Martingale solutions and Markov selections for stochastic partial differential equations

We present a general framework for solving stochastic porous medium equations and stochastic Navier–Stokes equations in the sense of martingale solutions. Following Krylov [N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 691–708] and Flandoli–Romito [F. Flandoli, N. Romito, Markov selections for the 3D stochastic Navier–Stokes equations, Probab. Theory Related Fields 140 (2008) 407–458], we also study the existence of Markov selections for stochastic evolution equations in the absence of uniqueness.     

On Closability of Directional Gradients

Let be a centred Gaussian measure on a separable real Banach space E, and let H be a Hilbert subspace of E. We provide necessary and sufficient conditions for closability in L ^p (E,) of the gradient D _H in the direction of H. These conditions are further elaborated in case when the gradient D _H corresponds to a bilinear form associated with a certain nonsymmetric Ornstein–Uhlenbeck operator. Some natural examples of closability and nonclosability are presented.     

Hypercontractivity of Solutions to Hamilton-Jacobi Equations

We show that solutions to some Hamilton-Jacobi Equations associated to the problem of optimal control of stochastic semilinear equations enjoy the hypercontractivity property.     

Improving performance of resonant tunneling devices in asymmetric structures

Based on the global coherent tunneling model, we present a self-consistent calculation and show that structural asymmetry of double barrier resonant tunneling structures (DBRTSs) significantly modifies the current–voltage characteristics compared to the symmetric structures. Within the framework of the dielectric continuum model, we further investigate the phonon-assisted tunneling (PAT) current in symmetric and asymmetric DBRTSs. Both the interface modes and the confined bulk-like longitudinal-optical phonons are considered. The results indicate that the four higher-frequency interface phonon modes (especially the one which has the largest electron–phonon interaction at either interface of the emitter barrier) dominate the PAT processes. We show that a suitably designed asymmetric structure can produce much larger peak current and absolute value of the negative differential conductivity than its commonly used symmetric counterpart.     

Parameter Estimation for Controlled Semilinear Stochastic Systems: Identifiability and Consistency

We consider a controlled stochastic semilinear evolution equation with the drift depending on the unknown parameter. We show that the maximum likelihood estimator is strongly consistent for a class of bounded predictable controls.     

The Ornstein–Uhlenbeck bridge and applications to Markov semigroups

For an arbitrary Hilbert space-valued Ornstein–Uhlenbeck process we construct the Ornstein–Uhlenbeck bridge connecting a given starting point x$x$ and an endpoint y$y$ provided y$y$ belongs to a certain linear subspace of full measure. We derive also a stochastic evolution equation satisfied by the OU bridge and study its basic properties. The OU bridge is then used to investigate the Markov transition semigroup defined by a stochastic evolution equation with additive noise. We provide an explicit formula for the transition density and study its regularity. These results are applied to show some basic properties of the transition semigroup. Given the strong Feller property and the existence of invariant measure we show that all L^p$L^p$ functions are transformed into continuous functions, thus generalising the strong Feller property. We also show that transition operators are q$q$-summing for some q>p>1$q>p>1$, in particular of Hilbert–Schmidt type.     

Dynamic programming principle for stochastic control problems driven by general Lévy noise

We extend the proof of the dynamic programming principle (DPP) for standard stochastic optimal control problems driven by general Lévy noise. Under appropriate assumptions, it is shown that the DPP still holds when the state process fails to have any moments at all.