Dissipative Stochastic Equations Research Articles

Ergodicity of Stochastic Dissipative Equations Driven by α-Stable Process

It is shown that the transition semigroup (P t ) t≥0 corresponding to stochastic dissipative equations driven by α-stable is strong Feller and irreducible for α ∈ (1, 2). This result ensures the ergodicity for the equation.

Onsager principle for nonlinear mechanical systems modeled by stochastic dissipative equations

A classical mechanical system subjected to frictional forces is considered in the limit of large frictional coefficient. Random white noise is also introduced in conformity to the fluctuation-dissipation theorem. The velocity is split into a deterministic component plus a random stochastic component consequently, the evolution operator (generator) for the probability density in configuration space is evaluated recalling previous work by the same author, by stochastically averaging the flux of particles. The averages depend upon the history of the system, but memory may be eliminated by suitably defining the drift, in the limit of large time. The fundamental solution of the diffusion equation is recast into the form of a Feynman path integral, and subsequently transformed into an Onsager–Machlup path integral, whose regressive stationary solutions satisfy the minimum entropy production principle. It is focused upon the role played by the appropriate definition of drift velocity adopted in this approach, allowing for interpretation of the Onsager–Machlup potential

Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups

We consider stochastic equations in Hilbert spaces with singular drift in the framework of [G. Da Prato, M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2) (2002) 261–303]. We prove a Harnack inequality (in the sense of [F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417–424]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure μ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure μ for noncontinuous drifts.

Singular dissipative stochastic equations in Hilbert spaces

Existence of solutions to martingale problems corresponding to singular dissipative stochastic equations in Hilbert spaces are proved for any initial condition. The solutions for the single starting points form a conservative diffusion process whose transition semigroup is shown to be strong Feller. Uniqueness in a generalized sense is proved also, and a number of applications is presented.

Dissipative stochastic equations in Hilbert space with time dependent coefficients

We prove existence and, under an additional assumption, uniqueness of an evolution system of measures (\nu_t)_{t\in \R} for a stochastic differential equation with time dependent dissipative coefficients. We prove that if P_{s,t} denotes the corresponding transition evolution operator, then P_{s,t}\varphi behaves asymptotically as t\to +\infty like a limit curve (which is independent of s ) for any continuous and bounded “observable” \varphi .

Power fluctuations in stochastic models of dissipative systems

We consider different models of stochastic dissipative equations and theoretically compute the probability distribution functions (actually the associated large deviation functions) of the time averaged injected power required to sustain a nontrivial stationary state. We discuss the results and in particular draw from our results some general features shared by these distributions in realistic dissipative systems.

Singular dissipative stochastic equations in Hilbert spaces

Existence of solutions to martingale problems corresponding to singular dissipative stochastic equations in Hilbert spaces are proved for any initial condition. The solutions for the single starting points form a conservative diffusion process whose transition semigroup is shown to be strong Feller. Uniqueness in a generalized sense is proved also, and a number of applications is presented.

Remarks on determining projections for stochastic dissipative equations

In this paper we consider the notion of determining projections fortwo classes of stochastic dissipative equations: a reaction-diffusion equation and a2-dimensional Navier-Stokes equation.We define certain finite dimensional objects that can capture the asymptoticbehavior of the related dynamical system. They are projections on a space of polynomial functions, generalizing the classical (but not very much studied in a stochasticcontext) concepts of determining modes, nodes and volumes.

On dissipative stochastic equations in a Hilbert space

A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.

On dissipative stochastic equations in a Hilbert space

On dissipative stochastic equations in a Hilbert space