Related papers: The Schroedinger Problem, Levy Processes Noise in …
The skew Brownian motion is a strong Markov process which behaves like a Brownian motion until hitting zero and exhibits an asymmetry at zero. We address the following question: what is a natural counterpart of the skew Brownian motion in…
We propose that the underlying context of holographic duality and the Ryu-Takayanagi formula is that the volume measure of spacetime is a probability measure constrained by quantum dynamics. We define quantum stochastic processes using…
This paper is concerned with the existence of invariant measure for 3D stochastic primitive equations driven by linear multiplicative noise under non-periodic boundary conditions. The common method is to apply Sobolev imbedding theorem to…
The buildup process of the probability density inside the quantum well of a double-barrier resonant structure is studied by considering the analytic solution of the time dependent Schr\"{o}dinger equation with the initial condition of a…
Through certain appropriate constructions, we establish periodic solutions in distribution for some stochastic differential equations with infinite-dimensional Levy noise. Additionally, we obtain the corresponding periodic measures and…
We consider an infinite-dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of Wiener processes. By studying how randomness is spread by the system we develop a counterpart of Hormander's…
In this initial paper in a series, we first discuss why classical motions of small particles should be treated statistically. Then we show that any attempted statistical description of any nonrelativistic classical system inevitably yields…
We explore the transient dynamics associated with the emergence of the classical signal in the full quantum system. We start our study from the instability which promotes the squeezing of the quantum system. This is often interpreted as the…
Environmental noise on a controlled quantum system is generally modeled by a dissipative Lindblad equation. This equation describes the average state of the system via the density matrix $\rho$. One way of deriving this Lindblad equation is…
The question of existence and properties of stationary solutions to Langevin equations driven by noise processes with stationary increments is discussed, with particular focus on noise processes of pseudo-moving-average type. On account of…
In this thesis, we develop analytical methods to study out-of-equilibrium stochastic processes driven by colored noise, i.e., noise with temporal correlations. These non-Markovian processes pose significant analytical challenges compared to…
We study a stochastic Schr{\"o}dinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension less than 3. When the Hurst index is large enough, we prove local well-posedness of the problem using…
I examine whether Nelson's stochastic formulation of Schr\"{o}dinger equation could be derived from a phase space process through a colored noise smoothing. If this conjecture is true, it would yield a local stochastic hidden variable…
Some intriging connections between the properties of nonlinear noise driven systems and the nonlinear dynamics of a particular set of Hamilton's equation are discussed. A large class of Fokker-Planck Equations, like the Schr\"odinger…
Do stochastic Schr\"odinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system {\em on average} obeys a master equation, the answer is yes. Markovian stochastic Schr\"odinger…
For an SDE driven by a rotationally invariant $\alpha$-stable noise we prove weak uniqueness of the solution under the balance condition $\alpha+\gamma>1$, where $\gamma$ denotes the Holder index of the drift coefficient. We prove existence…
Linear dynamical systems, driven by a non-white noise which has the Levy distribution, are analysed. Noise is modelled by a specific stochastic process which is defined by the Langevin equation with a linear force and the Levy distributed…
We study stochastic Euler equations in both compressible and incompressible regimes, on the whole space and on the torus, driven by genuinely mixed multiplicative noise: continuous Stratonovich/It\^o components and a discontinuous Marcus…
Many of the conceptual problems students have in understanding quantum mechanics arise from the way probabilities are introduced in standard (textbook) quantum theory through the use of measurements. Introducing consistent microscopic…
We use a probabilistic method to describe the effect of laser noise on the laser-atom interaction, in the case that the atom is a two level system without spontaneous emission. The stochastic differential equation for the laser-atom…