In this paper we introduce a new generalisation of the relative Fisher Information for Markov jump processes on a finite or countable state space, and prove an inequality which connects this object with the relative entropy and a large deviation rate functional. In addition to possessing various favourable properties, we show that this generalised Fisher Information converges to the classical Fisher Information in an appropriate limit. We then use this generalised Fisher Information and the aforementioned inequality to qualitatively study coarse-graining problems for jump processes on discrete spaces.Remark 1.1. The space P(X ) is a subset of 1 (X ), and the weak measure topology on P(X ) coincides with the σ( 1 , ∞ )-topology on 1 (X ). Recall that by Schur's theorem, weak and strong convergence on 1 (X ) are the same, even though the weak and strong topologies may be different; therefore functions f : [0, T ] → 1 (X ) are strongly continuous if and only they are weakly continuous. Since 'weak measure convergence' in P(X ) is the same as the σ( 1 , ∞ )-convergence in 1 (X ), we will omit the term 'weak' in our discussion and notation, and simply talk about 'continuous' functions from [0, T ] to P(X ) or to 1 (X ).

We consider a one-dimensional Swift-Hohenberg equation coupled to a conservation law, where both equations contain additional dispersive terms breaking the reflection symmetry x → −x. This system exhibits a Turing instability and we study the dynamics close to the onset of this instability. First, we show that periodic traveling waves bifurcate from a homogeneous ground state. Second, fixing the bifurcation parameter close to the onset of instability, we construct modulating traveling fronts, which capture the process of pattern-formation by modeling the transition from the homogeneous ground state to the periodic traveling wave through an invading front. The existence proof is based on center manifold reduction to a finite-dimensional system. Here, the dimension of the center manifold depends on the relation between the spreading speed of the invading modulating front and the linear group velocities of the system. Due to the broken reflection symmetry, the coefficients in the reduced equation are genuinely complex. Therefore, the main challenge is the construction of persistent heteroclinic connections on the center manifold, which correspond to modulating traveling fronts in the full system.

We consider traveling front solutions connecting an invading state to an unstable ground state in a Ginzburg–Landau equation with an additional conservation law. This system appears as the generic amplitude equation for Turing pattern forming systems admitting a conservation law structure such as the Bénard–Marangoni problem. We prove the nonlinear stability of sufficiently fast fronts with respect to perturbations which are exponentially localized ahead of the front. The proof is based on the use of exponential weights ahead of the front to stabilize the ground state. The main challenges are the lack of a comparison principle and the fact that the invading state is only diffusively stable, i.e. perturbations of the invading state decay polynomially in time.

Coarse-graining techniques play a central role in reducing the complexity of stochastic models, and are typically characterised by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyse an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example.

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