2013/10/25 by Yuval Peres, Peres, Yuval, Perla Sousi +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60J65 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1310.7002
openalex publication_date 2013/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a fractional Brownian motion in ℝd. For any Borel function f:[0,1] → ℝd, we express the Hausdorff dimension of the image and the graph of X+f in terms of f. This is new even for the case of Brownian motion and continuous f, where it was known that this dimension is almost surely constant. The expression involves an adaptation of the parabolic dimension previously used by Taylor and Watson to characterize polarity for the heat equation. In the case when the graph of f is a self-affine McMullen-Bedford carpet, we obtain an explicit formula for the dimension of the graph of X+f in terms of the generating pattern. In particular, we show that it can be strictly bigger than the maximum of the Hausdorff dimension of the graph of f and that of X. Despite the random perturbation, the Minkowski and Hausdorff dimension of the graph of X+f can disagree.