One-parameter convolution semigroups of rapidly decreasing distributions

The Petrovskii correctness and semigroups of operators

Let $P(\partial/\partial x)$ be an $m\times n$ matrix whose entries are PDO on $\bbR^n$ with constant coefficients, and let $\calS(\bbR^n)$ be the space of infinitely differentiable rapidly decreasing functions on $\bbR^n$. It is proved that $P(\partial/\partial x)|_{(\calS(\bbR^n))^m}$ is the infinitesimal generator of a $(C_0)$-semigroup $(S_t)_{t\ge0}\subset L((\calS(\bbR^n))^m)$ if and only if $P(\partial/\partial x)$ satisfies the Petrovski\uıcorrectness condition. Moreover, if it is the case, then $(S_t)_{t\ge0}$ is an exponential semigroup whose characteristic exponent is equal to the s…

随机卷积连续性的免费午餐不存在

No free lunch for continuity of stochastic convolutions

本文构造了指数稳定的 $C_0$-半群，使得随机卷积在正概率下无界，并利用必要条件与 Schauder 乘子给出反例，同时在更强条件下建立了连续性与极大估计。

Spatial asymptotics at infinity for heat kernels of integro-differential operators

We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits $$ \lim_{r \to \infty} \frac{p_t(rθ-y)}{t \, ν(rθ)}, \quad t \in T, \ \ θ\in E, \ \ y \in \mathbb R^d, $$ exist and can be effectively computed. Here $ν$ is the corresponding Lévy density, $T \subset (0,\infty)$ is a bounded time-set and $E$ is a subset of the unit sphere in $\mathbb R^d$, $d \geq 1$. Our results are local on the unit sphere. They apply to a wide class of convolution …

Linear and nonlinear convolution operator equations on the infinite strip

In the present paper we characterize the existence and uniqueness of maximal Lp-regular solutions of high order convolution operator equations. Particularly, we get coercive uniform estimates with respect to spectral parameter and we show that corresponding realization operator is R-positive and generates analytic semigroup in Lp. Then we apply these results to various problems of nonlinear integro-differential equations.

Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case

We introduce the notion of a family of convolution operators associated with a given elliptic partial differential operator. Such a convolution structure is shown to exist for a general class of Laplace-Beltrami operators on two-dimensional manifolds endowed with cone-like metrics. This structure gives rise to a convolution semigroup representation for the Markovian semigroup generated by the Laplace-Beltrami operator. In the particular case of the operator $\mathcal{L} = \partial_x^2 + {1 \over 2x} \partial_x + {1 \over x} \partial_θ^2$ on $\mathbb{R}^+ \times \mathbb{T}$, we deduce the exist…

在非局部设置中对地面态变换半群的$ L^1 $平滑性研究

Smoothing on $L^1$ for ground state transformed semigroups in non-local settings

本文研究了非局部设置中地面态变换半群的$ L^1 $平滑性，揭示了其在奥里奇空间中的作用，并通过实例展示了正则化效应随时间增强的特性。