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We consider the evolution problem associated to the infinity fractional Laplacian introduced by Bjorland, Caffarelli and Figalli (2012) as the infinitesimal generator of a non-Brownian tug-of-war game. We first construct a class of viscosity solutions of the initial-value problem for bounded and uniformly continuous data. An important result is the equivalence of the nonlinear operator in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially symmetric and monotone functions. Thanks to this and a comparison theorem between classical and viscosity solutions, we are able to establish a global Harnack inequality that, in particular, explains the long-time behavior of the solutions., 26 pages, 5 figures

We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump L\'evy processes with some $\sigma$-stable like behaviour. Included are $\sigma$-stable processes and fractional Laplace diffusion operators $(-\Delta)^{\frac{\sigma}2}$, tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order $\sigma\in(1,2)$. We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs uses pure PDE-methods and build on ideas of Lions et al. The new ingredients are fractional heat kernel estimates, regularity results for fractional Bellman, Fokker-Planck and coupled Mean Field Game equations, and a priori bounds and compactness of (very) weak solutions of fractional Fokker-Planck equations in the whole space. Our techniques requires no moment assumptions and uses a weaker topology than Wasserstein., Comment: Major update. A much larger class of anomalous diffusions, local coupling results are now complete, problems are posed in the whole space and not the compact torus. New equicontinuity, $L^1$, and $L^\infty$-results for the Fokker-planck equation are needed and proved by using pure PDE arguments. We now work in a more general metrics than Wasserstein

To demonstrate the existence of an evolutionarily stable strategy (ESS) in fluctuating ecological systems (such as predator−prey system with limit cycles) is important. Deriving the analytic conditions for such an ESS can be of great help when studying real biological systems. A main aim of our study (1) was just that. Rather than saying “extended the ESS … [↵][1]1To whom correspondence may be addressed. Email: n.c.stenseth{at}mn.uio.no. [1]: #xref-corresp-1-1

We prove a new generation result in $L^1$ for a large class of non-local operators with non-degenerate local terms. This class contains the operators appearing in Fokker-Planck or Kolmogorov forward equations associated with Levy driven SDEs, i.e. the adjoint operators of the infinitesimal generators of these SDEs. As a byproduct, we also obtain a new elliptic regularity result of independent interest. The main novelty in this paper is that we can consider very general Levy operators, including state-space depending coefficients with linear growth and general Levy measures which can be singular and have fat tails.

A result by Courr\`ege says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ where $$ \mathcal{L}^{\sigma,b}[u](x)=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} \mu(z). $$ This class of operators coincides with the infinitesimal generators of L\'evy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$ are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$. The proofs combine arguments from PDE and group theories. They are simple and short., Comment: This is an independent and substantial update of arXiv:1807.01843. Here we treat general operators which could be both local and nonlocal, symmetric and nonsymmetric. 13 pages. v3: Update according to the suggestions of the referees. To appear in Journal de Math\'ematiques Pures et Appliqu\'ees

We study monotone numerical schemes for nonlocal Isaacs equations, the dynamic programming equations of stochastic differential games with jump-diffusion state processes. These equations are fully-nonlinear non-convex equations of order less than $2$. In this paper they are also allowed to be degenerate and have non-smooth solutions. The main contribution is a series of new a priori error estimates: The first results for nonlocal Isaacs equations, the first general results for degenerate non-convex equations of order greater than $1$, and the first results in the viscosity solution setting giving the precise dependence on the fractional order of the equation. We also observe a new phenomena, that the rates differ when the nonlocal diffusion coefficient depend on $x$ and $t$, only on $x$, or on neither., Comment: 24 pages

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of L\'evy type and $\varphi$ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators $\mathfrak{L}^{\sigma,\mu}$ are the (fractional) Laplacians $\Delta$ and $-(-\Delta)^{\frac\alpha2}$ for $\alpha\in(0,2)$, discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L\'evy operators, allows us to give a unified and compact {\em nonlocal} theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions -- including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in \cite{DTEnJa17b}. We also present some numerical tests, but extensive testing is deferred to the companion paper \cite{DTEnJa18b} along with a more detailed discussion of the numerical methods included in our theory., Comment: 34 pages, 3 figures. To appear in SIAM Journal on Numerical Analysis

We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form $$\partial_tu-A\varphi(u)=0.$$ These equations are possibly degenerate nonlinear diffusion equations with a general nondecreasing continuous nonlinearity $\varphi$ and the largest class of linear symmetric nonlocal diffusion operators $A$ considered so far. The operators are defined from a bilinear energy form $\mathcal{E}$ and may be degenerate and have some $x$-dependence. The fractional Laplacian, symmetric finite differences, and any generator of symmetric pure jump Levy processes are included. The main results are (i) an Ole\uinik type uniqueness result for energy solutions; (ii) an existence (and uniqueness) result for distributional solutions with finite energy; and (iii) equivalence between the two notions of solution, and as a consequence, new well-posedness results for both notions of solutions. We also obtain quantitative energy and related $L^p$-estimates for distributional solutions. Our uniqueness results are given for a class of functions defined from test functions by completion in a certain topology. We study rigorously several cases where this space coincides with standard function spaces. In particular, for operators comparable to fractional Laplacians, we show that this space is a parabolic homogeneous fractional Sobolev space.

\noindent We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}$ is a general symmetric L\'evy type diffusion operator. Included are both local and nonlocal problems with e.g. $\mathfrak{L}=\Delta$ or $\mathfrak{L}=-(-\Delta)^{\frac\alpha2}$, $\alpha\in(0,2)$, and porous medium, fast diffusion, and Stefan type nonlinearities $\varphi$. By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are $L^p$-stable for $p\in[1,\infty]$, compact, and convergent in $C([0,T];L_{loc}^p(\mathbb{R}^N))$ for $p\in[1,\infty)$. The first part of this project is given in \cite{DTEnJa18a} and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of \cite{DTEnJa18a} apply and testing the schemes numerically. Our examples include fractional diffusions of different orders, and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems., Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway - ERCIM “Alain Bensoussan” Fellowship programme - “Juan de la Cierva - formación” program (FJCI-2016-30148)

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of $$\partial_tu-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=g(x,t)\quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\varphi$ is merely continuous and nondecreasing and $\mathfrak{L}^{\sigma,\mu}$ is the generator of a general symmetric L\'evy process. This means that $\mathfrak{L}^{\sigma,\mu}$ can have both local and nonlocal parts like e.g. $\mathfrak{L}^{\sigma,\mu}=\Delta-(-\Delta)^{\frac12}$. New uniqueness results for bounded distributional solutions of this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for $\mathfrak{L}^{\sigma,\mu}$. Existence and a priori estimates are deduced from a numerical approximation, and energy type estimates are also obtained., Comment: 6 pages. Minor revision. Added details to Step 2 of the proof of Theorem 3.1

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $\varphi:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations., Comment: To appear in "Advances in Mathematics"

We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p-Laplace, ∞-Laplace, mean curvature of graph, and even strongly degenerate operators, in addition to some nonlocal quasilinear operators appearing in the existing literature. Our main results are comparison, uniqueness, and existence results for viscosity solutions of linear and fully nonlinear equations involving these operators. Because of the structure of our operators, especially the existence proof is highly non-trivial and non-standard. We also identify the conditions under which the nonlocal operators converge to local quasilinear operators, and show that the solutions of the corresponding nonlocal equations converge to the solutions of the local limit equations. Finally, we give a (formal) stochastic representation formula for the solutions and provide many examples. This is a submitted manuscript of an article published by Elsevier Ltd in Journal of Differential Equations, 12 December 2016.

We study a numerical method to compute probability density functions of solutions of stochastic differential equations. The method is sometimes called the numerical path integration method and has been shown to be fast and accurate in application oriented fields. In this paper we provide a rigorous analysis of the method that covers systems of equations with unbounded coefficients. Working in a natural space for densities, L1, we obtain stability, consistency, and new convergence results for the method, new well-posedness and semigroup generation results for the related Fokker-Planck-Kolmogorov equation, and a new and rigorous connection to the corresponding probability density functions for both the approximate and the exact problems. To prove the results we combine semigroup and PDE arguments in a new way that should be of independent interest. © Springer Verlag. This is the authors' submitted manuscript to the article. The final publication is available at https://link.springer.com/article/10.1007%2Fs10444-017-9558-4

We obtain new L1 contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or nonlocal diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local L1 contraction results for scalar conservation laws. A key ingredient in the proofs is a new and nontrivial construction of a subsolution of a fully nonlinear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and, in the nonlocal case, new existence and uniqueness results. © 2014 Society for Industrial and Applied Mathematics

First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods--even when the principal derivatives have any fractional order between 1 and 2! The class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general Levy diffusion. Special cases are conservation laws, fractional conservation laws, certain fractional porous medium equations, and new strongly degenerate equations.

We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.

We obtain estimates on the continuous dependence on the coefficient for second-order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et al., Jakobsen and Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman–Isaacs equations and certain quasilinear equation.

Error estimates are derived for a class of finite difference-quadrature schemes approximating viscosity solutions of nonlinear degenerate parabolic integro-PDEs with variable diffusion coefficients. The relevant equations can be viewed as Bellman equations associated to a class of controlled jump-diffusion (Lévy) processes. The results cover both finite and infinite activity cases.

We formulate and prove a non-local "maximum principle for semi- continuous functions" in the setting of fully nonlinear and degenerate elliptic integro-partial dierential equations with integro operators of second order. Similar results have been used implicitly by several researchers to obtain com- parison/uniqueness results for integro-partial dierential equations, but proofs have so far been lacking.

We study a semi-discrete splitting method for computing approximate viscosity solutions of the initial value problem for a class of nonlinear degenerate parabolic equations with source terms. It is fairly standard to prove that the semi-discrete splitting approximations converge to the desired viscosity solution as the splitting step Δt tends to zero. The purpose of this paper is, however, to consider the more difficult problem of providing a precise estimate of the convergence rate. Using viscosity solution techniques we establish the L∞ convergence rate $\mathcal{O}(\sqrt{\Delta t})$ for the approximate solutions, and this estimate is robust with respect to the regularity of the solutions. We also provide an extension of this result to weakly coupled systems of equations, and in the case of more regular solutions we recover the “classical” rate $\mathcal{O}(\Delta t)$ . Finally, we analyze in an example a fully discrete splitting method.

In this paper we provide and analyze a nonlinear degenerate parabolic equation in one space dimension with the following smoothing property: If the initial data is only uniformly continuous, at positive times, the solution has bounded second derivatives (it belongs to W 2, ∞). We call this surprising phenomenon a W 2, ∞ regularizing effect. So far, such phenomena have only been observed in uniformly parabolic equations.

In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one space-dimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to certain finite difference methods and control schemes based on the dynamic programming principle. Recently, Krylov, Barles, and Jakobsen obtained similar estimates for convex Hamilton–Jacobi–Bellman equations in arbitrary space-dimensions. Our results are only valid in one space-dimension, but they are the first results of this type for non-convex second-order equations.

We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the nonlinear case.

Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.

The fractal conservation law \(\partial _t u + \partial _x(f(u))+(-\varDelta )^{{\alpha }/2}u=0\) changes characteristics as \({\alpha }\rightarrow 2\) from non-local and weakly diffusive to local and strongly diffusive. In this paper we present a corrected finite difference quadrature method for \((-\varDelta )^{{\alpha }/2}\) with \({\alpha }\in [0,2]\), combined with usual finite volume methods for the hyperbolic term, that automatically adjusts to this change and is uniformly convergent with respect to \({\alpha }\in [\eta ,2]\) for any \(\eta >0\). We provide numerical results which illustrate this asymptotic-preserving property as well as the non-uniformity of previous finite difference or finite volume type of methods.

We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, $\Delta^{\alpha/2}$ for $\alpha \in (0,2)$. Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on $\alpha$ in the $BV$-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits $\alpha \downarrow 0$ and $\alpha \uparrow 2$. In the limit $\alpha \uparrow 2$, $\Delta^{\alpha/2}$ converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231-251, 1999) for local degenerate parabolic equations (thus providing an alternative proof)., Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-014-0737-x