Large Deviations at Level 2.5 for Markovian Open Quantum Systems: Quantum Jumps and Quantum State Diffusion

We consider quantum stochastic processes and discuss a level 2.5 large deviation formalism providing an explicit and complete characterisation of fluctuations of time-averaged quantities, in the large-time limit. We analyse two classes of quantum stochastic dynamics, within this framework. The first class consists of the quantum jump trajectories related to photon detection; the second is quantum state diffusion related to homodyne detection. For both processes, we present the level 2.5 functional starting from the corresponding quantum stochastic Schrödinger equation and we discuss connections of these functionals to optimal control theory.

Analysis of long-term solution of chemotactic model with indirect signal consumption in three-dimensional case

In this paper, we consider the chemotaxis model u_t&=\Delta u-\nabla\cdot(u\nabla v),& \qquad x\in\Omega,\,t>0,v_t&=\Delta v-vw,& \qquad x\in\Omega,\,t>0,w_t&=-\delta w+u,& \qquad x\in\Omega,\,t>0,under homogeneous Neumann boundary conditions in a bounded and convex domain $\Om\subset \mathbb{R}^3$ with smooth boundary, where $\delta>0$ is a given parameter. It is shown that for arbitrarily large initial data, this problem admits at least one global weak solution for which there exists $T>0$ such that the solution $(u,v,w)$ is bounded and smooth in $\Om\times(T,\infty)$. Furthermore, it is asserted that such solutions approach spatially constant equilibria in the large time limit.

Large time behavior in a predator-prey system with pursuit-evasion interaction

<p style='text-indent:20px;'>This work considers a pursuit-evasion model</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1000">\begin{document}$\begin{equation} \left\{ \begin{split} &amp;u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &amp;v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &amp;w_t = \Delta w-w+v,\\ &amp;z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1) \end{equation}$\end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>with positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> in a bounded domain <inline-formula><tex-math id="M7">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M8">\begin{document}$ N $\end{document}</tex-math></inline-formula> is the dimension of the space) with smooth boundary. We prove that if <inline-formula><tex-math id="M9">\begin{document}$ a&lt;2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}&gt;\max\{\chi,\xi\} $\end{document}</tex-math></inline-formula>, (1) possesses a global bounded classical solution with a positive constant <inline-formula><tex-math id="M11">\begin{document}$ C_{\frac{N}{2}+1} $\end{document}</tex-math></inline-formula> corresponding to the maximal Sobolev regularity. Moreover, it is shown that if <inline-formula><tex-math id="M12">\begin{document}$ b\mu&lt;\lambda $\end{document}</tex-math></inline-formula>, the solution (<inline-formula><tex-math id="M13">\begin{document}$ u,v,w,z $\end{document}</tex-math></inline-formula>) converges to a spatially homogeneous coexistence state with respect to the norm in <inline-formula><tex-math id="M14">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula> in the large time limit under some exact smallness conditions on <inline-formula><tex-math id="M15">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>. If <inline-formula><tex-math id="M17">\begin{document}$ b\mu&gt;\lambda $\end{document}</tex-math></inline-formula>, the solution converges to (<inline-formula><tex-math id="M18">\begin{document}$ \mu,0,0,\mu $\end{document}</tex-math></inline-formula>) with respect to the norm in <inline-formula><tex-math id="M19">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M20">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula> under some smallness assumption on <inline-formula><tex-math id="M21">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> with arbitrary <inline-formula><tex-math id="M22">\begin{document}$ \xi $\end{document}</tex-math></inline-formula>.</p>

Nonnegative solutions to a doubly degenerate nutrient taxis system

<p style='text-indent:20px;'>This paper deals with the doubly degenerate nutrient taxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = (uv u_x)_x - (u^2 vv_x)_x + \ell uv, \qquad &amp; x\in \Omega, \ t&gt;0, \\ v_t = v_{xx} -uv, \qquad &amp; x\in \Omega, \ t&gt;0, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in an open bounded interval <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R} $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ \ell \ge0 $\end{document}</tex-math></inline-formula>, which has been proposed to model the formation of diverse morphological aggregation patterns observed in colonies of <i>Bacillus subtilis</i> growing on the surface of thin agar plates.</p><p style='text-indent:20px;'>It is shown that under the mere assumption that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{l} u_0\in W^{1,\infty}( \Omega) \mbox{ is nonnegative with } u_0\not\equiv 0 \qquad \mbox{and} \\ v_0\in W^{1,\infty}( \Omega) \mbox{ is positive in } \overline{\Omega}, \end{array} \right. \qquad \qquad (\star) \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>an associated no-flux initial boundary value problem possesses a globally defined and continuous weak solution <inline-formula><tex-math id="M3">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ u\ge 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ v&gt;0 $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M6">\begin{document}$ \overline{\Omega}\times [0,\infty) $\end{document}</tex-math></inline-formula>, and that moreover there exists <inline-formula><tex-math id="M7">\begin{document}$ u_\infty\in C^0( \overline{\Omega}) $\end{document}</tex-math></inline-formula> such that the solution <inline-formula><tex-math id="M8">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> approaches the pair <inline-formula><tex-math id="M9">\begin{document}$ (u_\infty,0) $\end{document}</tex-math></inline-formula> in the large time limit with respect to the topology <inline-formula><tex-math id="M10">\begin{document}$ (L^{\infty}( \Omega)) ^2 $\end{document}</tex-math></inline-formula>. This extends comparable results recently obtained in [<xref ref-type="bibr" rid="b17">17</xref>], the latter crucially relying on the additional requirement that <inline-formula><tex-math id="M11">\begin{document}$ \int_\Omega \ln u_0&gt;-\infty $\end{document}</tex-math></inline-formula>, to situations involving nontrivially supported initial data <inline-formula><tex-math id="M12">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>, which seems to be of particular relevance in the addressed application context of sparsely distributed populations.</p>

Determinantal Expressions in Multi-Species TASEP

Consider an inhomogeneous multi-species TASEP with drift to the left, and define a height function which equals the maximum species number to the left of a lattice site. For each fixed time, the multi-point distributions of these height functions have a determinantal structure. In the homogeneous case and for certain initial conditions, the fluctuations of the height function converge to Gaussian random variables in the large-time limit. The proof utilizes a coupling between the multi-species TASEP and a coalescing random walk, and previously known results for coalescing random walks.

Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

The chemotaxis-growth system(⋆)\left\{\begin{aligned} \displaystyle u_{t}&\displaystyle=D\Delta u-\chi\nabla\cdot(u\nabla v)+\rho u-\mu u^{\alpha},\\ \displaystyle v_{t}&\displaystyle=d\Delta v-\kappa v+\lambda u\end{aligned}\right.is considered under homogeneous Neumann boundary conditions in smoothly bounded domains \Omega\subset\mathbb{R}^{n}, n\geq 1. For any choice of \alpha>1, the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of (⋆), the present work shows that, whenever \alpha\geq 2-\frac{2}{n}, under an appropriate smallness assumption on 𝜒, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state \bigl{(}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}},\frac{\lambda}{\kappa}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}}\bigr{)} in the large time limit.

The transition rates of an isotropic quantum charged oscillator in the presence of external fields

In this work, explicit expressions for the transition rates of an isotropic quantum charged harmonic oscillator in the vicinity of a perfectly conducting half-space under the influence of an external classical source are obtained. In the absence of external sources, it is shown that the decay rate of an initially exited state of the oscillator is a periodic function in terms of the normalized distance to the plate. The modified transition rates in the presence of external classical sources are obtained in the large-time limit indicating a contribution proportional to the squared module of the Fourier transform of the external source. In the absence of the conducting plate and external sources, the results are in agreement with the free space case. The problem is generalized to the case of a real conducting half-space.

Open XXZ chain and boundary modes at zero temperature

We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.

Energetics of Poisson–Kac Stochastic Processes Possessing Finite Propagation Velocity

A local fluctuation–dissipation theorem for the power delivered by a stochastic forcing is derived for Ornstein–Uhlenbeck processes driven by smooth, i. e. almost everywhere (a. e.)-differentiable stochastic perturbations (Poisson–Kac processes). An analytic expression for the probability density function of the fluctuational power is obtained in the large time limit. As these processes converge, in the Kac limit, toward classical Langevin equations driven by Wiener processes, a coarse-grained analysis of the statistical properties of the fluctuational work is developed.