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An asymptotic evaluation of heat kernel for short time. (French) Zbl 0869.60056

Azéma, J. (ed.) et al., Séminaire de probabilités XXX. Berlin: Springer. Lect. Notes Math. 1626, 104-107 (1996).
Consider the following heat equation (1) \(\partial u/ \partial t= (\Delta/2 +V)u\), where \(\Delta\) is the Laplacian operator on \(\mathbb{R}^d\) and \(V\) is a continuous function on \(\mathbb{R}^d\). Under mild assumptions on \(V\) the fundamental solution of equation (1) exists and can be expressed by the Feynman-Kac formula [see the author, Stochastic Processes Appl. 54, No. 2, 215-232 (1994; Zbl 0812.60052)]. This fundamental solution is called the heat kernel. The purpose of this paper is to prove the following theorem, which gives an asymptotic evaluation of the heat kernel for short time: Let \(V\) be a continuous function on \(\mathbb{R}^4\). Assume there exist positive constants \(C\), \(C_1\) and \(C_2\) such that \(V(x)^+ \leq C(1+|x |^2)\), \(V(x)^- \leq C_1e^{C_2 |x|^2}\). Let \(q(t,x,y)\) be the fundamental solution of the heat equation (1). Then we have \[ \lim_{t\downarrow 0} {1\over t} \log {q(t,x,y) \over p(t,x,y)} =\int^1_0V \bigl((1-s)x+ sy\bigr) ds, \] where \(p(t,x,y)\) is the transition density of a standard Brownian motion.
For the entire collection see [Zbl 0840.00041].

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Citations:

Zbl 0812.60052
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