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| = Problems for integro-differential equations with rough coefficients or nonlinear equations =
| | 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. |
| == Hölder estimates for singular integro-differential equations ==
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| Consider an integro-differential equation of the form
| | Informally speaking, a Lévy process is a random trajectory, generalizing the concept of Brownian motion, which may contain jump discontinuities. 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. |
| \[ \int_{\R^d} \left(u(x+y) - u(x) \right) \mathrm{d} \mu_x(y) = 0 \qquad \text{for all } x \in B_1.\]
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| (An extra gradient correction term may be necessary if the measure $\mu_x$ is too singular at the origin) | |
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| [[Hölder estimates]] are known to hold under certain 'ellipticity' assumptions for the measures $\mu_x(y)$. In many cases, we consider the absolutely continuous version $\mathrm{d} \mu_x(y) = K(x,y) \mathrm{d}y$ and write the assumptions in terms of the kernel $K$. One would expect that the estimates should hold every time the measures $\mu_x$ satisfy.
| | == Definition == |
| \[ \int_{B_{2R} \setminus B_R} (x \cdot e)^2 \mathrm{d} \mu_x(y) \approx R^{2-\alpha}, \]
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| for all radius $R>0$ and $x \in B_1$, for some given constant $\alpha \in (0,2)$. This is the sharp assumption for stable operators that are independent of $x$ <ref name="ros2014regularity" />.
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| [[Hölder estimates]] are not known to hold under such generality. For the current methods, singular measures $\mu_x$ (without an absolutely continuous part) are out of reach. A new idea is needed in order to solve this problem.
| | 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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| Note that a key part of this problem is that the measures $\mu_x$ should not have any regularity assumption respect to $x$.
| | 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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| == An integral ABP estimate ==
| | 2.For any positive times $s\leq t$ the random variables $X(t-s)$ and $X(t)-X(s)$ have the same probability law. |
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| The nonlocal version of the [[Alexadroff-Bakelman-Pucci estimate]] holds either for a right hand side in $L^\infty$ <ref name="CS"/> (in which the integral right hand side is approximated by a discrete sum) or under very restrictive assumptions on the kernels <ref name="GS"/>. Would the following result be true?
| | 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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| Assume $u_n \leq 0$ outside $B_1$ and for all $x \in B_1$,
| | == Lévy-Khintchine Formula == |
| \[ \int_{\R^n} (u(x+y)-u(x)) K(x,y) \mathrm d y \geq \chi_{A_n}(x). \]
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| Where $\chi_{A_n}$ stands for the characteristic function of the sets $A_n$. Assume that the kernels $K$ satisfy symmetry and a uniform ellipticity condition
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| \begin{align*}
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| K(x,y) &= K(x,-y) \\
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| \lambda |y|^{-n-s} \leq K(x,y) &\leq \Lambda |y|^{-n-s} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2).
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| \end{align*}
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| If $|A_n|\to 0$ as $n \to +\infty$, is it true that $\sup u_n^+ \to 0$ as well?
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| This type of estimate is currently known only under strong structural hypothesis on the kernels $K$.<ref name="GS"/>
| | 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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| == Holder estimates for parabolic equations with variable order == | | \[X(t) = \left( X(h)-X(0)\right)+\left( X(2h)-X(h)\right)+...+\left( X(t)-X((n-1)h)\right)\] |
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| [[Holder estimates]] are known for elliptic and parabolic integro-differential equations with rough kernels. For elliptic equations, these estimates are available even when the order of the equation changes from point to point <ref name="BK"/> <ref name="S" />. Such estimate is not available for parabolic equations and it is not clear whether it holds.
| | 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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| More precisely, we would like to study a parabolic equation of the form
| | \[ \mathbb{E} \left [ e^{i\xi\cdot X_t}\right ] = e^{t\eta(\xi)}\] |
| \[ u_t(t,x) = \int_{\R^n} (u(t,x+y) - u(t,x)) K(t,x,y) dy.\] | |
| Here $K$ is symmetric (i.e. $K(t,x,y) = K(t,x,-y)$) and satisfies the bounds
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| \[ \frac \lambda {|y|^{n+s(t,x)}} \leq K(t,x,y) \leq \frac \Lambda {|y|^{n+s(t,x)}}.\]
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| The order of the equation $s(t,x) \in (0,1)$ changes from point to point and it should stay strictly away from zero. It would also make sense to study other families of [[linear integro-differential operators]]. Does a parabolic [[Holder estimate]] hold in this case?
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| == A [[comparison principle]] for $x$-dependent nonlocal equations which are '''not''' in the Levy-Ito form ==
| | the function $\eta(\xi)$ given by |
| Consider two continuous functions $u$ and $v$ such that
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| \begin{align*} | |
| u(x) &\leq v(x) \qquad \text{for all $x$ outside some set } \Omega,\\
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| F(x,\{I_\alpha u(x)\}) &\geq F(x,\{I_\alpha v(x)\})\qquad \text{for all $x \in \Omega$}.
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| \end{align*}
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| Is it true that $u \leq v$ in $\Omega$ as well?
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| It is natural to expect this result to hold if $F$ is continuous respect to $x$ and the [[linear integro-differential operators]] $I_\alpha$ satisfy some nondegeneracy condition and continuity respect to $x$, e.g.
| | \[\eta(\xi)=i y\cdot b -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{i \xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y) \] |
| \begin{align*} | |
| I[u] = \int (u(x+z) - u(x) - Du(x)\cdot z 1_{B}(z))\mu_x(dz)
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| \end{align*} | |
| where $(\mu_x)_x$ is a family of L\'evy measures, H\"older continous with respect to $x$?
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| Currently the comparison principle is only known if the kernels are continuous when written in the Levy-Ito form.<ref name="BI"/>
| | where $b$ is a vector, $A$ is a positive matrix, $B_1$ is the unit ball and $\mu$ is a Lévy measure, that is, a Borel measure in $\mathbb{R}^d$ such that |
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| == Holder estimates for drift-diffusion equations (sharp assumptions for $b$ in the case $s>1/2$) ==
| | \[ \int_{\mathbb{R}^d}\frac{|y|^2}{1+|y|^2}d\mu(y) <+\infty. \] |
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| Consider a [[drift-diffusion equation]] of the form
| | The interpretation of this measure $\mu$ is that ''jumps'' from some point $x$ to $x+y$ with $y$ in some set $A$ occur as a Poisson process with intensity $\mu(A)$. |
| \[ u_t + b \cdot \nabla u + (-\Delta)^s u = 0.\] | |
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| The solution $u$ is known to become Holder continuous under a variety of assumptions on the vector field $b$. If we assume that $\mathrm{div}\, b = 0$, we may expect that the required assumptions are slightly more flexible. Indeed, if $s=1/2$, the solution $u$ becomes Holder for positive time if $b \in L^\infty(BMO)$ <ref name="CV"/>. On the other hand, if $s=1$, the solution $u$ becomes Holder continuous for positive time if $b \in L^\infty(BMO^{-1})$ (if $b$ is the sum of derivatives of $BMO$ functions) <ref name="FV"/> <ref name="SSSZ"/>. A natural conjecture would be that the same result applies for $s \in (1/2,1)$ if $b \in L^\infty(BMO^{2s-1})$ (meaning that $(-\Delta)^{1-2s} b \in L^\infty(BMO)$).
| | == Connection with linear integro-differential operators == |
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| The case $s < 1/2$ is completely understood and the assumption $\mathrm{div}\, b =0$ is not even necessary. For $s \in (1/2,1)$, only some perturbative results seem to be known under stronger assumptions. It is conceivable that the approach of Caffarelli and Vasseur <ref name="CV"/> can be worked out assuming that $b \in L^\infty(L^p)$ for a critical power $p$.
| | Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows |
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| = Open problems for equations related to fluids = | | \[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t)) \right ]\] |
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| == Well posedness of the supercritical [[surface quasi-geostrophic equation]] and related problems ==
| | 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 $U_t \circ U_s = U_{t+s}$. |
| Let $\theta_0 : \R^2 \to \R$ be a smooth function either with compact support or periodic. Let $s \in (0,1/2)$. Is there a global classical solution $\theta :\R^2 \to \R$ for the SQG equation?
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| \begin{align*}
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| \theta(x,0) &= \theta_0(x) \\
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| \theta_t + u \cdot \nabla \theta &= 0 \qquad \text{in } \R^2 \times (0,+\infty)
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| \end{align*}
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| where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
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| This is a very difficult open problem. It is believed that a solution would be a major step towards the understanding of Navier-Stokes equation. In the supercritical regime $s\in (0,1/2)$, the effect if the drift term is larger than the diffusion in small scales. Therefore, it seems unlikely that a proof of well posedness could be achieved with the methods currently known and listed in this wiki.
| | 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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| Note that if the relation between $u$ and $\theta$ was changed by $u = R\theta$, then the equation is ill posed. This suggests that the divergence free nature of $u$ must play an important role, unlike the critical and subcritical cases $s \geq 1/2$.
| | \[\partial_t f = Lf \;\;\;\mbox{ for all } (x,t)\in\mathbb{R}^d\times \mathbb{R}_+\] |
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| There are difficult open problems related to simpler active scalar equations as well. The ''Hilbert flow problem'' refers to the equation
| | where for any smooth function $\phi$, we have |
| \[ \theta_t + H\theta \, \theta_x + (-\Delta)^s \theta = 0.\]
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| Here $\theta(t,x)$ is a function of $t \in [0,\infty)$ and $x \in \R$. The equation is known to be well posed for $s \geq 1/2$ and it is known to develop singularities in finite time when $s < 1/4$. For $s$ in the interval $s \in [1/4,1/2)$, it is not known whether singularities in finite time may occur.<ref name="cordobacordoba2005" /><ref name="li2011one" /><ref name="silvestre2014transport" />
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| = Open problems related to minimal surfaces and free boundaries = | | \[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \] |
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| == Regularity of [[nonlocal minimal surfaces]] == | | == [[Stochastic control]] and fully non-linear integro-differential operators == |
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| A nonlocal minimal surface that is sufficiently flat is known to be smooth <ref name="CRS"/>. The possibility of singularities in the general case reduces to the analysis of a possible existence of nonlocal minimal cones. The problem can be stated as follows. | | A similar connection holds via the [[Isaacs equation|Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it. |
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| For any $s \in (0,1)$, and any natural number $n$, is there any set $A \in \R^n$, other than a half space, such that
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| # $A$ is a cone: $\lambda A = A$ for any $\lambda > 0$.
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| # If $B$ is any set in $\R^n$ which coincides with $A$ outside of a compact set $C$, then the following inequality holds
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| \[ \int_C \int_{C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2 \int_C \int_{\R^n \setminus C} \frac{|\chi_A(x) - \chi_A(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y \leq \int_C \int_{C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y + 2\int_C \int_{\R^n \setminus C} \frac{|\chi_B(x) - \chi_B(y)|}{|x-y|^{n+s}} \mathrm d x \mathrm d y. \]
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| When $s$ is sufficiently close to one, such set does not exist if $n < 8$.
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| == Optimal regularity for the [[obstacle problem]] for a general integro-differential operator ==
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| Let $u$ be the solution to the [[obstacle problem for the fractional laplacian]],
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| \begin{align*}
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| u &\geq \varphi \qquad \text{in } \R^n, \\
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| (-\Delta)^{s/2} u &\geq 0 \qquad \text{in } \R^n, \\
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| (-\Delta)^{s/2} u &= 0 \qquad \text{in } \{u>\varphi\}, \\
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| \end{align*}
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| where $\varphi$ is a smooth compactly supported function. It is known that $u \in C^{1,s/2}$ (where $s$ coincides with the order of the fractional Laplacian). This regularity is optimal in the sense that one can construct solutions that are not in $C^{1,s/2+\varepsilon}$ for any $\varepsilon>0$. One can consider the same problem replacing the fractional Laplacian by any other nonlocal operator. In fact, this problem corresponds to the [[optimal stopping problem]] in stochastic control, with applications to mathematical finance. The fractional Laplacian is just the particular case when the [[Levy process]] involved is $\alpha$-stable and radially symmetric. The optimal regularity for the general problem is currently an open problem. Even in the linear case with constant coefficients this is nontrivial. If $u$ is a solution of
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| \begin{align*}
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| u &\geq \varphi \qquad \text{in } \R^n, \\
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| L u &\leq 0 \qquad \text{in } \R^n, \\
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| L u &= 0 \qquad \text{in } \{u>\varphi\}, \\
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| \end{align*}
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| where $L$ is a [[linear integro-differential operator]], then what is the optimal regularity we can obtain for $u$?
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| The optimal regularity would naturally depend on some assumptions on the linear operator $L$. If $L$ is a purely integro-differential with a kernel $K$ satisfying the usual ellipticity conditions
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| \begin{align*}
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| K(y) &= K(-y) \\
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| \frac{\lambda(2-s)}{ |y|^{n+s}} \leq K(y) &\leq \frac{\Lambda(2-s)}{ |y|^{n+s}} \qquad \text{for some } 0<\lambda<\Lambda \text{ and } s \in (0,2),
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| \end{align*}
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| it is natural to expect the solution $u$ to be $C^s$, but this regularity is not optimal. Is the optimal regularity going to be $C^{1,s/2}$ as in the fractional Laplacian case? Most probably some extra assumption on the kernel will be needed.
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| A solution to this problem would be very interesting if it provides an optimal regularity result for a natural family of kernels. If the assumption is something hard to check (like for example that there exists an extension problem whose Dirichlet to Neumann map is $L$), then the result may not be that interesting.
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| UPDATE: This problem has been recently solved by Caffarelli, Ros-Oton, and Serra <ref name="CRS16" />.
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| == Complete understanding of free boundary points in the [[fractional obstacle problem]] ==
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| Some free boundary points of the [[fractional obstacle problem]] are classified as regular and the free boundary is known to be smooth around them <ref name="CSS"/>. Other points on the free boundary are classified as singular, and for $s=\frac12$ they are shown to be contained in a lower dimensional differentiable surface, and therefore to be rare <ref name="GP"/>. However, there may be other points on the free boundary that do not fall under those two categories. Two questions need to be answered.\
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| # Can there be any point on the free boundary that is neither regular nor singular? It is easy to produce examples in the [[thin obstacle problem]], using the [[extension technique]]. However, it is not clear if such examples can be made in the original formulation of the [[fractional obstacle problem]] because of the decay at infinity requirement.
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| # In case that a point of a third category exist, is the free boundary smooth around these points in the ''third category''?
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| == References ==
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| {{reflist|refs=
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| <ref name="CS">{{Citation | last1=Caffarelli | first1=Luis | last2=Silvestre | first2=Luis | title=Regularity theory for fully nonlinear integro-differential equations | url=http://dx.doi.org/10.1002/cpa.20274 | doi=10.1002/cpa.20274 | year=2009 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=62 | issue=5 | pages=597–638}}</ref>
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| <ref name="CV">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Vasseur | first2=Alexis | title=Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation | url=http://dx.doi.org/10.4007/annals.2010.171.1903 | doi=10.4007/annals.2010.171.1903 | year=2010 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=171 | issue=3 | pages=1903–1930}}</ref>
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| <ref name="SSSZ">{{Citation | last1=Seregin | first1=G. | last2=Silvestre | first2=Luis | last3=Sverak | first3=V. | last4=Zlatos | first4=A. | title=On divergence-free drifts | year=2010 | journal=Arxiv preprint arXiv:1010.6025}}</ref>
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| <ref name="FV">{{Citation | last1=Friedlander | first1=S. | last2=Vicol | first2=V. | title=Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics | year=2011 | journal=Annales de l'Institut Henri Poincare (C) Non Linear Analysis}}</ref>
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| <ref name="CRS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Roquejoffre | first2=Jean Michel |last3= Savin | first3= Ovidiu | title= Nonlocal Minimal Surfaces | url=http://onlinelibrary.wiley.com/doi/10.1002/cpa.20331/abstract | doi=10.1002/cpa.20331 | year=2010 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0003-486X | volume=63 | issue=9 | pages=1111–1144}}</ref>
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| <ref name="CRS16">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Ros-Oton | first2=Xavier |last3= Serra | first3= Joaquim | title= Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries | year=2016 | journal=[[preprint arXiv (2016)]]}}</ref>
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| <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>
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| <ref name="CSS">{{Citation | last1=Caffarelli | first1=Luis A. | last2=Salsa | first2=Sandro | last3=Silvestre | first3=Luis | title=Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian | url=http://dx.doi.org/10.1007/s00222-007-0086-6 | doi=10.1007/s00222-007-0086-6 | year=2008 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=171 | issue=2 | pages=425–461}}</ref>
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| <ref name="GP">{{Citation | last1=Petrosyan | first1=A. | last2=Garofalo | first2=N. | title=Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2009 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=177 | issue=2 | pages=415–461}}</ref>
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| <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>
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| <ref name="BI">{{Citation | last1=Barles | first1=Guy | last2=Imbert | first2=Cyril | title=Second-order elliptic integro-differential equations: viscosity solutions' theory revisited | url=http://dx.doi.org/10.1016/j.anihpc.2007.02.007 | doi=10.1016/j.anihpc.2007.02.007 | year=2008 | journal=Annales de l'Institut Henri Poincaré. Analyse Non Linéaire | issn=0294-1449 | volume=25 | issue=3 | pages=567–585}}</ref>
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| <ref name="BK">{{Citation | last1=Bass | first1=Richard F. | last2=Kassmann |
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| first2=Moritz | title=Hölder continuity of harmonic functions with respect to
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| operators of variable order | url=http://dx.doi.org/10.1080/03605300500257677 |
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| doi=10.1080/03605300500257677 | year=2005 | journal=Communications in Partial
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| Differential Equations | issn=0360-5302 | volume=30 | issue=7 |
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| pages=1249–1259}}</ref>
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| <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=Hölder
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| estimates for solutions of integro-differential equations like the fractional
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| Laplace | url=http://dx.doi.org/10.1512/iumj.2006.55.2706 |
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| doi=10.1512/iumj.2006.55.2706 | year=2006 | journal=Indiana University
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| Mathematics Journal | issn=0022-2518 | volume=55 | issue=3 |
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| pages=1155–1174}}</ref>
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| <ref name="silvestre2014transport">{{Citation | last1=Silvestre | first1= Luis | last2=Vicol | first2= Vlad | title=On a transport equation with nonlocal drift | journal=arXiv preprint arXiv:1408.1056}}</ref>
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| <ref name="li2011one">{{Citation | last1=Li | first1= Dong | last2=Rodrigo | first2= José L | title=On a one-dimensional nonlocal flux with fractional dissipation | journal=SIAM Journal on Mathematical Analysis | year=2011 | volume=43 | pages=507--526}}</ref>
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| <ref name="cordobacordoba2005">{{Citation | last1=Córdoba | first1= Antonio | last2=Córdoba | first2= Diego | last3=Fontelos | first3= Marco A. | title=Formation of singularities for a transport equation with nonlocal velocity | url=http://dx.doi.org/10.4007/annals.2005.162.1377 | journal=Ann. of Math. (2) | issn=0003-486X | year=2005 | volume=162 | pages=1377--1389 | doi=10.4007/annals.2005.162.1377}}</ref>
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| <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>
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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 random trajectory, generalizing the concept of Brownian motion, which may contain jump discontinuities. 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
\[\eta(\xi)=i y\cdot b -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{i \xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y) \]
where $b$ is a vector, $A$ is a positive matrix, $B_1$ is the unit ball and $\mu$ is a Lévy measure, that is, a Borel measure in $\mathbb{R}^d$ such that
\[ \int_{\mathbb{R}^d}\frac{|y|^2}{1+|y|^2}d\mu(y) <+\infty. \]
The interpretation of this measure $\mu$ is that jumps from some point $x$ to $x+y$ with $y$ in some set $A$ occur as a Poisson process with intensity $\mu(A)$.
Connection with linear integro-differential operators
Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a linear 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
\[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \]
Stochastic control and fully non-linear integro-differential operators
A similar connection holds via the Isaacs-Bellman equation arising in stochastic control problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it.