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Recently, computational modelling became a very important research tool that enables us to study problems that for decades evaded scientific analysis. Evolutionary systems are certainly examples of such problems: they are composed of many…
The issue of the parameterization of small-scale dynamics is addressed in the context of passive-scalar turbulence. The basic idea of our strategy is to identify dynamical equations for the coarse-grained scalar dynamics starting from…
We give sharp conditions for global in time existence of gradient flow solutions to a Cahn-Hilliard-type equation, with backwards second order degenerate diffusion, in any dimension and for general initial data. Our equation is the…
Using a newly introduced connection between the local and non-local description of open quantum system dynamics, we investigate the relationship between these two characterisations in the case of quantum semi-Markov processes. This class of…
We propose and analyze an optimal mass transport method for a random genetic drift problem driven by a Moran process under weak-selection. The continuum limit, formulated as a reaction-advection-diffusion equation known as the Kimura…
Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard energies. We modify these energies, mollifying the singularities to stabilize the…
We consider the $H^{-m}$-gradient flow of length for closed plane curves. This flow is a generalization of curve diffusion flow. We investigate the large-time behavior assuming the global existence of the flow. Then we show that the…
The original Miura transformation, considered as a nonlinear potential transformation, is applicable to a continual class of evolution equations, not only to discrete integrable equations and their hierarchies. The same continual class of…
We consider a biological population evolving under the joint action of selection, mutation and random genetic drift. The evolutionary dynamics are described by a one-dimensional Fokker-Planck equation whose eigenfunctions obey a confluent…
We propose a general method to identify nonlinear Fokker--Planck--Kolmogorov equations (FPK equations) as gradient flows on the space of probability measures on $\mathbb{R}^d$ with a natural differential geometry. Our notion of gradient…
In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential…
To model discrete sequences such as DNA, proteins, and language using diffusion, practitioners must choose between three major methods: diffusion in discrete space, Gaussian diffusion in Euclidean space, or diffusion on the simplex. Despite…
The master equation of one-dimensional three-species reaction-diffusion processes is mapped onto an imaginary-time Schr\"odinger equation. In many cases the Hamiltonian obtained is that of an integrable quantum chain. Within this approach…
We propose a fully discrete variational scheme for nonlinear evolution equations with gradient flow structure on the space of finite Radon measures on an interval with respect to a generalized version of the Wasserstein distance with…
The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied. To this end, a rescaled zero-range…
Individual constituent balance equations are often used to derive expressions for species-specific segregation velocities in flows of dense granular mixtures. We propose a semiempirical expression for the interspecies momentum exchange in…
In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions…
We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this paper we…
By constructing successful couplings for degenerate diffusion processes, explicit derivative formula and Harnack type inequalities are presented for solutions to a class of degenerate Fokker-Planck equations on $\R^m\times\R^{d}$. The main…
We consider the task of filtering a dynamic parameter evolving as a diffusion process, given data collected at discrete times from a likelihood which is conjugate to the marginal law of the diffusion, when a generic dual process on a…