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| 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 | | The Isaacs equation is the equality |
| | \[ \sup_{a \in \mathcal{A}} \ \inf_{b \in \mathcal{B}} \ L_{ab} u(x) = f(x), \] |
| | where $L_{ab}$ is some family of linear integro-differential operators with two indices $a \in \mathcal A$ and $b \in \mathcal B$. |
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| \[ Lu(x) = \mathrm{tr} \, A(x) \cdot D^2 u + b(x) \cdot \nabla u + c(x) u + d(x) + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, \mathrm{d} \mu_x(y) \]
| | The equation appears naturally in zero sum stochastic games with [[Levy processes]]. |
| where $A(x)$ is a nonnegative matrix for all $x$, and $\mu_x$ is a nonnegative measure for all $x$ satisfying
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| \[ \int_{\R^n} \min(y^2 , 1) \mathrm{d} \mu_x(y) < +\infty. \]
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| The above definition is very general. In most cases we are interested in some subclass of linear operators. The simplest of all is the [[fractional Laplacian]]. We list below several extra assumptions that are usually made. | | The equation is [[uniformly elliptic]] with respect to any class $\mathcal{L}$ that contains all the operators $L_{ab}$. |
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| == Absolutely continuous measure == | | Note that any second order fully nonlinear uniformly elliptic PDE $F(D^2 u)=0$ can be written as an Isaacs equation by the following two steps: |
| | # $F(X)$ is Lipschitz with constant $\Lambda$, so it is the infimum of all cones $C_{X_0}(x) = F(X_0) + \Lambda|X-X_0|$. |
| | # Each cone $C(X)$ is the supremum of all linear functions of the form $L(X) = F(X_0) + \mathrm{tr} \, A \cdot (X-X_0)$ for $||A||\leq \Lambda$. |
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| In most cases, the nonnegative measure $\mu$ is assumed to be absolutely continuous: $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$.
| | A more general second order fully nonlinear uniformly elliptic PDE $F(D^2 u, Du, u, x)=0$ can also be written as an Isaacs equation if it is Lipschitz with respect to all parameters. |
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| We keep this assumption in all the examples below.
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| == Purely integro-differential operator ==
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| In this case we neglect the local part of the operator
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| \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(x,y) \mathrm d y. \]
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| == Symmetric kernels ==
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| 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.
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| In the purely integro-differentiable case, it reads as
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| \[ Lu(x) = \frac 12 \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \, K(x,y) \mathrm d y. \]
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| The second order incremental quotient is sometimes abbreviated by $\delta u(x,y) := (u(x+y)+u(x+y)-2u(x))$.
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| == Translation invariant operators ==
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| In this case, all coefficients are independent of $x$.
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| \[ Lu(x) = \mathrm{tr} \, A \cdot D^2 u + b \cdot \nabla u + c u + d + \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x) \chi_{B_1}(y)) \, K(y) \mathrm{d}y. \]
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| == The fractional Laplacian ==
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| 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.
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| \[ -(-\Delta)^{s/2} u(x) = C_{n,s} \int_{\R^n} (u(x+y)+u(x+y)-2u(x)) \frac{1}{|y|^{n+s}} \mathrm d y. \]
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| == Uniformly elliptic of order $s$ ==
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| This corresponds to the assumption that the kernel is comparable to the one of the fractional Laplacian of the same order.
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| \[ \frac {(2-s)\lambda}{|y|^{n+s}} \leq K(x,y) \leq \frac {(2-s)\Lambda}{|y|^{n+s}}. \]
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| The normalizing factor $(2-s)$ is a normalizing factor which is only important when $s$ approaches two.
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| 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.
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| == Smoothness class $k$ of order $s$ ==
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| 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
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| \[ D^r K(x,y) \leq \frac {\Lambda}{|y|^{n+s+r}} \ \ \text{for all } r\leq k. \]
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| == Order strictly below one ==
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| If a non symmetric kernel $K$ satisfies the extra local integrability assumption
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| \[ \int_{\R^n} \min(|y|,1) K(x,y) \mathrm d y < +\infty, \]
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| then the extra gradient term is not necessary in order to define the operator.
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| \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x)) \, K(x,y) \mathrm d y. \]
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| The modification in the integro-differential part of the operator becomes an extra drift term.
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| A uniformly elliptic operator of order $s<1$ satisfies this condition.
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| == Order strictly above one ==
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| If a non symmetric kernel $K$ satisfies the extra integrability assumption on its tail.
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| \[ \int_{\R^n} \min(|y|^2,|y|) K(x,y) \mathrm d y < +\infty, \]
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| then the gradient term in the integral can be taken global instead of being cut off in the unit ball.
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| \[ Lu(x) = \int_{\R^n} (u(x+y) - u(x) - y \cdot \nabla u(x)) \, K(x,y) \mathrm d y. \]
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| The modification in the integro-differential part of the operator becomes an extra drift term.
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| A uniformly elliptic operator of order $s>1$ satisfies this condition.
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| == Indexed by a matrix ==
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| 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 $s$:
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| \[ K_A(y) = \frac{(2-s) \langle y , Ay \rangle}{|y|^{n+2+s}}. \]
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