arXiv subjects

Miłosz Krupski

Publications and source records attributed to Miłosz Krupski.

5 recordsLinked to original sources

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.

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arXiv2020

A framework for non-local, non-linear initial value problems

We study the Cauchy problem for non-linear non-local operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g. the porous medium equation with the fractional Laplacian and the parabolic fractional $p$-Laplacian. We show the existence, uniqueness of bounded solutions and study their further properties. Several new examples of non-local, non-linear operators are provided.

math.AP↗

arXiv2017

Convection-Diffusion Equations with Random Initial Conditions

We consider an evolution equation generalising the viscous Burgers equation supplemented by an initial condition which is a homogeneous random field. We develop a non-linear framework enabling us to show the existence and regularity of solutions as well as their long time behaviour.

math.AP↗

arXiv2015

Scaling Limits of Solutions of Linear Evolution Equations with Random Initial Conditions

We consider a linear equation $\partial_t u = \mathcal{L}u$, where $\mathcal{L}$ is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition $u(0)$ being a generalised stationary random field on $\mathbb{R}^d$. We show the existence and uniqueness of generalised solutions to such initial value problems. Then we investigate their scaling limits.

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