Diffusion Semigroups Research Articles

A Bakry–Émery criterion for weighted contractivity and L2-Hardy inequalities

We show a symmetric Markov diffusion semigroup satisfies a weighted contractivity condition if and only if a [Formula: see text]-Hardy inequality holds, and we give a Bakry–Émery-type criterion for the former. We then give some applications.

Quantitative estimates for bounded holomorphic semigroups

We revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood–Paley–Stein theory for symmetric diffusion semigroups.

A weak approximation for Bismut’s formula: An algorithmic differentiation method

The paper provides a novel algorithmic differentiation method by constructing a weak approximation for Bismut’s formula. A new operator splitting method based on Gaussian Kusuoka-approximation is introduced for an enlarged semigroup describing “differentiation of diffusion semigroup”. The effectiveness of the new algorithmic differentiation is checked through numerical examples.

A rational preconditioner for multi-dimensional Riesz fractional diffusion equations

We propose a rational preconditioner for an efficient numerical solution of linear systems arising from the discretization of multi-dimensional Riesz fractional diffusion equations. In particular, the discrete problem is obtained by employing finite difference or finite element methods to approximate the fractional derivatives of order α with α∈(1,2]. The proposed preconditioner is then defined as a rational approximation of the Riesz operator expressed as the integral of the standard heat diffusion semigroup. We show that, being the sum of k inverses of shifted Laplacian matrices, the resulting preconditioner belongs to the generalized locally Toeplitz class. As a consequence, we are able to provide the asymptotic description of the spectrum of the preconditioned matrices and we show that, despite the lack of clustering just as for the Laplacian, our preconditioner for α close to 1 and k≠1 reasonably small, provides better results than the Laplacian itself, while sharing the same computational complexity.

Stationary and oscillatory patterns of a food chain model with diffusion and predator‐taxis

In this paper, we investigate pattern dynamics in a reaction‐diffusion‐chemotaxis food chain model with predator‐taxis, which extends previous studies of reaction‐diffusion food chain model. By virtue of diffusion semigroup theory, we first prove global classical solvability and boundedness for the considered model over a bounded domain with smooth boundary for arbitrary predator‐taxis sensitivity coefficient. Then the linear stability analysis for the considered model shows that chemotaxis can induce the losing of stability of the unique positive spatially homogeneous steady state via Turing bifurcation and Turing‐spatiotemporal Hopf bifurcation. These bifurcations results in the formation of two kinds of important spatiotemporal patterns: stationary Turing pattern and oscillatory pattern. Simultaneously, the threshold values for Turing bifurcation and Turing‐spatiotemporal Hopf bifurcation are given explicitly. Finally, numerical simulations are performed to illustrate and support our theoretical findings, and some interesting non‐Turing patterns are found in temporal Hopf parameter space by numerical simulation.

Geometric approximations to transition densities of Jump-type Markov processes

Abstract This paper is concerned with the transition functions of symmetric Levy-type processes generated by a pseudo-differential operator with variable coefficients. We first give the general estimates of heat kernels of jump diffusion semigroups, which leads to diagonal estimates of transition function and subordination in the context of two-dimensional Cauchy semigroup. Then off-diagonal estimates of special classes of Levy-type processes where transition function can be expressed using the diagonal estimation results and related metrics are derived. Furthermore, we show geometric approximation of the general two-dimensional Levy processes, and graphical experiments have been made by freezing the coefficients of the generators.

AN EXPLICIT CHARACTERIZATION OF THE DOMAIN OF THE INFINITESIMAL GENERATOR OF A SYMMETRIC DIFFUSION SEMIGROUP ON LP OF A COMPLETE POSITIVE SIGMA-FINITE MEASURE SPACE

Let X be a complete positive σ–finite measure space and {At}t≥0 be a symmetric diffusion semigroup of contraction operators on Lp(X). We prove that for 1<p<∞, the domain of the infinitesimal generator of the semigroup is precisely the space ∫01Ash ds:h∈Lp(X). We also establish that for 1<p<∞, the function spaces 2n-1∫02-(n-1)Ash ds|h∈Lp(X) are equal for every n∈ℤ.

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.

A unified construction of product formulas and convolutions for Sturm–Liouville operators

We establish a positive product formula for the solutions of the Sturm–Liouville equation $$\ell (u) = \lambda u$$ , where $$\ell $$ belongs to a general class which includes singular and degenerate Sturm–Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm–Liouville operator. We show that the product formula gives rise to a probability preserving convolution algebra structure on the space of finite measures which satisfies the basic axioms for developing harmonic analysis on the convolution algebra. Unlike previous works, our framework includes a subfamily of Sturm–Liouville operators for which the support of the convolution of Dirac measures is noncompact. The connection with hypergroup theory is discussed. Convolution-type integral equations on weighted Lebesgue spaces are also discussed, and a solvability condition is established.

Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups

We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires (i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and (ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both (i) and (ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, (i) and (ii). Conditions for (ii) to hold are studied in the literature. Here we produce sufficient conditions for (i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.

Operator splitting around Euler–Maruyama scheme and high order discretization of heat kernels

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker–Campbell–Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler–Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials

<p style='text-indent:20px;'>Some weighted inequalities for the maximal operator with respect to the discrete diffusion semigroups associated with exceptional Jacobi and Jacobi-Dunkl polynomials are given. This setup allows to extend the corresponding results obtained for discrete heat semigroup recently to richer class of differential-difference operators.

An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family

Let X be a topological space equipped with a complete positive σ -finite measure and T a subset of the reals with 0 as an accumulation point. Let a t x , y be a nonnegative measurable function on X × X which integrates to 1 in each variable. For a function f ∈ L 2 X and t ∈ T , define A t f x ≡ ∫ a t x , y f y d y . We assume that A t f converges to f in L 2 , as t ⟶ 0 in T . For example, A t is a diffusion semigroup (with T = 0 , ∞ ). For W a finite measure space and w ∈ W , select real-valued h w ∈ L 2 X , defined everywhere, with h w L 2 X ≤ 1 . Define the distance D by D x , y ≡ h w x − h w y L 2 W . Our main result is an equivalence between the smoothness of an L 2 X function f (as measured by an L 2 -Lipschitz condition involving a t · , · and the distance D ) and the rate of convergence of A t f to f .

Asymptotic behaviour of fast diffusions on graphs

We study a diffusion process on a finite graph with semipermeable membranes on vertices. We prove, in L^1 and L^2-type spaces that for a large class of boundary conditions, describing communication between the edges of the graph, the process is governed by a strongly continuous semigroup of operators, and we describe asymptotic behaviour of the diffusion semigroup as the diffusions’ speed increases at the same rate as the membranes’ permeability decreases. Such a process, in which communication is based on the Fick law, was studied by Bobrowski (Ann. Henri Poincaré 13(6):1501–1510, 2012) in the space of continuous functions on the graph. His results were generalized by Banasiak et al. (Semigroup Forum 93(3):427–443, 2016). We improve, in a way that cannot be obtained using a very general tool developed recently by Engel and Kramar Fijavž (Evolut. Equ. Control Theory 8(3)3:633–661, 2019), the results of J. Banasiak et al.

Central Limit Theorems for Non-Symmetric Random Walks on Nilpotent Covering Graphs: Part II

In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup is generated by the homogenized sub-Laplacian with a constant drift of the asymptotic direction on the nilpotent Lie group, which is equipped with the Albanese metric associated with the symmetrized random walk. We next prove a functional CLT (i.e., Donsker-type invariance principle) in a Holder space over the nilpotent Lie group by combining the semigroup CLT, standard martingale techniques, and a novel pathwise argument inspired by rough path theory. Applying the corrector method, we finally extend these CLTs to the case where the realizations are not necessarily harmonic.

Some Extensions of E. Stein’s Work on Littlewood–Paley Theory Applied to Symmetric Diffusion Semigroups

Discrete diffusion semigroups have proven to be highly effective tools for machine learning and data analysis due to the interplay between diffusion processes (of inferences), and their naturally associated geometries. Inspired by the harmonic analysis machinery that E. Stein developed for symmetric diffusion semigroups acting on $$L_p$$ spaces, we show that a correspondence between the rate of diffusion approximation and a Besov-type version of smoothness exists in the general continuous case, even without a local kernel representation. Specifically, let $$\left\{ A_t\right\} _{t\ge 0}$$ be a symmetric diffusion semigroup on $$L_p(X)$$, for X a complete positive $$\sigma $$-finite measure space. We first establish that for $$1<p<\infty $$ the semigroup $$\left\{ A_t\right\} _{t\ge 0}$$ acting on $$L_p(X)$$ is strongly continuous and holomorphic. We then show that for $$f\in L_p(X)$$ and $$0<\alpha <1$$, the following are equivalent: We also present some extensions, including results for the $$p=\infty $$ case.

Gradient estimates for nonlinear diffusion semigroups by coupling methods

In this paper, we obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First, we derive uniform gradient estimates for certain semi-linear PDEs based on the coupling method introduced by Wang in 2011 and the theory of backward SDEs. Then we generalize Wang's coupling to the G-expectation space and obtain gradient estimates for nonlinear diffusion semigroups, which correspond to the solutions of certain fully nonlinear PDEs.

Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Let $P_t$ be the (Neumann) diffusion semigroup $P_t$ generated by a weighted Laplacian on a complete connected Riemannian manifold $M$ without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant $\ll>0$ if and only if $$W_p(\mu_1P_t, \mu_2P_t)\le \e^{-\ll t} W_p (\mu_1,\mu_2),\ \ t\ge 0, p\ge 1 $$ holds for all probability measures $\mu_1$ and $\mu_2$ on $M$, where $W_p$ is the $L^p$ Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction $$W_p(\mu_1P_t, \mu_2P_t)\le c\e^{-\ll t} W_p (\mu_1,\mu_2),\ \ p\ge 1, t\ge 0$$ for some constants $c,\ll>0$ for a class of diffusion semigroups with negative curvature where the constant $c$ is essentially larger than $1$. Similar results are derived for SDEs with multiplicative noise by using explicit conditions on the coefficients, which are new even for SDEs with additive noise.

Gradient bounds for Kolmogorov type diffusions

Nous étudions des bornes de gradient et autres inégalités fonctionnelles pour des semigroupes de diffusion engendrés par des opérateurs de type Kolmogorov. Nous nous concentrons sur deux méthodes : couplage et $\Gamma$-calculus. Les avantages et inconvénients de ces méthodes sont discutés.

Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

<p style="text-indent:20px;">The paper is devoted to analysis of <em>far-from-equilibrium</em> pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular processes and diffusing growth factors. Pattern formation results from hysteresis in the dependence of the quasi-stationary solution of the ODE on the diffusive component. Bistability alone, without hysteresis, does not result in stable patterns. We provide a systematic description of the hysteresis-driven stationary solutions, which may be monotone, periodic or irregular. We prove existence of infinitely many stationary solutions with jump discontinuity and their asymptotic stability for a certain class of reaction-diffusion-ODE systems. Nonlinear stability is proved using direct estimates of the model nonlinearities and properties of the strongly continuous diffusion semigroup.