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Stalls should provide a horse as much visual stimulation as possible, such as windows and a clear view of other horses (if any others share the same premises). In addition a horse that must be kept within a stall for a long period of time should receive plenty of hay throughout the day so that he does not become miserable with his circumstances.
The most common cause of stall walking and horse weaving comes with separation anxiety. Horses are herd animals and therefore they can become very unhappy if they are isolated from all forms of companionship. Studies have shown that surrounding a horse with others, even if each are in individual stalls, is usually enough to keep walking and weaving at bay. You will know your horse is having separation anxiety if he neighs loudly as he circles or weaves; this neighing is a cry out to other horses.
Many horsemen are afraid that walking and weaving are contagious behaviors that can spread to other horses in the barn, but unlike cribbing, this is a vice that appears to have far more to do with a genetic disposition than a learned habit. The majority of horses will not weave, so the only time you really need to worry about exposing a weaver to other horses is if any of the others share the same genetics as the weaver. A colt might pick up the weaving habit from his mother due to being genetically predisposed to it.
Compared to other horse vices, walking and weaving are generally less harmful to the horse. The repetitive weaving can create an unnatural wear in your stall floor and the horse's hooves, as well as place stress on the horse's joints.
There are no guarantees that an ingrained habit can ever be broken, so it's possible that a long-time walker or weaver will find it difficult to drop the vice entirely. That being said, because walking and weaving are developed due to very specific causes, addressing those causes will make the habit far easier to cure than cribbing and other such problems.
Allowing a horse to eat more throughout the day as well as experience time in the pasture can cure a horse that is simply unhappy with his feed and surroundings. Imagine how you would feel if you were trapped within a prison cell almost all day, every day. Such an environment is even more abnormal for horses than it is for us, so allowing them more freedom can go a long way.
As mentioned earlier, a lack of companionship is probably the most common cause for these vices, so if we can alleviate their separation anxiety we can often cure their weaving entirely. One way to do this is ensure the horse can see other horses in the stable at all times just by looking through a window. Although seemingly a small thing, studies have shown this to be incredibly effective.
If your horse is not located in a stable and therefore has no other horses to watch or communicate with, consider purchasing another type of animal to keep him company. An old trick for racehorses is to purchase goats that hang out with the horse in his stall… and it does work! Cats, chickens and all sorts of other animals can also work, but goats are generally preferred due to their size; they are easier for a horse to see (and avoid stepping on) plus they aren't able to "escape" the stall and therefore abandon their equine buddy.
Finally, if other live forms of companionship just aren't in the cards you can consider purchasing a mirror and hanging it inside the horse's stall. The mirror must not be created from glass since that can create a hazard for the horse, but there are plastic mirrors that are specifically made for weavers - ones incorporating the right size and safety standards for horses. A mirror too small isn't enough, whereas a mirror too large can actually be intimidating. Ask your vet or other experts for advice on where you might be able to obtain these mirrors locally.
As odd as it may sound, these mirrors have been shown to be very effective against weaving; the horse gains some type of reassurance just being able to see its reflection since it appears another horse is nearby.
None of the above techniques can guarantee a long-term walker or weaver will be broken of the habit, but they do provide a very strong chance of kicking it once and for all.