PhD Advisor: Kostya Khanin
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This thesis studies global solutions to the semidiscrete stochastic heat equation and the associated Cauchy problem known as Parabolic Anderson Model. Via a Feynman-Kac formula, it is linked with the analysis of directed polymers in random environment, and this thesis establishes a number of results for the corresponding partition function.
We consider a continuous-time simple symmetric random walk on the integer lattice ${\Bbb Z}^d$ in dimension $d \geq 3$, subject to a random potential given by two-sided Wiener processes. In the high-temperature regime, we prove the existence of the $L^2$- and almost sure limit of the partition function as time $t \to \pm \infty$. We show that the $L^2$-convergence rate is at least polynomial and that the limiting partition function is positive almost surely. Furthermore, we show that this limiting partition function defines a global stationary solution to the semidiscrete stochastic heat equation which is unique up to a rescaling, and which in some sense attracts solutions to the Parabolic Anderson Model for any subexponentially growing initial data. One of the primary tools in the proof of this uniqueness and attraction result is a factorization formula for the point-to-point partition function, which is related to the ones obtained by Sinai (1995) and Kifer (1997) for other polymer models, but valid not only on the diffusive scale but up to any sub-ballistic scale. This factorization formula allows us to obtain a uniqueness result for physical invariant probability measures of a certain skew product that can be naturally associated with the semidiscrete stochastic heat equation, which in turns gives uniqueness of global stationary solutions.