One mistake I’ve seen a lot of beginners make in Jiu-Jitsu (including myself when I started) is holding their breath. It often seems to occur when they’re attempting a technique, as if they’re concentrating so much on what they’re doing that they forget to breathe, and they’re probably not even aware they’re doing it. I’ve also seen people hold their breath when they’re taking an impact, such as hitting the ground, or when someone drops weight on them. One of the reasons they teach people to kiai (yell) when being thrown in Judo is because it forces you to exhale, which is much better on the body than holding your breath. If you don’t believe me, have someone punch you in the stomach while you hold your breath, and then again while you exhale upon contact, and tell me which one feels better!
In Karate we’re not just reminded to keep breathing, but we’re also taught to breathe through our bellies, not our chest. Diaphragmatic breathing has a more calming effect on the body, which can help in many ways. This kind of breathing still doesn’t come naturally for me, but I’ve noticed that when I remind myself to do it while I’m grappling, I’m able to stay more relaxed, and I don’t tire out as easily. It’s especially helpful when I find myself in a bad position, perhaps because the focus on my breathing distracts my mind from the panic I might otherwise feel. I’ve even successfully used belly-breathing to help stay calm in other stressful situations outside of martial arts.
Whether it’s through your belly or chest, in through the nose and out through the mouth or straight-up panting, the most important thing is to pay attention to your breathing, and make sure you never stop. You can’t relax if you’re not breathing, and holding your breath doesn’t conserve any air, it just means you’ll run out of it sooner. Males actually have a 20-25% higher lung capacity than females, but one of the ways I’m able to beat white belt men who are younger, bigger, and stronger than me is by waiting until they run out of breath!