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Joint Applied Math and Probability Seminar
Diffusion in random environment and the renewal theorem
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Consider a diffusion satisfying the formal SDE dX(t)=dB(t)-(1/2) W'(X(t))dt, X(0)=0 where W is a two sided Brownian motion and B a standard Brownian motion independent of W. It is known that (X_t-b(log t))/log^2 t goes to zero in probability as t goes to infinity, where b is a process having an explicit description and depending only on the environment W. We compute the distribution of the sign changes of b on any compact interval of (0,+\infty). What is important is the method of proof used. This method puts into a mathematical framework a renormalization group analysis proposed in the non-rigorous physics paper of Le Dousal et al [1]. The tools we use is a path decomposition for W, relying on excursion theory, and the renewal theorem. Then we consider the case where W is a spectrally negative stable process with index in (1,2] and we compute the distribution of b(t) (fixed t). [1] P. Le Doussal, C. Monthus and D. Fisher. Random walkers in one-dimensional random environments: Exact renormalization group analysis. Physical Review E 59, 5.(1999), 4795-4840. |