Potential Analysis最新文献

In the setting of a metric space equipped with a doubling measure supporting a (1, 1)-Poincaré inequality, we study the problem of minimizing the BV-energy in a bounded domain (Omega ) of functions bounded between two obstacle functions inside (Omega ), and whose trace lies between two prescribed functions on the boundary. If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when (Omega ) is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Malý, Shanmugalingam, and Speight (2019). While such solutions are not unique in general, we show the existence of unique minimal solutions. Since candidate functions need not agree outside of the domain, standard compactness arguments fail to provide existence of weak solutions as they are defined for the problem with single boundary condition. To overcome this, we introduce a class of ( varepsilon )-weak solutions as an intermediate step. Our existence results generalize those of Ziemer and Zumbrun (1999), who studied this problem in the Euclidean setting with a single obstacle and single boundary condition.

We study the long time behavior of the standard linear implicit Euler scheme for the discretization of a class of erdogic parabolic semilinear SPDEs driven by additive space-time white noise. When the nonlinearity is a gradient, the invariant distribution is of Gibbs form, but it cannot be approximated in the total variation sense by the standard Euler scheme. We prove that the numerical scheme gives an approximation in the total variation sense of a modified Gibbs distribution, which is the invariant distribution of a modified SPDE. The modified distribution and the modified equation depend on the time-step size. This original result goes beyond existing results in the literature where the weak error estimates for the approximation of the invariant distribution do not imply convergence in total variation when the time-step size vanishes. The proof of the main result requires regularity properties of associated infinite dimensional Kolmogorov equations.

Abstract

By the probabilistic coupling approach which combines a new refined basic coupling with the synchronous coupling for Lévy processes, we obtain explicit exponential contraction rates in terms of the standard (L^1) -Wasserstein distance for the following Langevin dynamic ((X_t,Y_t)_{tge 0}) of McKean-Vlasov type on (mathbb R^{2d}) : $$begin{aligned} left{ begin{array}{l} dX_t=Y_t,dt, dY_t=left( b(X_t)+displaystyle int _{mathbb R^d}tilde{b}(X_t,z),mu ^X_t(dz)-{gamma }Y_tright) ,dt+dL_t,quad mu ^X_t=textrm{Law}(X_t), end{array} right. end{aligned}$$ where ({gamma }>0) , (b:mathbb R^drightarrow mathbb R^d) and (tilde{b}:mathbb R^{2d}rightarrow mathbb R^d) are two globally Lipschitz continuous functions, and ((L_t)_{tge 0}) is an (mathbb R^d) -valued pure jump Lévy process. The proof is also based on a novel distance function, which is designed according to the distance of the marginals associated with the constructed coupling process. Furthermore, by applying the coupling technique above with some modifications, we also provide the propagation of chaos uniformly in time for the corresponding mean-field interacting particle systems with Lévy noises in the standard (L^1) -Wasserstein distance as well as with explicit bounds.

The parabolic integro-differential Cauchy problem with spatially dependent coefficients is considered in generalized Bessel potential spaces where smoothness is defined by Lévy measures with O-regularly varying profile. The coefficients are assumed to be bounded and Hölder continuous in the spatial variable. Our results can cover interesting classes of Lévy measures that go beyond those comparable to (dy/left| yright| ^{d+alpha }.)

In this paper, we are concerned with stochastic susceptible-exposed-infected-removed epidemics on complete graphs with vertex-dependent transition rates. Large and moderate deviations of empirical density fields of our models are given. Proofs of our main results utilize exponential martingale strategies. In the proof of the moderate deviation principle, we introduce an iteration approach to check the exponential tightness of scaled density fields of our processes. As an application of our main results, moderate deviations of a family of hitting times of our processes are also given.

We characterize weakly harmonic maps with respect to non-local Dirichlet forms by Markov processes and martingales. In particular, we can obtain discontinuous martingales on Riemannian manifolds from the image of symmetric stable processes under fractional harmonic maps in a weak sense. Based on this characterization, we also consider the continuity of weakly harmonic maps along the paths of Markov processes and describe the condition for the continuity of harmonic maps by quadratic variations of martingales in some situations containing cases of energy minimizing maps.

In this paper, we prove that stochastic porous media equations over (sigma )-finite measure spaces ((E,mathcal {B},mu )), driven by time-dependent multiplicative noise, with the Laplacian replaced by a self-adjoint transient Dirichlet operator L and the diffusivity function given by a maximal monotone multi-valued function (Psi ) of polynomial growth, have a unique solution. This generalizes previous results in that we work on general measurable state spaces, allow non-continuous monotone functions (Psi ), for which, no further assumptions (as e.g. coercivity) are needed, but only that their multi-valued extensions are maximal monotone and of at most polynomial growth. Furthermore, an (L^p(mu ))-Itô formula in expectation is proved, which is not only crucial for the proof of our main result, but also of independent interest. The result in particular applies to fast diffusion stochastic porous media equations (in particular self-organized criticality models) and cases where E is a manifold or a fractal, and to non-local operators L, as e.g. (L=-f(-Delta )), where f is a Bernstein function.

A formula which expresses logarithmic energy of Borel measures on (mathbb {R}^n) in terms of the Fourier transforms of the measures is established and some applications are given. In addition, using similar techniques a (known) formula for Riesz energy is reinvented.

We study properties of the boundary trace operator on the Sobolev space (W^1_1(Omega )). Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator (Tr: W^1_1(Omega _K)rightarrow X(Omega _K)), where (Omega _K) is von Koch’s snowflake and (X(Omega _K)) is a trace space with the quotient norm. Since (Omega _K) is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr, i.e. a linear operator (S: X(Omega _K) rightarrow W^1_1(Omega _K)) such that (Tr circ S= Id_{X(Omega _K)}). In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch’s snowflake. Moreover we identify the isomorphism class of the trace space as (ell _1). As an additional consequence of our approach we obtain a simple proof of the Peetre’s theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain (Omega ) with regular boundary, which explains Banach space geometry cause for this phenomenon.