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.