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In this paper, the stochastic modified Camassa-Holm (MCH) equation is concerned. Firstly, the local well-posedness for this equation is established by the trilinear estimates to the approximate solutions. Then the large deviation principle (LDP) for the regularized stochastic MCH is obtained by the weak convergence approach. To get the LDP for stochastic MCH, some exponentially equivalents of the probability measures are proved. The regularization method plays an crucial role.

For the extended Dirichlet space $\mathcal {F}_{e}$ of a general irreducible recurrent regular Dirichlet form $(\mathcal {E},\mathcal {F})$ on L 2(E;m), we consider the family $\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}$ of centered Gaussian random variables defined on a probability space $({\Omega }, \mathcal {B}, \mathbb {P})$ indexed by the elements of $\mathcal {F}_{e}$ and possessing the Dirichlet form $\mathcal {E}$ as its covariance. We formulate the Markov property of the Gaussian field $\mathbb {G}(\mathcal {E})$ by associating with each set A ⊂ E the sub-σ-field σ(A) of $\mathcal {B}$ generated by X u for every $u\in \mathcal {F}_{e}$ whose spectrum s(u) is contained in A. Under a mild absolute continuity condition on the transition function of the Hunt process associated with $(\mathcal {E}, \mathcal {F})$ , we prove the equivalence of the Markov property of $\mathbb {G}(\mathcal {E})$ and the local property of $(\mathcal {E},\mathcal {F})$ . One of the key ingredients in the proof is in that we construct potentials of finite signed measures of zero total mass and show that, for any Borel set B with m(B) > 0, any function $u\in \mathcal {F}_{e}$ with s(u) ⊂ B can be approximated by a sequence of potentials of measures supported by B.

Let Pt be the (Neumann) diffusion semigroup Pt 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 ≪> 0 if and only if $$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq e^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ t\geq 0, p\geq 1 $$ holds for all probability measures μ1 and μ2 on M, where Wp is the Lp Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction $$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq ce^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ p \geq 1, t\geq 0$$ for some constants c,≪> 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.

In this paper, we obtain some interesting reproducing kernel estimates and some Carleson properties that play an important role. We characterize the bounded and compact Toeplitz operators on the weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, we estimate the essential norm of them assuming that they are bounded.

The Harnack inequality established in Röckner and Wang (J Funct Anal 203:237–261, 2003) for generalized Mehler semigroup is improved and generalized. As applications, the log-Harnack inequality, the strong Feller property, the hyper-bounded property, and some heat kernel inequalities are presented for a class of O-U type semigroups with jump. These inequalities and semigroup properties are indeed equivalent, and thus sharp, for the Gaussian case. As an application of the log-Harnack inequality, the HWI inequality is established for the Gaussian case. Perturbations with linear growth are also investigated.

Le but de cet article est d'estimer la capacité d'un pétit segment [a(1−ε), a(1+ε)]×{z=0} du démi-plan П := {x = (r, z)|r > 0, z ∈ ℝ} de ℝ2 par rapport à la capacité définie par la norme $$\left\| u \right\|_{H_L }^2 = \int_\prod u Lu r dr dz: = \int_\prod \{ u_r^2 + u_z^2 \} r^{ - k} dr dz$$ . Ce résultat est ensuite utilisé pour estimer le diamètre de la surface libre de certains problèmes elliptiques non linéaires.

In this note we show that for f ∈ C((0,∞); R+) ∩ C1 ((0,∞)) with support in [0,∞), if a function u ∈ C1(R2) is such that support (u+) is compact and u(x) = ∫R2 f(u(y)) log 1/(|x-y|)dy ∀ x, then u is radial. This result is important for some free boundary problems in R2 or some axisymmetric ones in Rn.

Let Ω a open subset of ℝ n , n⩾3, and Ω⊂ $$\overline \omega$$ ⊂Ω an open. Existence and unicity are proved for the Dirichlet problem $$\left\{ \begin{gathered} \mathcal{L}u: = - \sum\nolimits_j {\frac{\partial }{{\partial x_j }}} \left( {\sum\nolimits_j {\alpha _{ij} \frac{{\partial u}}{{\partial x_i }} + \delta _j u} } \right) + \mathcal{B}( \cdot ,u,\nabla u) = 0, {\text{in }}\omega ; \hfill \\ u = g, {\text{on }}\partial \omega {\text{.}} \hfill \\ \end{gathered} \right.$$ It is assumed that the linear part of ℒ satisfy the conditions of Hervé, ℬ(·,u,∇u): Ω×ℝ×ℝ n →ℝ satisfy Carathéodory's condition and structure conditions (H1), (H2) and (H3) below. Let H denote the sheaf of L-solutions, we prove that (Ω,H) is a nonlinear Bauer harmonic space.