ASP.NET and Internet Browser SSL Certificates

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When using ASP.NET over an SSL connection I can gather information from the Request.ClientCertificate. I want to use this information in future requests to identify a computer.
Is this infromation reliable? Does the certificate change frequently? Is this certificate typically generated by the user(their computer, OS, Browser)? Should I have asked any additional questions.
For anyone curious I would be using this as an automatic logon. Like we would do with a cookie. Is it better or safer to use the cookie? A colleague is concerned that the cookie could be copied from one computer to another(I don't know if that's even possible).