Initial states are randomly distributed and not all in $\Delta_0$. The RHS of your equation is counting the fraction of states in the chosen $\Delta_0$ interval. Density function is simply a function that when integrated over an interval gives that fraction.

This is standard result in linear PDE theory. E.g., the solution of heat equation $\partial_tf(x,t)=\Delta f(x,t)$ is $e^{\Delta t}f(x,0)$. Numerous proofs...you can start by reading up Green's functions

See this paper: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction , Nonlinear Dyn (2016) 86:1493–1534

As t grows, the system will settle in one of the attractors (if any) of the system. But that attractor may or may not have the periodicity of the BCs. In general, if the nonlinearity or the forcing amplitutde is small in some sense, the nonlinear system may have a solution like the one you desire. However, this needs to be confirmed on a case by case basis. In the applied community, this is studied under the umbrella of nonlinear normal modes.

Apriori, there isn't any reason for this to be true. This is false in general for any nonlinear system.

Thanks, I have started going through these papers. They all seem to be in the "class" of the examples similar to the one that I mention in the OP. Are there qualitatively other ways of "certifying/guaranteeing" or even guessing the sparsity of the invariant distribution of a given Markov chain ?

@πr8 Yes, thats the spirit of the question.

You might want to google "metriplectic". IIRC, this is a way to make "non-Hamiltonian" systems have some kind of symplectic structure. P J Morrison 2009 J. Phys.: Conf. Ser. 169 012006

@SteveHuntsman Can you elaborate more. I understand MCMC constructs a Markov Chain with a given distribution. But if you are given such a chain, can you tell (without actually computing the stationary distribution) that the stationary distribution will be localized ?

do you mind writing down what you are not able to prove ?

Doesn't this imply the function is not Lipschitz and hence even uniqueness is not guaranteed ?

Are you asking there exists such a $k$ for any given neighborhood $U$? Thats false since linearity will be dominated by nonlinear terms at some size of perturbation.

@dohmatob The field of equivariant dynamical systems studies this. For PDEs, these symmetries really show up when discussing bifurcations. A nice book is :The symmetry perspective: from equilibrium to chaos in phase space and physical space. Also see:scholarpedia.org/article/Equivariant_dynamical_systems

Is this simply a consequence of equivariance of the give PDE, or is the gradient structure important here ? My point being that if a given PDE is equivariant under some group operations, we know the solutions will lie the symmetry subspaces...is that whats happening here ? This fact is regardless of whether a PDE is gradient form or not.

The methods you need come from "spatial dynamics". Look up Evans function, e.g. in this book: Kapitula, Todd, and Keith Promislow. Spectral and dynamical stability of nonlinear waves. Vol. 457. New York: Springer, 2013. Another keyword: exponential dichotomy

Thanks, this is helpful and yes, the isolated eigenvalue case is the one that I am interested in. Is there a source for this material other than the wiki links you provided ?