We primarily provide several estimates for the heat kernel defined on the 2-dimensional simple random walk. Additionally, we offer an estimate for the heat kernel on high-dimensional random walks, demonstrating that t...We primarily provide several estimates for the heat kernel defined on the 2-dimensional simple random walk. Additionally, we offer an estimate for the heat kernel on high-dimensional random walks, demonstrating that the heat kernel in higher dimensions converges rapidly. We also compute the constants involved in the estimate for the 1-dimensional heat kernel. Furthermore, we discuss the general case of on-diagonal estimates for the heat kernel.展开更多
Let be the simple random walk in zd, and SupPers f(n) is an integer-valued function and increases to infinity as n tends to infinity, and In this paper,a necessary and sufficient condition to ensure or 1 is derived fo...Let be the simple random walk in zd, and SupPers f(n) is an integer-valued function and increases to infinity as n tends to infinity, and In this paper,a necessary and sufficient condition to ensure or 1 is derived for d=3,4. This problem was first studied by P. Erdos and S.J. Taylor.展开更多
In this paper,we prove an explicit formula of the heat kernel on the circle.As a consequence,we establish the monotonicity of the heat kernel.It is well known that the heat kernel can be viewed as the transition proba...In this paper,we prove an explicit formula of the heat kernel on the circle.As a consequence,we establish the monotonicity of the heat kernel.It is well known that the heat kernel can be viewed as the transition probability of random walk on graphs.We also give the definition of the simple lazy random walk on graphs.The transition probabilities of simple lazy random walk on Z and cycle are derived.展开更多
文摘We primarily provide several estimates for the heat kernel defined on the 2-dimensional simple random walk. Additionally, we offer an estimate for the heat kernel on high-dimensional random walks, demonstrating that the heat kernel in higher dimensions converges rapidly. We also compute the constants involved in the estimate for the 1-dimensional heat kernel. Furthermore, we discuss the general case of on-diagonal estimates for the heat kernel.
文摘Let be the simple random walk in zd, and SupPers f(n) is an integer-valued function and increases to infinity as n tends to infinity, and In this paper,a necessary and sufficient condition to ensure or 1 is derived for d=3,4. This problem was first studied by P. Erdos and S.J. Taylor.
基金Supported by NSFC (Nos.10671182,12061020)NSF of Guizhou Province (Nos.QKH[2019]1123,QKHKY[2021]088,QKHKY[2022]301,QKH-ZK[2021]331)Ph.D. Project of Guizhou Education University (No.2021BS005)。
文摘In this paper,we prove an explicit formula of the heat kernel on the circle.As a consequence,we establish the monotonicity of the heat kernel.It is well known that the heat kernel can be viewed as the transition probability of random walk on graphs.We also give the definition of the simple lazy random walk on graphs.The transition probabilities of simple lazy random walk on Z and cycle are derived.
