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| The fractional Laplacian $(-\Delta)^s$ is a classical operator which gives the standard Laplacian when $s=1$. One can think of $-(-\Delta)^s$ as the most basic [[elliptic linear integro-differential operator]] of order $2s$ and can be defined in several equivalent ways (listed below). A range of powers of particular interest is $s \in (0,1)$, in which case for $u \in \mathcal{S}(\mathbb{R}^n)$ we can write the operator as
| | A Lévy process is an important type of [[stochastic process]] (namely, a family of $\mathbb{R}^d$ valued random variables each indexed by a positive number $t\geq 0$). In the context of parabolic integro-differential equations they play the same role that Brownian motion and more general diffusions play in the theory of second order parabolic equations. |
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| \[-(-\Delta)^su(x) = c_{n,s} \int_{\mathbb{R}^d}\frac{\delta u (x,y) }{|y|^{d+2s}}dy\]
| | Informally speaking, a Lévy process is a Brownian motion which may jump, the times, length and direction of the jumps being random variables. A prototypical example would be $X(t)=B(t)+N(t)$ where $B(t)$ is the standard [[Brownian motion]] and $N(t)$ is a [[Compound Poisson process]], the trajectory described by typical sample path of this process would look like the union of several disconnected Brownian motion paths. |
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| where $c_{n,s}$ is a universal constant and $\delta u(x,y):= u(x+y)+u(x-y)-2u(x)$. This particular expression shows that in this range of $s$ the operator enjoys the following monotonicity property: if $u$ has a global maximum at $x$, then $(-\Delta)^s u(x) \geq 0$, with equality only if $u$ is constant. From this monotonicity, a [[comparison principle]] can be derived for equations involving the fractional Laplacian.
| | == Definition == |
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| == Definitions == | | A stochastic process $X=\{X(t)\}_{t \geq 0}$ with values in $\mathbb{R}^d$ is said to be a Lévy process if |
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| All the definitions below are equivalent.
| | 1.For any sequence $0 \leq t_1 < t_2 <...<t_n$ the random variables $X(t_0),X(t_1)-X(t_0),...,X(t_n)-X(t_{n-1})$ are independent. |
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| === As a pseudo-differential operator ===
| | 2.For any positive times $s\leq t$ the random variables $X(t-s)$ and $X(t)-X(s)$ have the same probability law. |
| The fractional Laplacian is the pseudo-differential operator with symbol $|\xi|^{2s}$. In other words, the following formula holds
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| \[ \widehat{(-\Delta)^s f}(\xi) = |\xi|^{2s} \hat f(\xi).\]
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| for any function (or tempered distribution) for which the right hand side makes sense.
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| This formula is the simplest to understand and it is useful for problems in the whole space. On the other hand, it is hard to obtain local estimates from it.
| | 3.Almost surely, the trajectory of $X(t)$ is continuous from the right, with limit from the left also known as "càdlàg" for its acronym in french. |
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| === From functional calculus === | | == Lévy-Khintchine Formula == |
| Since the operator $-\Delta$ is a self-adjoint positive definite operator in a dense subset $D$ of $L^2(\R^n)$, one can define $F(-\Delta)$ for any continuous function $F:\R^+ \to \R$. In particular, this serves as a more or less abstract definition of $(-\Delta)^s$.
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| This definition is not as useful for practical applications, since it does not provide any explicit formula.
| | It follows from the first two properties above that if $X$ is a Lévy process and we further assume $X(0)=0$ a.s. then for each fixed positive $t$ the random variable $X(t)$ is infinitely divisible, that is, it can be written as the sum of $n$ independent and identically distribued random variables, for all $n\in\mathbb{N}$. Indeed, let $h=\tfrac{t}{n}$, then |
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| === As a singular integral === | | \[X(t) = \left( X(h)-X(0)\right)+\left( X(2h)-X(h)\right)+...+\left( X(t)-X((n-1)h)\right)\] |
| If $f$ is regular enough and $s \in (0,1)$, $(-\Delta)^s f(x)$ can be computed by the formula
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| \[ (-\Delta)^s f(x) = c_{n,s} \int_{\R^n} \frac{f(x) - f(y)} {|x-y|^{n+2s}} \mathrm d y .\]
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| Where $c_{n,s}$ is a constant depending on dimension and $s$.
| | and by the above definition the differences $X(kh)-X((k-1)h)$ are independent and distributed the same as $X(h)$. From the infinite divisibility of $X(t)$ it follows by a theorem of Lévy and Khintchine that for any $\xi \in \mathbb{R}^d$ we have |
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| This formula is the most useful to study local properties of equations involving the fractional Laplacian and regularity for critical semilinear problems.
| | \[ \mathbb{E} \left [ e^{i\xi\cdot X_t}\right ] = e^{t\eta(\xi)}\] |
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| === As a generator of a [[Levy process]] ===
| | the function $\eta(\xi)$ given by |
| The operator can be defined as the generator of $\alpha$-stable Lévy processes. More precisely, if $X_t$ is the isotropic $\alpha$-stable Lévy process starting at zero and $f$ is a smooth function, then
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| \[ (-\Delta)^{\alpha/2} f(x) = \lim_{h \to 0^+} \frac 1 {h} \mathbb E [f(x) - f(x+X_h)]. \]
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| This definition is important for applications to probability.
| | \[\eta(\xi)=y\cdot b i -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{\xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y) \] |
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| == Inverse operator ==
| | where $b$ is a vector, $A$ is a positive matrix and $\mu$ is a Lévy measure, that is, a Borel measure in $\mathbb{R}^d$ such that |
| The inverse of the $s$ power of the Laplacian is the $-s$ power of the Laplacian $(-\Delta)^{-s}$. For $0<s<n/2$, there is an integral formula which says that $(-\Delta)^{-s}u$ is the convolution of the function $u$ with the ''Riesz potential'':
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| \[ (-\Delta)^{-s} u(x) = C_{n,s} \int_{\R^n} u(x-y) \frac{1}{|y|^{n-2s}} \mathrm d y,\]
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| which holds as long as $u$ is integrable enough for the right hand side to make sense.
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| == Heat kernel ==
| | \[ \int_{\mathbb{R}^d}\frac{|y|^2}{1+|y|^2}d\mu(y) <+\infty \] |
| The fractional heat kernel $p(t,x)$ is the fundamental solution to the [[fractional heat equation]]. It is the function which solves the equation
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| \begin{align*} | |
| p(0,x) &= \delta_0 \\
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| p_t(t,x) + (-\Delta)^s p &= 0
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| \end{align*} | |
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| The kernel is easy to compute in Fourier side as $\hat p(t,\xi) = e^{-t|\xi|^{2s}}$. There is no explicit formula in physical variables, but the following inequalities are known to hold for some constant $C$
| | == Connection to linear integro-differential operators == |
| \[ C^{-1} \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right) \leq p(t,x) \leq C \left( t^{-\frac n {2s}} \wedge \frac{t}{|x|^{n+2s}} \right). \]
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| Moreover, the function $p$ is $C^\infty$ in $x$ for $t>0$ and the following identity follows by scaling
| | Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows |
| \[ p(t,x) = t^{-\frac n {2s}} p \left( 1 , t^{-\frac 1 {2s}} x \right). \] | |
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| == Poisson kernel ==
| | \[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t) \right ]\] |
| Given a function $g : \R^n \setminus B_1 \to \R$, there exists a unique function $u$ which solves the Dirichlet problem
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| \begin{align*}
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| u(x) &= g(x) \qquad \text{if } x \notin B_1 \\
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| (-\Delta)^s u(x) &= 0 \qquad \text{if } x \in B_1. | |
| \end{align*}
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| The solution can be computed explicitly using the Poisson kernel
| | Given the initial assumption on $X(0)$ it is clear that $U_0$ is the identity, and given that $X(t)-X(s)$ is distributed as $X(t-s)$ it follows that $U_t \circ U_s = U_{t+s}$. |
| \[ u(x) = \int_{\R^n \setminus B_1} g(y) P(y,x) \mathrm d y,\]
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| where<ref name="R"/>
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| \[ P(y,x) = C_{n,s} \left( \frac{1-|x|^2}{|y|^2-1}\right)^s \frac 1 {|x-y|^n}.\]
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| The justification of this Poisson kernel can be found in the classical book of Landkof (1.6.11')<ref name="L"/>.
| | As a semigroup, $U_t$ has an infinitesimal generator which turns out to be a [[Linear integro-differential operator]]. More precisely, if we let $f(x,t):=(U_tf)(x)$, then, assuming that $f(x,t)$ has enough regularity it can be checked that |
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| == Green's function for the ball == | | \[\partial_t f = Lf \;\;\;\mbox{ for all } (x,t)\in\mathbb{R}^d\times \mathbb{R}_+\] |
| For a function $g \in L^2(B_1)$, there exists a unique function $u \in H^s(\R^n)$ such that
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| \begin{align*} | |
| u(x) &= 0 && \text{if } x \notin B_1 \\
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| (-\Delta)^s u &= g(x) && \text{if } x \in B_1.
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| \end{align*}
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| The solution is given explicitly using the Green's function,
| | where for any smooth function $\phi$, we have |
| \[ u(x) = \int_{B_1} G_{B_1}(x, y) g(y) \mathrm d y, \]
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| where<ref name="R"/> | |
| \[ G_{B_1}(x, y) = C_{n,s} |x - y|^{2 s - n} \int_0^{r_0(x, y)} \frac{r^{s-1}}{(r+1)^{n/2}} \, \mathrm d r \]
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| with
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| \[ r_0(x, y) = \frac{(1 - |x|^2) (1 - |y|^2)}{|x - y|^2} . \]
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| The above formula holds for all $s \in (0, 1)$ also for $n = 1$.<ref name="BGR"/>
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| == Regularity issues ==
| | \[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \] |
| Any function $u$ which satisfies $(-\Delta)^s u=0$ in any open set $\Omega$, then $u \in C^\infty$ inside $\Omega$. This follows from the smoothness of the Poisson kernel for balls.
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| More generally, one has the estimate
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| \[\|u\|_{C^{\alpha+2s}(B_{1/2})}\leq C\left( | |
| \|(-\Delta)^s u\|_{C^{\alpha}(B_1)}+\|u\|_{L^{\infty}(B_1)}+\int_{\R^n\setminus B_1}|u(y)|\frac{dy}{|y|^{n+2s}}\right)\] | |
| for any $\alpha\geq0$ such that $\alpha+2s$ is not an integer.
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| === Full space regularization of the Riesz potential ===
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| If $(-\Delta)^s u = f$ in $\R^n$, then of course $u = (-\Delta)^{-s}f$. It is simple to see that the operator $(-\Delta)^{-s}$ regularizes the functions ''up to $2s$ derivatives''. In Fourier side, $\hat u(\xi) = |\xi|^{-2s} \hat f(\xi)$, thus $\hat u$ has a stronger decay than $\hat f$. More precisely, if $f \in C^\alpha$, then $u \in C^{2s+\alpha}$ as long as $2s+\alpha$ is not an integer (A proof of this using only the integral representation of $(-\Delta)^{-s}$ was given in the preliminaries section of <ref name="S"/>, but the result is presumably very classical). More generally, if $f$ belongs to the Besov space $B_{p,q}^r$, then $u \in B_{p,q}^{r+2s}$, $s>0$. However, if $f$ belongs to $L^p$ then it does not follow that $u\in W^{2s,p}$; this is true only for $p\geq2$. For $1<p<2$ one only have $u\in B^{2s}_{p,2}\supset W^{2s,p}$ ---see Chapter V in Stein<ref name="Stein"/>.
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| === Boundary regularity ===
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| From the Poisson formula, one can observe that if the boundary data $g$ of the Dirichlet problem in $B_1$ is bounded and smooth, then $u \in C^s(\overline B_1)$ and in general no better. The singularity of $u$ occurs only on $\partial B_1$, the solution $u$ would be $C^\infty$ in the interior of the unit ball (which is also a consequence of the explicit Poisson kernel).
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| Even if $u$ is not $C^\infty$ up to the boundary, we have the following: consider the solution $u$ to the Dirichlet problem
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| \[\left\{ \begin{array}{rcll}
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| (-\Delta)^s u &=&g&\textrm{in }\Omega \\ | |
| u&=&0&\textrm{in }\R^n\backslash \Omega.
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| \end{array}\right.\]
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| If $\Omega$ is $C^\infty$, then
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| \[g\in C^\infty(\overline\Omega)\qquad \Longrightarrow \qquad u/d^s\in C^\infty(\overline\Omega),\]
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| where $d(x)$ is (a smoothed version of) the distance to $\partial\Omega$; see <ref name="Grubb"/> and also <ref name="RS"/>.
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| If $\Omega$ is $C^{2,\alpha}$ and $g$ is $C^\alpha$, then $u/d^s$ is $C^{\alpha+s}$ up to the boundary <ref name="RS-K"/>.
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| Related to this, if $g$ is not bounded but only in $L^p(\Omega)$ then $u\in L^q$ with $q=\frac{np}{n-2ps}$ in case $p<n/(2s)$, while $u\in L^\infty(\Omega)$ in case $p>n/(2s)$ ---see for example Proposition 1.4 in <ref name="RS2"/>.
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| == References ==
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| {{reflist|refs=
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| <ref name="BGR">{{Citation | last1=Blumenthal | first1=R. M. | last2=Getoor | first2=R. K. | last3=Ray | first3=D. B. | title=On the distribution of first hits for the symmetric stable processes | url=http://www.jstor.org/stable/1993561 | year=1961 | journal=Trans. Amer. Math. Soc. | issn=0002-9947 | volume=99 | pages=540–554}}</ref>
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| <ref name="L">{{Citation | last1=Landkof | first1=N. S. | title=Foundations of modern potential theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1972}}</ref>
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| <ref name="R">{{Citation | last1=Riesz | first1=M. | title=Intégrales de Riemann-Liouville et potentiels | url=http://acta.fyx.hu/acta/showCustomerArticle.action?id=5634&dataObjectType=article | year=1938 | journal=Acta Sci. Math. Szeged | issn=0001-6969 | volume=9 | issue=1 | pages=1–42}}</ref>
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| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Regularity of the obstacle problem for a fractional power of the Laplace operator | url=http://dx.doi.org/10.1002/cpa.20153 | doi=10.1002/cpa.20153 | year=2007 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=60 | issue=1 | pages=67–112}}</ref>
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| <ref name="RS">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The Dirichlet problem for the fractional Laplacian: regularity up to the boundary | url=http://arxiv.org/abs/1207.5985 | year=2012 | journal=[[J. Math. Pures Appl.]] | volume=101 | pages=275-302 }}</ref>
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| <ref name="Stein">{{Citation | last1=Stein | first1=E. | title=Singular Integrals And Differentiability Properties Of Functions | publisher=[[Princeton Mathematical Series]] | year=1970}}</ref>
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| <ref name="RS2">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=The extremal solution for the fractional Laplacian | url=http://arxiv.org/abs/1305.2489 | year=2013 | journal=[[Calc. Var. Partial Differential Equations]] | pages=to appear }}</ref>
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| <ref name="Grubb">{{Citation | last1=Grubb | first1=G. | title=Fractional Laplacians on domains, a development of Hormander's theory of $mu$-transmission pseudodifferential operators | url=http://arxiv.org/abs/1310.0951 | year=2014 | journal=[[arXiv]] | pages=1-43 }}</ref>
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| <ref name="RS-K">{{Citation | last1=Ros-Oton | first1=X. | last2=Serra | first2=J. | title=Boundary regularity for fully nonlinear integro-differential equations | year=2014 | journal=[[preprint arXiv]] }}</ref>
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| }}
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A Lévy process is an important type of stochastic process (namely, a family of $\mathbb{R}^d$ valued random variables each indexed by a positive number $t\geq 0$). In the context of parabolic integro-differential equations they play the same role that Brownian motion and more general diffusions play in the theory of second order parabolic equations.
Informally speaking, a Lévy process is a Brownian motion which may jump, the times, length and direction of the jumps being random variables. A prototypical example would be $X(t)=B(t)+N(t)$ where $B(t)$ is the standard Brownian motion and $N(t)$ is a Compound Poisson process, the trajectory described by typical sample path of this process would look like the union of several disconnected Brownian motion paths.
Definition
A stochastic process $X=\{X(t)\}_{t \geq 0}$ with values in $\mathbb{R}^d$ is said to be a Lévy process if
1.For any sequence $0 \leq t_1 < t_2 <...<t_n$ the random variables $X(t_0),X(t_1)-X(t_0),...,X(t_n)-X(t_{n-1})$ are independent.
2.For any positive times $s\leq t$ the random variables $X(t-s)$ and $X(t)-X(s)$ have the same probability law.
3.Almost surely, the trajectory of $X(t)$ is continuous from the right, with limit from the left also known as "càdlàg" for its acronym in french.
Lévy-Khintchine Formula
It follows from the first two properties above that if $X$ is a Lévy process and we further assume $X(0)=0$ a.s. then for each fixed positive $t$ the random variable $X(t)$ is infinitely divisible, that is, it can be written as the sum of $n$ independent and identically distribued random variables, for all $n\in\mathbb{N}$. Indeed, let $h=\tfrac{t}{n}$, then
\[X(t) = \left( X(h)-X(0)\right)+\left( X(2h)-X(h)\right)+...+\left( X(t)-X((n-1)h)\right)\]
and by the above definition the differences $X(kh)-X((k-1)h)$ are independent and distributed the same as $X(h)$. From the infinite divisibility of $X(t)$ it follows by a theorem of Lévy and Khintchine that for any $\xi \in \mathbb{R}^d$ we have
\[ \mathbb{E} \left [ e^{i\xi\cdot X_t}\right ] = e^{t\eta(\xi)}\]
the function $\eta(\xi)$ given by
\[\eta(\xi)=y\cdot b i -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{\xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y) \]
where $b$ is a vector, $A$ is a positive matrix and $\mu$ is a Lévy measure, that is, a Borel measure in $\mathbb{R}^d$ such that
\[ \int_{\mathbb{R}^d}\frac{|y|^2}{1+|y|^2}d\mu(y) <+\infty \]
Connection to linear integro-differential operators
Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows
\[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t) \right ]\]
Given the initial assumption on $X(0)$ it is clear that $U_0$ is the identity, and given that $X(t)-X(s)$ is distributed as $X(t-s)$ it follows that $U_t \circ U_s = U_{t+s}$.
As a semigroup, $U_t$ has an infinitesimal generator which turns out to be a Linear integro-differential operator. More precisely, if we let $f(x,t):=(U_tf)(x)$, then, assuming that $f(x,t)$ has enough regularity it can be checked that
\[\partial_t f = Lf \;\;\;\mbox{ for all } (x,t)\in\mathbb{R}^d\times \mathbb{R}_+\]
where for any smooth function $\phi$, we have
\[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \]