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Using the theory of stochastic integration developed recently by the authors, in this article is proved an It\^{o} formula for Hilbert space-valued It\^{o} processes defined with respect to a cylindrical-martingale valued measure. This formula is applied to prove an integration by parts formula for It\^{o} processes, to prove a Burkholder inequality for the stochastic integral, and finally to prove an It\^{o} formula for Hilbert space-valued It\^{o} processes defined with respect to a L\'evy process.

Anyons are exotic low-dimensional quasiparticles whose unconventional quantum statistics extends the binary particle division into fermions and bosons. The fractional quantum Hall regime provides a natural host, with first convincing anyon signatures recently observed through interferometry and cross-correlations of colliding beams. However, the fractional regime is rife with experimental complications, such as an anomalous tunneling density of states, which impede the manipulation of anyons. Here we show experimentally that the canonical integer quantum Hall regime can provide a robust anyon platform. Exploiting the Coulomb interaction between two co-propagating quantum Hall channels, an electron injected into one channel splits into two fractional charges behaving as abelian anyons. Their unconventional statistics is revealed by negative cross-correlations between dilute quasiparticle beams. Similarly to fractional quantum Hall observations, we show that the negative signal stems from a time-domain braiding process, here involving the incident fractional quasiparticles and spontaneously generated electron-hole pairs. Beyond the dilute limit, a theoretical understanding is achieved via the edge magnetoplasmon description of interacting integer quantum Hall channels. Our findings establish that, counter-intuitively, the integer quantum Hall regime provides a platform of choice for exploring and manipulating quasiparticles with fractional quantum statistics., Comment: 6 pages, 4 figures, Methods, Supplemental Information

Let $\Phi'$ denote the strong dual of a nuclear space $\Phi$ and let $C_{\infty}(\Phi')$ be the collection of all continuous mappings $x:[0,\infty) \rightarrow \Phi'$ equipped with the topology of local uniform convergence. In this paper we prove sufficient conditions for tightness of probability measures on $C_{\infty}(\Phi')$ and for weak convergence in $C_{\infty}(\Phi')$ for a sequence of $\Phi'$-valued processes. We illustrate our results with two applications. First, we show the central limit theorem for local martingales taking values in the dual of an ultrabornological nuclear space. Second, we prove sufficient conditions for the weak convergence in $C_{\infty}(\Phi')$ for a sequence of solutions to stochastic partial differential equations driven by semimartingale noise., Comment: The original submission has been split into two separated works. Tightness and weak uniform convergence covered in this work, and UCP convergence is studied in the separated work arXiv:2303.17082

We develop a theory of Hilbert-space valued stochastic integration with respect to cylindrical martingale-valued measures. As part of our construction, we expand the concept of quadratic variation, introduced by Veraar and Yaroslavtsev (2016), to the case of cylindrical martingale-valued measures that are allowed to have discontinuous paths (this is carried out within the context of separable Banach spaces). Our theory of stochastic integration is applied to address the existence and uniqueness of solutions to stochastic partial differential equations in Hilbert spaces., Comment: This is an updated version with new examples and remarks

A constructive definition of the supremum of a family of set functions is exploited in the context of Riesz spaces of signed measures and finitely additive functions (signed charges) on semi-rings. We explore applications, particularly to establish a Jordan decomposition for signed charges on semi-rings, whether the structure of Riesz space is present or not.

Let $\Phi'$ denote the strong dual of a nuclear space $\Phi$. In this paper we introduce sufficient conditions for the convergence uniform on compacts in probability for a sequence of $\Phi'$-valued processes with continuous or c\`{a}dl\`{a}g paths. We illustrate the usefulness of our results by considering two applications to stochastic analysis. First, we introduce a topology on the space of $\Phi'$-valued semimartingales which are good integrators and show that this topology is complete and that the stochastic integral mapping is continuous on the integrators. Second, we introduce sufficient conditions for the convergence uniform on compacts in probability of the solutions to a sequence of linear stochastic evolution equations driven by semimartingale noise., Comment: The original submission has been split into two separated works. UCP convergence covered in this work, and weak uniform convergence is studied in a separated work

Squeezing of the quadratures of the electromagnetic field has been extensively studied in optics and microwaves. However, previous works focused on the generation of squeezed states in a low impedance ($Z_0 \approx 50 \Omega$) environment. We report here on the demonstration of the squeezing of bosonic edge magnetoplasmon modes in a quantum Hall conductor whose characteristic impedance is set by the quantum of resistance ($R_K \approx 25 k \Omega$), offering the possibility of an enhanced coupling to low-dimensional quantum conductors. By applying a combination of dc and ac drives to a quantum point contact, we demonstrate squeezing and observe a noise reduction 18\% below the vacuum fluctuations. This level of squeezing can be improved by using more complex conductors, such as ac driven quantum dots or mesoscopic capacitors., Comment: 6+2 pages, 3+1 figures

Let $\Phi$ a locally convex space and $\Psi$ be a quasi-complete, bornological, nuclear space (like spaces of smooth functions and distributions) with dual spaces $\Phi'$ and $\Psi'$. In this work we introduce sufficient conditions for time regularity properties of the $\Psi'$-valued stochastic convolution $\int_{0}^{t} \int_{U} S(t-r)'R(r,u) M(dr,du)$, $t \in [0,T]$, where $(S(t): t \geq 0)$ is a $C_{0}$-semigroup on $\Psi$, $R(r,\omega,u)$ is a suitable operator form $\Phi'$ into $\Psi'$ and $M$ is a cylindrical-martingale valued measure on $\Phi'$. Our result is latter applied to study time regularity of solutions to $\Psi'$-valued stochastic evolutions equations.

Let $\Phi$ be a nuclear space and let $\Phi'$ denote its strong dual. In this paper we introduce sufficient conditions for the almost surely uniform convergence on bounded intervals of time for a sequence of $\Phi'$-valued processes having continuous (respectively c\`{a}dl\`{a}g) paths. The main result is formulated first in the general setting of cylindrical processes but later specialized to other situations of interest. In particular, we establish conditions for the convergence to occur in a Hilbert space continuously embedded in $\Phi'$. Furthermore, in the context of the dual of an ultrabornological nuclear space (like spaces of smooth functions and distributions) we also include applications to the convergence of a series of independent c\`{a}dl\`{a}g process and to the convergence of solutions to linear evolution equations driven by L\'{e}vy noise.

In this article we give sufficient and necessary conditions for the existence of a weak and mild solution to stochastic evolution equations with (general) L\'{e}vy noise taking values in the dual of a nuclear space. As part of our approach we develop a theory of stochastic integration with respect to a L\'{e}vy process taking values in the dual of a nuclear space. We also derive further properties of the solution such as the existence of a solution with square moments, the Markov property and path regularity of the solution. In the final part of the paper we give sufficient conditions for the weak convergence of the solutions to a sequence of stochastic evolution equations with L\'{e}vy noises.

We observe and comprehend the dynamical Coulomb blockade suppression of the electrical conductance across an electronic quantum channel submitted to a temperature difference. A broadly tunable, spin-polarized Ga(Al)As quantum channel is connected on-chip, through a micron-scale metallic node, to a linear $RC$ circuit. The latter is made up of the node's geometrical capacitance $C$ in parallel with an adjustable resistance $R\in \{1/2,1/3,1/4\}\times h/e^2$ formed by 2--4 quantum Hall channels. The system is characterized by three temperatures: a temperature of the electrons in the large electrodes ($T$) and in the node ($T_\mathrm{node}$), and a temperature of the electromagnetic modes of the $RC$ circuit ($T_\mathrm{env}$). The temperature in the node is selectively increased by local Joule dissipation, and characterized from current fluctuations. For a quantum channel in the tunnel regime, a close match is found between conductance measurements and tunnel dynamical Coulomb blockade theory. In the opposite near ballistic regime, we develop a theory that accounts for different electronic and electromagnetic bath temperatures, again in very good agreement with experimental data. Beyond these regimes, for an arbitrary quantum channel set in the far out-of-equilibrium situation where the temperature in the node significantly exceeds the one in the large electrodes, the equilibrium (uniform temperature) prediction for the conductance is recovered, albeit at a rescaled temperature $\alpha T_\mathrm{node}$., Comment: Submitted

In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on a locally convex space $\Phi$. Our construction of the stochastic integral is based on the theory of tensor products of topological vector spaces and the property of good integrators of real-valued semimartingales. This theory is further developed in the case where $\Phi$ is a complete, barrelled, nuclear space, where we obtain a complete description of the class of integrands as $\Phi$-valued locally bounded and weakly predictable processes. Several other properties of the stochastic integral are proven, including a Riemann representation, a stochastic integration by parts formula and a stochastic Fubini theorem. Our theory is then applied to provide sufficient and necessary conditions for existence and uniqueness of solutions to linear stochastic evolution equations driven by semimartingale noise taking values in the strong dual $\Phi'$ of $\Phi$. In the last part of this article we apply our theory to define stochastic integrals with respect to a sequence of real-valued semimartingales.

In this work we introduce a theory of stochastic integration for operator-valued integrands with respect to some classes of cylindrical martingale-valued measures in Hilbert spaces. The integral is constructed via the radonification of cylindrical martingales by a Hilbert-Schmidt operator theorem and unifies several other theories of stochastic integration in Hilbert spaces. In particular, our theory covers the theory of stochastic integration with respect to a Hilbert space valued L\'{e}vy process (which is not required to satisfy any moment condition), with respect to a cylindrical L\'{e}vy processes with (weak) second moments and with respect to a L\'{e}vy-valued random martingale measures with finite second moment. As an application of our theory of integration we prove existence and uniqueness of solutions for stochastic stochastic partial differential equations driven by multiplicative cylindrical martingale-valued measure noise with rather general coefficients.

Let $\Phi$ be a locally convex space and let $\Phi'$ denote its strong dual. In this paper we introduce sufficient conditions for the existence of a continuous or a c\`{a}dl\`{a}g $\Phi'$-valued version to a cylindrical process defined on $\Phi$. Our result generalizes many other known results on the literature and their different connections will be discussed. As an application, we use our results to show the existence of a $\Phi'$-valued c\`{a}dl\`{a}g L\'{e}vy process version to a given cylindrical L\'{e}vy process in $\Phi'$., Comment: arXiv admin note: text overlap with arXiv:1701.06630, arXiv:1902.03981, arXiv:1806.10231, arXiv:1511.08443

This work is devoted to the study of semimartingales on the dual of a general nuclear space. We start by establishing conditions for a cylindrical semimartingale in the strong dual $\Phi'$ of a nuclear space $\Phi$ to have a $\Phi'$-valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples are also provided. Later, we will show that under some general conditions every semimartingale taking values in the dual of a nuclear space has a canonical representation. The concept of predictable characteristics is introduced and is used to establish necessary and sufficient conditions for a $\Phi'$-valued semimartingale to be a $\Phi'$-valued L\'{e}vy process.

Let $\Phi'_{\beta}$ denotes the strong dual of a nuclear space $\Phi$ and let $D_{T}(\Phi'_{\beta})$ be the Skorokhod space of right-continuous with left limits (c\`{a}dl\`{a}g) functions from $[0,T]$ into $\Phi'_{\beta}$. In this article we introduce the concepts of cylindrical random variables and cylindrical measures on $D_{T}(\Phi'_{\beta})$, and prove analogues of the regularization theorem and Minlos theorem for extensions of these objects to bona fide random variables and probability measures on $D_{T}(\Phi'_{\beta})$ respectively. Later, we establish analogues of L\'{e}vy's continuity theorem to provide necessary and sufficient conditions for uniform tightness of families of probability measures on $D_{T}(\Phi'_{\beta})$ and sufficient conditions for weak convergence of a sequence of probability measures on $D_{T}(\Phi'_{\beta})$. Extensions of the above results to the space $D_{\infty}(\Phi'_{\beta})$ of c\`{a}dl\`{a}g functions from $[0,\infty)$ into $\Phi'_{\beta}$ are also given. Afterwards, we apply our results to study weak convergence of $\Phi'_{\beta}$-valued c\`{a}dl\`{a}g processes and in particular to L\'{e}vy processes. We finalize with an application of our theory to the study of tightness and weak convergence of probability measures on the Skorokhod space $D_{\infty}(H)$ where $H$ is a Hilbert space., Comment: To appear in Studia Mathematica

We deal with the dynamical system properties of a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation with mean-field Hamiltonian that models a simple laser by applying a mean field approximation to a quantum system describing a single-mode optical cavity and a set of two level atoms, each coupled to a reservoir. We prove that the mean field quantum master equation has a unique regular stationary solution. In case a relevant parameter $C_\mathfrak{b} $, i.e., the cavity cooperative parameter, is less than $1$, we prove that any regular solution converges exponentially fast to the equilibrium, and so the regular stationary state is a globally asymptotically stable equilibrium solution. We obtain that a locally exponential stable limit cycle is born at the regular stationary state as $C_\mathfrak{b} $ passes through the critical value $1$. Then, the mean-field laser equation has a Poincar\'e-Andronov-Hopf bifurcation at $C_\mathfrak{b} =1 $ of supercritical-like type. Namely, we derive rigorously, at the level of density matrices --for the first time--, the transition from a global attractor quantum state, where the light is not emitted, to a locally stable set of coherent quantum states producing coherent light. Moreover, we establish the local exponential stability of the limit cycle in case a relevant parameter is between the first and second laser thresholds appearing in the semiclassical laser theory. Thus, we get that the coherent laser light persists over time under this condition. In order to prove the exponential convergence of the quantum state, we develop a new technique for proving the exponential convergence in open quantum systems that is based in a new variation of constant formula. Applying our main results we find the long-time behavior of the von Neumann entropy, the photon-number statistics, and the quantum variance of the quadratures.

We develop a new approach for solving stochastic quantum master equations with mixed initial states. First, we obtain that the solution of the jump-diffusion stochastic master equation is represented by a mixture of pure states satisfying a system of stochastic differential equations of Schr\"odinger type. Then, we design three exponential schemes for these coupled stochastic Schr\"odinger equations, which are driven by Brownian motions and jump processes. Hence, we have constructed efficient numerical methods for the stochastic master equations based on quantum trajectories. The good performance of the new numerical integrators is illustrated by simulations of two quantum measurement processes.

Quantum phase transitions are ubiquitous in many exotic behaviors of strongly-correlated materials. However the microscopic complexity impedes their quantitative understanding. Here, we observe thoroughly and comprehend the rich strongly-correlated physics in two profoundly dissimilar regimes of quantum criticality. With a circuit implementing a quantum simulator for the three-channel Kondo model, we reveal the universal scalings toward different low-temperature fixed points and along the multiple crossovers from quantum criticality. Notably, an unanticipated violation of the maximum conductance for ballistic free electrons is uncovered. The present charge pseudospin implementation of a Kondo impurity opens access to a broad variety of strongly-correlated phenomena., Comment: Article (6 figures) with supplementary (4 figures); http://science.sciencemag.org/content/360/6395/1315

We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general L\'{e}vy processes in this context.

Let $\Phi$ be a nuclear space and let $\Phi'_{\beta}$ denote its strong dual. In this work we establish the one-to-one correspondence between infinitely divisible measures on $\Phi'_{\beta}$ and L\'{e}vy processes taking values in $\Phi'_{\beta}$. Moreover, we prove the L\'{e}vy-It\^{o} decomposition, the L\'{e}vy-Khintchine formula and the existence of c\`{a}dl\`{a}g versions for $\Phi'_{\beta}$-valued L\'{e}vy processes. A characterization for L\'{e}vy measures on $\Phi'_{\beta}$ is also established. Finally, we prove the L\'{e}vy-Khintchine formula for infinitely divisible measures on $\Phi'_{\beta}$., Comment: To appear in Journal of Theoretical Probability

In this paper, a weak Local Linearization scheme for Stochastic Differential Equations (SDEs) with multiplicative noise is introduced. First, for a time discretization, the solution of the SDE is locally approximated by the solution of the piecewise linear SDE that results from the Local Linearization strategy. The weak numerical scheme is then defined as a sequence of random vectors whose first moments coincide with those of the piecewise linear SDE on the time discretization. The rate of convergence is derived and numerical simulations are presented for illustrating the performance of the scheme.

We use the linear scalar SDE as a test problem to show that it is possible to construct almost sure stable first-order weak balanced schemes based on the addition of stabilizing functions to the drift terms. Then, we design balanced schemes for multidimensional bilinear SDEs achieving the first order of weak convergence, which do not involve multiple stochastic integrals. To this end, we follow two methodologies to find appropriate stabilizing weights; through an optimization procedure or based on a closed heuristic formula. Numerical experiments show a promising performance of the new numerical schemes.

We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution $X_t$ by integrating the system of coupled SDEs that describes the evolution of the norm of $X_t$ and its projection on the unit sphere. This allows us to develop an explicit scheme for stiff SDEs with multiplicative noise that shows a solid performance in various numerical experiments. Under general conditions, the new integrator preserves the almost sure stability of the solutions for any step-size, as well as the property of being distant from $0$. The scheme also has linear rate of weak convergence for a general class of SDEs with locally Lipschitz coefficients,and one-half strong order of convergence., Comment: SIAM Journal on Numerical Analysis (to appear)