for any \(b\in \mathbb {R}\),
We also set
and we use (2.59) and the substitution 𝜗 := ω (t − τ) to calculate that
where Γ denotes here the Euler’s Gamma Function.
Exploiting this and (T.1), and making use also of (2.60), we can write the expression \(\partial ^s_{C,t} u=f\) in terms of the Laplace Transform as
with b := u(0). Hence, dividing by ω s and inverting the Laplace Transform, we obtain that
which is (2.61).
Appendix U: Proof of (2.62)
We take G to be the fundamental solution of the operator “identity minus Laplacian”, namely
being δ 0 the Dirac’s Delta. The study of this fundamental solution can be done by Fourier Transform in the sense of distributions, and this leads to an explicit representation in dimension 1 recalling (I.1); we give here a general argument, valid in any dimension, based on the heat kernel
Notice that ∂ t g = Δg and g(⋅, 0) = δ 0(⋅). Let also
Notice that
hence G, as defined in (U.2) solves (U.1).
Notice also that G is positive and bounded, due to (U.2). We also claim that
for some c, C > 0. To this end, let us fix \(x\in \mathbb {R}^n\setminus B_1\) and distinguish two regimes. If t ∈ [0, |x|], we have that \(\frac {|x|{ }^{2}}{t}\geqslant |x|\) and thus
Consequently, using the substitution \(\rho :=\frac {|x|{ }^{2}}{8t}\),
for some C > 0 possibly varying from line to line. Furthermore
for some C > 0. This and (U.4) give that
up to renaming C, which implies (U.3) in view of (U.2).
Now we compute the Laplace Transform of t s: namely, by (2.59),
We compare this result with the Laplace Transform of the mean squared displacement related to the diffusion operator in (2.62). For this, we take u to be as in (2.62) and, in the light of (2.42), we consider the function
In addition, by taking the Laplace Transform (in the variable t, for a fixed \(x\in \mathbb {R}^n\)) of the equation in (2.62), making use of (2.60) we find that
Now, we let
From (U.7), we have that
and so, comparing with (U.1), we have that W(x, ω) = G(x).
Accordingly, by (U.8),
We insert this information into (U.6) and we conclude that
We remark that the latter integral is finite, thanks to (U.3), hence we can write that
for some C > 0.
Therefore, we can compare this result with (U.5) and use the inverse Laplace Transform to obtain that the mean squared displacement in this case is proportional to t s, as desired.
Appendix V: Memory Effects of Caputo Type
It is interesting to observe that the Caputo derivative models a simple memory effect that the classical derivative cannot comprise. For instance, integrating a classical derivative of a function u with u(0) = 0, one obtains the original function “independently on the past”, namely if we set
we just obtain in this case that M u(t) = u(t) − u(0) = u(t). On the other hand, an expression as in (V.1) which takes into account the Caputo derivative does “remember the past” and takes into account the preceding events in such a way that recent events “weight” more than far away ones. To see this phenomenon, we can modify (V.1) by defining, for every s ∈ (0, 1),
To detect the memory effect, for the sake of concreteness, we take a large time \(t:=N\in \mathbb {N}\) and we suppose that the function u is constant on unit intervals, that is u = u k in [k − 1, k), for each k ∈{1, …, N}, with \(u_k\in \mathbb {R}\), and u(0) = u 1 = 0. We see that \(M_u^s\) in this case does not produce just the final outcome u N, but a weighted average of the form
To check this, we notice that, for all τ ∈ (0, N),
and hence we exploit (2.56) and (V.2) to see that
with
This completes the proof of the memory effect claimed in (V.3).
Appendix W: Proof of (3.7)
Since M is bounded and positive and u is bounded, it holds that
Moreover, for y ∈ B 1,
To simplify the notation, we now fix \(x\in \mathbb {R}^n\) and we define \({\mathcal {M}}(y):= M(x-y,y)\). Then, for y ∈ B 1, we have that
and so
Consequently, since \({\mathcal {M}}(0)=M(x,0)\) is non-degenerate, we can write
and
Hence (for smooth and bounded functions u, and y ∈ B 1) we obtain that
Thus, since the map \(y\mapsto \frac {\nabla u(x)\cdot y}{|{\mathcal {M}}(0)\,y|{ }^{n+2s} }\) is odd, recalling (W.2) we conclude that
Now we observe that, for any α⩾0,
Indeed, using polar coordinates and the fact that φ(ρω) = ρ 2+α φ(ω), for any ρ⩾0 and ω ∈ S n−1, thanks to the homogeneity, we see that
which implies (W.5).
Using (W.5) (with α := 0 and α := 2), we obtain that
and
Thanks to this, (W.1) and (W.4), we find that
with
We observe that
To check this, we first compute that
Now, we write a Taylor expansion of M(x, y) in the variable y of the form
for some \(A_{\ell m}:\mathbb {R}^n\to \mathbb {R}\) and \(B_{\ell m}:\mathbb {R}^n\to \mathbb {R}^n\). We notice that
Also,
Furthermore, we use the structural assumption (3.6), and we see that
Comparing the linear terms, this gives that
This and (W.10) imply that
Comparing this with (W.9), we see that
So, we insert this information into (W.8) and we conclude that
This establishes (W.7), as desired.
Then, plugging (W.7) into (W.6), we obtain the equation in divergence formFootnote 12 which was claimed in (3.7).
A nice representation of nonlocal effects
Appendix X: Proof of (3.12)
First we observe that
Furthermore, for y ∈ B 1,
Consequently,
and so, from the non-degeneracy of M(⋅, ⋅),
Using this and the expansion in (W.2), we see that, for y ∈ B 1,
Thus, since, in the light of (3.11), we know that the map \(y\mapsto \frac { \nabla u(x)\cdot y }{|M(x,y)\,y|{ }^{n+2s}}\) is odd, we can write that
where the last identity follows by using (W.5) (with α := 0). From this and (X.1) we obtain that
which gives (3.12).
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Abatangelo, N., Valdinoci, E. (2019). Getting Acquainted with the Fractional Laplacian. In: Dipierro, S. (eds) Contemporary Research in Elliptic PDEs and Related Topics. Springer INdAM Series, vol 33. Springer, Cham. https://doi.org/10.1007/978-3-030-18921-1_1
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