Integral Equation for Weighted Signature Kernels Stochastic Analysis Factorial Decay Stochastic Analysis Non-Polishness of $(\mathcal{C}_1, \chi_{\mathrm{pr}})$ Stochastic Analysis Global Error of Euler Scheme Stochastic Analysis MMD Metrizes Weak Convergence Stochastic Analysis Prohorov's Theorem Stochastic Analysis Brownian Motion as a Rough Path Stochastic Analysis Duality of Tensor Algebras Stochastic Analysis