# Linear integro-differential operator

(Difference between revisions)
 Revision as of 20:15, 8 April 2015 (view source)Luis (Talk | contribs) (→More singular/irregular kernels)← Older edit Latest revision as of 20:21, 17 November 2016 (view source)Luis (Talk | contribs) (→Symmetric kernels) (8 intermediate revisions not shown) Line 22: Line 22: In the purely integro-differentiable case, it reads as In the purely integro-differentiable case, it reads as - $Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y.$ + $Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x-y)-2u(x)) \, K(x,y) \mathrm d y.$ - The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$. + 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 39: Line 39: These are the operators whose kernel is homogeneous in $y$ These are the operators whose kernel is homogeneous in $y$ - $K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.$ + $K(y)=\frac{a(y/|y|)}{|y|^{n+s}}\qquad\textrm{or}\qquad K(x,y)=\frac{a(x,y/|y|)}{|y|^{n+s}}.$ - They are the generators of stable Lévy processes. + 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 81: Line 81: == More singular/irregular kernels == == More singular/irregular kernels == - The concept of uniform ellipticity can be relaxed in various ways. The following family of kernels was considered in the paper of Silvestre and Schwab. + 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.. 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 $K: \R^d \to \R$ is assumed to satisfy the following assumptions. + 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 $x \in \Omega$, - * $K(h) \geq 0$ for all $h\in\R^d$. + * $K(x,h) \geq 0$ for all $h\in\R^d$. This is a basic assumption for the integral operator to be a legit diffusion operator. This is a basic assumption for the integral operator to be a legit diffusion operator. - * For every $r>0$, $\int_{B_{2r} \setminus B_r} K(h) \mathrm{d} h \leq (2-\alpha) \Lambda r^{-\alpha}$ + * 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 $K(y) \leq (2-\alpha) \Lambda |y|^{-n-\alpha}$. Indeed, it only requires the upper bound on average. + 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 * For every $r>0$, there exists a set $A_r$ such that Line 94: Line 99: ** $A_r$ is symmetric in the sense that $A_r = -A_r$. ** $A_r$ is symmetric in the sense that $A_r = -A_r$. ** $|A_r| \geq \mu |B_{2r} \setminus B_r|$. ** $|A_r| \geq \mu |B_{2r} \setminus B_r|$. - ** $K(h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$. + ** $K(x,h) \geq (2-\alpha) \lambda r^{-d-\alpha}$ in $A_r$. - Equivalently $\left\vert \left\{y \in B_{2r} \setminus B_r: K(h) \geq (2-\alpha) \lambda r^{-d-\alpha} \text{ and } K(-h) \geq (2-\alpha) \lambda r^{-d-\alpha} \right\} \right\vert \geq \mu |B_{2r} \setminus B_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 $K(y) \geq (2-\alpha) \lambda |y|^{-n-\alpha}$ takes place on sets of positive density in certain scale invariant sense. + 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 $r>0$, $\left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}.$ * For all $r>0$, $\left\vert \int_{B_{2r} \setminus B_r} h K(h) \mathrm{d} h \right\vert \leq \Lambda |1-\alpha| r^{1-\alpha}.$ Line 107: Line 112: 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 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. == 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 + $a_{ij} e_i e_j = \int_{\partial B_1} (x \cdot e)^2 a(x) \, \mathrm d S(x), \qquad \text{for vectors } e \in \R^d.$ - 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. 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 + 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. 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 == == Characterization via global maximum principle == Line 154: Line 164: {{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}} {{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}} {{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}} {{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}} + {{Citation | last1=Ros-Oton | first1= Xavier | last2=Serra | first2= Joaquim | title=Regularity theory for general stable operators | journal=arXiv preprint arXiv:1412.3892}} + {{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}} }} }}

## 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.

4. 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
5. 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