Wiener measure in a space of functions of two variables

An Arc-Sine Law for Last Hitting Points in the Two-Parameter Wiener Space

“…In this section we briefly introduce the two-parameter Wiener process for background knowledge of our development as it is far less accessible than the Wiener process. This section is mostly based on Reference [10]. For readers who are not familiar with the two-parameter Wiener process, a more detailed digest of it can be found in the author's recent work [11].…”

“…The completed measure space of the two-parameter Wiener measure m 2 is denoted by (C 2 (R), M 2 , m 2 ) and called the two-parameter Wiener space. The space is also called the Yeh-Wiener space in some literature because the space was first introduced by Yeh in Reference [10] and further work [12]. One can see the measure space is a probability space since m 2 (C 2 (R)) = 1.…”

“…The equality means that if one side exists then the other side also exists and both sides coincide. It tells us that the integral over the infinite dimensional function space C 2 (R) on the left is reduced to an ordinary Lebesgue integral over the finite dimensional space R mn on the right [10].…”

“…The function X s,t is a random variable and the collection {X s,t : s, t ≥ 0} was originally called a two-time parameter Wiener process by Yeh [10]; later, it is called the Yeh-Wiener process in the literature, such as References [13][14][15]. We keep the name, the two-parameter Wiener process, throughout this paper.…”

Numerical solutions of some stochastic hyperbolic wave equations including sine-Gordon equation

“…These applications have often employed two-parameter Wiener processes, or Brownian motion, {W (x, t), x ∈ X, t ∈ T }, with mean zero and covariance Cov[W (x, s), W (y, t)] = min(x, y) min(s, t) where X and T are sub-intervals of R or R + (or their formal derivatives, space-time white noise {w(x, t)}). Such two-parameter processes first appeared over 50 years ago in works by Kitagawa (1951Kitagawa ( ),Čencov, (1956 and Yeh (1960). Zimmerman (1972) constructed a stochastic integral with respect to W and obtained a solution to what that author called a diffusion equation,…”