In this paper, we investigate the long time asymptotics of the exponential moment of the time-space Brownian Hamiltonian, whose temporal kernel is a non-negative definite integrable function, and spatial kernel is a non-negative definite generalized function with homogeneity. Our work is a generalization of [M. D. Donsker and S. R. S. Varadhan, Asymptotics for the polaron, Comm. Pure Appl. Math. 36 (1983), no. 4, 505--528] on the asymptotics for the polaron.
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Chengbo Sun (2026) studied this question.
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