Off diagonal short time asymptotics for fundamental solution of diffusion equation
Off diagonal short time asymptotics for fundamental solution of diffusion equation
复制标题
扩散方程基本解的非对角短时渐近
DOI：
10.1080/03605307708820048
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发表时间：
1977
影响因子：
1.9
通讯作者：
Y. Kannai
中科院分区：
文献类型：
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作者：
Y. Kannai
人们应该注意到（1.2）中各项的指数递减（对于 x* y），以及该公式中左侧与右侧前 N 项之和之间的差值（与 Colin de Verdiere [6l] 对比）。因此，(1.2)（以及类似的全局结果）似乎不能从伪微分算子（甚至傅里叶积分算子）的通常微积分中导出，因为这种微积分中的项和余数表现得像幂而不像指数。 JK Cohen 和 RM Lewis [4] 得到了诸如 (1.2) 的展开式；然而，他们的讨论只是形式上的，并没有证明所获得的级数确实代表了渐近的实际解决方案。 JK Cohen、FG Hagin 和 JB Keller 在 [51 对于一维情况；该证明涉及一些仔细的估计，并且不能轻易推广到更高维度的情况。另一方面，Varadhan [21] 已经证明（在对系数的弱得多的假设下）对于所有 x、y、m 都是任意的。但u(x,y,t)完全渐近展开的问题无法用Varadhan的方法解决。马斯洛夫在他的名著《Cl61》的结尾（没有证明）描述了 u (x, y, t) 的全局渐近展开式的首项，假设度量 d 仅具有非共轭割点（参见第 4 节）。 Buslaev C31使用了连续积积分的概念
One should note the exponential decrease as t&(for x* y) of the individual terms in (1.2) and of the difference between the left-hand side and the sum of the first N terms on the right in that formula (contrast with Colin de Verdiere [6l). It appears, therefore, that (1.2)(and similar global results) cannot be derived from a usual calculus of pseudo-differential operators (or even of Fourier integral operators) as the terms and the remainders in such a calculus behave like powers and not like exponentials. Expansions such as (1.2) have been obtained by JK Cohen and RM Lewis [4]; their discussion, however, was only formal and contained no proof that the series obtained does represent asymptotically an actual solution. Such a proof was supplied by JK Cohen, FG Hagin and JB Keller in [51 for the one dimensional case; this proof involves some careful estimates and cannot be generalized easily to higher dimensional cases. On the other hand, Varadhan [21] has proved (under much weaker assumptions on the coefficients) that for all x, y, m arbitrary. But the problem of the complete asymptotic expansion of u (x, y, t) could not be solved by Varadhan's method. Maslov described at the end of his famous book Cl61 (without proof) the leading term of a global asymptotic expansion for u (x, y, t) assuming that the metric d possesses only non-conjugate cut points (see section 4). Buslaev C31 used the concept of continuum product integral