This paper is a natural continuation of Krylov (2020 and 2021) where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(\mathbb {R}^{d+1})$ and some properties of their Green’s functions and probability of passing through narrow tubes are investigated. On the basis of this here we study some further properties of these processes such as Harnack inequality, Hölder continuity of potentials, Fanghua Lin estimates and so on.

In this paper, we consider the large deviations principles (LDPs) for the stochastic linear Schrödinger equation and its symplectic discretizations. These numerical discretizations are the spatial semi-discretization based on the spectral Galerkin method, and the further full discretizations with symplectic schemes in temporal direction. First, by means of the abstract Gärtner–Ellis theorem, we prove that the observable $B_{T}=\frac {u(T)}{T}$ , T > 0 of the exact solution u is exponentially tight and satisfies an LDP on $L^{2}(0, \pi ; \mathbb C)$ . Then, we present the LDPs for both $\{{B^{M}_{T}}\}_{T>0}$ of the spatial discretization $\{u^{M}\}_{M\in \mathbb N}$ and $\{{B^{M}_{N}}\}_{N\in \mathbb N}$ of the full discretization $\{{u^{M}_{N}}\}_{M,N\in \mathbb N}$ , where ${B^{M}_{T}}=\frac {u^{M}(T)}{T}$ and ${B^{M}_{N}}=\frac {{u^{M}_{N}}}{N\tau }$ are the discrete approximations of BT. Further, we show that both the semi-discretization $\{u^{M}\}_{M\in \mathbb N}$ and the full discretization $\{{u^{M}_{N}}\}_{M,N\in \mathbb N}$ based on temporal symplectic schemes can weakly asymptotically preserve the LDP of {BT}T> 0. These results show the ability of symplectic discretizations to preserve the LDP of the stochastic linear Schrödinger equation, and first provide an effective approach to approximating the large deviations rate function in infinite dimensional space based on the numerical discretizations.

Under suitable conditions on a family (I(t))t≥ 0 of Lipschitz mappings on a complete metric space, we show that, up to a subsequence, the strong limit S(t):=lim _{nto infty }(I(t 2^{-n}))^{2^{n}} exists for all dyadic time points t, and extends to a strongly continuous semigroup (S(t))t≥ 0. The common idea in the present approach is to find conditions on the generating family (I(t))t≥ 0, which can be transferred to the semigroup. The construction relies on the Lipschitz set, which is invariant under iterations and allows to preserve Lipschitz continuity to the limit. Moreover, we provide a verifiable condition which ensures that the infinitesimal generator of the semigroup is given by lim _{hdownarrow 0}tfrac {I(h)x-x}{h}, whenever this limit exists. The results are illustrated with several examples of nonlinear semigroups such as robustifications and perturbations of linear semigroups.

In this paper, we establish small time large deviation principles for the quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone.

We prove Poincaré and Logβ-Sobolev inequalities for a class of probability measures on step-two Carnot groups.

In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk overline {B}(1/(1-k^{4}),k^{2}/(1-k^{4})). This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension 1. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a 1-dimensional subset of a line under a quasiconformal mapping.

We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.

In this article, well-posedness of stochastic anisotropic $p$-Laplace equation driven by L\'evy noise is shown. Such an equation in deterministic setting was considered by Lions [7]. The results obtained in this article can be applied to solve a large class of semilinear and quasilinear stochastic partial differential equations.