## 2 Dynamic Lognormal Shadowing Framework
### Dynamic Model  
We consider the received SNR from \(m\) base stations, at \(k\) locations of interest. We denote by \(\gamma_{i}(t)\), \(i=1,\ldots,n=mk\) (The notations \(\gamma(t)\) or \(\gamma_{t}\) is used interchangeably for the time dependence), the received SNR in linear scale, i.e., every index corresponds to a specific location for a specific base station. According to (1) expressed in terms of SNR (dB) the SNR in linear scale \(\gamma_{i}\) can be modeled as a lognormal r. v. for each time instant. Its long-term statistical parameters are \(\sigma_{i}\) and \(\ln(\gamma_{m_{i}})\). We are introducing the following multi-dimensional stochastic model. We assume that the resulting shadowing vector process \(\mathbf{X}(t)=[X_{1}(t),\ldots,X_{n}(t)]\) after applying the nonlinear transformation:  
\[X_{i}(t)=\xi\ln(\gamma_{i}(t)/\gamma_{m_{i}}),1\leq i\leq n \tag{15}\]  
to each component process \(\gamma_{i}(t)\), is a solution to a linear \(n\)-dimensional SDE of Ornstein-Uhlenbeck type [39]:  
\[d\mathbf{X_{t}}=\mathbf{A}\cdot\mathbf{X_{t}}dt+\mathbf{B}\cdot d\mathbf{W_{t} },\,\mathbf{X_{0}}=\mathbf{x}_{0} \tag{16}\]  
where \(\mathbf{A}\) is the diagonal matrix \(\mathbf{A}=\left[a_{ij}\right]_{1\leq i,j\leq n}\) with elements:  
\[a_{ij}=-a_{i}\cdot\delta_{ij} \tag{17}\]  
\(\delta_{ij}\) is the Kronecker delta function and \(a_{i}\), \(1\leq i\leq n\) are the dynamic parameters of shadowing, in principle different for each link and \(\mathbf{X}(0)=\mathbf{x}_{0}\) is the initial condition of the stochastic differential equation. The solution to the \(n\)-dimensional SDE (16) is straightforward and is given [39]:  
\[\mathbf{X}_{t}=e^{t\cdot\mathbf{A}}\cdot\mathbf{X}_{0}+e^{t\cdot\mathbf{A}} \cdot\int\limits_{0}^{t}e^{-s\cdot\mathbf{A}}\cdot\mathbf{B}\cdot d\mathbf{W_ {s}} \tag{18}\]  
where  
\[e^{t\cdot\mathbf{A}}=\sum\limits_{n=0}^{\infty}\frac{t^{n}}{n!}\mathbf{A}^{n} \tag{19}\]  
Now due to the fact that the matrix \(\mathbf{A}\) is diagonal (see (17)) it can be easily verified that  
\[\left[e^{t\cdot\mathbf{A}}\right]_{ij}=e^{-a_{i}\cdot t}\delta_{ij} \tag{20}\]The solution stochastic process \(\mathbf{X}_{t}\) as given in (18) is a Gaussian process if \(\mathbf{X}_{0}\) follows an \(n\)-variate normal distribution, including the degenerate case. The mean vector of \(\mathbf{X}_{t}\), \(\mathbf{M}_{\mathbf{X}}(t)\) is given by:  
\[\mathbf{M}_{\mathbf{X}}(t)=e^{t\cdot\mathbf{A}}\cdot\mathbf{M}_{\mathbf{X}}(0) \tag{21}\]  
The covariance matrix \(\mathbf{C}_{\mathbf{X}}(t)\) of the vector process \(\mathbf{X}_{t}\) for each time instant \(t\) is given [39]:  
\[\mathbf{C}_{\mathbf{X}}(t)=e^{t\cdot\mathbf{A}}\Bigg{[}\mathbf{C}_{\mathbf{X} }(0)+\int\limits_{0}^{t}e^{-s\cdot\mathbf{A}}\mathbf{B}\mathbf{B}^{T}e^{-s \cdot\mathbf{A}^{T}}ds\Bigg{]}e^{t\cdot\mathbf{A}^{T}} \tag{22}\]  
which is the solution to the following linear differential equation  
\[\overset{\bullet}{\mathbf{C}}_{\mathbf{X}}(t)=\mathbf{A}\mathbf{C}_{\mathbf{X }}(t)+\mathbf{C}_{\mathbf{X}}(t)\mathbf{A}^{T}+\mathbf{B}\mathbf{B}^{T} \tag{23}\]  
The matrix \(\mathbf{A}\) has by definition all of its eigenvalues real and negative (equal to \(-a_{i}\)), so the convergence of the following integral is assured. \[\mathbf{C}_{\mathbf{X}}=\int\limits_{0}^{\infty}e^{s\cdot\mathbf{A}}\mathbf{ B}\mathbf{B}^{T}e^{s\cdot\mathbf{A}^{T}}ds \tag{24}\]  
It is easy to verify that if \(\mathbf{C}_{\mathbf{X}}(0)=\mathbf{C}_{\mathbf{X}}\) then \(\mathbf{C}_{\mathbf{X}}(t)=\mathbf{C}_{\mathbf{X}}\) is a solution to (23). This means that a stationary solution of (16) exists and in this case for (23) we are leading to the following algebraic matrix equation:  
\[d\mathbf{X_{t}}=\mathbf{A}\cdot\mathbf{X_{t}}dt+\mathbf{B}\cdot d\mathbf{W_{t} },\,\mathbf{X_{0}}=\mathbf{x}_{0} \tag{16}\]0  
This equation forms the physical basis of the model. The stationary covariance matrix \(\mathbf{C}_{\mathbf{X}}\) of \(\mathbf{X}_{t}\) is equal to the covariance matrix of the shadowing process \(\mathbf{S}(t)\).