Your problem is a seemingly weird interpretation of "random(ness)".

A process may be deterministic or indeterministic. This is a philosophical thing. We have no means to distinguish determinism from indeterminism.

If we describe observations we model relationships between variables. Quantitative models use well-defined functions to express these relationships.This has nothing to do with determinism, these are concepts on completely different worlds. The functions (or fitted models) are practical, useful tools.

Functions map values from (a set of) variable(s) to another variable. If we compare the mapped values with actual real observations we may find that the result is wrong: the function does not (correctly) predict what we observe. However, the predictions may be better or worse in terms of the frequency of correct predictions or the size of the deviations between predicted and observed values.

This means that a model seems to be able of having/using a certain amount of information from the given variables to make a perfect prediction. If the information is completely sufficient, then the correct prediction is made. Some lack of information distorts a perfect prediction. This "lack of information" can be described quantitatively by a probability distribution. Instead of dealing with predictions we then work with our "expectations". A process or model with such an assigned probability distribution/model is called a "stochastic" model.

It does not matter why we have insufficient information. It may be that the missing information is just practically not available or that it is in principle not existent (like from a process that is called "indeterministic"). The probability model only represents our kind and amount of uncertainty, it does not care *why* we are uncertain.

So: Brownian motion may or may not be deterministic. If there is any part of the whole story indeterministic, then the whole thing is indeterministic. However, this is a philosophical question, and this can not be decided based on data. A completely different question is what models we have to describe Brownian motion and how we can use them. And how we can obtain the required information to make a particular model useful. For instance, a stochastic model over the average impulses is enough to predict diffusion properties with sufficient precision, but it is not at all useful to predict a single-particle trace. It might be that we could build a model that would allow it, but I daubt that we would have the (practical or theoretical) possibility to get enough information to make it useful (i.e. precise enough for any practical purpose).

Your problem is a seemingly weird interpretation of "random(ness)".

A process may be deterministic or indeterministic. This is a philosophical thing. We have no means to distinguish determinism from indeterminism.

If we describe observations we model relationships between variables. Quantitative models use well-defined functions to express these relationships.This has nothing to do with determinism, these are concepts on completely different worlds. The functions (or fitted models) are practical, useful tools.

Functions map values from (a set of) variable(s) to another variable. If we compare the mapped values with actual real observations we may find that the result is wrong: the function does not (correctly) predict what we observe. However, the predictions may be better or worse in terms of the frequency of correct predictions or the size of the deviations between predicted and observed values.

This means that a model seems to be able of having/using a certain amount of information from the given variables to make a perfect prediction. If the information is completely sufficient, then the correct prediction is made. Some lack of information distorts a perfect prediction. This "lack of information" can be described quantitatively by a probability distribution. Instead of dealing with predictions we then work with our "expectations". A process or model with such an assigned probability distribution/model is called a "stochastic" model.

It does not matter why we have insufficient information. It may be that the missing information is just practically not available or that it is in principle not existent (like from a process that is called "indeterministic"). The probability model only represents our kind and amount of uncertainty, it does not care *why* we are uncertain.

So: Brownian motion may or may not be deterministic. If there is any part of the whole story indeterministic, then the whole thing is indeterministic. However, this is a philosophical question, and this can not be decided based on data. A completely different question is what models we have to describe Brownian motion and how we can use them. And how we can obtain the required information to make a particular model useful. For instance, a stochastic model over the average impulses is enough to predict diffusion properties with sufficient precision, but it is not at all useful to predict a single-particle trace. It might be that we could build a model that would allow it, but I daubt that we would have the (practical or theoretical) possibility to get enough information to make it useful (i.e. precise enough for any practical purpose).

I agree with Jochen.

If you define the location of a particle at position x,y,z at time t, then Brownian motion is deterministic as no other set of conditions would allow that position. 

The idea of random is another matter. The use of random has become almost mythological in statistics. Random simply means that you have no way of determining, knowing, or influencing. Stochastic is a better term for molecular motion and/or the indeterminate nature of the multiple body problem. Herein lies the problem. If the motion is indeterminate (unpredictable), how can an exact location position be deterministic?