2017/10/02 by Yannic Broeker, Broeker, Yannic, Chiranjib Mukherjee +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1710.00631
openalex publication_date 2017/10/02 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We continue with the study of the mollified stochastic heat equation in\nd\≥ 3 given by d u\ε,t= frac 12\Δ u\ε,t+ \β\n\ε(d-2)/2 ,u\ε,t ,d B\ε,t with spatially\nsmoothened cylindrical Wiener process B, whose (renormalized) Feynman-Kac\nsolution describes the partition function of the continuous directed polymer.\nIn an earlier work ( citeMSZ16), a phase transition was obtained, depending\non the value of \β>0 in the limiting object of the smoothened solution\nu_\ε as the smoothing parameter \ε\→ 0 This partition function\nnaturally defines a quenched polymer path measure and we prove that as long as\n\β>0 stays small enough while u_\ε converges to a strictly\npositive non-degenerate random variable, the distribution of the diffusively\nrescaled Brownian path converges under the aforementioned polymer path measure\nto standard Gaussian distribution.\n