Heat kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces

In this paper, we consider a symmetric pure jump Markov process $X$ on a general metric measure space that satisfies volume doubling conditions. We study estimates of the transition density $p(t,x,y)$ of $X$ and their stabilities when the jumping kernel for $X$ has general mixed polynomial growths. Unlike [24], in our setting, the rate function which gives growth of jumps of $X$ may not be comparable to the scale function which provides the borderline for $p(t,x,y)$ to have either near-diagonal estimates or off-diagonal estimates. Under the assumption that the lower scaling index of scale function is strictly bigger than $1$, we establish stabilities of heat kernel estimates. If underlying metric measure space admits a conservative diffusion process which has a transition density satisfying a general sub-Gaussian bounds, we obtain heat kernel estimates which generalize [2, Theorems 1.2 and 1.4]. In this case, scale function is explicitly given by the rate function and the function $F$ related to walk dimension of underlying space. As an application, we proved that the finite moment condition in terms of $F$ on such symmetric Markov process is equivalent to a generalized version of Khintchine-type law of iterated logarithm at the infinity

HEAT KERNEL ESTIMATES FOR SYMMETRIC JUMP PROCESSES WITH MIXED POLYNOMIAL GROWTHS

Bae J, Kang J, Kim P, Lee J. HEAT KERNEL ESTIMATES FOR SYMMETRIC JUMP PROCESSES WITH MIXED POLYNOMIAL GROWTHS. ANNALS OF PROBABILITY. 2019;47(5):2830-2868.In this paper, we study the transition densities of pure-jump symmetric Markov processes in R-d, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of the transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Levy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at infinity

Additive manufacturing-based combinatorial approach to improve bonding strength and heat transfer performance in wrought-cast Al compound casting

We improved the interfacial bonding strength and heat transfer performance between wrought and cast Al alloy parts by implementing an interfacial Al-Si-Zn-based alloy coating layer. An additive manufacturing (AM)-based combinatorial experiment was employed to determine the optimal composition of coating layers. Various compositional combinations of Al-Si-Zn layers were deposited onto an Al plate using a multi-nozzle direct energy deposition system, and the wetting behavior of the cast materials onto the combinatorial samples was examined to select the best composition of the coating layer. The selected coating layer was sprayed on the cooling pipes, and an actual-scale motor housing integrated by the pipe was produced by a compound die casting. The Al-Si-Zn coating layer caused the surface of the coating layers on pipes to partially melt when in contact with the molten alloy during the casting. Consequently, a reactive layer was rapidly formed between the coating and cast materials before solidification. As a result, the interfacial bonding strength increased to 13 MPa and heat transferability between the wrought and cast materials was improved. Therefore, we described the AM-based combinatorial exploring method for the interfacial coating layer to overcome the interfacial issues between the wrought and cast materials of compound casting