The Proof Technique of Unique Solutions of Contractions

International audienceWe review some recent work aimed at understanding proof techniques for behavioural equivalence on processes based on the concept of unique solution of equations. The schema of equations is refined to that of contraction, based on partial orders rather than equalities

Unique Solutions of Contractions, CCS, and their HOL Formalisation

The unique solution of contractions is a proof technique for bisimilarity that overcomes certain syntactic constraints of Milner's "unique solution of equations" technique. The paper presents an overview of a rather comprehensive formalisation of the core of the theory of CCS in the HOL theorem prover (HOL4), with a focus towards the theory of unique solutions of contractions. (The formalisation consists of about 20,000 lines of proof scripts in Standard ML.) Some refinements of the theory itself are obtained. In particular we remove the constraints on summation, which must be weakly-guarded, by moving to rooted contraction, that is, the coarsest precongruence contained in the contraction preorder.Comment: In Proceedings EXPRESS/SOS 2018, arXiv:1808.0807

Unique solutions of contractions, CCS, and their HOL formalisation

International audienceThe unique solution of contractions is a proof technique for (weak) bisimilarity that overcomes certainsyntactic limitations of Milner’s “unique solution of equations” theorem. This paper presents an overview ofa comprehensive formalisation of Milner’s Calculus of Communicating Systems (CCS) in the HOL theoremprover (HOL4), with a focus towards the theory of unique solutions of equations and contractions. Theformalisation consists of about 24,000 lines (1MB) of code in total. Some refinements of the “unique solutionof contractions” theory itself are obtained. In particular we remove the constraints on summation, whichmust be guarded, by moving from contraction to rooted contraction. We prove the “unique solution ofrooted contractions” theorem and show that rooted contraction is the coarsest precongruence contained inthe contraction preorder

Completion, extension, factorization, and lifting of operators with a negative index

The famous results of M.G. Kre\u{\i}n concerning the description of selfadjoint contractive extensions of a Hermitian contraction $T_1$ and the characterization of all nonnegative selfadjoint extensions $\widetilde A$ of a nonnegative operator $A$ via the inequalities $A_K\leq \widetilde A \leq A_F$, where $A_K$ and $A_F$ are the Kre\u{\i}n-von Neumann extension and the Friedrichs extension of $A$, are generalized to the situation, where $\widetilde A$ is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators $I-T_1^*T_1$ and $A$, respectively; these conditions are automatically satisfied if $T_1$ is contractive or $A$ is nonnegative, respectively. The approach developed in this paper starts by establishing first a generalization of an old result due to Yu.L. Shmul'yan on completions of $2\times 2$ nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M.G. Kre\u{\i}n and, in addition, to solve some related lifting problems for $J$-contractive operators in Hilbert, Pontryagin and Kre\u{\i}n spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.Comment: 29 page

A limitation on Long's model in stratified fluid flows

The flow of a continuously stratified fluid into a contraction is examined, under the assumptions that the dynamic pressure and the density gradient are constant upstream (Long's model). It is shown that a solution to the equations exists if and only if the strength of the contraction does not exceed a certain critical value which depends on the internal Froude number. For the flow of a stratified fluid over a finite barrier in a channel, it is further shown that, if the barrier height exceeds this same critical value, lee-wave amplitudes increase without bound as the length of the barrier increases. The breakdown of the model, as implied by these arbitrarily large amplitudes, is discussed. The criterion is compared with available experimental results for both geometries

Interior feedback stabilization of wave equations with dynamic boundary delay

In this paper we consider an interior stabilization problem for the wave equation with dynamic boundary delay.We prove some stability results under the choice of damping operator. The proof of the main result is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent

The evolution to equilibrium of solutions to nonlinear Fokker-Planck equation

One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, \ t\geq0, \ x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and $b$ the convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$, $L^1(\mathbb{R}^d)$, respectively, for some $t_n\to\infty$ of the solution $u(t_n)$ to an equilibrium state of the equation for a large set of nonnegative initial data in $L^1$. These results are new in the literature on nonlinear Fokker-Planck equations arising in the mean field theory and are also relevant to the theory of stochastic differential equations. As a matter of fact, by the above convergence result, it follows that the solution to the McKean-Vlasov stochastic differential equation corresponding to (1), which is a nonlinear distorted Brownian motion, has this equilibrium state as its unique invariant measure. Keywords: Fokker-Planck equation, $m$-accretive operator, probability density, Lyapunov function, $H$-theorem, McKean-Vlasov stochastic differential equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject Classification: 35B40, 35Q84, 60H10

Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications

We study regularity properties for invariant measures of semilinear diffusions in a separable Hilbert space. Based on a pathwise estimate for the underlying stochastic convolution, we prove a priori estimates on such invariant measures. As an application, we combine such estimates with a new technique to prove the $L^1$-uniqueness of the induced Kolmogorov operator, defined on a space of cylindrical functions. Finally, examples of stochastic Burgers equations and thin-film growth models are given to illustrate our abstract result.Comment: 19 page