When executing the test below the memory consumption in the Task Manager grows up to 2GB within seconds on my test system - the issue does not occur when the stage style is changed from TRANSPARENT to UNDECORATED.

    public class TransparentStageMemoryIssueTest extends Application
    {
      public static void main(String[] args)
      {
        Application.launch(args);
      }
      @Override
      public void start(Stage stage)
      {
        System.err.println(System.getProperty("javafx.runtime.version"));
        //stage.initStyle(StageStyle.UNDECORATED);
        stage.initStyle(StageStyle.TRANSPARENT);
        final BorderPane root = new BorderPane();
        root.setStyle("-fx-background-color:#808080;-fx-background-radius:30;-fx-border-radius:30;-fx-border-width:14;-fx-border-color:blue;");
        Scene scene = new Scene(root, 800, 600);
        stage.setScene(scene);
        stage.setX(10);
        stage.setY(10);
        stage.show();
        new Thread()
        {
          public void run()
          {
            for (int i = 0; i < 1000; i++)
            {
              Platform.runLater(new Runnable()
              {
                @Override
                public void run()
                {
                  int minSize = 500;
                  int maxSize = 1000;
                  int w = minSize + new Random().nextInt(maxSize - minSize);
                  int h = minSize + new Random().nextInt(maxSize - minSize);
                  Window win = root.getScene().getWindow();
                  win.setWidth(w);
                  win.setHeight(h);
                }
              });
              try
              {
                sleep(50);
              }
              catch (InterruptedException e)
              {
                e.printStackTrace();
              }
            }
          };
        }.start();
      }
    }