Perturbed random walks and Brownian motions, and local times (Q1383517)

The paper is based on two talks given by the author on the self-interacting random walks and the local times of the standard Brownian motion \(b_t\), respectively. The special case of the reinforced random walk considered by the author [Probab. Theory Relat. Fields 84, No. 2, 203-229 (1990; Zbl 0665.60077)] is the integer-valued stochastic process \(X_0= 0\), \(X_1,X_2,\dots\) with \(| X_{i+ 1}- X_i|= 1\), where the probabilities of the transitions from \(j\) to \(j+1\) (or \(j-1\)) depend on the event whether \(j+1\) (or \(j-1\), respectively) is already visited or not by the random walk. The author presents answers to those questions that arise on general theory of Markov processes and random walks. For instance, the problems of recurrence properties, the weak convergence, the width of the interval of visited states are addressed. The second part concerns the local time \(L_f\) spent by \(b_t\) on the graph of a smooth function \(f\). The theorem announced shows that among all nondecreasing functions \(f\), the local time \(L_0\) is not the largest in terms of distribution. More precisely, equality \(P\{L_f< x\}\geq P\{2L_0< x\}\) holds and 2 cannot be replaced by a smaller number.