# Linear integro-differential operator

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In the purely integro-differentiable case, it reads as | In the purely integro-differentiable case, it reads as | ||

- | \[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x | + | \[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \, K(x,y) \mathrm d y. \] |

- | The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x | + | The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x-y)-2u(x))$. |

== Translation invariant operators == | == Translation invariant operators == | ||

Line 32: | Line 32: | ||

== The fractional Laplacian == | == The fractional Laplacian == | ||

- | The [[fractional | + | The [[fractional Laplacian]] is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous. |

\[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \] | \[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \] | ||

+ | |||

+ | == Stable operators == | ||

+ | |||

+ | These are the operators whose kernel is homogeneous in $y$ | ||

+ | \[ K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.\] | ||

+ | They are the generators of stable Lévy processes. The function $a$ cound be any $L^1$ function on $S^{n-1}$, or even any measure. | ||

== Uniformly elliptic of order $s$ == | == Uniformly elliptic of order $s$ == | ||

Line 47: | Line 53: | ||

== Smoothness class $k$ of order $s$ == | == Smoothness class $k$ of order $s$ == | ||

This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded | This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded | ||

- | \[ | + | \[ |\partial_y^r K(x,y)| \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \] |

== Order strictly below one == | == Order strictly below one == | ||

Line 72: | Line 78: | ||

A uniformly elliptic operator of order $s>1$ satisfies this condition. | A uniformly elliptic operator of order $s>1$ satisfies this condition. | ||

+ | |||

+ | == More singular/irregular kernels == | ||

+ | |||

+ | The concept of uniform ellipticity can be relaxed in various ways. The following family of operators was considered in the paper of Silvestre and Schwab.<ref name="schwab2014regularity" />. The integro-differential operators have the form | ||

+ | \begin{align*} | ||

+ | \int_{\R^d} \big(u(x+y) - u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\ | ||

+ | \int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x)\chi_{B_1}(y) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha = 1, \\ | ||

+ | \int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha > 1. | ||

+ | \end{align*} | ||

+ | |||

+ | For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \Omega \times \R^d \to \R$ is assumed to satisfy the following assumptions for all $x \in \Omega$, | ||

+ | * $K(x,h) \geq 0$ for all $h\in\R^d$. | ||

+ | This is a basic assumption for the integral operator to be a legit diffusion operator. | ||

+ | |||

+ | * For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(x,h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\] | ||

+ | This assumption is more general than $K(x,h) \leq (2-\alpha) \Lambda |h|^{-n-\alpha}$. Indeed, it only requires the upper bound on average. | ||

+ | |||

+ | * For every $r>0$, there exists a set $A_r$ such that | ||

+ | ** $A_r \subset B_{2r} \setminus B_r$. | ||

+ | ** $A_r$ is symmetric in the sense that $A_r = -A_r$. | ||

+ | ** $|A_r| \geq \mu |B_{2r} \setminus B_r|$. | ||

+ | ** $K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$. | ||

+ | Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(x,-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu |B_{2r} \setminus B_r|. | ||

+ | \] | ||

+ | |||

+ | This assumption says that the lower bound $K(x,h) \geq (2-\alpha) \lambda |h|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense. | ||

+ | |||

+ | * For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}. \] | ||

+ | |||

+ | This last assumption is a technical restriction which measures the contribution of the non-symmetric part of $K$ and is only relevant for the limit $\alpha \to 1$. | ||

== Indexed by a matrix == | == Indexed by a matrix == | ||

In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$: | In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$: | ||

\[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \] | \[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \] | ||

+ | This family of kernels has the outstanding property that the corresponding linear operator $L$ coincides with $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{(s-2)/2} u \right] (x)$ for some coefficients $a_{ij}$. This property has been exploited in order to obtain an ABP estimate in integral form. <ref name="GS"/> | ||

== Second order elliptic operators as limits of purely integro-differential ones == | == Second order elliptic operators as limits of purely integro-differential ones == | ||

Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators | Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators | ||

- | |||

\[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{|y|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \] | \[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{|y|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \] | ||

+ | define in the limit $\sigma \to 2^-$ a second order linear elliptic operator (possibly degenerate). This can be checked for any fixed $C^2$ function $u$ by a straightforward computation using the second order Taylor expansion. We get | ||

+ | \[ \lim_{\sigma \to 2^-} L_\sigma u(x) = a_{ij} \partial_{ij} u(x),\] | ||

+ | where $a_{ij}$ is determined by the quadratic form | ||

+ | \[ a_{ij} e_i e_j = \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x), \qquad \text{for vectors } e \in \R^d.\] | ||

- | + | A class of kernels that is big enough to recover all translation invariant elliptic operators of the form $Lu(x) = Tr ( A \cdot D^2u(x) )$ is given by the kernels | |

- | + | ||

\[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\] | \[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\] | ||

- | |||

where $A$ is an invertible symmetric matrix. | where $A$ is an invertible symmetric matrix. | ||

+ | |||

+ | Conversely, the condition | ||

+ | \[ \lambda |e|^2 \leq \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x) \leq \Lambda |e|^2, \qquad \text{for all vectors } e \in \R^d,\] | ||

+ | defines the largest class of stable operators that may be considered uniformly elliptic. Indeed, this is the condition that ensures regularity of solutions to translation invariant integro-differential equations.<ref name="ros2014regularity" /> If we let the operator depend on $x$, it is an outstanding [[Open problems#Hölder estimates for singular integro-differential equations|open problem]] whether this condition alone (for every value of $x$) ensures [[Holder estimates]] for the integro-differential equation. | ||

+ | |||

+ | == Characterization via global maximum principle == | ||

+ | |||

+ | A bounded linear operator | ||

+ | |||

+ | \[ L: C^2_0(\mathbb{R}^n) \to C(\mathbb{R}^n) \] | ||

+ | |||

+ | is said to satisfy the global maximum principle if given any $u \in C^2_0(\mathbb{R}^n)$ with a global maximum at some point $x_0$ we have | ||

+ | |||

+ | \[ (Lu)(x_0) \leq 0 \] | ||

+ | |||

+ | It turns out this property imposes strong restrictions on the operator $L$, and we have the following theorem due to Courrège <ref name="C65"/> <ref name="C64"/>: if $L$ satisfies the global maximum principle then it has the form | ||

+ | |||

+ | \[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \] | ||

+ | |||

+ | where again $A(x)$ is a nonnegative matrix for all $x$, $c(x)\leq 0$ and $\mu_x$ is a nonnegative measure for all $x$ satisfying | ||

+ | |||

+ | \[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \] | ||

+ | |||

+ | and $A(x),c(x)$ and $b(x)$ are bounded. | ||

+ | |||

+ | |||

+ | == See also == | ||

+ | |||

+ | * [[Fractional Laplacian]] | ||

+ | * [[Levy processes]] | ||

+ | * [[Dirichlet form]] | ||

+ | |||

+ | |||

+ | == References == | ||

+ | {{reflist|refs= | ||

+ | <ref name="C64">{{Citation | last1=Courrège | first1=Philippe | title=Générateur infinitésimal d'un semi-groupe de convolution sur $R^n$, et formule de Lévy-Khinchine | year=1964 | journal=Bulletin des Sciences Mathématiques. 2e Série | issn=0007-4497 | volume=88 | pages=3–30}}</ref> | ||

+ | <ref name="C65">{{Citation | last1=Courrège | first1=P. | title=Sur la forme intégro-différentielle des opéateurs de $C_k^\infty(\mathbb{R}^n)$ dans $C(\mathbb{R}^n)$ satisfaisant au principe du maximum | journal=Sém. Théorie du potentiel (1965/66) Exposé | volume=2}}</ref> | ||

+ | <ref name="schwab2014regularity">{{Citation | last1=Schwab | first1= Russell W | last2=Silvestre | first2= Luis | title=Regularity for parabolic integro-differential equations with very irregular kernels | journal=arXiv preprint arXiv:1412.3790}}</ref> | ||

+ | <ref name="ros2014regularity">{{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=arXiv preprint arXiv:1412.3892}}</ref> | ||

+ | <ref name="GS">{{Citation | last1=Guillen | first1=N. | last2=Schwab | first2=R. | title=Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations | year=2010 | journal=Arxiv preprint arXiv:1101.0279}}</ref> | ||

+ | }} |

## Latest revision as of 20:21, 17 November 2016

The linear integro-differential operators that we consider *in this wiki* are the generators of Levy processes. According to the Levy-Kintchine formula, they have the general form

\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \] where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying \[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]

The above definition is very general. Many theorems, and in particular regularity theorems, require extra assumptions in the kernels $K$. These assumptions restrict the study to certain sub-classes of linear operators. The simplest of all is the fractional Laplacian. We list below several extra assumptions that are usually made.

## Absolutely continuous measure

In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.

We keep this assumption in all the examples below.

## Purely integro-differential operator

In this case we neglect the local part of the operator \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]

## Symmetric kernels

If the kernel is symmetric $K(x,y) = K(x,-y)$, then we can remove the gradient term from the integral and replace the difference by a second order quotient.

In the purely integro-differentiable case, it reads as \[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \, K(x,y) \mathrm d y. \]

The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x-y)-2u(x))$.

## Translation invariant operators

In this case, all coefficients are independent of $x$. \[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \]

## The fractional Laplacian

The fractional Laplacian is the simplest and most common purely integro-differential operator. It corresponds to a translation invariant operator for which $K(y)$ is radially symmetric and homogeneous.

\[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]

## Stable operators

These are the operators whose kernel is homogeneous in $y$ \[ K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.\] They are the generators of stable Lévy processes. The function $a$ cound be any $L^1$ function on $S^{n-1}$, or even any measure.

## Uniformly elliptic of order $s$

This corresponds to the assumption that the kernel is comparable to the one of the fractional Laplacian of the same order. \[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]

The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.

An operator of variable order can be either one for which $s$ depends on $x$, or one for which there are two values $s_1<s_2$, one for the left hand side and another for the right hand side.

## Smoothness class $k$ of order $s$

This class (sometimes denoted as $\mathcal L_k^s$) corresponds to kernels that are uniformly elliptic of order $s$ and, moreover, their derivatives are also bounded \[ |\partial_y^r K(x,y)| \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \]

## Order strictly below one

If a non symmetric kernel $K$ satisfies the extra local integrability assumption \[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \] then the extra gradient term is not necessary in order to define the operator.

\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]

The modification in the integro-differential part of the operator becomes an extra drift term.

A uniformly elliptic operator of order $s<1$ satisfies this condition.

## Order strictly above one

If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail. \[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \] then the gradient term in the integral can be taken global instead of being cut off in the unit ball.

\[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]

The modification in the integro-differential part of the operator becomes an extra drift term.

A uniformly elliptic operator of order $s>1$ satisfies this condition.

## More singular/irregular kernels

The concept of uniform ellipticity can be relaxed in various ways. The following family of operators was considered in the paper of Silvestre and Schwab.^{[1]}. The integro-differential operators have the form
\begin{align*}
\int_{\R^d} \big(u(x+y) - u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha < 1, \\
\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x)\chi_{B_1}(y) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha = 1, \\
\int_{\R^d} \big(u(x+y) - u(x) - y \cdot \nabla u(x) \big) K(x,h) \mathrm{d} h \qquad \text{ if } \alpha > 1.
\end{align*}

For $\lambda$, $\Lambda$, $\mu$ and $\alpha$ given, the kernel $K: \Omega \times \R^d \to \R$ is assumed to satisfy the following assumptions for all $x \in \Omega$,

- $K(x,h) \geq 0$ for all $h\in\R^d$.

This is a basic assumption for the integral operator to be a legit diffusion operator.

- For every $r>0$, \[ \int_{B_{2r} \setminus B_r} K(x,h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}\]

This assumption is more general than $K(x,h) \leq (2-\alpha) \Lambda |h|^{-n-\alpha}$. Indeed, it only requires the upper bound on average.

- For every $r>0$, there exists a set $A_r$ such that
- $A_r \subset B_{2r} \setminus B_r$.
- $A_r$ is symmetric in the sense that $A_r = -A_r$.
- $|A_r| \geq \mu |B_{2r} \setminus B_r|$.
- $K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$.

Equivalently \[ \left\vert \left\{y \in B_{2r} \setminus B_r: K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(x,-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu |B_{2r} \setminus B_r|. \]

This assumption says that the lower bound $K(x,h) \geq (2-\alpha) \lambda |h|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense.

- For all $r>0$, \[ \left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}. \]

This last assumption is a technical restriction which measures the contribution of the non-symmetric part of $K$ and is only relevant for the limit $\alpha \to 1$.

## Indexed by a matrix

In some cases, it is interesting to study a family of kernels $K$ that are indexed by a matrix. For example, given the matrix $A$, one can consider the kernel of order $s$:
\[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]
This family of kernels has the outstanding property that the corresponding linear operator $L$ coincides with $Lu(x) = a_{ij} \partial_{ij}\left[(-\Delta)^{(s-2)/2} u \right] (x)$ for some coefficients $a_{ij}$. This property has been exploited in order to obtain an ABP estimate in integral form. ^{[2]}

## Second order elliptic operators as limits of purely integro-differential ones

Given any bounded, even, positive function $a: \mathbb{R}^n\to \mathbb{R}$, the family of operators \[L_\sigma u(x) = (2-\sigma) \int_{\mathbb{R}^n} (u(x+y)+u(x-y)-2u(x))\frac{a(y)}{|y|^{n+\sigma}}dy,\;\; \sigma \in (0,2), \] define in the limit $\sigma \to 2^-$ a second order linear elliptic operator (possibly degenerate). This can be checked for any fixed $C^2$ function $u$ by a straightforward computation using the second order Taylor expansion. We get \[ \lim_{\sigma \to 2^-} L_\sigma u(x) = a_{ij} \partial_{ij} u(x),\] where $a_{ij}$ is determined by the quadratic form \[ a_{ij} e_i e_j = \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x), \qquad \text{for vectors } e \in \R^d.\]

A class of kernels that is big enough to recover all translation invariant elliptic operators of the form $Lu(x) = Tr ( A \cdot D^2u(x) )$ is given by the kernels \[ K_A(y) = (2-\sigma) \frac{1}{|Ay|^{n+\sigma}},\] where $A$ is an invertible symmetric matrix.

Conversely, the condition
\[ \lambda |e|^2 \leq \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x) \leq \Lambda |e|^2, \qquad \text{for all vectors } e \in \R^d,\]
defines the largest class of stable operators that may be considered uniformly elliptic. Indeed, this is the condition that ensures regularity of solutions to translation invariant integro-differential equations.^{[3]} If we let the operator depend on $x$, it is an outstanding open problem whether this condition alone (for every value of $x$) ensures Holder estimates for the integro-differential equation.

## Characterization via global maximum principle

A bounded linear operator

\[ L: C^2_0(\mathbb{R}^n) \to C(\mathbb{R}^n) \]

is said to satisfy the global maximum principle if given any $u \in C^2_0(\mathbb{R}^n)$ with a global maximum at some point $x_0$ we have

\[ (Lu)(x_0) \leq 0 \]

It turns out this property imposes strong restrictions on the operator $L$, and we have the following theorem due to Courrège ^{[4]} ^{[5]}: if $L$ satisfies the global maximum principle then it has the form

\[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]

where again $A(x)$ is a nonnegative matrix for all $x$, $c(x)\leq 0$ and $\mu_x$ is a nonnegative measure for all $x$ satisfying

\[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]

and $A(x),c(x)$ and $b(x)$ are bounded.

## See also

## References

- ↑ Schwab, Russell W; Silvestre, Luis, "Regularity for parabolic integro-differential equations with very irregular kernels",
*arXiv preprint arXiv:1412.3790* - ↑ Guillen, N.; Schwab, R. (2010), "Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations",
*Arxiv preprint arXiv:1101.0279* - ↑ Ros-Oton, Xavier; Serra, Joaquim, "Regularity theory for general stable operators",
*arXiv preprint arXiv:1412.3892* - ↑ Courrège, P., "Sur la forme intégro-différentielle des opéateurs de $C_k^\infty(\mathbb{R}^n)$ dans $C(\mathbb{R}^n)$ satisfaisant au principe du maximum",
*Sém. Théorie du potentiel (1965/66) Exposé***2** - ↑ Courrège, Philippe (1964), "Générateur infinitésimal d'un semi-groupe de convolution sur $R^n$, et formule de Lévy-Khinchine",
*Bulletin des Sciences Mathématiques. 2e Série***88**: 3–30, ISSN 0007-4497