Markov Semigroups Research Articles

Covariant Ergodic Quantum Markov Semigroups via Systems of Imprimitivity

We construct relativistic quantum Markov semigroups from covariant completely positive maps. We proceed by generalizing a step in Stinespring's dilation to a general system of imprimitivity and basing it on Poincare group. The resulting noise channels are relativistically consistent and the method is applicable to any fundamental particle, though we demonstrate it for the case of light-like particles. The Krauss decomposition of the relativistically consistent completely positive identity preserving maps (our set up is in Heisenberg picture) enables us to construct the covariant quantum Markov semigroups that are uniformly continuous. We induce representations from the little groups to ensure the quantum Markov semigroups that are ergodic due to transitive systems imprimitivity.

Quantitative spectral gaps for hypoelliptic stochastic differential equations with small noise

We study the convergence rate to equilibrium for a family of Markov semigroups $\{\mathcal{P}_t^{\epsilon}\}_{\epsilon > 0}$ generated by a class of hypoelliptic stochastic differential equations on $\mathbb{R}^d$, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter $\epsilon$. By scaling, this regime implies corresponding optimal results both for fixed dissipation and large noise limits or fixed noise and vanishing dissipation limits. As part of the proof, and of independent interest, we obtain uniform-in-$\epsilon$ upper and lower bounds on the density of the stationary measure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lower bounds by a de Giorgi-type iteration (both uniform in $\epsilon$). The spectral gap estimate on the semigroup is obtained by a weak Poincar\'e inequality argument combined with quantitative hypoelliptic regularization of the time-dependent problem.

Invariant Markov semigroups on quantum homogeneous spaces

Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected co-ideal *-subalgebras. Several one-to-one correspondences between such invariant functionals are established. Adding a positivity condition, this yields one-to-one correspondences of invariant quantum Markov semigroups acting on expected co-ideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces.

Regularity of Solutions to Kolmogorov Equations with Perturbed Drifts

We prove that a probability solution of the stationary Kolmogorov equation generated by a first order perturbation v of the Ornstein–Uhlenbeck operator L possesses a highly integrable density with respect to the Gaussian measure satisfying the non-perturbed equation provided that v is sufficiently integrable. More generally, a similar estimate is proved for solutions to inequalities connected with Markov semigroup generators under the curvature condition \(CD(\theta ,\infty )\). For perturbations from Lp an analog of the Log-Sobolev inequality is obtained. It is also proved in the Gaussian case that the gradient of the density is integrable to all powers. We obtain dimension-free bounds on the density and its gradient, which also covers the infinite-dimensional case.

Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

Complete Gradient Estimates of Quantum Markov Semigroups

In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.

Stochastic effects on an HIV/AIDS infection model with incomplete diagnosis

In this paper, we establish a stochastic differential equation model of the HIV/AIDS infection including four classes: susceptible individuals, undiagnosed HIV infected, diagnosed HIV infected and AIDS infected individuals. For the deterministic model, we obtain that the basic reproduction number R0 can determine whether there is a unique disease-free equilibrium or an endemic equilibrium, and show the solution of the model is positive and global. The mathematical analysis of the stochastic model (SDE) corresponding to the deterministic model (ODE) is investigated to determine the conditions for the stability of the random equilibria. The reproduction number R0s of the SDE model is derived, and by applying the Markov semigroups, it shows that the threshold parameter R0s can be used to control the stochastic dynamics of the model. What’s more, by solving the Fokker-Planck equation, we derive an expression of the probability density function of the SDE model around the quasi-endemic equilibrium of the ODE model under a certain condition. A series of numerical simulations are performed to illustrate the analytical results. Our study demonstrates that (i) reducing the recruitment rate of susceptible population and increasing diagnosis rate are the effective methods for controlling the HIV/AIDS disease; and (ii) large environmental noise intensity and small valid disease contact rate can suppress the outbreak of disease.

Transfer of Regularity for Markov Semigroups by Using an Interpolation Technique

Transfer of Regularity for Markov Semigroups by Using an Interpolation Technique

Compactness of Semigroups Generated by Symmetric Non-Local Dirichlet Forms with Unbounded Coefficients

Let \((\mathcal {E},\mathcal {F})\) be a symmetric non-local Dirichlet from with unbounded coefficient on \(L^{2}(\mathbb {R}^{d};\mathrm {d} x)\) defined by $$ \mathcal{E}(f,g)=\iint_{\mathbb{R}^{d}\times \mathbb{R}^{d}} (f(y)-f(x))(g(x)-g(y)){W(x,y)} J(x,\mathrm{d} y) \mathrm{d} x, \quad f,g\in \mathcal{F}, $$ where J(x,dy) is regarded as the jumping kernel for a pure-jump symmetric Lévy-type process with bounded coefficients, and W(x,y) is seen as a weighted (unbounded) function. We establish sharp criteria for compactness and non-compactness of the associated Markovian semigroup (Pt)t≥ 0 on \(L^{2}(\mathbb {R}^{d};\mathrm {d} x)\). In particular, we prove that if J(x,dy) = |x − y|−d−αdy with α ∈ (0,2), and $$W(x,y)=\left\{\begin{array}{ll} (1+|x|)^{p}+(1+|y|)^{p}, \ & |x-y|< 1 \\ (1+|x|)^{q}+(1+|y|)^{q}, \ & |x-y|\geq 1 \end{array}\right. $$ with \(p\in [0,\infty )\) and q ∈ [0,α), then (Pt)t≥ 0 is compact, if and only if p > 2. This indicates that the compactness of \((\mathcal {E},\mathcal {F})\) heavily depends on the growth of the weighted function W(x,y) only for |x − y| < 1. Our approach is based on establishing the essential super Poincaré inequality for \((\mathcal {E},\mathcal {F})\). Our general results work even if the jumping kernel J(x,dy) is degenerate or is singular with respect to the Lebesgue measure.

Generalized Dobrushin Coefficients on Banach Spaces

The asymptotic behavior of iterates of bounded linear operators (not necessarily positive), acting on Banach spaces, is studied. Through the Dobrushin ergodicity coefficient, we generalize some ergodic theorems obtained earlier for classical Markov semigroups acting on L^1 (or positive operators on abstract state spaces).

The stochastic tamed MHD equations: existence, uniqueness and invariant measures

We study the tamed magnetohydrodynamics equations, introduced recently in a paper by the author, perturbed by multiplicative Wiener noise of transport type on the whole space {mathbb {R}}^{3} and on the torus {mathbb {T}}^{3}. In a first step, existence of a unique strong solution are established by constructing a weak solution, proving that pathwise uniqueness holds and using the Yamada–Watanabe theorem. We then study the associated Markov semigroup and prove that it has the Feller property. Finally, existence of an invariant measure of the equation is shown for the case of the torus.

Log-Harnack inequality for reflected SPDEs driven by multiplicative noises and its applications

We mainly investigate the log-Harnack inequality for the reflected stochastic partial differential equation driven by multiplicative noises based on the gradient estimate of the associated Markov semigroup. To do it, the penalization method and the comparison principle for stochastic partial differential equations are adopted. As its applications, we also establish the entropy-cost inequality and study some important estimates of the transition density relative to its invariant measure.

Spectral expansions of non-self-adjoint generalized Laguerre semigroups

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.

On infinitesimal generators of sublinear Markov semigroups

We establish a Dynkin formula and a Courrege-von Waldenfels theorem for sublinear Markov semigroups. In particular, we show that any sublinear operator $A$ on $C_c^{\infty}(\mathds{R}^d)$ satisfying the positive maximum principle can be represented as supremum of a family of pseudo-differential operators: \begin{equation*} Af(x) = \sup_{\alpha \in I} (-q_{\alpha}(x,D) f)(x). \end{equation*} As an immediate consequence, we obtain a representation formula for infinitesimal generators of sublinear Markov semigroups with a sufficiently rich domain. We give applications in the theory of non-linear Hamilton-Jacobi-Bellman equations and Levy processes for sublinear expectations.

Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $\mathbb {L}^{1}$

It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in $L^p$ for $p>1$. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in $\mathbb{R}^n$, as well as for the $M/M/\infty$ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.

Uniform and Completely Nonequilibrium Invariant States for Weak Coupling Limit Type Quantum Markov Semigroups Associated with Eulerian Cycles

We define the uniform and completely nonequilibrium invariant states, which are associated with Eulerian cycles; once we did this, we use the Hierholzer’s algorithm to obtain a canonical Euler-Hierholzer cycle, and for it, characterize the invariant state. For the simplest case of nonequilibrium, we give sufficient conditions for these states to be invariant and write its eigenvalues explicitly.

Solution semiflow to the compressible Euler equations with damping

This paper is concerned with the 3D isentropic compressible Euler equations with damping in periodic domain. Inspired by the theory of Markov semigroups, we show the existence of solution semiflow which satisfies the standard semigroup property and minimizes the energy (maximizes the energy dissipation) among all dissipative solutions.

RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

AbstractOne of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups.We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient-${\mathcal {S}}_2$condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

Heat kernel upper bounds for symmetric Markov semigroups

It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of [6] for Nash-type inequalities with power order considerably and also extend that of [26] for second order differential operators on a complete non-compact manifold.

Risk‐neutral pricing techniques and examples

AbstractTractability and flexibility are among the two most attractive features of models in mathematical finance. For the pricing of derivative products, to avoid arbitrage opportunities, the fundamental theorem of asset pricing requires the existence of an equivalent martingale measure under which the discounted price process is a (local) martingale. In the semimartingale setting, the traditional approach of risk‐neutral valuation uses a change of measure invoking Girsanov's theorem. This approach often destroys the tractability of the process which is undesirable for quantitative finance applications. To overcome these limitations, we employ a transformation based on the concept of intertwining relationships that allows us to convert Markovian semigroups into a pricing semigroup while keeping its tractability. In order to illustrate the usefulness of this approach, we apply it to exponential Lévy, positive self‐similar, and generalized CIR processes that all fall within the class of polynomial processes introduced by Cuchiero et al. (2012). Furthermore, for the jump CIR class, relying on recent work of Patie and Savov (2019), we provide an eigenvalues expansion of the pricing semigroup and carry out some numerical experiments.