Singular perturbations to semilinear stochastic heat equations

Local $L^{p}$-solution for semilinear heat equation with fractional noise

We study the $L^{p}$-solutions for the semilinear heat equation with unbounded coefficients and driven by a infinite dimensional fractional Brownian motion with self-similarity parameter $H > 1/2$. Existence and uniqueness of local mild solutions are showed.

Mixed Boundary Value Problems of Semilinear Elliptic PDEs and BSDEs with Singular Coefficients

In this paper, we prove that there exists a unique weak solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.

关于半线性热方程分支扩散求解器的稳定性分析

Stability analysis of a branching diffusion solver for semilinear heat equations

本文研究了半线性热方程分支扩散求解器的稳定性，通过分析随机分支过程的可积性条件，证明了解的非爆炸性和弱解的唯一性。

On a singular semilinear elliptic boundary value problem and the boundary Harnack principle

We consider the singular boundary-value problem Δu = f(u) in D; u|_dD= phi, where 1. D is a bounded C^2-domain of R^d, d >= 3 2. f: (0,1) -> (0,1) is a locally Hölder continuous function such that f(u) -> 1 as u -> 0 at the rate u^{-α}, for some αin (0,1), 3. and phi is a positive continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in D-bar. Such solutions are shown to satisfy a boundary Harnack principle. Probabilistic techniques are used in proving the main results.

Existence and non-existence results for a class of semilinear nonlocal operators with exterior condition

We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.

Support theorem for a singular semilinear stochastic partial differential equation

We consider the generalized parabolic Anderson equation (gPAM) in 2 dimensions with periodic boundary. This is an example of a singular semilinear stochastic partial differential equations, solutions of which require renormalization and have only be understood recently via Hairer's regularity structures and, in some cases equivalently, paracontrollled distributions due to Gubinelli, Imkeller and Perkowski. In the present paper we describe the law of gPAM, by establishing a Stroock{Varadhan type support theorem in suitable Holder{Besov spaces.