If someone invents something new (and unique), does Judaism recognize that he has ownership over the invention [for x years]?
If I understand correctly, a patent means you've told the world your idea, but the law protects others from using it to compete against you for a period of time. (As opposed to actually stealing someone's secret formula, which is an entirely different issue.) Based on my understanding of this article, there are two schools of thought among poskim about intellectual property as applies to copyrights -- I'd assume the same applies to patents, unless someone says otherwise.
Inherently, your using my idea is not "stealing." However, a society has the right to enact all sorts of laws; in this case, it might encourage innovation by creating temporary, artificial ownership on ideas. Violating those terms of ownership is an unfair infringement of another person's business (hasagat g'vul) and a transgression of the law of the land (dina d'malchuta dina). This is the opinion of the late Rabbis Moshe Feinstein in America and Shlomo Zalman Auerbach in Israel (zt'l).
Ideas can be exclusively owned -- and therefore stolen -- just as an ox, lamb, or piece of clothing can be stolen. Any stealing of ideas is Halachichally theft. This is the opinion of R' Elyashiv shlit'a.
For a related description of these two points of a view from a secular perspective, try this piece from the Economist.
Lastly, I heard a recording (sorry, don't recall which one) from R' Hershel Shachter about times when, to encourage a printing company to risk their money in printing a major Judaic text, the rabbis banned all others for several years from printing the same work. Not exactly a patent, but fairly close. It sounds like this is an enactment, and not the default halacha. (The ban would also be voided if the printing was wildly successful and the printer wound up easily "in the black.")
If I recall, the first major Halachic case of copyright infringement involved the Maharam Padua in the mid-1500s, in which he wrote to his cousin R' Moshe Issreles for psak. Could be worth studying the discussion there for more details.
Regarding patents, the law of the land is to be followed as stated above. This is usually dispositive of the question of patent infringement. Halacha also recognizes that infringement is a tort that has remedies. There is the theft of an idea and there is also confusion in the public’s mind regarding the article. The confusion may arise if the infringer palms off the infringing article as being made by the inventor and it may damage the inventor’s reputation if the infringing article is inferior in quality to the infringed article. The infringer may have to pay the inventor and/or patent holder a share in the cost of development and registering the article infringed. There will also be an injunction issued by Beth Din to cease the infringement; Beth Din can also assess money damages.
This is a sefer Feldheim published about copyright. It's a translation of a Hebrew sefer on the topic. The concept in Halacha boils down to 'Charamim' which were issued on books, enacting penalties against anyone who would publish the same work within a certain period of time. The intention in doing so is to give the publisher the ability to gain a profit. This is similar to the motivation for such legislation in Jewish law (cf. the Constitution).
Note that as mentioned by @Shalom there is nothing explicit about this topic until relatively recently (mainly because of the advent of the printing press). The institution of Charamim is an exercise of Rabbinic communal authority, not a mandate from the Tanakh, Gemara, or even Rishonim.