Transition density estimates for subordinated reflected Brownian motion on simple nested fractals

Stochastic processes on fractals, more generally on irregular sets, have been studied for over 40 years. The Brownian motion is the first process constructed in various spaces, such as the Sierpiński carpet [2], the Sierpiński gasket [3], post critically-finite sets [9], as well as on more general sets [16, 11, 9, 6].

Suposse that the Brownian motion on an unbounded nested fractal $\mathcal{K}^{\left\langle\infty\right\rangle}$ has been constructed. For $M\in\mathbb{Z_{+}}$, we construct the reflected Brownian motion on those compact fractals $\mathcal{K}^{\left\langle M\right\rangle}$ that obey the good labeling property (see Section 2.3 for the definitions of $\mathcal{K}^{\left\langle\infty\right\rangle},\mathcal{K}^{\left\langle M\right\rangle}$ and the good labeling property). This has been achieved in [7] via a suitable projection procedure. The reflected Brownian motion on $\mathcal{K}^{\langle M\rangle}$ is a conservative diffusion process, whose transition density function satisfies (see [12]):

where $L$ is the scaling factor of $\mathcal{K}^{\left\langle\infty\right\rangle}$ and $\mathcal{K}^{\left\langle M\right\rangle},$ parameters $d,d_{w},d_{J}$ depend on the geometry of the fractal, $c_{1},\ldots c_{6}>0$ are absolute constants. It is worth noting that when the time is large, then this transition density is comparable with $L^{-Md}$, which means that the process is roughly uniformly distributed over the $M-$ complex.

In this paper we would like to obtain estimates on the transition density function for the subordinate reflected Brownian motion on $\mathcal{K}^{\left\langle M\right\rangle}.$ We will consider two classes of subordinate processes: $\alpha-$stable processes and $\alpha-$stable relativistic processes. When $\mathcal{K}^{\left\langle M\right\rangle}$ is the Sierpiński gasket, it has been proven in [8] that the subordination and the reflection commute, and this property holds in present case too. Therefore it convenient to understand the subordinate reflected process as the process subordinate to the reflected Brownian motion via the given subordinator (stable or relativistic). In this paper, we prove the following result. For the transition density of the $\alpha-$stable reflected Brownian motion on $\mathcal{K}^{\left\langle M\right\rangle}$, denoted $p^{M}_{S}(t,x,y)$ (Theorem 3.1): there exist positive constants $B_{1},B_{2},B_{3},B_{4},$ such that for $M\in\mathbb{Z}_{+}$ and $x,y\in\mathcal{K}^{\left\langle M\right\rangle}$

In the case of subordination via the relativistic subordinator i.e. the relativistic $\alpha-$stable Brownian motion, we get that for any $M\in\mathbb{Z}_{+}$, the transition density of the reflected relativistic $\alpha-$stable process, denoted $p_{R}^{M}(t,x,y)$ on $\mathcal{K}^{\langle M\rangle},$ satisfies (Theorem 4.1):

1. 1)

   for $t\geq L^{Md_{w}},\ x,y\in\mathcal{K}^{\left\langle M\right\rangle}$ there exist constants $C_{1},C_{2}$ such that

   $$C_{1}L^{-Md}\leq p^{M}_{R}(t,x,y)\leq C_{2}L^{-Md}$$

   (1.1)

2. 2)

   for $t<L^{Md_{w}},\ x,y\in\mathcal{K}^{\left\langle M\right\rangle}$ there exists constant $H_{1}$ such that

   $$p_{R}(t,x,y)\leq p_{R}^{M}(t,x,y)\leq p_{R}(t,H_{1}x,H_{1}y).$$

Taking into account the estimates on the relativistic stable process on $\mathcal{K}^{\left\langle\infty\right\rangle}$ from [1] (cf. formulas (2.18),(2.19),(2.20) below) we get that there exist constants $C_{3},\ldots C_{12}>0$ such that

1. 1)

   for $1\leq t<L^{Md_{w}},\ x,y\in\mathcal{K}^{\left\langle M\right\rangle}$

   $$C_{3}t^{-d/d_{w}}\exp\left\{-C_{4}\min\left(|x-y|^{\frac{d_{w}}{d_{J}}},\left({|x-y|}{t^{-\frac{1}{d_{w}}}}\right)^{\frac{d_{w}}{d_{J}-1}}\right)\right\}\leq p^{M}_{R}(t,x,y)\\ \leq C_{5}t^{-d/d_{w}}\exp\left\{-C_{6}\min\left(|x-y|^{\frac{d_{w}}{d_{J}}},\left({|x-y|}{t^{-\frac{1}{d_{w}}}}\right)^{\frac{d_{w}}{d_{J}-1}}\right)\right\}$$

   (1.2)

2. 2)

   for $t\in(0,1),|x-y|\geq 1$

   $$C_{7}t{\rm e}^{-C_{8}|x-y|^{\frac{d_{w}}{d_{J}}}}\leq p^{M}_{R}(t,x,y)\leq C_{9}t{\rm e}^{-C_{10}|x-y|^{\frac{d_{w}}{d_{J}}}}$$

   (1.3)

3. 3)

   for $t\in(0,1),|x-y|<1$

   $$C_{11}t^{\frac{-d}{\alpha d_{w}}}\left(\left(\frac{t^{\frac{1}{\alpha d_{w}}}}{|x-y|}\right)^{d+\alpha d_{w}}\wedge 1\right)\leq p^{M}_{R}(t,x,y)\leq C_{12}t^{\frac{-d}{\alpha d_{w}}}\left(\left(\frac{t^{\frac{1}{\alpha d_{w}}}}{|x-y|}\right)^{d+\alpha d_{w}}\wedge 1\right).$$

   (1.4)

And again, these results show that those processes initially behave similarly to the ’original’ ones and in large times they are almost uniformly distributed over entire $M-$complex.

The paper is organized as follows. In Section 2 we provide definitions and notations regarding unbounded simple nested fractals, subordination and reflected Brownian motion. Section 3 contains the proof of the estimates of the transition density for subordinated reflected Brownian motion via the $\alpha-$stable subordinator, and Section 4 - via the relativistic subordinator.

Notation. Throughout the paper, upper- and lowercase, numbered constants, $A_{i},K_{i},C_{i},c_{i}$ denote constants whose values, once fixed, will not change. Constants that are not numbered, i.e. $c,C,c^{\prime},C^{\prime},\ldots$ can change their value inside the proofs. For two functions defined on a common domain, $f\asymp g$ means that there is an absolute(independent of $t,x,y,M$) constant $C>0$ s.t. $\frac{1}{C}f(\cdot)\leq g(\cdot)\leq Cf(\cdot)$ , also $f\gtrsim g$ means that there is an absolute constant $C>0$ s.t. $f(\cdot)\geq Cg(\cdot).$

We stand with the simpler case of the reflected $\alpha-$stable Brownian motion, obtained by the subordination of the reflected Brownian motion on $\mathcal{K}^{\left\langle\infty\right\rangle}.$ Let $M\in\mathbb{Z_{+}}$ be fixed. We have the following.