scholarly journals Minimal thinness for subordinate Brownian motion in half-space

2012 ◽  
Vol 62 (3) ◽  
pp. 1045-1080 ◽  
Author(s):  
Panki Kim ◽  
Renming Song ◽  
Zoran Vondraček
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Subordinate Brownian Motion ◽  
Minimal Thinness

scholarly journals Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

2020 ◽  
Vol 30 (5) ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Terhi Kaarakka
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Laplace Operator ◽  
Harmonic Functions ◽  
Riemannian Metric ◽  
Hyperbolic Metric ◽  
Hyperbolic Functions ◽  
The Hyperbolic Metric ◽  
Hyperbolic Brownian Motion ◽  
Hyperbolic Harmonic Functions

Abstract We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$ d s 2 = d x 1 2 + ⋯ + d x n 2 x n 2 α n - 2 in the upper half space $$\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}$$ R + n = { x 1 , … , x n ∈ R n : x n > 0 } . They are called $$\alpha $$ α -hyperbolic harmonic. An important result is that a function f is $$\alpha $$ α -hyperbolic harmonic íf and only if the function $$g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) $$ g x = x n - 2 - n + α 2 f x is the eigenfunction of the hyperbolic Laplace operator $$\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}$$ △ h = x n 2 ▵ - n - 2 x n ∂ ∂ x n corresponding to the eigenvalue $$\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0$$ 1 4 α + 1 2 - n - 1 2 = 0 . This means that in case $$\alpha =n-2$$ α = n - 2 , the $$n-2$$ n - 2 -hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of $$\alpha $$ α -hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.

scholarly journals Half-Space Stationary Kardar–Parisi–Zhang Equation

2020 ◽  
Vol 181 (4) ◽  
pp. 1149-1203 ◽  
Author(s):  
Guillaume Barraquand ◽  
Alexandre Krajenbrink ◽  
Pierre Le Doussal
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Large Time ◽  
Fredholm Determinant ◽  
Exact Formula ◽  
Kpz Equation ◽  
Stationary Growth ◽  
Painlevé Ii ◽  
Positive Half ◽  
The Continuum

Abstract We study the solution of the Kardar–Parisi–Zhang (KPZ) equation for the stochastic growth of an interface of height h(x, t) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at $$x=0$$ x = 0 . The boundary condition $$\partial _x h(x,t)|_{x=0}=A$$ ∂ x h ( x , t ) | x = 0 = A corresponds to an attractive wall for $$A<0$$ A < 0 , and leads to the binding of the polymer to the wall below the critical value $$A=-1/2$$ A = - 1 / 2 . Here we choose the initial condition h(x, 0) to be a Brownian motion in $$x>0$$ x > 0 with drift $$-(B+1/2)$$ - ( B + 1 / 2 ) . When $$A+B \rightarrow -1$$ A + B → - 1 , the solution is stationary, i.e. $$h(\cdot ,t)$$ h ( · , t ) remains at all times a Brownian motion with the same drift, up to a global height shift h(0, t). We show that the distribution of this height shift is invariant under the exchange of parameters A and B. For any $$A,B > - 1/2$$ A , B > - 1 / 2 , we provide an exact formula characterizing the distribution of h(0, t) at any time t, using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters A, B. In particular, when $$(A, B) \rightarrow (-1/2, -1/2)$$ ( A , B ) → ( - 1 / 2 , - 1 / 2 ) , the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik–Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea–Ferrari–Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.

scholarly journals PATH INTEGRAL REPRESENTATION FOR SCHRÖDINGER OPERATORS WITH BERNSTEIN FUNCTIONS OF THE LAPLACIAN

2012 ◽  
Vol 24 (06) ◽  
pp. 1250013 ◽  
Author(s):  
FUMIO HIROSHIMA ◽  
TAKASHI ICHINOSE ◽  
JÓZSEF LŐRINCZI
Keyword(s):  
Brownian Motion ◽  
Path Integral ◽  
Schrödinger Operators ◽  
Integral Representations ◽  
Schrodinger Operators ◽  
Kato Class ◽  
Path Integral Representation ◽  
Bernstein Functions ◽  
Subordinate Brownian Motion ◽  
The One

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

A mechanical model of Brownian motion in half-space

1989 ◽  
Vol 55 (3-4) ◽  
pp. 649-693 ◽  
Author(s):  
Paola Calderoni ◽  
Detlef D�rr ◽  
Shigeo Kusuoka
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Mechanical Model
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