Related papers: Quantitative heat kernel estimates for diffusions …

We use a Harnack-type inequality on exit times and spectral bounds to characterize upper bounds of the heat kernel associated with any regular Dirichlet form without killing part, where the scale function may vary with position. We further…

Let $d\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sqrt{2}{\rm d} W_t, $$ where $b$ belongs to the space ${\mathbb…

We give a direct proof of the sharp two-sided estimates, recently established in [4,9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1, 1}$ open sets by using Duhamel formula. We also obtain a…

The presence of fluctuations and non-linear interactions can lead to scale dependence in the parameters appearing in stochastic differential equations. Stochastic dynamics can be formulated in terms of functional integrals. In this paper we…

We establish uniform pointwise estimates for the densities of a family of $\alpha$-stable processes with respect to the index $\alpha \in [\alpha_0,2]$ for some $\alpha_0>0$. In addition, we estimate the difference between the heat kernels…

We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter H (1/3 < H <= 1/2) under the ellipticity assumption at the starting point. In such a case, the law…

We introduce a fractional stochastic heat equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize…

We consider random walks in a balanced i.i.d. random environment in $Z^d$ for $d\ge2$ and the corresponding discrete non-divergence form difference operators. We first obtain an exponential integrability of the heat kernel bounds. We then…

We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, $F(t)$, has a nontrivial quartic variation. This process,…

We study boundary values of holomorphic functions in translation-invariant distribution spaces of type $\mathcal{D}'_{E'_{\ast}}$. New edge of the wedge theorems are obtained. The results are then applied to represent…

We consider the stochastic heat equation with multiplicative noise $u_t={1/2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where $\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise…

We study the numerical approximation of the stochastic heat equation with a distributional reaction term. Under a condition on the Besov regularity of the reaction term, it was proven recently that a strong solution exists and is unique in…

Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter H in the interval (1/2, 1), is investigated. An averaged SDE for the original SDE is proposed, and their…

We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. We establish existence of martingale solutions which are strong in the PDE sense and study their large-time behavior. Our…

This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the L\'{e}vy-type…

Let $(W,H,\mu)$ be the classical Wiener space on $\R^d$. Assume that $X=(X_t)$ is a diffusion process satisfying the stochastic differential equation $dX_t=\sigma(t,X)dB_t+b(t,X)dt$, where $\sigma:[0,1]\times C([0,1],\R^n)\to \R^n\otimes…

By using Duhamel's formula, we prove sharp two-sided estimates for the heat kernel of spectral fractional Laplacian with time-dependent gradient perturbation in bounded $C^{1,1}$ domains. Moreover, we also obtain gradient estimate as well…

We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system…

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given $\sigma$-field $\mathcal{Q}$. In our framework, we recall well-known results about Markov--Wiener diffusions. We…

The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop…