In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation $${{\cal S}^b}: = {\overline {\rm{\Delta }} ^{\alpha /2}} + b \cdot \nabla $$ where $${\overline {\rm{\Delta }} ^{\alpha /2}}$$ is the truncated fractional Laplacian, α ∈ (1, 2) and b ∈ K−1 . In the second part, for a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance ∣x − y∣ in short time.

中文翻译：

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非对称有限范围跳跃过程的热核估计

在本文中，我们首先建立梯度扰动下截断分数拉普拉斯算子的尖锐两侧热核估计和梯度估计$${{\cal S}^b}: = {\overline {\rm{\Delta } } ^{\alpha /2}} + b \cdot \nabla $$ 其中 $${\overline {\rm{\Delta }} ^{\alpha /2}}$$ 是截断的分数拉普拉斯算子，α ∈ ( 1, 2) 和 b ∈ K−1 。在第二部分，对于更一般的有限范围跳跃过程，我们提出了一些充分条件，使热核的两侧估计在短时间内与大距离 ∣x − y∣ 的泊松型函数相当。