岡田 いず海

We consider the heat equation with a dynamic potential [Formula presented]where N>2. Here the potential V is given by a Hardy-type function V(x,t)=λ|x−ξ(t)|−μ with constants λ,μ>0, and the singular point ξ(t) is a path of the N-dimensional fractional Brownian motion with the Hurst exponent 0<H<1/2. By using the Feynman–Kac formula, it is shown that the initial value problem is solvable if and only if μ<(1/H)∧N. We also study the heat equation with a singular nonhomogeneous term.

We consider solutions of the linear heat equation in RN with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent α∈(0,1). We show that any isolated singularity is removable if it is weaker than a certain order depending on α. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent H∈(0,1/2]. It turns out that H=1/N is critical.

We investigate a problem suggested by Dembo, Peres, Rosen, and Zeitouni, which states that the growth exponent of favorite points associated with a simple random walk in Z2 coincides, on average and almost surely, with those of late points and high points associated with the discrete Gaussian free field.

As Dembo (In Lectures on Probability Theory and Statistics (2005) 1-101 Springer, and International Congress of Mathematicians, Vol. III (2006) 535-558, Eur. Math. Soc.) suggested, we consider the problem of late points for a simple random walk in two dimensions. It has been shown that the exponents for the number of pairs of late points coincide with those of favorite points and high points in the Gaussian free field, whose exact values are known. We determine the exponents for the number of j-tuples of late points on average.

Herein, we describe multidimensional Brownian motions for d = 2; 3 with drifts whose order is the same as that of the mean displacement of a Brownian motion. We consider the probabilities that the processes remain in specific cones for a considerable amount of time. We obtain exponents expressing the probabilities, which are different from that of the ordinary Brownian motion. Finally, we suggest an open problem concerning the exact values.

Herein, we present a study on frequently visited sets in a simple random walk in Z2. We estimate the expectation of numbers of j-tuples of favorite points and obtain an exact exponent.

This paper considers the question: how many times does a simple random walk revisit the most frequently visited site among the inner boundary points? It is known that in ℤ2, the number of visits to the most frequently visited site among all of the points of the random walk range up to time n is asymptotic to π-1(logn)2, while in ℤd(d≥3), it is of order log n. We prove that the corresponding number for the inner boundary is asymptotic to βdlogn for any d≥2, where βd is a certain constant having a simple probabilistic expression.

In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If Ln be the number of the inner boundary points of random walk range in the n steps, we prove lim n→∞(L n/n) exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on the two dimensional square lattice is of the same order as n/(log n)2.

In this paper we consider the winding number, θ(s), of planar Brownian motion and study asymptotic behavior of the process of the maximum time, the time when θ(s) attains the maximum in the interval 0 ≤ s ≤ t. We find the limit law of its logarithm with a suitable normalization factor and the upper growth rate of the maximum time process itself. We also show that the process of the last zero time of θ(s) in [0; t] has the same law as the maximum time process.