First Passage Time Densities through Hölder curves

Abstract

Abstract. We prove that for a standard Brownian motion, there exists a firstpassage-time density function through a Hölder curve with exponent greater than 1/2. With a property of local time of a standard Brownian motion and adopting the theories of partial differential equations in Cannon (1984) and the strategies in Fasano (2005) and Carinci et al. (2016), we find a sufficient condition for existence of the density function. We also show that this density function is proportional to the space derivative of the Green function of the heat equation with Dirichlet boundary condition at the moving boundary.

Non local branching Brownian motions with annihilation and free boundary problems

“…To prove this it is enough to show that the classical solution is squeezed in between the lower and the upper barriers which is done in Theorem 4.9 below. The proof of the theorem is similar to others for analogous models, they all exploit the representation of solutions of the heat equation in terms of Brownian motions, in particular that the hitting probability of a Brownian at a curve γ t has a density with respect to Lebesgue, a property which is well known for C 1 curves but which extends to Holder curves with parameter > 1/2, see [12]. Assume (ρ(·, t), γ t ) t∈[0,T ] is a classical solution of the free boundary problem P (see (3.7) recalling that the initial datum is a probability density ρ 0 ∈ L ∞ (R, R + ).…”

The first passage time density of Brownian motion and the heat equation with Dirichlet boundary condition in time dependent domains

“…For the purpose of extending [4] into multidimensional domains, we show that there exists a continuous first-passage-time density function of standard d-dimensional Brownian motion in moving boundaries in R d , d ≥ 2, under a C 3 -diffeomorphism. Similarly as in [4], by using a property of local time of standard d-dimensional Brownian motion and the heat equation with Dirichlet boundary condition, we find a sufficient condition for the existence of the continuous density function.…”

“…First passage time (FPT) problem, which is also called boundary crossing problem, is the one of classical subjects in probability which has also many applications to other fields, for example, finance and biology. There are a bunch of articles studying this problem, but especially we mention one of them, [4], which is about we can have a continuous first-passage-time density function of one dimensional standard Brownian motion when the boundary is Hölder continuous with exponent greater than 1/2. The purpose of this paper is that we extend the result and the general strategy of [4] into the multidimensional domain, precisely, to find a continuous density function of the first hitting time in a time varying domain in R d , d ≥ 2, by investigating a relation between the first passage time density and the derivative at the boundary of the solution of the heat equation with Dirichlet boundary condition.…”

“…There are a bunch of articles studying this problem, but especially we mention one of them, [4], which is about we can have a continuous first-passage-time density function of one dimensional standard Brownian motion when the boundary is Hölder continuous with exponent greater than 1/2. The purpose of this paper is that we extend the result and the general strategy of [4] into the multidimensional domain, precisely, to find a continuous density function of the first hitting time in a time varying domain in R d , d ≥ 2, by investigating a relation between the first passage time density and the derivative at the boundary of the solution of the heat equation with Dirichlet boundary condition. Thus this article is concerned with standard d-dimensional Brownian motion killed on the boundary of a deterministic moving domain which corresponds to the heat equation with Dirichlet boundary condition.…”

“…

In [4], it is proved that we can have a continuous first-passage-time density function of one dimensional standard Brownian motion when the boundary is Hölder continuous with exponent greater than 1/2. For the purpose of extending [4] into multidimensional domains, we show that there exists a continuous first-passage-time density function of standard d-dimensional Brownian motion in moving boundaries in R d , d ≥ 2, under a C 3 -diffeomorphism.

…”