2010/10/14 by Yuval Peres, Peres, Yuval, Perla Sousi +1
Mathematics · Economics, Econometrics and Finance · #Stochastic processes and statistical mechanics #Stochastic processes and financial applications #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1010.2987
By the Cameron--Martin theorem, if a function f is in the Dirichlet space\nD, then B+f has the same a.s. properties as standard Brownian motion, B.\nIn this paper we examine properties of B+f when f \∉ D. We start by\nestablishing a general 0-1 law, which in particular implies that for any fixed\nf, the Hausdorff dimension of the image and the graph of B+f are constants\na.s. (This 0-1 law applies to any L 'evy process.) Then we show that if the\nfunction f is H "older(1/2), then B+f is intersection equivalent to B.\nMoreover, B+f has double points a.s. in dimensions d\≤ 3, while in d\≥\n4 it does not. We also give examples of functions which are H "older with\nexponent less than 1/2, that yield double points in dimensions greater than\n4. Finally, we show that for d \≥ 2, the Hausdorff dimension of the image of\nB+f is a.s. at least the maximum of 2 and the dimension of the image of f.\n