POST from jQuery to PHP

When you load your process.php file directly, it’s not within the context of the WordPress environment, so no WordPress functions are available. WordPress has a native AJAX API that should be used for this sort of thing.

First, enqueue your javascript file, then use wp_localize_script to pass the location of admin-ajax.php, which will be processing the requests:

function wpa_scripts() {
    wp_enqueue_script(
        'wpa_script',
        get_template_directory_uri() . '/js/script.js',
        array('jquery'),
        null,
        true
    );
    $script_data = array(
        'admin_ajax' => admin_url( 'admin-ajax.php' )
    );
    wp_localize_script(
        'wpa_script',
        'wpa_data',
        $script_data
    );
}
add_action( 'wp_enqueue_scripts', 'wpa_scripts' );

Next, add an action to your ajax data, which will map to the function hooked to process the request:

$.ajax({
    type: "POST",
    url: wpa_data.admin_ajax,
    dataType: "json",
    data: {
        "action":"some_action",
    },
    success: function() {
        //get permalink for post from php and go to it
    }
});

Last, add the action in php and hook it to the function which will receive and process it, then move the contents of process.php inside this function

add_action( 'wp_ajax_some_action', 'your_process_function' );
add_action( 'wp_ajax_nopriv_some_action', 'your_process_function' );