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Register a new userFokker--Planck and Kolmogorov Backward Equations for Continuous Time Random Walk scaling limits
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It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck Equations for diffusion processes. In contrast to previous such results, it is not assumed that the underlying process has absolutely continuous laws. Moreover, governing equations in the backward variables are derived. Three examples of anomalous diffusion processes illustrate the theory.
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In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Holder regularity estimate obtained recently in cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non density-distribution dependent case, it seems that this is the first result about the well-posedness of SDEs with measurable diffusion coefficients.
We prove two new results connected with elliptic Fokker-Planck-Kolmogorov equations with drifts integrable with respect to solutions. The first result answers negatively a long-standing question and shows that a density of a probability measure satisfying the Fokker-Planck-Kolmogorov equation with a drift integrable with respect to this density can fail to belong to the Sobolev class~$W^{1,1}(mathbb{R}^d)$. There is also a version of this result for densities with respect to Gaussian measures. The second new result gives some positive information about properties of such solutions: the solution density is proved to belong to certain fractional Sobolev classes.
We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these coefficients can even be zero. Also a new existence result is proved. Some applications to Fokker-Planck-Kolmogorov equations associated with SPDEs are presented.
We prove a generalization of the known result of Trevisan on the Ambrosio-Figalli-Trevisan superposition principle for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation, according to which such a solution is generated by a solution to the corresponding martingale problem. The novelty is that in place of the integrability of the diffusion and drift coefficients $A$ and $b$ with respect to the solution we require the integrability of $(|A(t,x)|+|langle b(t,x),xrangle |)/(1+|x|^2)$. Therefore, in the case where there are no a priori global integrability conditions the function $|A(t,x)|+|langle b(t,x),xrangle |$ can be of quadratic growth. Moreover, as a corollary we obtain that under mild conditions on the initial distribution it is sufficient to have the one-sided bound $langle b(t,x),xrangle le C+C|x|^2 log |x|$ along with $|A(t,x)|le C+C|x|^2 log |x|$.
Continuous time random Walk model has been versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as disordered or porous media. We are studying the continuous limits of Heterogeneous Continuous Time Random Walk model, when a random walk is making jumps on a graph within different time-length. We apply the concept of a generalized master equation to study heterogeneous continuous-time random walks on networks. Depending on the interpretations of the waiting time distributions the generalized master equation gives different forms of continuous equations.
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