# controlled random walk

consider a controlled random walk in 2 dimensions, with state vector $X = x_{1},x_{2}$. Whenever $x_{1}>x_{2}$ a control is applied with a drift and variance such that drift of $x_{1}$ is more than that of $x_{2}$.(The outcome of this control vector is independent of the state vector.) Similarly for the other case. The mean of these two drifts is such that the random walk stays at the line $x_{1} = x_{2}$ in the limit, i.e. for every $r$ positive integer, there exists an $N(r)$ such that for all $t>N(r)$, $E(|x_{1}(t)-x_{2}(t)|^{r})$ is bounded.

Question which I am interested in (and which leads to a strong characterization of growth of |X(t)|) as follows: How much time does this random walk spend on each of these 2 spaces (each space separated by $x_{1}=x_{2}$). Not the expected time, but is there almost sure limit of $\frac{\sum_{s=1}^{T}1_{X_{s}\in R_{1}}}{T}$ where $R_{1}$ and $R_{2}$ are two regions. This problem is trivial in 2 dimensions (balancing the total time and mean drift to zero gives two equations and there are two unknowns.) I am interested in the similar problem for general dimension.

An alternate problem which will be equally useful will be: what impact does the condition for every $r$ positive integer, there exists an $N(r)$ such that for all $t>N(r)$, $E(|x_{1}(t)-x_{2}(t)|^{r})$ is bounded, has on the occupations $\frac{\sum_{s=1}^{T}1_{X_{s}\in R_{1}}}{T}$. Modelling the controlled random walk as a markov chain, does this imply anything on the above stated quantity?

-