Brownian Motion on a Pseudo Sphere in Minkowski Space $\mathbb {R}^l_v$

Abstract

For a Brownian motion moving on a pseudo sphere in Minkowski space $\mathbb {R}^l_v$ of radius r starting from point X, we obtain the distribution of hitting a fixed point on this pseudo sphere with $l\ge 3$ by solving Dirichlet problems. The proof is based on the method of separation of variables and the orthogonality of trigonometric functions and Gegenbauer polynomials.

Appendix: Calculations of the Laplacian–Beltrami Operator

This appendix is devoted to the calculations of the Laplacian–Beltrami operator restricted on pseudo spheres. First, we consider Laplacian–Beltrami operator restricted on a spacelike pseudo sphere of $\mathbb {R}^3_1$, namely, on the spacelike pseudo sphere

$$\begin{aligned} \mathbb {S}_r^{3,1}=\{X=(x_0,x_1,x_2)\in \mathbb {R}^3_1|-x_0^2+x_1^2+x_2^2=r^2\} \end{aligned}$$

under the relationship given by

$$\begin{aligned} x_0=\eta \sinh \alpha ,\ x_1=\eta \cos \beta \cosh \alpha ,\ x_2=\eta \sin \beta \cosh \alpha , \end{aligned}$$

(4.64)

or

$$\begin{aligned} \eta = \Vert X\Vert ,\ \tanh \alpha =\frac{x_0}{\sqrt{x_1^2+x_2^2}},\ \tan \beta =\frac{x_2}{x_1}. \end{aligned}$$

(4.65)

Thanks to multivariable calculus, we have the expression of $\Delta$ restricted on $\mathbb {S}^{3,1}$. That is,

$$\begin{aligned} \begin{array}{ll} \Delta _{\mathbb {S}}^{3,1}= &{} \left( \frac{\partial ^2\eta }{\partial x_1^2} + \frac{\partial ^2\eta }{\partial x_2^2} - \frac{\partial ^2\eta }{\partial x_0^2} \right) \frac{\partial }{\partial \eta } + \left( \frac{\partial ^2\alpha }{\partial x_1^2} + \frac{\partial ^2\alpha }{\partial x_2^2} -\frac{\partial ^2\alpha }{\partial x_0^2} \right) \frac{\partial }{\partial {\alpha }}\\ &{} + \left( \frac{\partial ^2\beta }{\partial x_1^2} + \frac{\partial ^2\beta }{\partial x_2^2} -\frac{\partial ^2\beta }{\partial x_0^2} \right) \frac{\partial }{\partial \beta } + \left( \left( \frac{\partial \eta }{\partial x_1}\right) ^2+ \left( \frac{\partial \eta }{\partial x_2}\right) ^2 -\left( \frac{\partial \eta }{\partial x_0}\right) ^2 \right) \frac{\partial ^2}{\partial \eta ^2}\\ &{} + \left( \left( \frac{\partial {\alpha }}{\partial x_1}\right) ^2+ \left( \frac{\partial {\alpha }}{\partial x_2}\right) ^2 -\left( \frac{\partial {\alpha }}{\partial x_0}\right) ^2 \right) \frac{\partial ^2}{\partial {\alpha }^2} \\ &{} + \left( \left( \frac{\partial \beta }{\partial x_1}\right) ^2+ \left( \frac{\partial \beta }{\partial x_2}\right) ^2 -\left( \frac{\partial \beta }{\partial x_0}\right) ^2 \right) \frac{\partial ^2}{\partial \beta ^2}\\ &{} +\, 2 \left( \frac{\partial \eta \partial {\alpha }}{\partial x_1^2} + \frac{\partial \eta \partial \alpha }{\partial x_2^2} -\frac{\partial \eta \partial \alpha }{\partial x_0^2} \right) \frac{\partial ^2}{\partial \eta \partial \alpha } + 2 \left( \frac{\partial \alpha \partial \beta }{\partial x_1^2} + \frac{\partial \alpha \partial \beta }{\partial x_2^2} -\frac{\partial \alpha \partial \beta }{\partial x_0^2} \right) \frac{\partial ^2}{\partial \alpha \partial \beta }\\ &{} +\, 2 \left( \frac{\partial \beta \partial \eta }{\partial x_1^2} + \frac{\partial \beta \partial \eta }{\partial x_2^2} -\frac{\partial \beta \partial \eta }{\partial x_0^2} \right) \frac{\partial ^2}{\partial \beta \partial \eta }. \end{array} \end{aligned}$$

(4.66)

To calculate the Laplacian $\Delta$ in spacelike coordinates, we only need to calculate the partial differentials on the right hand side of equality (4.66). By (4.65), we get

$$\begin{aligned} \begin{array}{lll} \frac{\partial \eta }{\partial x_0}=-\frac{x_0}{\eta }, &{}\frac{\partial \eta }{\partial x_1}=\frac{x_1}{\eta }, &{}\frac{\partial \eta }{\partial x_2}=\frac{x_2}{\eta }, \\ \frac{\partial \alpha }{\partial x_0}=\frac{\sqrt{\eta ^2+x_0^2}}{\eta ^2}, &{}\frac{\partial \alpha }{\partial x_1}=-\frac{x_0x_1\cosh ^2\alpha }{(\eta ^2+x_0^2)^{3/2}}, &{}\frac{\partial \alpha }{\partial x_2}=-\frac{x_0x_2\cosh ^2\alpha }{(\eta ^2+x_0^2)^{3/2}}, \\ \frac{\partial \beta }{\partial x_0}=0, &{}\frac{\partial \beta }{\partial x_1}=-\frac{x_2}{\eta ^2+x_0^2}, &{}\frac{\partial \beta }{\partial x_2}=\frac{x_1}{\eta ^2+x_0^2}, \\ \frac{\partial ^2\eta }{\partial x_0^2}=-\frac{\eta ^2+x_0^2}{\eta ^3}, &{}\frac{\partial ^2\eta }{\partial x_1^2}=\frac{\eta ^2-x_1^2}{\eta ^3}, &{}\frac{\partial ^2\eta }{\partial x_2^2}=\frac{\eta ^2-x_2^2}{\eta ^3}, \\ \frac{\partial ^2\alpha }{\partial x_0^2}=\frac{2x_0\sqrt{\eta ^2+x_0^2}}{\eta ^4}, &{}\frac{\partial ^2\alpha }{\partial x_1^2}=\frac{2x_0x_1^2(\eta ^2+x_0^2)-x_0x_2^2\eta ^2}{\eta ^4(\eta ^2+x_0^2)^{3/2}}, &{}\frac{\partial ^2\alpha }{\partial x_2^2}=\frac{2x_0x_2^2(\eta ^2+x_0^2)-x_0x_1^2\eta ^2}{\eta ^4(\eta ^2+x_0^2)^{3/2}}, \\ \frac{\partial ^2\beta }{\partial x_0^2}=0, &{}\frac{\partial ^2\beta }{\partial x_1^2}=\frac{2x_1x_2}{(\eta ^2+x_0^2)^2}, &{}\frac{\partial ^2\beta }{\partial x_2^2}=-\frac{2x_1x_2}{(\eta ^2+x_0^2)^2}. \end{array} \end{aligned}$$

Thus, the coefficients in the partial differential operators in (4.66) are

$$\begin{aligned} \frac{\partial ^2\eta }{\partial x_1^2}+\frac{\partial ^2\eta }{\partial x_2^2}-\frac{\partial ^2\eta }{\partial x_0^2}&=\frac{2}{\eta }, \\ \frac{\partial ^2\alpha }{\partial x_1^2}+\frac{\partial ^2\alpha }{\partial x_2^2}-\frac{\partial ^2\alpha }{\partial x_0^2}&=-\frac{\tanh \alpha }{\eta ^2}, \\ \frac{\partial ^2\beta }{\partial x_1^2}+\frac{\partial ^2\beta }{\partial x_2^2}-\frac{\partial ^2\beta }{\partial x_0^2}&=0,\\ \left( \frac{\partial \eta }{\partial x_1}\right) ^2+ \left( \frac{\partial \eta }{\partial x_2}\right) ^2 -\left( \frac{\partial \eta }{\partial x_0}\right) ^2&=1,\\ \left( \frac{\partial \alpha }{\partial x_1}\right) ^2+ \left( \frac{\partial \alpha }{\partial x_2}\right) ^2 -\left( \frac{\partial \alpha }{\partial x_0}\right) ^2&=\frac{1}{\eta ^2\cosh ^2\alpha },\\ \left( \frac{\partial \beta }{\partial x_1}\right) ^2+ \left( \frac{\partial \beta }{\partial x_2}\right) ^2 -\left( \frac{\partial \beta }{\partial x_0}\right) ^2&=-\frac{1}{\eta ^2}, \end{aligned}$$

while the coefficients of the cross terms are zeros.

Then the Laplacian–Beltrami operator $\Delta$ restricted on $\mathbb {S}^{3,1}$ can be expressed in coordinates $(\eta ,\alpha ,\beta )_{\mathbb {S}}$ as

$$\begin{aligned} \Delta _{\mathbb {S}}^{3,1}=\left( \frac{2}{\eta }\frac{\partial }{\partial \eta }+\frac{\partial ^2}{\partial \eta ^2}\right) -\left( \frac{\tanh \alpha }{\eta ^2}\frac{\partial }{\partial \alpha }+\frac{1}{\eta ^2}\frac{\partial ^2}{\partial \alpha ^2}\right) +\frac{1}{\eta ^2\cosh ^2\alpha }\frac{\partial ^2}{\partial \beta ^2}, \end{aligned}$$

while the Laplacian–Beltrami operator restricted on $\mathbb {S}^{3,1}_r$ is expressed as

$$\begin{aligned} \Delta _{sr}^{3,1}=-\frac{1}{r^2}\left( \tanh \alpha \frac{\partial }{\partial \alpha }+\frac{\partial ^2}{\partial \alpha ^2}\right) +\frac{1}{r^2}\frac{1}{\cosh ^2\alpha }\frac{\partial ^2}{\partial \beta ^2}. \end{aligned}$$

(4.67)

Now return to the general case $\mathbb {R}^l_v$. Letting $x=(x_1,x_2,\ldots ,x_v)$ and $y=(y_1,y_2,\ldots ,y_{l-v})$, for any point $X\in \mathbb {S}^{l,v}$ we have

$$\begin{aligned} -\sum _{i=1}^vx_i^2+\sum _{j=1}^{l-v}y_j^2=\eta ^2>0. \end{aligned}$$

Again, we use the coordinates similar to $(\cdot ,\cdot ,\cdot )_{\mathbb {S}}$, namely, a system of coordinates $(\eta ,\alpha ,\theta _y,\theta _x)_{\mathbb {S}}$ satisfying

$$\begin{aligned} \begin{array}{ll} &{} \eta ^2=-\sum _{i=1}^vx_i^2+\sum _{j=1}^{l-v}y_j^2,\\ &{} \tanh ^2\alpha =\frac{\sum _{i=1}^vx_i^2}{\sum _{j=1}^{l-v}x_j^2}, \end{array} \end{aligned}$$

(4.68)

and $\theta _x=(\theta _{x,1},\ldots ,\theta _{x,v-1})$, $\theta _y=(\theta _{y,1},\ldots ,\theta _{y,l-v-1})$ are the standard polar coordinates on the Euclidean unit sphere $\mathcal {S}^{v-1}$ and $\mathcal {S}^{l-v-1}$ respectively. Thus the Laplacian–Beltrami operator restricted on $\mathbb {S}^{l,v}$ with $l\ge 4$ can be expressed as

$$\begin{aligned} \begin{array}{ll} \Delta _{\mathbb {S}}^{l,v}= &{} \left( \frac{\partial ^2}{\partial \eta ^2}+\frac{l-1}{\eta }\frac{\partial }{\partial \eta }\right) \\ &{} -\,\frac{1}{\eta ^2}\left( \frac{\partial ^2}{\partial \alpha ^2}+[(l-v-1)\tanh \alpha +(v-1)\coth \alpha ]\frac{\partial }{\partial \alpha }\right) \\ &{} +\,\frac{1}{\eta ^2\cosh ^2\alpha }\Delta _{\mathcal {S}^{l-v-1}}-\frac{1}{\eta ^2\sinh ^2\alpha }\Delta _{\mathcal {S}^{v-1}}, \end{array} \end{aligned}$$

(4.69)

where $\Delta _{\mathcal {S}^n}$ is the Laplacian operator on the n-dimensional unit sphere.