arXiv subjects

Espen R. Jakobsen

Publications and source records attributed to Espen R. Jakobsen.

At least 19 recordsLinked to original sources

arXiv2026

ARMA approximation of a Non-separable Spatio-Temporal Model with Fractional Smoothnesses in Space and Time

The Matérn covariance model is ubiquitous in spatial modelling, but there is no default choice for spatio-temporal modelling. In this paper, we consider the recently proposed ``diffusion-based'' extension of the spatial Matérn covariance model to a spatio-temporal non-separable covariance model that allows fractional smoothnesses in space and in time. The model is described in terms of a space-time fractional stochastic partial differential equation, but currently proposed computational approaches have strong restrictions on the possible smoothnesses in time. We propose a discretization method based on rational approximations in time to handle arbitrary smoothnesses, which leads to a vector autoregressive moving average process (VARMA). We prove that the covariance function of the approximation converges pointwise, determine explicit convergence rates as a function of spatial and temporal resolutions and the accuracy of the rational approximation, and conduct numerical verification to demonstrate small pointwise error for low orders of the VARMA process. Through a simulation study, we demonstrate that the parameters can be estimated back and that correctly specifying the temporal smoothness is especially important for forecasting. The approach is illustrated for three months of daily mean temperatures in mainland France.

stat.ME↗

arXiv2025

A convergent finite difference-quadrature scheme for the porous medium equation with nonlocal pressure

We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and Vázquez: \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-Δ)^{-σ}u) \qquad \text{for} \qquad m\geq2 \quad \text{and} \quad σ\in(0,1). \] Our scheme is for one space dimension and positive solutions $u$. It consists of solving numerically the equation satisfied by $v(x,t)=\int_{-\infty}^xu(x,t)dx$, the quasilinear non-divergence form equation \[ \partial_t v= -|\partial_x v|^{m-1} (- Δ)^{s} v \qquad \text{where} \qquad s=1-σ, \] and then computing $u=v_x$ by numerical differentiation. Using upwinding ideas in a novel way, we construct a new and simple, monotone and $L^\infty$-stable, approximation for the $v$-equation, and show local uniform convergence to the unique discontinuous viscosity solution. Using ideas from probability theory, we then prove that the approximation of $u$ converges weakly-$*$, or more precisely, up to normalization, in $C(0,T; P(\mathbb{R}))$ where $P(\mathbb{R})$ is the space of probability measures under the Rubinstein-Kantorovich metric.The analysis include also fundamental solutions where the initial data for $u$ is a Dirac mass. Numerical tests are included to confirm the results. Our scheme seems to be the first numerical scheme for this type of problems.

math.NA↗

arXiv2024

A strongly degenerate fully nonlinear mean field game with nonlocal diffusion

There are few results on mean field game (MFG) systems where the PDEs are either fully nonlinear or have degenerate diffusions. This paper introduces a problem that combines both difficulties. We prove existence and uniqueness for a strongly degenerate, fully nonlinear MFG system by using the well-posedness theory for fully nonlinear MFGs established in our previous paper. It is the first such application in a degenerate setting. Our MFG involves a controlled pure jump (nonlocal) Lévy diffusion of order less than one, and monotone, smoothing couplings. The key difficulty is obtaining uniqueness for the corresponding degenerate, non-smooth Fokker-Plank equation: since the regularity of the coefficient and the order of the diffusion are interdependent, it holds when the order is sufficiently low. Viscosity solutions and a non-standard doubling of variables argument are used along with a bootstrapping procedure.

math.AP↗

arXiv2024

On fully nonlinear parabolic mean field games with nonlocal and local diffusions

We introduce a class of fully nonlinear mean field games posed in $[0,T]\times\mathbb{R}^d$. We justify that they are related to controlled local or nonlocal diffusions, and more generally in our setting, to a new control interpretation involving time change rates of stochastic (Lévy) processes. The main results are existence and uniqueness of solutions under general assumptions. These results are applied to non-degenerate equations - including both local second order and nonlocal with fractional Laplacians. Uniqueness holds under monotonicity of couplings and convexity of the Hamiltonian, but neither monotonicity nor convexity need to be strict. We consider a rich class of nonlocal operators and processes and develop tools to work in the whole space without explicit moment assumptions.

math.AP↗

arXiv2024

Towards a Schauder theory for fractional viscous Hamilton--Jacobi equations

We survey some results on Lipschitz and Schauder regularity estimates for viscous Hamilton--Jacobi equations with subcritical Lévy diffusions. The Schauder estimates, along with existence of smooth solutions, are obtained with the help of a Duhamel formula and $L^1$ bounds on the spatial derivatives of the heat kernel. Our results cover very general nonlocal and mixed local-nonlocal diffusions, including strongly anisotropic, nonsymmetric, mixed order, and spectrally one-sided models.

math.AP↗

arXiv2023

Precise Error Bounds for Numerical Approximations of Fractional HJB Equations

We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton-Jacobi-Bellman (HJB) equations. We consider diffusion corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations which are (formally) fractional order and 2nd order methods. It is well-known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) Strongly degenerate problems with Lipschitz solutions, and (ii) weakly non-degenerate problems where we show that solutions have bounded fractional derivatives of order between 1 and 2. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly non-degenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than 1/2.

math.AP↗

arXiv2022

Evolution Driven by the Infinity Fractional Laplacian

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.

math.AP↗

arXiv2022

Uniform tail estimates and $L^p(\mathbb{R}^N)$-convergence for finite-difference approximations of nonlinear diffusion equations

We obtain new equitightness and $C([0,T];L^p(\mathbb{R}^N))$-convergence results for finite-difference approximations of generalized porous medium equations of the form $$ \partial_tu-\mathfrak{L}[φ(u)]=g\qquad\text{in $\mathbb{R}^N\times(0,T)$}, $$ where $φ:\mathbb{R}\to\mathbb{R}$ is continuous and nondecreasing, and $\mathfrak{L}$ is a local or nonlocal diffusion operator. Our results include slow diffusions, strongly degenerate Stefan problems, and fast diffusions above a critical exponent. These results improve the previous $C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$-convergence obtained in a series of papers on the topic by the authors. To have equitightness and global $L^p(\mathbb{R}^N)$-convergence, some additional restrictions on $\mathfrak{L}$ and $φ$ are needed. Most commonly used symmetric operators $\mathfrak{L}$ are still included: the Laplacian, fractional Laplacians, and other generators of symmetric Lévy processes with some fractional moment. We also discuss extensions to nonlinear possibly strongly degenerate convection-diffusion equations.

math.AP↗

arXiv2021

On Numerical approximations of fractional and nonlocal Mean Field Games

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.

math.AP↗

arXiv2020

The Liouville theorem and linear operators satisfying the maximum principle

A result by Courrège says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{σ,b}+\mathcal{L}^μ$ where $$ \mathcal{L}^{σ,b}[u](x)=\text{tr}(σσ^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^μ[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} μ(z). $$ This class of operators coincides with the infinitesimal generators of Lévy 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.

math.AP↗

arXiv2019

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{σ,μ}[φ(u)]=f \quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}^{σ,μ}$ is a general symmetric diffusion operator of Lévy type and $φ$ 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 $\mathfrak{L}^{σ,μ}$ are the (fractional) Laplacians $Δ$ and $-(-Δ)^{\frac\alpha2}$ for $α\in(0,2)$, discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal Lévy 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.

math.NA↗

arXiv2019

Improved order 1/4 convergence for piecewise constant policy approximation of stochastic control problems

In N.V. Krylov, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electron. J. Probab., 4(2), 1999, it is proved under standard assumptions that the value functions of controlled diffusion processes can be approximated with order 1/6 error by those with controls which are constant on uniform time intervals. In this note we refine the proof and show that the provable rate can be improved to 1/4, which is optimal in our setting. Moreover, we demonstrate the improvements this implies for error estimates derived by similar techniques for approximation schemes, bringing these in line with the best available results from the PDE literature.

math.PR↗

arXiv2018

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments

We develop unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}[φ(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}$ is a general symmetric Lévy type diffusion operator. Included are both local and nonlocal problems with e.g. $\mathfrak{L}=Δ$ or $\mathfrak{L}=-(-Δ)^{\frac\alpha2}$, $α\in(0,2)$, and porous medium, fast diffusion, and Stefan type nonlinearities $φ$. 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 $C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$ 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.

math.NA↗

arXiv2018

Characterization of nonlocal diffusion operators satisfying the Liouville theorem. Irrational numbers and subgroups of $\mathbb{R}^d$

We investigate the characterization of generators $\mathcal{L}$ of Lévy processes satisfying the Liouville theorem: Bounded functions $u$ solving $\mathcal{L}[u]=0$ are constant. These operators are degenerate elliptic of the form $\mathcal{L}=\mathcal{L}^{σ,b}+\mathcal{L}^μ$ for some local part $\mathcal{L}^{σ,b}[u]=\text{tr}(σσ^{\texttt{T}} D^2u)+b \cdot Du$ and nonlocal part $$ \mathcal{L}^μ[u](x)=\int \big(u(x+z)-u(x)-z \cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \, \text{d} μ(z), $$ where $μ\geq 0$ is a so-called Lévy measure possibly unbounded for small $z$. In this paper, we focus on the pure nonlocal case $σ=0$ and $b=0$, where we assume in addition that $μ$ is symmetric which corresponds to self-adjoint pure jump Lévy operators $\mathcal{L}=\mathcal{L}^μ$. The case of general Lévy operators $\mathcal{L}=\mathcal{L}^{σ,b}+\mathcal{L}^μ$ will be considered in the forthcoming paper \cite{AlDTEnJa18}. In our setting, we show that $\mathcal{L}^μ[u]=0$ if and only if $u$ is periodic wrt the subgroup generated by the support of $μ$. Therefore, the Liouville property holds if and only if this subgroup is dense, and in space dimension $d=1$ there is an equivalent condition in terms of irrational numbers. In dimension $d \geq 1$, we have a clearer view of the operators \textit{not} satisfying the Liouville theorem whose general form is precisely identified. The proofs are based on arguments of propagation of maximum.

math.AP↗

arXiv2017

On distributional solutions of local and nonlocal problems of porous medium type

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of $$\partial_tu-\mathfrak{L}^{σ,μ}[φ(u)]=g(x,t)\quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $φ$ is merely continuous and nondecreasing and $\mathfrak{L}^{σ,μ}$ is the generator of a general symmetric Lévy process. This means that $\mathfrak{L}^{σ,μ}$ can have both local and nonlocal parts like e.g. $\mathfrak{L}^{σ,μ}=Δ-(-Δ)^{\frac12}$. 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 $\mathfrak{L}^{σ,μ}$. Existence and a priori estimates are deduced from a numerical approximation, and energy type estimates are also obtained.

math.AP↗

arXiv2017

On the rate of convergence for monotone numerical schemes for nonlocal Isaacs' equations

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.

math.AP↗

arXiv2017

On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type

We study well-posedness and equivalence of different notions of solutions with finite energy for nonlocal porous medium type equations of the form $$\partial_tu-Aφ(u)=0.$$ These equations are possibly degenerate nonlinear diffusion equations with a general nondecreasing continuous nonlinearity $φ$ 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 Lévy processes are included. The main results are (i) an Ole\uınik 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.

math.AP↗

arXiv2016

Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^μ[φ(u)]=0$. Here $\mathcal{L}^μ$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $φ:\mathbb{R} \to \mathbb{R}$ 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.

math.AP↗