Multi-point Lyapunov exponents of the Stochastic Heat Equation

We obtain the multi-point positive integer Lyapunov exponents of the Stochastic Heat Equation (SHE) and provide three expressions for them. We prove the result by matching the upper and lower bounds for the Lyapunov exponents. The upper bound is obtained by analyzing the contour integral formula in [4]. For the lower bound, we apply an induction argument, relying on a tree recently appeared in [71]. The tree is related to the optimal trajectories of the Brownian motions in the Feynman-Kac formula.