We prove the large deviation principle for the law of the one-dimensional semi-linear stochastic partial differential equations driven by a nonlinear multiplicative noise. Firstly, combining the energy estimate and approximation procedure, we obtain the existence of the global solution. Secondly, the large deviation principle is obtained via the weak convergence method.

Understanding treatment heterogeneity plays a key role in many contemporary applications arising from different areas. Although there is a growing literature on subgroup analysis based on the heterogeneous univariate regression model, little work has been done for the heterogeneous multi-response regression model. In contrast to the existing methods which are based on subject-specific intercepts, we introduce a heterogeneous multi-response regression method which allows the coefficients for treatment variables to be subject-dependent with unknown grouping information, thus applicable to a wider range of situations. Moreover, we provide an efficient algorithm based on concave pairwise fusion penalization and establish the oracle property of the proposed estimator. The effectiveness of the suggested method is demonstrated through simulation examples and an empirical study.

Jie Wu and Wenjie Gao are co-first authors.

The online version contains supplementary material available at https://doi.org/10.1007/s40304-023-00342-w.

In this paper, we first study the complete convergence for arrays of rowwise widely orthant dependent random variables under sub-linear expectations. The complete convergence theorems are established in sense of sub-additive capacities under some mild conditions. As an application of the main results, we investigate the strong consistency for the weighted estimator in a nonparametric regression model based on widely orthant dependent errors under sub-linear expectations. In addition, we also obtain the rate of strong consistency for the estimator in a nonparametric regression model based on widely orthant dependent errors under sub-linear expectations.

By using the notion of Fischer–Marsden equation on real hypersurfaces in the complex hyperbolic quadric ${{Q^m}^*} = {{\text {SO}}^o_{2,m}/{\text {SO}}_2 {\text {SO}}_m}$, we can assert that there does not exist a non-trivial solution $(g,{\nu })$ of Fischer–Marsden equation on real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadric ${Q^m}^*$. Next, as an application we also show that there does not exist a non-trivial solution $(g,{\nu })$ of the Fischer–Marsden equation on contact real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*$. Consequently, the Fischer–Marsden conjecture is true on these two kinds of real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*$.

A new version of the cluster expansion is proposed without breaking the translation and rotation invariance. As an application of this technique, we construct the connected Schwinger functions of the regularized $\phi ^4$ theory in a continuous way.

This paper is concerned with the linear-quadratic social optima for a class of N weakly coupled backward system with partial information structure. The system dynamics are governed by linear backward stochastic differential equations, and the objective is to minimize a social cost. The stochastic filtering Hamiltonian system is obtained from variational analysis. By virtue of the stochastic filtering technique and backward decoupling method, the feedback form of optimal control is derived. Aiming to overcome the curse of dimensionality and reduce the information requirements, we design a set of decentralized control laws, which is further shown to be asymptotic. Finally, an example of the scalar-valued case is studied.

Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, the canonical Euler Maruyama discretization of the Langevin diffusion process, referred as unadjusted Langevin algorithm (ULA), studied mostly in the context of smooth (gradient Lipschitz) and strongly log-concave densities, is a considerable hindrance for its deployment in many sciences, including statistics and machine learning. In this paper, we establish several theoretical contributions to the literature on such sampling methods for non-convex distributions. Particularly, we introduce a new mixture weakly smooth condition, under which we prove that ULA will converge with additional log-Sobolev inequality. We also show that ULA for smoothing potential will converge in $L_{2}$-Wasserstein distance. Moreover, using convexification of nonconvex domain (Ma et al. in Proc Natl Acad Sci 116(42):20881–20885, 2019) in combination with regularization, we establish the convergence in Kullback–Leibler divergence with the number of iterations to reach $\epsilon $-neighborhood of a target distribution in only polynomial dependence on the dimension. We relax the conditions of Vempala and Wibisono (Advances in Neural Information Processing Systems, 2019) and prove convergence guarantees under isoperimetry, and non-strongly convex at infinity.

This paper presents a characterization of equilibrium in a game theoretic description of discounting stochastic consumption, investment and reinsurance problem, in which the controlled state process evolves according to a multi-dimensional linear stochastic differential equation, when the noise is driven by a Brownian motion under the effect of a Markovian regime switching. The running and the terminal costs in the objective functional, are explicitly depended on some general discount functions, which create the time inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and sufficient equilibrium conditions as well as a verification result. A state feedback equilibrium strategy is achieved via certain partial differential-difference equation. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for some particular cases of power and logarithmic utility functions. A numerical example is provided to demonstrate the efficacy of theoretical results.