Let $\epsilon > 0$ be a (small) fixed constant, and let $v\colon \mathbb{R}^n \to \mathbb{R}^n$ be twice continuously differentiable and (spatially) nonconstant. I'm looking to approximate solutions to the variable-coefficient advection-diffusion PDE $$ \frac{\partial \rho}{\partial t}(x,t) = \nabla_x \cdot (\epsilon \nabla_x \rho) (x,t) - \nabla_x \cdot (\rho v)(x,t), \quad \rho(\cdot,0) = \rho_0, $$ on a domain $\Omega \subseteq \mathbb{R}^n$, with $n$ large (too large for grid-based methods like finite difference). I'd like to assume that $\rho_0$ is a probability density function and to employ zero-flux boundary conditions so that $\rho(\cdot,t)$ is a probability density function for all $t$. I'm okay with assuming that $\Omega$ is a compact convex set if it makes things easier, or, alternatively, assuming that $\Omega = \mathbb{R}^n$ if that makes things easier instead.
The two methods that keep coming up in my search for numerical schemes for high-dimensional PDEs are deterministic particle/blob methods, and stochastic/Monte Carlo/random walk methods. However, most of the methods that appear in my searches are recent research articles that are catered towards nonlinear PDEs, which makes me assume that such methods have already been developed for the more classical linear PDEs such as the one I'm considering. However, I cannot seem to find them. I am not well-versed in either of these realms (deterministic and stochastic particle methods), and am also not aware if there are potentially better alternative methods for my particular high-dimensional problem.
In general, I'm looking for suggestions and references on particular methods to use. Here are a few more specific questions as well:
I've seen lecture notes and papers that assert that Monte Carlo-based methods for PDEs avoid the curse of dimensionality. Do deterministic particle/blob methods also avoid the curse of dimensionality? In other words, which of these particle-based methods work better in high dimensions?
I'm hoping for a method with provable convergence guarantees, i.e., one that yields an approximate solution that provably converges to the true solution as the "computational effort" increases (e.g., the number of particles used). Ideally this convergence would be in some strong sense, e.g., in $L^\infty$, but really all I need is that the expectation $\int_\Omega x \bar{\rho}(x,t) dx$ of the approximate solution $\bar{\rho}$ converges to $\int_\Omega x \rho(x,t) dx$ as the computational effort increases (that is, I'm hoping to show that the mean of the approximate solution well-approximates the mean of the true distribution). Are there any methods catered towards such provable convergence guarantees?
I'm primarily interested in the long-time solution $\rho(\cdot,t)$ as $t\to\infty$. Does this allow for any other numerical methods?
Any suggestions and/or references would be greatly appreciated.