Transition Semigroup Research Articles

Approximation of a semigroup model of anomalous diffusion in a bounded set

The convergence is established for a sequence of operator semigroups, where the limiting object is the transition semigroup for a reflected stable processes. For semilinear equations involving the generators of these transition semigroups, an approximation method is developed as well. This makes it possible to derive an a priori bound for solutions to these equations, and therefore prove global existence of solutions. An application to epidemiology is also given.

On Wiener-Poisson type multivalued stochastic differential equations with non-Lipschitz coefficients

In this paper, we prove local uniqueness for multivalued stochastic differential equations with Poisson jumps. Then existence and uniqueness of global solutions is obtained under the conditions that the coefficients satisfy locally Lipschitz continuity and one-sided linear growth of b. Moreover, we also prove the Markov property of the solution and the existence of invariant measures for the corresponding transition semigroup.

TIME‐CHANGED ORNSTEIN–UHLENBECK PROCESSES AND THEIR APPLICATIONS IN COMMODITY DERIVATIVE MODELS

This paper studies subordinate Ornstein–Uhlenbeck (OU) processes, i.e., OU diffusions time changed by Lévy subordinators. We construct their sample path decomposition, show that they possess mean‐reverting jumps, study their equivalent measure transformations, and the spectral representation of their transition semigroups in terms of Hermite expansions. As an application, we propose a new class of commodity models with mean‐reverting jumps based on subordinate OU processes. Further time changing by the integral of a Cox–Ingersoll–Ross process plus a deterministic function of time, we induce stochastic volatility and time inhomogeneity, such as seasonality, in the models. We obtain analytical solutions for commodity futures options in terms of Hermite expansions. The models are consistent with the initial futures curve, exhibit Samuelson's maturity effect, and are flexible enough to capture a variety of implied volatility smile patterns observed in commodities futures options.

Coupling property and gradient estimates of Lévy processes via the symbol

We derive explicitly the coupling property for the transition semigroup of a L\'{e}vy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively. Our results can be applied to a large class of L\'{e}vy processes, including stable L\'{e}vy processes, layered stable processes, tempered stable processes and relativistic stable processes.

Norm Transitive Semigroups on Operator Algebra

Let \(\mathcal A \) be a semigroup of bounded linear operators on the Banach algebra \(B(X)\) for a separable Banach space \(X\). We show the transitivity of \(\mathcal A \) with the operator norm topology, implies hypercyclicity with the strong operator topology (SOT) while the converse may not be true. As a consequence, SOT-transitive semigroup of left multiplication operators on \(B(X)\) is characterized.

Perturbation of strong Feller semigroups and well-posedness of semilinear stochastic equations on Banach spaces

We prove a Miyadera–Voigt type perturbation theorem for strong Feller semigroups. Using this result, we prove well-posedness of the semilinear stochastic equation on a separable Banach space E, assuming that F is bounded and measurable and that the associated linear equation, i.e. the equation with , is well-posed and its transition semigroup is strongly Feller and satisfies an appropriate gradient estimate. We also study existence and uniqueness of invariant measures for the associated transition semigroup.

Exponential Ergodicity for SDEs with Jumps and Non-Lipschitz Coefficients

In this paper we show irreducibility and the strong Feller property for transition probabilities of stochastic differential equations with jumps and monotone coefficients. Thus, exponential ergodicity and the spectral gap for the corresponding transition semigroups are obtained.

Kolmogorov equation associated to a stochastic Kuramoto–Sivashinsky equation

This paper investigates the relation between the Kolmogorov operator associated to a stochastic Kuramoto–Sivashinsky equation and the infinitesimal generator for the corresponding transition semigroup. We prove that the infinitesimal generator is the closure of Kolmogorov operator in the space of continuous functions with kth-polynomial growth with respect to π-convergence topology and the space L2(H,ν) respectively. The proof depends on various estimates on the solution, invariant measure and transition semigroup. As a product, we also obtains smoothing properties of the transition semigroup.

Limit theorems for stationary Markov processes with L2-spectral gap

Let $(X_t, Y_t)_{t\in T}$ be a discrete or continuous-time Markov process with state space $X \times R^d$ where $X$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_t, Y_t)_{t\in T}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_t)_{t\in T}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $(Y_t)_{t\in T}$ is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have sup{t\in(0,1]\cap T : E{\pi,0}[|Y_t| ^{\alpha}] < 1 with the expected order with respect to the independent case (up to some $\varepsilon > 0$ for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $(X_t)_{t\in T}$ has an invariant probability distribution $\pi$, is stationary and has the $L^2(\pi)$-spectral gap property (that is, $(X_t)t\in N}$ is $\rho$-mixing in the discrete-time case). The case where $(X_t)_{t\in T}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with $\rho$-mixing Markov chains.

ON THE TRANSITION SEMIGROUPS OF CENTRALLY LABELED RAUZY GRAPHS

Rauzy graphs of subshifts are endowed with an automaton structure. For Sturmian subshifts, it is shown that its transition semigroup is the syntactic semigroup of the language recognized by the automaton. An inverse limit of the partial semigroups of nonzero regular elements of their transition semigroups is described. If the subshift is minimal, then this inverse limit is isomorphic, as a partial semigroup, to the [Formula: see text]-class associated to it in the free pro-aperiodic semigroup.

The Stochastic Reflection Problem in Hilbert Spaces

We construct a Markov process X associated with the stochastic reflection problem on a closed convex subset with non empty interior and smooth boundary in a Hilbert space, as a solution to a random convex control problem. The transition semigroup corresponding to X is exactly that defined by the Kolmogorov equation with Neumann homogeneous boundary conditions (see [3]).

Bounded Variation with Respect to a Log-Concave Measure

Let H be a separable Hilbert space and let A: 𝒟(A) ⊂ H → H be a self-adjoint operator with A ≤ ωI, ω > 0 and Tr(−A −1) < ∞. We endow H with the centered Gaussian measure μ with covariance operator and consider a function U ∈ C 3(H) with bounded second and third order derivatives, the SDE dX = (AX − DU(X))dt + dW(t), X(0) = x and the associated transition semigroup P t . We define the class BV(H, γ) of bounded variation functions with respect to the probability measure γ(dx) = Z −1 e −2U(x)μ(dx), where Z is the normalization constant, through an integration by parts formula and prove that P t u ∈ W 1, 1(H, γ) for t > 0, u ∈ BV(H, γ), and that u ∈ BV(H, γ) if and only if the limit of ‖DP t u‖ L 1(H, γ) as t → 0 is finite.

Interacting superprocesses with discontinuous spatial motion

A class of interacting superprocesses arising from branching particle systems with continuous spatial motions, called superprocesses with dependent spatial motion (SDSMs), has been introduced and studied by Wang and by Dawson, Li and Wang. In this paper, we extend the model to allow discontinuous spatial motions. Under Lipschitz condition for coefficients, we show that under a proper rescaling, branching particle systems with jump-diffusion underlying motions in a random medium converge to a measure-valued process, called stable SDSM. We further characterize this stable SDSM as a unique solution of a well-posed martingale problem. To prove the uniqueness of the martingale problem, we establish the C2+γ-regularity for the transition semigroup of a class of jump-diffusion processes, which may be of independent interest.

On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes

Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein–Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein–Uhlenbeck processes. Then we present the explicit coupling property of Ornstein–Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein–Uhlenbeck processes via the symbol.

Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations

We consider some reaction–diffusion equations perturbed by white noise and prove Schauder estimates for the elliptic problem associated with the generator of the corresponding transition semigroup, defined in the Banach space of continuous functions. This requires the proof of some new interpolation result.

Coupling for Ornstein–Uhlenbeck processes with jumps

Consider the linear stochastic differential equation (SDE) on $\mathbb{R}^n$: \[\mathrm {d}{X}_t=AX_t\,\mathrm{d}t+B\,\mathrm{d}L_t,\] where $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ real matrix and $L_t$ is a L\'{e}vy process with L\'{e}vy measure $\nu$ on $\mathbb{R}^d$. Assume that $\nu(\mathrm {d}{z})\ge \rho_0(z)\,\mathrm{d}z$ for some $\rho_0\ge 0$. If $A\le 0,\operatorname {Rank}(B)=n$ and $\int_{\{|z-z_0|\le\varepsilon\}}\rho_0(z)^{-1}\,\mathrm{d}z<\infty$ holds for some $z_0\in \mathbb{R}^d$ and some $\varepsilon>0$, then the associated Markov transition probability $P_t(x,\mathrm {d}{y})$ satisfies \[\|P_t(x,\cdot)-P_t(y,\cdot)\|_{\mathrm{var}}\le \frac{C(1+|x-y|)}{\sqrt{t}}, x,y\in \mathbb{R}^d,t>0,\] for some constant $C>0$, which is sharp for large $t$ and implies that the process has successful couplings. The Harnack inequality, ultracontractivity and the strong Feller property are also investigated for the (conditional) transition semigroup.

On the coupling property of Lévy processes

We give necessary and sufficient conditions guaranteeing that the coupling for L\'evy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall-Rogers on couplings of random walks. In particular, we obtain that a L\'{e}vy process admits a successful coupling, if it is a strong Feller process or if the L\'evy (jump) measure has an absolutely continuous component.

Lp‐uniqueness of Kolmogorov operators associated with 2D‐stochastic Navier‐Stokes‐Coriolis equations

AbstractWe prove that the Kolmogorov operator associated with stochastic Navier‐Stokes‐Coriolis equations on the 2D‐Torus is m‐dissipative in the space Lp(μ) for any p ∈ [1, ∞[, where μ is an infinitesimally invariant measure. The proof is based on exponential moment estimates on μ related to the enstrophy and pointwise gradient estimates on the transition semigroup induced by finite dimensional approximations.

Evolution systems of measures for stochastic flows

A new concept of an evolution system of measures for stochastic flows is considered. It corresponds to the notion of an invariant measure for random dynamical systems (or cocycles). The existence of evolution systems of measures for asymptotically compact stochastic flows is obtained. For a white noise stochastic flow, there exists a one to one correspondence between evolution systems of measures for a stochastic flow and evolution systems of measures for the associated Markov transition semigroup. As an application, an alternative approach for evolution systems of measures of 2D stochastic Navier–Stokes equations with a time-periodic forcing term is presented.

EXPONENTIAL MIXING FOR SOME SPDEs WITH LÉVY NOISE

We show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in infinite dimensions. We concentrate on semilinear systems driven by cylindrical α-stable noises introduced in [18], α ∈ (0,2). We first prove that if the nonlinearity is bounded, then the system is ergodic and strong mixing. Then we show that the system is exponentially mixing provided that the nonlinearity, or its Lipschitz constant, are sufficiently small.