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We study the extremal behavior of a stochastic integral driven by a multivariate L\'{e}vy process that is regularly varying with index $\alpha>0$. For predictable integrands with a finite $(\alpha+\delta)$-moment, for some $\delta>0$, we…
We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both classical and fractional derivatives are proved. The fundamental…
We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built…
English version of abstract: The dynamic optimization problems treated by the calculus of variations are usually solved with the help of the 2nd order Euler-Lagrange differential equations. These equations are, generally speaking,…
We introduce a discrete-time fractional calculus of variations on the time scale $h\mathbb{Z}$, $h > 0$. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and…
This paper presents sufficient conditions for the existence of stationary optimal policies for average-cost Markov Decision Processes with Borel state and action sets and with weakly continuous transition probabilities. The one-step cost…
Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary…
We investigate the dynamics of elastic microstructures within a fluid that are subjected to thermal fluctuations. We perform analysis to obtain systematically simplified descriptions of the mechanics in the limiting regimes when (i) the…
General turbulent mean statistics are shown to be characterized by a variational principle. The variational functionals, or ``effective actions'', have experimental consequences for turbulence fluctuations and are subject to realizability…
In this work, we consider optimal control problems for mechanical systems on vector spaces with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an…
In this paper, we derive sufficient conditions ensuring the existence of a weak solution $u$ for a tempered fractional Euler-Lagrange equations $$ \frac{\partial L}{\partial x}(u,{^C}\mathbb{D}_{a^+}^{\alpha, \sigma} u, t) +…
In a recent paper (Abe S 2013 Phys. Rev. E 88 022142), a variational principle has been formulated for spatiotemporally-fractional Fokker-Planck equations and applied to derivations of their approximate analytic solutions based on the…
A conventional derivation of motion equations in mechanics and field equations in field theory is based on the principle of least action with a proper Lagrangian. With a time-independent Lagrangian, a function of coordinates and velocities…
Time-irreversible stochastic processes are frequently used in natural sciences to explain non-equilibrium phenomena and to design efficient stochastic algorithms. Our main goal in this thesis is to analyse their dynamics by means of large…
We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary…
We investigate symmetry reduction of optimal control problems for left-invariant control systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a…
Fractional variational approach has gained much attention in recent years. There are famous fractional derivatives such as Caputo derivative, Riesz derivative and Riemann-Liouville derivative. Several versions of fractional variational…
In this paper we explain how the notion of ''weak Dirichlet process'' is the suitable generalization of the one of semimartingale with jumps. For such a process we provide a unique decomposition which is new also for semimartingales: in…
Starting from the most general formulation of stochastic thermodynamics---i.e. a thermodynamically consistent nonautonomous stochastic dynamics describing systems in contact with several reservoirs---, we define a procedure to identify the…
We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are…