Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow

We establish moment estimates for the invariant measure of a stochastic partial differential equation describing motion by mean curvature flow in (1+1) dimension, leading to polynomial stability of the associated Markov semigroup. We also prove maximal dissipativity for the related Kolmogorov operator

Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms

Let E be an unbounded open (or closed) domain in Euclidean space of dimension greater or equal to two. We present conservativeness criteria for (possibly reflected) diffusions with state space E that are associated to fairly general perturbed divergence form operators. Our main tool is a recently extended forward and backward martingale decomposition, which reduces to the well-known Lyons-Zheng decomposition in the symmetric case.Comment: Corrected typos, minor modification

A Relative Value Iteration Algorithm for Non-degenerate Controlled Diffusions

The ergodic control problem for a non-degenerate controlled diffusion controlled through its drift is considered under a uniform stability condition that ensures the well-posedness of the associated Hamilton-Jacobi-Bellman (HJB) equation. A nonlinear parabolic evolution equation is then proposed as a continuous time continuous state space analog of White's `relative value iteration' algorithm for solving the ergodic dynamic programming equation for the finite state finite action case. Its convergence to the solution of the HJB equation is established using the theory of monotone dynamical systems and also, alternatively, by using the theory of reverse martingales.Comment: 17 page

Existence and approximation of Hunt processes associated with generalized Dirichlet forms

We show that any strictly quasi-regular generalized Dirichlet form that satisfies the mild structural condition D3 is associated to a Hunt process, and that the associated Hunt process can be approximated by a sequence of multivariate Poisson processes. This also gives a new proof for the existence of a Hunt process associated to a strictly quasi-regular generalized Dirichlet form that satisfies SD3 and extends all previous results.Comment: Revised, shortened and improved versio

N/V-limit for Langevin dynamics in continuum

We construct an infinite particle/infinite volume Langevin dynamics on the space of configurations in $\R^d$ having velocities as marks. The construction is done via a limiting procedure using $N$-particle dynamics in cubes $(-\lambda,\lambda]^d$ with periodic boundary conditions. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of $N$-particle systems in $(-\lambda,\lambda]^d$ with periodic boundary conditions. After proving tightness of the laws of finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space (and their weak limit) fulfilling a uniform Ruelle bound. Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for general repulsive interaction potentials $\phi$ of Ruelle type (e.g. the Lennard-Jones potential) and all temperatures, densities and dimensions $d\geq 1$

Mutation, selection, and ancestry in branching models: a variational approach

We consider the evolution of populations under the joint action of mutation and differential reproduction, or selection. The population is modelled as a finite-type Markov branching process in continuous time, and the associated genealogical tree is viewed both in the forward and the backward direction of time. The stationary type distribution of the reversed process, the so-called ancestral distribution, turns out as a key for the study of mutation-selection balance. This balance can be expressed in the form of a variational principle that quantifies the respective roles of reproduction and mutation for any possible type distribution. It shows that the mean growth rate of the population results from a competition for a maximal long-term growth rate, as given by the difference between the current mean reproduction rate, and an asymptotic decay rate related to the mutation process; this tradeoff is won by the ancestral distribution. Our main application is the quasispecies model of sequence evolution with mutation coupled to reproduction but independent across sites, and a fitness function that is invariant under permutation of sites. Here, the variational principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure

Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions

Bogachev VI, Röckner M, Stannat W. Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Sbornik: Mathematics. 2002;193(7-8):945-976.Let M be a complete connected Riemannian manifold of dimension d and let L be a second order elliptic operator on M that has a representation L = a(ij)partial derivative(xi)partial derivative(xj) + b(i)partial derivative(xi) in local coordinates, where a(ij) is an element of H-loc(p,1), b(i) is an element of L-loc(p) for some p > d, and the matrix (a'j) is non-singular. The aim of the paper is the study of the uniqueness of a solution of the elliptic equation L*mu = 0 for probability measures mu, which is understood in the weak sense: integralLphif dmu = 0 for all phi is an element of C-0(infinity)(M). In addition, the uniqueness of invariant probability measures for the corresponding semigroups (T-t(mu))tgreater than or equal to0 generated by the operator L is investigated. It is proved that if a probability measure it on M satisfies the equation L*mu = 0 and (L - I) (C-0(infinity)(M)) is dense in L-1 (M,p), then it is a unique solution of this equation in the class of probability measures. Examples are presented (even with a(ij) = delta(ij) and smooth b(i)) in which the equation L*mu = 0 has more than one solution in the class of probability measures. Finally, it is shown that if p > d+2, then the semigroup (T-t)(tgreater than or equal to0) generated by L has at most one invariant probability measure