arXiv subjects

Espen Robstad Jakobsen

Publications and source records attributed to Espen Robstad Jakobsen.

10 recordsLinked to original sources

arXiv2026

Long time behaviour of Mean Field Games with fractional diffusion

In this paper we study the long time behaviour of mean field games systems with fractional diffusion, modeling the case that the individual dynamics of the players is driven by independent jump processes and controlled through the drift term, while being confined by an external field in order to guarantee ergodicity. In the case of globally Lipschitz, locally uniformly convex Hamiltonian, and weakly coupled costs satisfying the Lasry-Lions monotonicity condition, we prove that there is a unique solution $(u_T,m_T)$ to the mean field game problem in $(0,T)$ and we show that, if $T$ is sufficiently large, $(u_T,m_T)$ satisfies the so-called turnpike property, namely it is exponentially close to the (unique) stationary ergodic state for any proportionally long intermediate time.

math.AP↗

arXiv2026

A Schauder regularity theory for nonlocal and mixed local-nonlocal viscous Hamilton$\unicode{x2013}$Jacobi equations

We prove space-time Schauder estimates $\unicode{x2013}$ optimal regularity estimates in Hölder spaces $\unicode{x2013}$ and well-posedness results for mild and classical solutions of viscous Hamilton$\unicode{x2013}$Jacobi equations with subcritical nonlocal and mixed local-nonlocal diffusions in $\mathbb{R}^d$. Our spatial Schauder estimates hold under mild assumptions on the nonlocal/mixed operators and Hamiltonians. The Laplacian, fractional Laplacians, nonsymmetric, spectrally one-sided, and strongly anisotropic integral operators, as well as sums of such operators are covered. We observe an interplay between the regularity of the initial data and the growth of the Hamiltonian in the gradient, and develop a spatial Schauder theory for two canonical cases: (i) Lipschitz initial data and general Hamiltonians that are Hölder in space and merely locally Lipschitz in the gradient, and (ii) Hölder initial data and Hamiltonians that are Hölder in space and locally Lipschitz with power growth in the gradient. We compute explicit blow-up rates for $C^1$ and higher order Hölder norms as $t\to 0$. The results include short and long time existence of mild solutions, optimal regularity in Hölder spaces and corresponding Schauder a priori estimates, and that spatially smooth mild solutions are regular in time and pointwise classical solutions. Under further assumptions on the diffusion operator, we then prove time and space-time Schauder regularity estimates in optimal Hölder spaces which respect the natural fractional parabolic scaling. These results generalize classical linear local and fractional Schauder estimates to our non-linear fractional, possibly anisotropic and nonsymmetric setting.

math.AP↗

arXiv2025

The master equation for mean field game systems with fractional and nonlocal diffusions

We prove existence and uniqueness of classical solutions of the master equation for mean field game (MFG) systems with fractional and nonlocal diffusions. We cover a large class of Lévy diffusions of order greater than one, including purely nonlocal, local, and even mixed local-nonlocal operators. In the process we prove refined well-posedness results for the MFG systems, results that include the mixed local-nonlocal case. We also show various auxiliary results on viscous Hamilton-Jacobi equations, linear parabolic equations, and linear forward-backward systems that may be of independent interest. This includes a rigorous treatment of certain equations and systems with data and solutions in the duals of Hölder spaces $C^γ_b$ on the whole of $\mathbb{R}^d$. We do not assume existence of any moments for the initial distributions of players. In a future work we will use the results of this paper to prove the convergence of $N$-player games to mean field games as $N\to\infty$.

math.AP↗

arXiv2025

Finite element approximation of parabolic SPDEs with Whittle--Matérn noise

We propose and analyse a new type of fully discrete finite element approximation of a class of linear stochastic parabolic evolution equations with additive noise. Our discretization differs from previous ones in that we use a finite element approximation of the noise, as opposed to an $L^2$ projection. This approximation is tailored for equations where the noise has covariance operator defined in terms of (negative powers of) elliptic operators, like Whittle--Matérn random fields. Strong convergence rates up to order $2$ in space and $1$ in time are shown and verified by numerical experiments in dimension $1$ and $2$.

math.NA↗

arXiv2024

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $σ\in(0,2)$ since they involve fractional Laplace operators $(-Δ)^{σ/2}$. They arise e.g. in control and game theory as dynamic programming equations -- HJB and Isaacs equation -- and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $σ$. The accuracy of previous approximations of fractional fully nonlinear equations depend on $σ$ and are worse when $σ$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

math.NA↗

arXiv2024

A Deep BSDE approximation of nonlinear integro-PDEs with unbounded nonlocal operators

Machine learning for partial differential equations (PDEs) is a hot topic. In this paper we introduce and analyse a Deep BSDE scheme for nonlinear integro-PDEs with unbounded nonlocal operators -problems arising in e.g. stochastic control and games involving infinite activity jump-processes. The scheme is based on a stochastic forward-backward SDE representation of the solution of the PDE and (i) approximation of small jumps by a Gaussian process, (ii) simulation of the forward part, and (iii) a neural net regression for the backward part. Unlike grid-based schemes, it does not suffer from the curse of dimensionality and is therefore suitable for high dimensional problems. The scheme is designed to be convergent even in the infinite activity/unbounded nonlocal operator case. A full convergence analysis is given and constitutes the main part of the paper.

math.AP↗

arXiv2024

Optimal stability results and nonlinear duality for $L^\infty$ entropy and $L^1$ viscosity solutions

We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality: \begin{equation}\int |S_t u_0-S_t v_0| φ_0 \mathrm{d}x\leq \int |u_0-v_0| G_t φ_0 \mathrm{d}x, \quad \forall φ_0 \geq 0, \forall u_0, \forall v_0, \qquad(\star)\end{equation} where $S_t$ is the entropy solution semigroup of the anisotropic degenerate parabolic equation \begin{equation*} \partial_t u+\mathrm{div} F(u) = \mathrm{div} (A(u) D u),\end{equation*} and where we look for the smallest semigroup $G_t$ satisfying ($\star$). This amounts to finding an optimal weighted $L^1$ contraction estimate for $S_t$. Our main result is that $G_t$ is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation\begin{equation*} \partial_t φ= \mathrm{sup}_ξ\{F'(ξ) \cdot D φ+\mathrm{tr}(A(ξ) D^2φ)\}.\end{equation*} Since weighted $L^1$ contraction results are mainly used for possibly nonintegrable $L^\infty$ solutions $u$, the natural spaces behind this duality are $L^\infty$ for $S_t$ and $L^1$ for $G_t$. We therefore develop a corresponding $L^1$ theory for viscosity solutions $φ$. But $L^1$ itself is too large for well-posedness, and we rigorously identify the weakest $L^1$ type Banach setting where we can have it -- a subspace of $L^1$ called $L^\infty_{\mathrm{int}}$. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].

math.AP↗

arXiv2021

On fractional and nonlocal parabolic Mean Field Games in the whole space

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 Lévy processes with some $σ$-stable like behaviour. Included are $σ$-stable processes and fractional Laplace diffusion operators $(-Δ)^{\fracσ2}$, 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 $σ\in(1,2)$. 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.

math.AP↗

arXiv2018

$L^1$ semigroup generation for Fokker-Planck operators associated with general Lévy driven SDEs

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 Lévy 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 Lévy operators, including state-space depending coefficients with linear growth and general Lévy measures which can be singular and have fat tails.

math.DS↗

arXiv2016

On numerical density approximations of solutions of SDEs with unbounded coefficients

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, $L^1$, 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.

math.DS↗